mm-4889 === Subject: Re: A Proof of the Collatz Conjecture I think Lemma 2 is wrong... (I have to double check it) But for the moment... Which would make the proof as not solving the entire Collatz problem, but giving the following conclusion: All non-cyclic trajectories end into -1, 0, 1 Roupam === Subject: Re: A Proof of the Collatz Conjecture I think Lemma 2 is wrong... (I have to double check it) > But for the moment... > Which would make the proof as not solving the entire Collatz problem, > but giving the following conclusion: > All non-cyclic trajectories end into -1, 0, 1 Huh? ALL tracectories are cyclic. Stopping at 1 is a red herring. 4 -> 2 -> 1 -> 4 -> 2 -> 1 ... > 0 -> 0 -> 0 -> 0 -> 0 -> 0 ... (this is not really a cycle due to > parity lock) > -1 -> -2 -> -1 -> -2 -> -1 ... and also the two I mentioned @ -5 and -17. What you need to do is accept that every trajectory reaches a cycle > (ignore 0) and to be able to prove EXACTLY what the cycle is. > For instance, the cycles at +1, -1 and -5 can all be shown to be > TRIVIAL (they result from a rational number whose denominator is > a unit). In this case, the denominator is the difference between > a power of two and a power of three. Are there any other possible > cases? Not likely, as an n-bit power of three would need a pop-count > of exactly 2 or exactly n, and except for the three known cases > (2-3, 4-3, 8-9), the pop-count of powers of three diverges from > n=2 and also diverges from n. That leaves the loop @ -17 to account for. The rational here is NOT > trivial (17*139/-139) yet it reduces to an integer and thus becomes > a non-Trivial cycle. This proves the set of non-Trivial cycles is > not empty, so you certainly will NOT be able to come up with a proof > that it is. You have to prove that the set of non-Trivial cycles > contains only one member or prove that all such members are in the > negative domain. I have no idea how you would do that. Roupam I think I have come up with a proper conclusion of this paper... I will make another post and give the link below... According to my observation, I would say (conservatively)... if G^n(x) ends in a limit with increasing n, the possible values of the limits are -1, 1 But since G(x) is a reduced form of the Collatz function, we can say, if for the Collatz function T(x), T^n(x) ends in a cycle containing only one odd integer then, the possible cycles are (-1,-2) (1,4,2) Roupam === Subject: Re: A Proof of the Collatz Conjecture AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) > I think Lemma 2 is wrong... (I have to double check it) > But for the moment... > Which would make the proof as not solving the entire Collatz problem, > but giving the following conclusion: > All non-cyclic trajectories end into -1, 0, 1 Huh? ALL tracectories are cyclic. Stopping at 1 is a red herring. 4 -> 2 -> 1 -> 4 -> 2 -> 1 ... > 0 -> 0 -> 0 -> 0 -> 0 -> 0 ... (this is not really a cycle due to > parity lock) > -1 -> -2 -> -1 -> -2 -> -1 ... and also the two I mentioned @ -5 and -17. What you need to do is accept that every trajectory reaches a cycle > (ignore 0) and to be able to prove EXACTLY what the cycle is. > For instance, the cycles at +1, -1 and -5 can all be shown to be > TRIVIAL (they result from a rational number whose denominator is > a unit). In this case, the denominator is the difference between > a power of two and a power of three. Are there any other possible > cases? Not likely, as an n-bit power of three would need a pop-count > of exactly 2 or exactly n, and except for the three known cases > (2-3, 4-3, 8-9), the pop-count of powers of three diverges from > n=2 and also diverges from n. That leaves the loop @ -17 to account for. The rational here is NOT > trivial (17*139/-139) yet it reduces to an integer and thus becomes > a non-Trivial cycle. This proves the set of non-Trivial cycles is > not empty, so you certainly will NOT be able to come up with a proof > that it is. You have to prove that the set of non-Trivial cycles > contains only one member or prove that all such members are in the > negative domain. I have no idea how you would do that. Roupam I think I have come up with a proper conclusion of this paper... > I will make another post and give the link below... According to my observation, I would say (conservatively)... if G^n(x) ends in a limit with increasing n, the possible values of > the limits are -1, 1 > But since G(x) is a reduced form of the Collatz function, we can say, > if for the Collatz function T(x), T^n(x) ends in a cycle containing > only one odd integer then, the possible cycles are (-1,-2) (1,4,2) Roupam Check the URL for the updated paper === Subject: Re: Is an arbitrary multivariate polynomial equation over GF[2] ?NP-complete? <4a04330d$0$318$b45e6eb0@senator-bedfellow.mit.edu> posting-account=pbWZ8QoAAACI9zXocLL9zZCrDIYrXvvv 5.1),gzip(gfe),gzip(gfe) >Since 1979 it is reported in the literature the NP-complete character >of the title problem. Yes, the reduction from 3SAT is an easy undergraduate exercise. It's telling > that not only are you unable to see the error in your own arguments even when > they are pointed out to you, but you can't even do this easy exercise. Hmm, it would be good to state exactly the problem. I agree > that showing NP-completness of solvability of in (GF[2])^n of > systems of sparse n-variate polynomials is an easy exercise. > Similarely for solvability in (GF[2])^n of a single n-variate > polynomial given as stright-line program. However, if one looks > at single sparse polynomial situation may to be different -- I do > not see how to prove that solving such equation in (GF[2])^n is > NP-complete. At first glance what Valls describe may work for single > equation. -- > Waldek Hebisch > hebi...@math.uni.wroc.pl I have a doubt about the notation that you are using. What do you mean by (GF[2])^n ? I denote the Galois Field GF[k] with k=2^n elements as GF[2^n], but I opened this thread referring only to the finite field GF[2] with 2 elements. The problem that I refer is the solvability of a single arbitrary multivariate polynomial equation over GF[2]. Is that problem an NP- complete one? That's all what I want to know with absolute security. RVHG (Rafael Valls Hidalgo-Gato) === Subject: Re: Is an arbitrary multivariate polynomial equation over GF[2] ??NP-complete? >Since 1979 it is reported in the literature the NP-complete character >of the title problem. Yes, the reduction from 3SAT is an easy undergraduate exercise. It's telling > that not only are you unable to see the error in your own arguments even when > they are pointed out to you, but you can't even do this easy exercise. Hmm, it would be good to state exactly the problem. I agree > that showing NP-completness of solvability of in (GF[2])^n of > systems of sparse n-variate polynomials is an easy exercise. > Similarely for solvability in (GF[2])^n of a single n-variate > polynomial given as stright-line program. However, if one looks > at single sparse polynomial situation may to be different -- I do > not see how to prove that solving such equation in (GF[2])^n is > NP-complete. At first glance what Valls describe may work for single > equation. -- > Waldek Hebisch > hebi...@math.uni.wroc.pl I have a doubt about the notation that you are using. What do you mean > by > (GF[2])^n ? I denote the Galois Field GF[k] with k=2^n elements as > GF[2^n], but I opened this thread referring only to the finite field > GF[2] with 2 elements. > Consider polynomial P(X_1,...,X_n) over GF[2] (that is coefficients of the polynomial belong to GF[2]). By definition, this polynomial have zero in (GF[2])^n <=> there exists x_1 in GF[2], x_2 in GF[2], ..., x_n in GF[2] such that P(x_1,...,x_n) = 0. Point of this notation is to specify where you look for solutions. Single polynomial always have zero in an algebraic extension, so I assume that you want solutions in (GF[2])^n (that is having all coordinates from GF[2]), but it is better to state this explicitely. > The problem that I refer is the solvability of a single arbitrary > multivariate polynomial equation over GF[2]. Is that problem an NP- > complete one? That's all what I want to know with absolute security. RVHG (Rafael Valls Hidalgo-Gato) > Sorry, I can not give you definite anwer (almost surely not, but...). Note that references talk about _equations_ (in plural). It is an easy excercise to show that solvability (in (GF[2])^n) of _system of equations_ given by sparse polynomials in n-variables in NP-complete -- if you want I can provide details. Working over GF[2] it is possible to replace system S of equations by a single polynomial, however if done naively using sparse representation the single polynomial may be exponentially bigger than S. One can avoid this growth using different representation of polynomials, as so called stright line programs -- again if you wish I can give more details and some references. Given a single sparse polynomial P over GF[2] solvability in (GF[2])^n is easy: first replace all higher powers of variables by first power getting new polynomial P_1. Due to equation x^2 = x valid in GF[2] P and P_1 have the same solutions in (GF[2])^n. Now, if P_1 is constant equal one, than there is no solutions, otherwise P is solvable in (GF[2])^n. AFAICS this is essentially procedure you propose. To summarize: we have simple argument that problem is polynomial and no evidence that it is NP-complete... -- Waldek Hebisch hebisch@math.uni.wroc.pl === Subject: Re: Question about Complex Made Simple posting-account=aLpfCwoAAACh4BOs3HOlQBCoxUpEgyxc Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) [...]It seems like I need to know that the >partial derivatives of u_n converge to zero uniformly on compact sets Not that this is the easiest way to do that exercise, but yes, this > is true. Let's say -> means uniformly on compact sets: It's a fact that if u_n is harmonic and u_n -> 0 then all the > partial derivatives of u_n also -> 0. Possibly the easiest > way to prove this in the present context is to use various > tricks you've seen in this thread to derive it from the > analogous fact for holomorphic functions. For an > argument that also works in higher dimensions where > the complex tricks don't exist: It's enough to show that if u_n -> 0 uniformly > in a neighborhood of the closure of D(a,r) then > the partials -> 0 uniformly in D(a,r). To simplify > the notation WLOG D(a,r) = D(0,1). Now > u_n is given by a Poisson integral; differentiate > under the integral sign. === Subject: Re: Ordinal numbers - some doubts posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > It seems rather pointless to define cardinals as initial ordinals in > this context. But pointlessness is not the point. Whether pointless or not, that is the definition. And there is no harm in stating the definition in Z while the definition is not very frutiful until we have at least Z +numeration theorem which is usually obtained via ZFC. > We cannot prove any uncountable ones exist, but all > cardinalities aleph n for finite n do. Certainly Kuratowski in the > original post cannot have been using that definition. I don't recall the context Kuratowski gave, but the ordinary definition of a cardinal is as I mentioned. > Similarly, von > Neumann ordinals are useless as canonical representatives of > order-types. omega + omega may not exist, but well-orderings with > cardinality aleph n exist for each finite n. Again, whether useful or not, I'm simply giving the ordinary definition. MoeBlee === Subject: Re: NY Times math problem ... The solution for the maximal value v given by Dave is for the rabbit to run a semicircle, then to continue straight (in the positive x direction). .. My question is this. For a given v, wouldn't the rabbit better its lead over the agent by the following alteration of the above strategy: Proceed as above until reaching a distance (1/sqrt(v)) from the center of the circle; then changing course to a new straight line, connecting its present position to a point, P', slightly counterclockwise to the original destination, P, on the circle. Provided P' is chosen sufficiently close to P, it seems that the extra time it takes the rabbit to reach the circle (at P' instead of at P) is less than the extra time it takes the agent to go form P to P'. If that is true, then it would seem the optimal strategy for the rabbit would involve coursing along some curve (that is not a straight line). === Subject: Re: NY Times math problem <9666204.91070.1242085163187.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=O9zR9AkAAACmp918j6u5m5plppeILcze Filter 1.2.0.72; GTB6; .NET CLR 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) ... The solution for the maximal value > v given by Dave is for the rabbit to run a semicircle, then to > continue straight (in the positive x direction). .. My question is this. For a given v, wouldn't the rabbit better its lead over the agent by the following alteration of the above strategy: Proceed as above until reaching a distance (1/sqrt(v)) from the center of the circle; then changing course to a new straight line, connecting its present position to a point, P', slightly counterclockwise to the original destination, P, on the circle. Provided P' is chosen sufficiently close to P, it seems that the extra time it takes the rabbit to reach the circle (at P' instead of at P) is less than the extra time it takes the agent to go form P to P'. If that is true, then it would seem the optimal strategy for the rabbit would involve coursing along some curve (that is not a straight line). Does this increase the maximum speed of the agent at which the rabbit can escape to above 4.6033388487517003525565820291030165130673...? If so, what is the new maximum speed? Dave === Subject: Re: NY Times math problem <9666204.91070.1242085163187.JavaMail.jakarta@nitrogen.mathforum.org>, ... The solution for the maximal value > v given by Dave is for the rabbit to run a semicircle, then to > continue straight (in the positive x direction). .. My question is this. For a given v, wouldn't the rabbit better its lead over > the agent by the following alteration of the above strategy: Proceed as above until reaching a distance (1/sqrt(v)) from the center of the > circle; then changing course to a new straight line, connecting its present position > to a point, P', slightly counterclockwise to the original destination, P, on > the circle. Provided P' is chosen sufficiently close to P, it seems that the extra time > it takes the rabbit to reach the circle (at P' instead of at P) is less than > the extra time it takes the agent to go form P to P'. If that is true, then it would seem the optimal strategy for the rabbit would > involve coursing along some curve (that is not a straight line). If the rabbit can escape at all, one successful strategy for the rabbit requires it at some point to get as far away from the agent as possible while still on the same diameter as the agent, then to head directly towards the nearest point on the edge of the pond regardless of what the agent does. As long as the rabbit moves to get on and then stay on the same diameter as the agent, the agent may as well continue circling the pond in the same direction at top speed, as neither slowing or reversing direction works any better. This combination leads to the rabbit moving along the semicircle mentioned of diameter v_rabbit/v_agent through the center of the pond until at one diameter of that semi-circle away from the center, then making its break for the edge, === Subject: Re: NY Times math problem <9666204.91070.1242085163187.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=O9zR9AkAAACmp918j6u5m5plppeILcze Filter 1.2.0.72; GTB6; .NET CLR 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > If the rabbit can escape at all, one successful strategy for the rabbit > requires it at some point to get as far away from the agent as possible > while still on the same diameter as the agent, then to head directly > towards the nearest point on the edge of the pond regardless of what the > agent does. That strategy will lead to the rabbit's escape if the agent's speed is less than pi + 1, but will fail if the agent's speed is pi + 1 or greater. A better strategy will let the rabbit escape if the agent's speed is less than 4.6033388487517003525565820291030165130673..., which, of course, is greater than pi + 1. Dave === Subject: Re: NY Times math problem DId anyone here see the problem presented in > the Science section of NY Times last week? > Quite startling, to see something so sophisticated > in a 'general readership' publication. Is it solvable without a calculus of variations approach? I don't get the NY times here. Could you transcribe the problem > here? (I'm at s.e.design, but crossposting to all of the above.) Rich SEE http://tierneylab.blogs.nytimes.com/2009/04/13/jimmy-carters-killer-rabbi t-puzzle/ === Subject: Re: Wikipedia-- the telemarketing-encyclopedia Re: Wikipedia is going rotten, with editor oneupmanship posting-account=yxbZkgkAAABQBvyYeebYQ-PAvi0uT3tG Gecko/20080829 Firefox/2.0.0.17,gzip(gfe),gzip(gfe) > If Wikipedia puts some judges on a kook list, it may just > happen this year in 2009, where a class action lawsuit against > Wikipedia lands them in the Supreme Court. An encyclopedia, driven by unemployed juvenile editors who volunteer > their free service, and in return we have a Internet encyclopedia full > of acid of negativity and belittling of most people. And we have > groups of people in organizations volunteering free editing and in > exchange they bias and slant the entries. Maybe what should have happened was that initially Wikipedia had > free editing but once they had body mass of a encyclopedia, they > should have eliminated the free editing and brought in professional > editors who made the hard cover encyclopedia's and let them improve > the work. You obviously still have no idea how Wikipedia works. All of your complaints would magically melt away if you could provide citations to reliable published sources that: 1) demonstrate that your ideas are /not/ considered eccentric or out of the mainstream of physics and mathematical theorists; or 2) demonstrate that your theories and ideas really are being debated in the pages of refereed physics and mathematics journals. Then you could present a cogent argument that someone out there does /not/ consider your ideas eccentric, and that fact could be noted on Wikipedia. But so far, this has not happened, has it? You really can't expect to have much control over what the majority of sci.math and sci.physics newsgroup users write about you. The simple (documented) fact is that they do not consider you an unrecognized genius. You know that. === Subject: Re: Wikipedia-- the telemarketing-encyclopedia Re: Wikipedia is going rotten, with editor oneupmanship posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse Trident/4.0; MathPlayer 2.10d; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > If Wikipedia puts some judges on a kook list, it may just > happen this year in 2009, where a class action lawsuit against > Wikipedia lands them in the Supreme Court. An encyclopedia, driven by unemployed juvenile editors who volunteer > their free service, and in return we have a Internet encyclopedia full > of acid of negativity and belittling of most people. And we have > groups of people in organizations volunteering free editing and in > exchange they bias and slant the entries. Maybe what should have happened was that initially Wikipedia had > free editing but once they had body mass of a encyclopedia, they > should have eliminated the free editing and brought in professional > editors who made the hard cover encyclopedia's and let them improve > the work. You obviously still have no idea how Wikipedia works. All of your complaints would magically melt away if you could > provide citations to reliable published sources that: 1) demonstrate that your ideas are /not/ considered eccentric > or out of the mainstream of physics and mathematical theorists; > or 2) demonstrate that your theories and ideas really are being > debated in the pages of refereed physics and mathematics > journals. Then you could present a cogent argument that someone > out there does /not/ consider your ideas eccentric, and that > fact could be noted on Wikipedia. But so far, this has not > happened, has it? You really can't expect to have much control over what > the majority of sci.math and sci.physics newsgroup users > write about you. The simple (documented) fact is that they > do not consider you an unrecognized genius. > You know that. As already noted by others, this is a logic of dominant opinions and marginalisation of differences: this approach should have stayed out of Wikipedia. Not to talk about legitimising the generalised ad hominem (in this case, by you and your peers): simply despicable. -LV === Subject: Re: Wikipedia is going rotten, with editor oneupmanship posting-account=B4ujLwoAAADyIxEERXnEcV1-gnuYMO8I Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) You goddamn pansies! You have any idea how unfair you're being? Wikipedia is the only public userbase website I can name which has a policy allowing you to ignore a rule if it's dipted and prevents a constructive edit. It's the only site that operates on common sense and not a jewish, rigid interpretation of the rules. I came across countless other places where the owner justifies being an incompetent dumb by saying it's my website, I'm the owner. It's my site... that's all he knows. Wikipedia is the only free encyclopedia that's uncensored and doesn't single anyone out. It is known for its lack of authority and lack of influence from any megacorp/government. Calling it an indoctrination machine is really ing low. Also, Wikipedia is a *reference* encyclopedia and a storm house Encyclopedia Dramatica. Learn the differences, fags. === Subject: Re: Wikipedia is going rotten, with editor oneupmanship posting-account=d-ESTAkAAAAG0l03yI1WJgsTVXx4ebeJ Gecko/20081217 Firefox/2.0.0.20,gzip(gfe),gzip(gfe) > You goddamn pansies! You have any idea how unfair you're being? [snip extremely well written rant defending the wiki editors and calling any criticism unfair] > Calling it an indoctrination machine is really ing low. Oh oh. Well, I guess we just heard from one of the Wikipedia editors. Known for lack of authority and and influence my ass. Christ. Just that POS the dumbest crap on the planet and totally refused to allow any modifications by persons with actual years of professional experience playing bass guitar. Lowest common denominator. So we are stuck at the high school newbie level forever. And that's just one > Also, Wikipedia is a *reference* encyclopedia and a storm house > Encyclopedia Dramatica. Learn the differences, fags. Ah. The old proof by name calling ploy. Sure. You really add to this discussion...NOT. You are just setting an example of the kind of attitude that has turned wiki into the conformist crap it is. Try to understand this numbnuts, [trying out your name-calling ploy for reenforcement] If an issue is controversial, you don't just take a vote to find out which side is most popular and then censor all other opinions, you carefully define the various positions. Got it? === Subject: Re: Wikipedia is going rotten, with editor oneupmanship > You goddamn pansies! You have any idea how unfair you're being? > Wikipedia is the only public userbase website I can name which has a > policy allowing you to ignore a rule if it's dipted and prevents a > constructive edit. It's the only site that operates on common sense > and not a jewish, rigid interpretation of the rules. I came across > countless other places where the owner justifies being an incompetent > dumb by saying it's my website, I'm the owner. It's my > site... that's all he knows. Wikipedia is the only free encyclopedia > that's uncensored and doesn't single anyone out. It is known for its > lack of authority and lack of influence from any megacorp/government. > Calling it an indoctrination machine is really ing low. Also, Wikipedia is a *reference* encyclopedia and a storm house > Encyclopedia Dramatica. Learn the differences, fags. Nasty little ing homosexual spick bastard, aren't you? *plonk* Do not reply to this generic message, it was automatically generated; you have been kill-filed, either for being boringly stupid, repetitive, unfunny, ineducable, repeatedly posting politics, religion or off-topic subjects to a sci. newsgroup, attempting cheapskate free advertising for profit, because you are a troll, simply insane or any combination or permutation of the aforementioned reasons; any reply will go unread. Boringly stupid is the most common cause of kill-filing, but because this message is generic the other reasons have been included. You are left to decide which is most applicable to you. There is no appeal, I have despotic power over whom I will electronically admit into my home and you do not qualify as a reasonable person I would wish to converse with or even poke fun at. Some weirdoes are not kill- filed, they amuse me and I retain them for their entertainment value as I would any chicken with two heads, either one of which enables the dumb bird to scratch dirt, step back, look down, step forward to the same spot and repeat the process eternally. This should not trouble you, many of those plonked find it a blessing that they are not required to think and can persist in their bigotry or crackpot theories without challenge. You have the right to free speech, I have the right not to listen. The kill-file will be cleared annually with spring cleaning or whenever I purchase a new computer or hard drive. I hope you find this explanation is satisfactory but even if you don't, damnly my frank, I don't give a dear. Have a nice day. === Subject: Re: A differential functional equation > f(x,y) is a real function R^2->R ,y in R - {0} , x in R And we' ve got f(x +1,y) = d/dy f(x,y) - 2*f(x,y)/y How can we manage such a thing? Any idea is welcome, > Alain Here's a fairly general solution: f(x, y) = Sum(n=2 to inf) g(x+n-2) y^n / (n-2)! where g(x) is any function of your choice. Bob I should have said: where g(x) is any function of your choice for which the infinite > series > converges absolutely and uniformly. Bob- Masquer le texte des messages pr.8ec.8edents - - Afficher le texte des messages pr.8ec.8edents - Bonjour Bob, Do you mind explaining how you did arrive at this > fairly general solution, I will give you my building Alain Alain, Assume: f(x, y) = Sum(n=0 to inf) a_n(x) y^n {1} Putting this into your equation gives: Sum(n=0 to inf) a_n(x+1) y^n = Sum(n=0 to inf) (n-2) a_n(x) y^(n-1) (2) so Sum(n=0 to inf) a_n(x+1) y^n = -2 a_0(x) y^(-1) - a_1(x) + Sum(n=3 to > inf) (n-2) a_n(x) y^(n-1) (3) Since there is no y^(-1) term on the left a_0(x) = 0 (all x) (4) So Sum(n=1 to inf) a_n(x+1) y^n = - a_1(x) + Sum(n=3 to inf) (n-2) a_n(x) > y^(n-1) (5) But now there is no y^0 = 1 term on the left, so a_1(x) = 0 (all x) (6) So: Sum(n=2 to inf) a_n(x+1) y^n = Sum(n=3 to inf) (n-2) a_n(x) y^(n-1) (7) On the right let n -> n+1 Sum(n=2 to inf) a_n(x+1) y^n = Sum(n=2 to inf) (n-1) a_(n+1)(x) y^n > (8) > a_n(x+1) = (n-1) a_(n+1)(x) n = 2, 3, ... (9) Rewrite this as: a_(n+1)(x) = a_n(x+1) / (n-1) n = 2, 3, ... (10) n = 2 > a_3(x) = a_2(x+1) = g(x+1) (so a_2(x) = g(x) ) n = 3 > a_4(x) = a_3(x+1) / 2 = g(x+2) / 2 n = 4 > a_5(x) = a_4(x+1) / 3 = g(x+3) / 3! . > . > . > so we see a_n(x) = g(x+n-2) / (n-2)! n = 2, 3, 4, ... and a_0(x) = a_1(x) = 0 Put this back into (1): f(x, y) = Sum(n=2 to inf) g(x+n-2) y^n / (n-2)! (11) and there's my fairly general solution. Bob- Masquer le texte des messages pr.8ec.8edents - - Afficher le texte des messages pr.8ec.8edents - Bonsoir Bob, I will propose you three different homemade solutions: > f1(x,y) = k*y^2*a^x*exp(a*y) , > f3(x,y) = k*y^2*sin(y+Pi*x/2) All built from a formal form using partial > derivation :f(x,y) = k*y^2*(d/dy)^(x) o g(y) k a constant , Alain Alain, In my solution, factor out y^2 and let n -> n+2: f(x, y) = y^2 Sum(n=0 to inf) g(x+n) y^n / n! (1) Take g(x) = k a^x Then f(x, y) = k y^2 Sum(n=0 to inf) a^(x+n) y^n / n! f(x, y) = k y^2 a^x Sum(n=0 to inf) (a*y)^n / n! f(x, y) = k y^2 a^x exp(a*y) (2) which is your f1(x, y). Next take: g(x) = k exp(i Pi*x/2) (3) This leads to: f(x, y) = k y^2 exp(i Pi*x/2) Sum(n=0 to inf) (y*exp(i Pi/2)^n / n! But exp(i Pi/2) = i , so f(x, y) = k y^2 exp(i Pi*x/2) Sum(n=0 to inf) (i*y)^n / n! f(x, y) = k y^2 exp(i Pi*x/2) exp(i*y) f(x, y) = k y^2 exp[i*(y + Pi*x/2)] f(x, y) = k y^2 cos(y + Pi*x/2) + i k y^2 sin(y + Pi*x/2) which gives two real solutions, your f3(x, y) and f4(x, y) = k y^2 cos(y + Pi*x/2) I don't immediately see what g(x) gives your f2(x, y). Bob- Masquer le texte des messages pr.8ec.8edents - - Afficher le texte des messages pr.8ec.8edents - > The continuous derivation solution I gave is a very > general one. > Bases I use are shared by many people: > (d/dy)^x o exp(a*y) = a^x*exp(a*y) > continuous extensions of integer known cases. A more general functional form could be: > f(x,y) = m(y)*(d/dy)^phi(x) o g(y) , with phi(h(x))=phi(x)+1 > Then we've got : > f(h(x),y)/m(y) = d/dy (f(x,y)/h(y)) > ............... > Your ideas,comments,views are welcome , Alain Alain, Your does not appear to be a solution to your equation. I get a y^(5-x) term on the right, and a y^(6-x) on the left. Bob === Subject: Re: A differential functional equation > > Delaney f(x,y) is a real function R^2->R ,y in R - {0} , x in R And we' ve got f(x +1,y) = d/dy f(x,y) - 2*f(x,y)/y How can we manage such a thing? Any idea is welcome, > Alain Here's a fairly general solution: f(x, y) = Sum(n=2 to inf) g(x+n-2) y^n / (n-2)! where g(x) is any function of your choice. Bob I should have said: where g(x) is any function of your choice for which the infinite > series > converges absolutely and uniformly. Bob- Masquer le texte des messages pr.8ec.8edents - - Afficher le texte des messages pr.8ec.8edents - Bonjour Bob, Do you mind explaining how you did arrive at this > fairly general solution, I will give you my building Alain Alain, Assume: f(x, y) = Sum(n=0 to inf) a_n(x) y^n {1} Putting this into your equation gives: Sum(n=0 to inf) a_n(x+1) y^n = Sum(n=0 to inf) (n-2) a_n(x) y^(n-1) > (2) so Sum(n=0 to inf) a_n(x+1) y^n = -2 a_0(x) y^(-1) - a_1(x) + Sum(n=3 to > inf) (n-2) a_n(x) y^(n-1) (3) Since there is no y^(-1) term on the left a_0(x) = 0 (all x) (4) So Sum(n=1 to inf) a_n(x+1) y^n = - a_1(x) + Sum(n=3 to inf) (n-2) > a_n(x) > y^(n-1) (5) But now there is no y^0 = 1 term on the left, so a_1(x) = 0 (all x) (6) So: Sum(n=2 to inf) a_n(x+1) y^n = Sum(n=3 to inf) (n-2) a_n(x) y^(n-1) > (7) On the right let n -> n+1 Sum(n=2 to inf) a_n(x+1) y^n = Sum(n=2 to inf) (n-1) a_(n+1)(x) y^n > (8) > a_n(x+1) = (n-1) a_(n+1)(x) n = 2, 3, ... (9) Rewrite this as: a_(n+1)(x) = a_n(x+1) / (n-1) n = 2, 3, ... (10) n = 2 > a_3(x) = a_2(x+1) = g(x+1) (so a_2(x) = g(x) ) n = 3 > a_4(x) = a_3(x+1) / 2 = g(x+2) / 2 n = 4 > a_5(x) = a_4(x+1) / 3 = g(x+3) / 3! . > . > . > so we see a_n(x) = g(x+n-2) / (n-2)! n = 2, 3, 4, ... and a_0(x) = a_1(x) = 0 Put this back into (1): f(x, y) = Sum(n=2 to inf) g(x+n-2) y^n / (n-2)! (11) and there's my fairly general solution. Bob- Masquer le texte des messages pr.8ec.8edents - - Afficher le texte des messages pr.8ec.8edents - Bonsoir Bob, I will propose you three different homemade solutions: > f1(x,y) = k*y^2*a^x*exp(a*y) , > f3(x,y) = k*y^2*sin(y+Pi*x/2) All built from a formal form using partial > derivation :f(x,y) = k*y^2*(d/dy)^(x) o g(y) k a constant , Alain Alain, In my solution, factor out y^2 and let n -> n+2: f(x, y) = y^2 Sum(n=0 to inf) g(x+n) y^n / n! (1) Take g(x) = k a^x Then f(x, y) = k y^2 Sum(n=0 to inf) a^(x+n) y^n / n! f(x, y) = k y^2 a^x Sum(n=0 to inf) (a*y)^n / n! f(x, y) = k y^2 a^x exp(a*y) (2) which is your f1(x, y). Next take: g(x) = k exp(i Pi*x/2) (3) This leads to: f(x, y) = k y^2 exp(i Pi*x/2) Sum(n=0 to inf) (y*exp(i Pi/2)^n / n! But exp(i Pi/2) = i , so f(x, y) = k y^2 exp(i Pi*x/2) Sum(n=0 to inf) (i*y)^n / n! f(x, y) = k y^2 exp(i Pi*x/2) exp(i*y) f(x, y) = k y^2 exp[i*(y + Pi*x/2)] f(x, y) = k y^2 cos(y + Pi*x/2) + i k y^2 sin(y + Pi*x/2) which gives two real solutions, your f3(x, y) and f4(x, y) = k y^2 cos(y + Pi*x/2) I don't immediately see what g(x) gives your f2(x, y). Bob- Masquer le texte des messages pr.8ec.8edents - - Afficher le texte des messages pr.8ec.8edents - > The continuous derivation solution I gave is a very > general one. > Bases I use are shared by many people: > (d/dy)^x o exp(a*y) = a^x*exp(a*y) > continuous extensions of integer known cases. A more general functional form could be: > f(x,y) = m(y)*(d/dy)^phi(x) o g(y) , with phi(h(x))=phi(x)+1 > Then we've got : > f(h(x),y)/m(y) = d/dy (f(x,y)/h(y)) > ............... > Your ideas,comments,views are welcome , Alain > Alain, Your > does not appear to be a solution to your equation. I get a y^(5-x) term > on the right, and a y^(6-x) on the left. Bob Sorry, I made a stupid mistake and forgot it was f(x+1, y) on the left. :-( Bob === Subject: Re: The complete infinite binary tree has only countably many Nntp-Posting-Host: hera.cwi.nl ... > Less than omega is not enough to evade the case that the diagonal > of the list > > 0.0 > 0.1 > 0.11 > 0.111 > ... > is not distinct from every line. > > Pray tell to which line it is equal. > > If all lines exist, then N is in a line. > > By which rule? (And we were talking about the diagonal...) > > By the rule that the diagonal cannot exist without lines. The diagonal > is made from the last digits of the lines. If the diagonal were longer > than every line, then it could not exist. Wrong. > The diagonal has more digits > than every line. Right. > Hence there must be a line that has more digits than > every line. Or actual infinity is wrong a concept. On what is the hence based? But as WM has left this thread (as is customary when it becomes clear that he cannot win the argument except by circularity), I will also leve the thread. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: EMIS has my old paper back up? Nntp-Posting-Host: hera.cwi.nl > I don't have PostScript - Last time I looked, it costs way too much. > > I don't understand what you mean. The language spec is free, isn't it? > > As far as software is concerned, > > http://pages.cs.wisc.edu/~ghost/gsview/index.htmhttp://en.wikipedia.org/w= > iki/Ghostscript > > As near as I can recall, I ran into the price considerations while > investigating: > > Using Acrobat Distiller as a > General Purpose PostScript Computer Using Acrobat Distiller as a postscript viewer is certainly overkill... It provided an expensive and cumbersome way to produce pdf files, and contained more than the standard share of bugs in Adobe software. Yes, at one time I was forced to maintain it at our institute. ghostscript/ghostview has been available on Unix from about 1986, so much so that even TeX does use it internally already quite a long time (at least on Unix/Linux systems). -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: EMIS has my old paper back up? > Using Acrobat Distiller as a postscript viewer is certainly overkill... > It provided an expensive and cumbersome way to produce pdf files, and > contained more than the standard share of bugs in Adobe software. Yes, > at one time I was forced to maintain it at our institute. For those stuck in Windows, PDFCreator is a handy ghostscript front-end. http://en.wikipedia.org/wiki/PDFCreator http://www.pdfforge.org/products/pdfcreator http://sourceforge.net/projects/pdfcreator/ === Subject: Re: JSH: EMIS has my old paper back up? > On May 9, 10:48=A0pm, Mariano Su=E1rez-Alvarez > > > Actually, the most common source of that particular problem occurs > when, while typing the *.tex source, one inadvertedly types > > instead of ., usually because the immediate prior character requires > the shift key. It happens to me a lot when a period follows an in- > line formula (which ends in $). > > Hmm. That may be the case in an USian keyboard layout, I guess. > > The usual keyboard (US) does not have an opening question mark. I saw your correction. But what Mariano meant was that possibly on a US keyboard the period was paired with the greater than sign. This is not universally true on all keyboards. And I think that Mariano's explamation initial position in the body of a mail message by >. This is due to the original Unix mailbox format where all messages are lumped after each other delivery software. (And I still do use such software...) -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: EMIS has my old paper back up? Nntp-Posting-Host: hera.cwi.nl ... > They trashed a decades worth of papers, which should be disquieting to > people who publish in electronic only journals. > > EMIS saved its archives though, and now it seems saw fit to save my > paper as well. Oh no. EMIS saved only two parts of a single issue of the journal. > That paper historically may be considered one of the biggest in > mathematical history. Probably because the author of it acknowledged that it contained errors before it was published, but refused to retract it. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Learning homology (Academic guidance will be appreciated) > On May 11, 2:14pm, Don I will really appreicate if you give me some > guidance on learning homology/cohomology. I was satisfied with the first half of Massey's A > basic course in algebraic topology. However, for the second part, the book starts with > a singular cubical homology, but I found it's hard to > find a reference for that part. For instance, the > book introduces degenerate cubes which I could not > find in Hatcher's or Vick's book. So, if I am stuck > in Massey's book, it is not easy to find other > references dealing the same ideas. For experts, I heard they see the singular cubical > homology and singular simplicial homology in a > consistent way, but they look two different things > for beginners like me. I think it is a safer way to start with Vick's or > Hatcher's book (homology/cohomology part) because > singular simplicial homology seems more standard way, > so I am able to use the other book as references. Any advice? Best wishes, > Don Have you tried Hatcher's book? It's treatment of > homology is quite > nice. -- m Not yet. To study homology part, I am planning to read the Hatcher's book in parallel with the Vick's book. Hope this combination works for me. Don. === Subject: Re: Olympic medals in sports related to Nobel medals in science Re: Alan Nntp-Posting-Host: hera.cwi.nl You are going over the top, Archie. > Now we ask, how many gold medals were won by Jews in the last Olympics > and how many were won by NonJews? How do you determine the religion of somebody when you do not know him? If you state by name, than I would state that you also are a jew because you also have a German name. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Nobel prizes have been slanted for over 100 years Re: Olympic medals Nntp-Posting-Host: hera.cwi.nl ... > Now we ask, how many gold medals were won by Jews in the last Olympics > and how many were won by NonJews? > > How do you determine the religion of somebody when you do not know him? > > The names of every gold medal winner in the 2008 Olympics is a public > record. That is not an answer to my question. > Because over 50% of the Nobel Science prizes are Jewish whereas the > Jewish > population is only 1% of world population, means the awarding system > is not fair. Or because more Jewish people were devoted to the sciences. Just like the majority comes from western europe... > Dik, I was also ignorant of what the definition of Jew was, until a > Jewish friend > informed me. It matters not whether you believe in Judiaism, but only > matters as > to whether you are a descendent of a Jewish mother in your lineage. I knew that rule. It is the rule according to the Jewish traditional laws. Not everybody goes with those rules. I know many people who would be Jewish according to those laws but do not consider themselves Jewish. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Nobel prizes have been slanted for over 100 years Re: Olympic medals in sports related to Nobel medals in science > Dik, I was also ignorant of what the definition of Jew was, until a > Jewish friend > informed me. It matters not whether you believe in Judiaism, but only > matters as > to whether you are a descendent of a Jewish mother in your lineage. Circular definition. How do you distinguish a Jewish mother from a non-Jewish mother? -- hz === Subject: origin of n5 lattice --- where in Dedekind's 1900 paper? posting-account=YftvUQoAAAD36OR-en6JDbmNKuS53Hzz Gecko/2008092417 Firefox/3.0.3,gzip(gfe),gzip(gfe) A lattice is modular if and only if it does not contain the N5 lattice. A lattice is distributive if and only if it does not contain the N5 lattice or the M3 lattice. My question is, what is the original source of the N5 lattice? Salii 1988 seems to indicate that it is Dedekind 1900. However, that paper is in German and I don't really understand German. Could anyone who does understand tell me where in that paper Dedekind describes the N5 lattice? The n5 and m3 lattices are illustrated here: http://banyan.cm.nctu.edu.tw/~dgreenhoe/n5origin.pdf Here are some references (see also n5origin.pdf): Dedekind, Richard: Ueber die von drei Moduln erzeugte Dualgruppe. Mathematische Annalen, 53 January 8 1900, 371[CapitalEth]403 ?URL: http://resolver.sub.uni-goettingen.de/purl/?GDZPPN002257947?, Regarding the Dual Group Produced by three-Modules SaliØi, V''a`cheslav Nikolaevich: Lattices with Unique Complements. Volume 69, Translations of mathematical monographs. Providence: American Mathematical Society, 1988 translation of Reshetki s edinstvennymi dopolneni''am` i, ISBN 0821845225 Dan === Subject: Re: origin of n5 lattice --- where in Dedekind's 1900 paper? posting-account=UEdI2woAAABTdf0LcYPp9Pe_P88Ck_vM Gecko/20011019 Netscape6/6.2,gzip(gfe),gzip(gfe) [...] Dedekind, Richard: Ueber die von drei Moduln erzeugte Dualgruppe. > Mathematische Annalen, 53 January 8 1900, 371[CapitalEth]403 > (URL: http://resolver.sub.uni-goettingen.de/purl/?GDZPPN002257947), > Regarding the Dual Group Produced by three-Modules > If the German title is correct, the English title should be Regarding the Dual Group Generated by Three Modules. The digitized version at the above URL seems to be inaccessible at present. Many university libraries should have bound issues of Mathematische Annalen, however. If your focus is on the historical development of a particular field of mathematics, you should acquire enough of German, French, etc. to read the original publications with the help of a dictionary. You will know that only a small subset of the grammar and vocabulary is required for this. Martin. === Subject: Re: origin of n5 lattice --- where in Dedekind's 1900 paper? posting-account=YftvUQoAAAD36OR-en6JDbmNKuS53Hzz Gecko/2008092417 Firefox/3.0.3,gzip(gfe),gzip(gfe) > (URL:http://resolver.sub.uni-goettingen.de/purl/?GDZPPN002257947), > The digitized version at the above URL seems to be > inaccessible at present. I just tried this link and it seems currently to work with no problem. In fact, that link gave me two other links, and I was able to download Dedekind's paper from both. If someone was interested in the paper and was not able to download before, maybe try again? > If the German title is correct, the English title should > be Regarding the Dual Group Generated by Three Modules. Yes, as in my original post, the title can be translated Regarding the Dual Group Produced by three-Modules. Or according to Google Translate, About three modules produced dual Group. But I think I like your Generated better than my Produced. > If your focus is on the historical development of a > particular field of mathematics ... Although I do value such books, it is not my particular focus. I never plan to write such a book. Dan [...] Dedekind, Richard: Ueber die von drei Moduln erzeugte Dualgruppe. > Mathematische Annalen, 53 January 8 1900, 371[CapitalEth]403 > (URL:http://resolver.sub.uni-goettingen.de/purl/?GDZPPN002257947), > Regarding the Dual Group Produced by three-Modules If the German title is correct, the English title should be Regarding > the Dual Group Generated by Three Modules. The digitized version at the above URL seems to be inaccessible at > present. Many university libraries should have bound issues of > Mathematische Annalen, however. If your focus is on the historical development of a particular field > of mathematics, you should acquire enough of German, French, etc. to > read the original publications with the help of a dictionary. You will > know that only a small subset of the grammar and vocabulary is > required for this. Martin. === Subject: Re: origin of n5 lattice --- where in Dedekind's 1900 paper? http://coding.derkeiler.com/pdf/Archive/General/comp.theory/2009-04/msg00122 .pdf === Subject: Re: polynomials closed under multiplication ? > most of you here are probably familiar with the > famous brahmagupta-fibonacci identity (a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2 which is the classical example of an integer > polynomial closed under multiplication. so , the logical question is : what integer polynomials of degree >= 3 are closed > under multiplication ? apart from the trivial cases ( e.g. representing all > integers by sum of cubes ) -> none ?? is this a FAQ ? > tommy1729 Of course I am familar. Have not read [P==NP]?http://coding.derkeiler.com/pdf/Archive/General/comp.theory/2009-04/ m sg00122.pdf :[Martin Musatov] === Subject: Re: polynomials closed under multiplication ? posting-account=Z3AipgkAAABkoMfyNwddSxsYhXHi5CDt MathPlayer 2.10d; .NET CLR 1.1.4322; PeoplePal 3.0),gzip(gfe),gzip(gfe) > most of you here are probably familiar with the famous brahmagupta-fibonacci identity (a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2 which is the classical example of an integer polynomial closed under multiplication. so , the logical question is : what integer polynomials of degree >= 3 are closed under multiplication ? apart from the trivial cases ( e.g. representing all integers by sum of cubes ) -> none ?? is this a FAQ ? > tommy1729 The formula you produce -- x^2 + y^2 is something called the norm form of the quadratic field Q(i) where i is the square root of -1. It is the norm of x + iy, where x and y are rational. If we restrict this to x and y integers, which you seem to want to do, then you have norm form of the Gaussian integers, the set of algebraic integers contained inside Q(i), since {1, i} is something called an integeral basis. Every algebraic number field produces a norm form - actually infinitely many, all connected by a particular equivalence relation. Examples of quadratic fields are x^2 + ry^2 for any square-free r, positive or negative. I believe that in a reply to my recent reply to a JSH thread, juandiego produced a cubic form. Norm forms always have same degee as the number of variables, and in these cases, there is actually a simple formula for how to produce the product. As you point out, something like x^2+y^2+z^2+w^2 works over the integers. Actually there is a formula for this one too, as it is the norm form for quaternions, and this norm is also multiplicative. Sums of 9 cubes also works over integers, since every integer is the sum 9 cubes. Another collection of examples comes from the work of Albrecht Pfister who showed that sums of 2^n squares are closed under multiplication in any field, which means for our purposes that you can add, subtract, multiply, and also divide by any non-zero element. However, it fails if you aren't allowed to divide. Hope this is of interest, Achava === Subject: Re: Cantor's argument is erroneous <87ab5yo00x.fsf@phiwumbda.org> <87tz46cflg.fsf@phiwumbda.org> <87r5z7thw5.fsf@phiwumbda.org> <87iqkjt4t3.fsf@phiwumbda.org> <87d4art0u0.fsf@phiwumbda.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > Specifically, rules of inference pertain *only* to formulas that are > *theorems*. But since theorems are defined from axioms, it isn't much > an exaggeration to say rules of inference pertains to *axioms*. [And that's > not just (in your words) ordinary convention but it's the very heart of the > definition of proofs - in FOL=]. > Perhaps the best example comes when there are *no* (non-logical) axioms at all. > Logic text books are full of examples asking students to show that > certain statements are theorems of FOL=, with *no* axioms. > (the language is implicitly or explicitly assumed to be such that > the formla in question is well formed). e.g., if wotsit is a > 1-place predicate, then > (all x. not wotsit(x)) -> not (some y wotsit(y)) > is provable in whatever system of FOL you want, from no > non-logical axioms. This sounds similar to the earlier comment by MoeBlee about statements involving thingamajig being provable. Both Smaill and MoeBlee argue that in the theory with no axioms, any tautology is provable, even those involving symbols in the language that aren't otherwise defined. Counterintuitive, yes -- but Smaill and MoeBlee appear to believe that this is the only way that makes sense. Once again, I accept that there can be vacuous truths, so that from the theory whose lone axiom is Ax (~wotsit(x)), one can prove Ax (wotsit(x) -> phi) for any formula phi (in the right language, of course). But what I still don't accept is that one can prove theorems, even if merely tautologies, with certain symbols if there is no axiom with that symbol. And even if we do allow tautologies to be proved from the null axiom theory, then I disagree that it's the only way to do it -- Nguyen's method makes sense as well. And of course, Ax (x+y=0) isn't even a tautology, so that's even worse than claiming that only tautologies with the symbol + can be proved if no axiom mentions +. I'd accept that the theory can prove Ax (x+y=x+y) well before I'd accept that it can prove Ax (x+y=0)! > You seem to be saying that without non-logical axioms, *nothing* > is provable in FOL= -- that's just no[t] so. I'd be more likely to accept this standard argument if the standard theorists didn't riducule Ross Finlayson when he mentions the null axiom theory in his posts. So far, what I've seen is that if RF (or any so-called crank) is the one proposing the null axiom theory, then nothing can be proved from it, but if the standard theorists are the ones proposing the null axiom theory, then suddenly it can prove infinitely many statements. === Subject: Re: Cantor's argument is erroneous <87tz46cflg.fsf@phiwumbda.org> <87r5z7thw5.fsf@phiwumbda.org> <87iqkjt4t3.fsf@phiwumbda.org> <87d4art0u0.fsf@phiwumbda.org> posting-account=yKimjgoAAACk5WwPVD4l9HmbpoR6Hmy4 Presto/2.1.1,gzip(gfe),gzip(gfe) > Specifically, rules of inference pertain *only* to formulas that are > *theorems*. But since theorems are defined from axioms, it isn't much > an exaggeration to say rules of inference pertains to *axioms*. [And that's > not just (in your words) ordinary convention but it's the very heart of the > definition of proofs - in FOL=]. > Perhaps the best example comes when there are *no* (non-logical) axioms at all. > Logic text books are full of examples asking students to show that > certain statements are theorems of FOL=, with *no* axioms. > (the language is implicitly or explicitly assumed to be such that > the formla in question is well formed). e.g., if wotsit is a > 1-place predicate, then > (all x. not wotsit(x)) -> not (some y wotsit(y)) > is provable in whatever system of FOL you want, from no > non-logical axioms. This sounds similar to the earlier comment by MoeBlee about > statements involving thingamajig being provable. Both Smaill > and MoeBlee argue that in the theory with no axioms, any > tautology is provable, even those involving symbols in the > language that aren't otherwise defined. Counterintuitive, yes -- but Smaill and MoeBlee appear to > believe that this is the only way that makes sense. How is this counter-intuitive? Also, they are not saying, as far as I understand, this is the only way that makes sense. They are saying this is how things are usually done. Both ways may make sense but they can't both be the way things are usually done. Can they? Also, what do you mean by the theory with no axioms? Is there only one? What do you think a theory is? A set of axioms? Isn't a theory a set of axioms and a language? > Once again, I accept that there can be vacuous truths, so that > from the theory whose lone axiom is Ax (~wotsit(x)), one > can prove Ax (wotsit(x) -> phi) for any formula phi (in the > right language, of course). But what I still don't accept is > that one can prove theorems, even if merely tautologies, with > certain symbols if there is no axiom with that symbol. Well, then it seems like you are simply mistaken. There is no reason that a symbol in the theory's language must be included in the axioms. And there's no reason that a symbol in the theory's language can't be used to form formulas or theorems. So what is your objection? L(anguage) = {0,+} A(xioms) = {(Ax)(Ay)(x=y)} T(heory) = (L,A) T |- (Ax)(Ay)x+y=0 Right? I mean, what exactly are you objecting to here?? I have literally no idea. > And > even if we do allow tautologies to be proved from the null > axiom theory, then I disagree that it's the only way to do > it -- Nguyen's method makes sense as well. Please explain preciesly what Nam's method is, and in what sense people have said it does not make sense. > And of course, Ax (x+y=0) isn't even a tautology, so that's > even worse than claiming that only tautologies with the > symbol + can be proved if no axiom mentions +. I'd accept > that the theory can prove Ax (x+y=x+y) well before I'd > accept that it can prove Ax (x+y=0)! Well it seems to me that what I posted above definitely proves both of those. Why not?? > You seem to be saying that without non-logical axioms, *nothing* > is provable in FOL= -- that's just no[t] so. I'd be more likely to accept this standard argument if the > standard theorists didn't riducule Ross Finlayson when he > mentions the null axiom theory in his posts. So far, what I've seen is that if RF (or any so-called > crank) is the one proposing the null axiom theory, then > nothing can be proved from it, but if the standard theorists > are the ones proposing the null axiom theory, then suddenly > it can prove infinitely many statements. You're got to be joking. RF proposes that a theory with no logical or non-logical axioms can prove things like all infinite sets are equivalent. Can you please acknowledge that this is not even wrong --- it just makes no sense at all! This has got nothing to do with anything being discussed here. === Subject: Re: Cantor's argument is erroneous <87r5z7thw5.fsf@phiwumbda.org> <87iqkjt4t3.fsf@phiwumbda.org> <87d4art0u0.fsf@phiwumbda.org> <87r5yv2hxn.fsf@phiwumbda.org> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) You don't accept it simply because there's a slew of folks saying Nam > is wrong and you want Nam to be right. Furthermore, the very reason that he wants Nam to be right is *because* there is a slew of folks saying he's wrong. Marshall === Subject: Re: Cantor's argument is erroneous > You don't accept it simply because there's a slew of folks saying Nam > is wrong and you want Nam to be right. > Furthermore, the very reason that he wants Nam to be right > is *because* there is a slew of folks saying [Nam's] wrong. At least a _consistent_ pattern of behavior. :-) Herb === Subject: Re: Cantor's argument is erroneous <87iqkjt4t3.fsf@phiwumbda.org> <87d4art0u0.fsf@phiwumbda.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > You don't accept it simply because there's a slew of folks saying Nam > is wrong and you want Nam to be right. > Furthermore, the very reason that he wants Nam to be right > is *because* there is a slew of folks saying he's wrong. Only half-right. It's because there's a slew of folks saying that Nam is wrong and there's a slew of folks saying that Ross Finlayson is wrong as well! RF once came up with a null axiom theory -- i.e., a theory with no axioms at all. The standard theorists then told RF that such a theory was pointless, because without axioms there could be no theorems. Now, not only can theorems about + such as Ax (Ay (x+y=0)) be proved in theories with no axioms mentioning +, suddenly theorems can be proved without axioms at all. RF can't prove theorems without axioms in his null axiom theory, but the standard theorists can. So now, I ask of the standard theorists, can theorems be proved in the null axiom theory, a theory without any axioms at all? If no, then Nguyen is right. If yes, then RF is right. The standard theorists would have it so that both Nguyen and RF are wrong. I'm sorry, but the standard theorists can't have it both ways. Either the null axiom theory has theorems and RF is right, or it has no theorems and Nguyen is right. I'll accept that either Nguyen or RF is wrong , as soon as the standard theorists accept that one is right . > Furthermore, the very reason that he wants Nam to be right > is *because* there is a slew of folks saying he's wrong. And the very reason that the standard theorists want Nguyen to be wrong is because there is a slew of folks saying he's wrong, and the very reason that the standard theorists want RF to be wrong is because there is a slew of folks saying he's wrong as well. The standard theorists always seem to stick together. Standard logic dictates that exactly one of Nguyen and RF is right, and exactly one of them is wrong. I'll let the standard theorists decide which is which. (Whichever one is wrong might be right using a different type of logic, such as one in which language must be defined before the theory. But exactly one of them is right using standard logic only.) === Subject: Re: Cantor's argument is erroneous <87d4art0u0.fsf@phiwumbda.org> posting-account=yKimjgoAAACk5WwPVD4l9HmbpoR6Hmy4 Presto/2.1.1,gzip(gfe),gzip(gfe) You don't accept it simply because there's a slew of folks saying Nam > is wrong and you want Nam to be right. > Furthermore, the very reason that he wants Nam to be right > is *because* there is a slew of folks saying he's wrong. Only half-right. It's because there's a slew of folks saying > that Nam is wrong and there's a slew of folks saying that > Ross Finlayson is wrong as well! RF once came up with a null axiom theory -- i.e., a theory > with no axioms at all. The standard theorists then told RF > that such a theory was pointless, because without axioms > there could be no theorems. If this is what they told him, then they were being sloppy and incorrect. However with you're track record you're most likely just misrepresenting things. > Now, not only can theorems about + such as Ax (Ay (x+y=0)) > be proved in theories with no axioms mentioning +, > suddenly theorems can be proved without axioms at all. RF > can't prove theorems without axioms in his null axiom > theory, but the standard theorists can. Mark well the distinction between logical and non-logical axioms, please. A theory with logical axioms but no non-logical axioms can prove some stuff (the consequences of the logical axioms). RF proposed a theory with neither , and claimed it was able to prove all sorts of fantastic things. > So now, I ask of the standard theorists, can theorems be > proved in the null axiom theory, a theory without any axioms > at all? What do you mean by THE null axiom theory? There are infinitely many theories with logical axioms but no non-logical axioms. There's only one theory with no axioms whatsoever, and it proves nothing. > If no, then Nguyen is right. > If yes, then RF is right. Or your question is badly posed because it is based on a misunderstanding or a misrepresentation! Color me surprised! > And the very reason that the standard theorists want Nguyen > to be wrong is because there is a slew of folks saying he's > wrong, and the very reason that the standard theorists want > RF to be wrong is because there is a slew of folks saying > he's wrong as well. The standard theorists always seem to > stick together. Will your head explode if you ever accept that someone can be wrong about something? === Subject: Re: Cantor's argument is erroneous <87iqkjt4t3.fsf@phiwumbda.org> <87d4art0u0.fsf@phiwumbda.org> <87tz3yniog.fsf@phiwumbda.org> <87r5yv2hxn.fsf@phiwumbda.org> said: > You don't accept it simply because there's a slew of folks saying > Nam is wrong and you want Nam to be right. > Furthermore, the very reason that he wants Nam to be right > is *because* there is a slew of folks saying he's wrong. Only half-right. It's because there's a slew of folks saying that Nam > is wrong _and_ there's a slew of folks saying that Ross Finlayson is > wrong as well! RF once came up with a null axiom theory -- i.e., a theory with no > axioms at all. The standard theorists then told RF that such a theory > was pointless, because without axioms there could be no theorems. That depends on exactly what the null axiom theory is supposed to be; RF was not terribly clear on the point as I recall. If, by the null axiom theory, RF means a theory that lacks even logical axioms, then of course nothing whatever be proved. However, if RF only means a theory with no *non-logical* axioms (e.g.,, the pure predicate calculus), then the theory has infinitely many theorems. > Now, not only can theorems about + such as Ax (Ay (x+y=0)) be proved > in theories with no axioms mentioning +, If, and only if, + is in the languages of the theories in question. > suddenly theorems can be proved without axioms at all. If, and only if, without axioms means without non-logical axioms. > RF can't prove theorems without axioms in his null axiom theory, but > the standard theorists can. Ross can too if, again, the null axiom theory simply means that his theory has no non-logical axioms. > So now, I ask of the standard theorists, can theorems be proved in the > null axiom theory, a theory without any axioms at all? As noted. === Subject: Re: Cantor's argument is erroneous > Cantor's argument is erroneous and its adoption leads > to unsound > mathematics. The basic idea in the argument is that there is no > bijection between > the set of counting numbers and the set of infinite > binary strings. > But such a bijection exists, it can be expressed in > terms of limit > points, or by transfinite induction; informally, it > can be defined as > the correspondence between the paths and the leaf > (i.e. limit) nodes > in the infinite binary tree. This invalidates all > results relating to > Cantor's transfinite. In particular, it is invalid to state that the set of > infinite binary > strings is uncountable. It is countable, being in > bijection with a > subset of a countable set, the set of nodes in the > infinite binary > tree. The other option is that one drops the > countability of infinite > sets completely, but I can see no advantage in > banning the > transfinite. The problem is possibly much broader, and deeper, > because it is the > very soundness of archimedean arithmetic that seems > at stake here. I > say soundness because from the archimedean > framework a tension > results, between computability and tractability, that > manifests a > deeper tension between sound and unsound mathematics, > and then even > logic. -LV Agreed. While I feel for Cantor and what he must have went through as a tortured genius we have to at some point stop marvelling at our ignorance. Perhaps if Cantor and Boole had known http://coding.derkeiler.com/pdf/Archive/General/comp.theory/2009-04/msg00122 . pdf[P==NP] we would have been much more advanced in our studies up to this point.[Martin Musatov]. === Subject: Re: Cantor's argument is erroneous <16993474.91393.1242092267155.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) On 12 Mai, 03:37, Martin Michael Musatov to unsound > mathematics. The basic idea in the argument is that there is no > bijection between > the set of counting numbers and the set of infinite > binary strings. > But such a bijection exists, it can be expressed in > terms of limit > points, or by transfinite induction; informally, it > can be defined as > the correspondence between the paths and the leaf > (i.e. limit) nodes > in the infinite binary tree. This invalidates all > results relating to > Cantor's transfinite. In particular, it is invalid to state that the set of > infinite binary > strings is uncountable. It is countable, being in > bijection with a > subset of a countable set, the set of nodes in the > infinite binary > tree. The other option is that one drops the > countability of infinite > sets completely, but I can see no advantage in > banning the > transfinite. The problem is possibly much broader, and deeper, > because it is the > very soundness of archimedean arithmetic that seems > at stake here. I > say soundness because from the archimedean > framework a tension > results, between computability and tractability, that > manifests a > deeper tension between sound and unsound mathematics, > and then even > logic. -LV Agreed. While I feel for Cantor and what he must have went through as a tortured genius we have to at some point stop marvelling at our ignorance. Perhaps if Cantor and Boole had knownhttp://coding.derkeiler.com/pdf/Archive/General/comp.theory/2009-04/m.. . [P==NP] we would have been much more advanced in our studies up to this point.[Martin Musatov]. In contrast to that so-called P=NP proof, the Cantor proof is at least a valid chain of arguments, even if some here seem unable to adopt their intuition to the logical consequences... === Subject: Re: Cantor's argument is erroneous <87tz4ab9s6.fsf@phiwumbda.org> <878wllxa2z.fsf@phiwumbda.org> <87ab5yo00x.fsf@phiwumbda.org> <87tz46cflg.fsf@phiwumbda.org> <87r5z7thw5.fsf@phiwumbda.org> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > The cornerstone of the the standard theorist Chris > Menzel's argument is that one either states a language > L first, then state a set of axioms in the language L, > or do an alternate that seems completely back@$$ward. > No. Chriz Menzel is simply passing comment, not making an argument. OK. And the comment he passed that that Nguyen's way of doing it is back@$$ward. > Your attempts to characterize the arguments in this subthread are > bizarrely mistaken. Try reading it again without assuming that one > side must be composed of standard theorist zealots trying to supress > anything non-standard. How about this: one side is composed of posters trying to Nguyen is wrong. It would be more balanced if some posters supported Nguyen's axioms-first method and some posters supported the language-first method, but no -- they all seem to be against Nguyen. Nguyen's opponents have made it appear that they are the side being open-minded to Nguyen's method and that Nguyen needs to be more open-minded, but really, it's Nguyen's opponents who need to be more open-minded. Comments such as Menzel's back@$$wards and Newman's calling Nguyen a crank, and earlier in this thread, an @$$hole, make it clear that their side is the side having trouble accepting Nguyen. === Subject: Re: Cantor's argument is erroneous <878wllxa2z.fsf@phiwumbda.org> <87ab5yo00x.fsf@phiwumbda.org> <87tz46cflg.fsf@phiwumbda.org> <87r5z7thw5.fsf@phiwumbda.org> posting-account=yKimjgoAAACk5WwPVD4l9HmbpoR6Hmy4 Presto/2.1.1,gzip(gfe),gzip(gfe) The cornerstone of the the standard theorist Chris > Menzel's argument is that one either states a language > L first, then state a set of axioms in the language L, > or do an alternate that seems completely back@$$ward. > No. Chriz Menzel is simply passing comment, not making an argument. OK. And the comment he passed that that Nguyen's way of doing > it is back@$$ward. Your attempts to characterize the arguments in this subthread are > bizarrely mistaken. Try reading it again without assuming that one > side must be composed of standard theorist zealots trying to supress > anything non-standard. How about this: one side is composed of posters trying to > Nguyen is wrong. It would be more balanced if some posters > supported Nguyen's axioms-first method and some posters > supported the language-first method, but no -- they all seem > to be against Nguyen. Is this an unacceptable state of affairs in your mind: someone says something, and everyone who responds thinks that they are wrong? What happens when someone really is wrong about something? Is it unacceptable for this to be pointed out? is it possible in your strange little world for someone to be simply mistaken? You think telling someone that you think they are mistaken is suppressing anything they write? Wow. > Nguyen's opponents have made it appear that they are the side > being open-minded to Nguyen's method and that Nguyen needs to > be more open-minded, but really, it's Nguyen's opponents who > need to be more open-minded. Comments such as Menzel's > back@$$wards and Newman's calling Nguyen a crank, and > earlier in this thread, an @$$hole, make it clear that > their side is the side having trouble accepting Nguyen. What precisely is it that you think that Nam is saying? === Subject: Re: Cantor's argument is erroneous > Hope the above helps. Hint: ...arguing with the crank is useless, because he will invariably dismiss > all evidence or arguments which contradict his unconventional belief. Source: > http://en.wikipedia.org/wiki/Crank_(person) Do you happen to know that those who were part of Police Vigilante, Commissar, or Inquisitors, would never consider themselves as criminals? In fact it wouldn't be a surprise if some of them would consider themselves as Saints! In the land of reasoning, the cranks are those whom we should fear less. What harm could they really bring to the foundations of reasoning? Those who have a mind of an Inquisitor should be the ones whom we should fear the most! -- To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician. (Shoenfield, Mathematical Logic) === Subject: Re: Cantor's argument is erroneous <87d4art0u0.fsf@phiwumbda.org> <%vONl.26977$0S.20110@newsfe22.iad> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > ...arguing with the crank is useless, because he will invariably dismiss > all evidence or arguments which contradict his unconventional belief. > Source: >http://en.wikipedia.org/wiki/Crank (person) > Do you happen to know that those who were part of Police Vigilante, > Commissar, or Inquisitors, would never consider themselves as criminals? > In fact it wouldn't be a surprise if some of them would consider themselves > as Saints! Lately, Newman has been calling everyone who disagrees with him a crank or an even less complimentary word. He has devoted most of his recent posts to telling these so-called cranks to shut up, and warning his allies of the futility of even arguing with us cranks (essentially telling them to shut up as well). Of course, Newman is considering anyone who disagrees with him to have an unconventional belief. Anyone who fails to accept Newman as being infallible on set theory and mathematics will receive this same treatment. > Those who have a mind of an Inquisitor should be the ones whom we should > fear the most! I couldn't agree more! === Subject: Re: Cantor's argument is erroneous <%vONl.26977$0S.20110@newsfe22.iad> posting-account=yKimjgoAAACk5WwPVD4l9HmbpoR6Hmy4 Presto/2.1.1,gzip(gfe),gzip(gfe) ...arguing with the crank is useless, because he will invariably dismiss > all evidence or arguments which contradict his unconventional belief. > Source: >http://en.wikipedia.org/wiki/Crank (person) > Do you happen to know that those who were part of Police Vigilante, > Commissar, or Inquisitors, would never consider themselves as criminals? > In fact it wouldn't be a surprise if some of them would consider themselves > as Saints! Lately, Newman has been calling everyone who disagrees with him a > crank or an even less complimentary word. He has devoted most > of his recent posts to telling these so-called cranks to shut > up, and warning his allies of the futility of even arguing with > us cranks (essentially telling them to shut up as well). Of course, Newman is considering anyone who disagrees with him to > have an unconventional belief. Anyone who fails to accept > Newman as being infallible on set theory and mathematics will > receive this same treatment. Oh, don't be asinine. === Subject: Re: Cantor's argument is erroneous > in most of these threads, Herb hasn't really entered into the > conversation at all, so no disagreement *could* take place. Right. You know, time is precious. Herb === Subject: Re: Cantor's argument is erroneous > said: > said: > As we surely agree, there is no ordinary convention that the > rules of inference pertain only to formulas with only symbols > mentioned in some non-logical set of axioms. If Nam disputes > that, then he's just wrong. > In so far as you and others have not been able to refute my > specific examples at all (other than saying something like > Shoenfield, Roger, etc... didn't say that, or invoking > off-the-topic non-FOL= logic frameworks) then I'm correct and you > and other opponents of mine are *not*. > Also you (MoeBlee) should have realized that as far as FOL= *proof > in a theory* is concerned, *rules of inference don't pertain to > just _any_ formula*! > They pertain to any formula in the language being considered; > Like I explained before, to any *theorem*-formulas, *not just* any > formulas you considered. For instance, let L = L(T = {Ax[Steel(x)]} > then Ex[Steel(x)] is pertained by rule(s) of inference. On the other > hand, e.g., as much as you'd like to consider F = Ex[Wood(s)] as > *any formula*, F is simply *not* pertained by the rule(s) as a theorem > of this T. > You're simply wrong here. > Not that Alan can't answer for himself, but I can't make any sense of > the preceding paragraph. In standard first-order logic, rules of > inference apply (i.e., pertain) to every formula of the language. For > instance, the rule of Modus Ponens is that, for *any* formulas A and B, > given A and (A -> B), you may infer B. Are you denying this? If so, > you're simply wrong here. > With due respect, no. Rules of inference are specific for theorems of > of a formal system. With due respect, no. You are completely confused. > If we use the rules for something else (which I suppose we could) then > we're *not* talking about syntactical proofs in FOL (as I've cautioned > before). Let me review with you what Shoenfield stated: > We need the third part of a formal system which will enable us to > conclude theorems from the axioms. This is provided by the *rules of > inference*, which we often call simply *rules*. Each rule of > inference states that under certain conditions, one formula, called > the *conclusion* of the rule, can be *inferred* from certain > formulas, called the *hypotheses* of the rule. > How should we define the *theorems* of a formal system F? Obviously > they should satisfy the two laws: > i) the axioms of F are theorems of F; > ii) if all the hypotheses of a rule of F are theorems of F, then the > conclusion of the rule is a theorem of F. > Moreover, we want a formula to be a theorem of F only if it follows > from these laws that it is a theorem. We can therefore define a > theorem of F to be a formula of F which can be seen to be a theorem > on the basis of laws (i) and (ii). > inference in the context of FOL proofs in a formal system (in which > is, again, the context of my argument), then the rules can't be > applied to just any formula. If any formula that is applicable to be > pertained by a rule of inference (whether as a hypothesis or a > conclusion) *it must be at least an axiom*: which means it can't be > just any formula, as you've alluded to above. No. Here is the source of your confusion. All Shoenfield is doing is > using the two laws above to *define* the notion of a theorem of system > F. Roughly, a theorem is either an axiom or something that you can > prove by means of the rules from previously established theorems. > You > are thus confusing the definition of theorem, which requires that you > restrict the rules of F to theorems if you want to generate *further > theorems*, with the (imaginary) stipulation that you must restrict the > rules of F to theorems generally in all contexts. No I'm not. I've said many times the context of my argument has been FOL syntactical proof w.r.t. an (axiom-set) formal system, in particular the formal system in the language that has *zero* non-logical symbols. Why is that a confusion on my part, as to what is being debated here? Didn't I just say above: > If we use the rules for something else (which I suppose we could) then > we're *not* talking about syntactical proofs in FOL (as I've cautioned > before). > [Note in law (ii) the phrase rule of F: he clearly meant the rule > that's being applied on the hypotheses of F, which clearly doesn't > mean *any* formula. No, it is talking, just as it says, about the hypotheses of a RULE of F, > which can be any formulas. Note that the claim is a conditional: IF all > the hypotheses of a rule of F are theorems of F, then the conclusion is > a theorem. This conditional does not say that you cannot apply the rule > to non-theorems; it simply says that you must apply it only to theorems > of F in order for a conclusion of the rule to count as a *theorem of F*. But as I've made caveats before, I'm not interested in applying the rules to non-theorems! Remember this all started out with the debate whether or not Axy[x+y=0] be a _theorem_ of T = {Axy[x=y]} where L(T) has zero nonlogical symbol. I.e. we've not been debating whether or not Axy[x+y=0] is a non theorem! In fact, of course, you *must* be able to apply the rules of F to any > formulas of the language of F, for this is the only way that you can > prove anything from a first-order theory like PA whose axioms are not > theorems of F. You misread my quotation of Shoenfield's book. In that section, F is a formal system like PA (as in How should we define the *theorems* of a formal system F). Then your prove anything from a first-order theory like PA whose axioms are not theorems of F is not parse-able: because in this context PA is F! (Read: in this context PA's axioms are F's theorems!). > In addition, we could easily see from his passage that rules of inference > can't be applied to *any* formula without violating the definition of > syntactical proof in FOL. For example, let L = L(T = {Ax[Sx=e]}. I guess this means that L is a language containing only the 1-place > function symbol S and the constant e. And that T is a first-order > theory whose sole non-logical axiom is Ax[Sx=e]. > Now, by Expansion Rule, we'd have: > (1) Ax[Sx=e] / ~(Ex[Sx=e]) Yes, that would be a theorem of the theory. > By the essence of what you've alluded above (and Allan, et al, said > before), if we just stipulate an extension of L which is, say, L' = > L(e,e',S), then we could have this theorem: > (1') Ax[Sx=e] / ~(Ex[Sx=e']) Sure. > then given the very same condition - same hypothesis - Ax[Sx=e], we could > derive 2 *different conclusions*, But we could already have done that above. Add any bunch of superfluous > conjuncts to (1) above. > a clear violation of what Shoenfield stated above: > under certain conditions, _one formula_, called the *conclusion* of > the rule, can be *inferred* from ... the *hypotheses* of _the_ rule. You are misreading this terribly. OK I made one mistake here. That's an incorrect example. The Expansion rule allows multiple conclusions from one hypothesis (due to the nature of the logical or). The example I should have given instead is that given the same T above, whatever rule(s) you'd use to prove Ex[Sx=e] from the lone axiom Ax[Sx=e], you wouldn't be able to prove Ex[Sx=e'], because e' is not in the signature of L = L(T = {Ax[Sx=e]}. > In any given *application* of a rule > there can of course only be one conclusion for any given set of > hypotheses. However, it is perfectly possible in a *further* > application of the same rule to the same hypotheses to generate a > different conclusion. Doesn't the rule you call Expansion above permit > exactly that? One can infer (1) and also, e.g., (1*) Ax[Sx=e] / (Ex[Sx=e]) So it's a correctable mistake on my part there. > In addition, as I've alluded to in previous posts, if we permit the > existence a nonlogical symbol *foreign* to the underlying (T = > {Ax[Sx=e]}, such as say e', then we'd open a floodgate to the entire > existences of all the elements of say a (non-empty) Grothendieck > universe (whose size would probably be more that just uncountable), to > be in theorems of T! One could demonstrate that I'm wrong here; but > from what I've gathered, that's not what FOL or FOL's syntactical > proof is about. Hope the above helps. No. You've pointed out one bad but correctable example. You still have *not* refuted my key point in this thread about Axy[x+y=0] *not* being a theorem of the lone axiom system T = {Axy[x=y]}, where L(T) has zero nonlogical symbols, using the rules of inference, and using the definitions of theorem proof in FOL. -- To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician. (Shoenfield, Mathematical Logic) === Subject: Re: Cantor's argument is erroneous <87tz46cflg.fsf@phiwumbda.org> <87r5z7thw5.fsf@phiwumbda.org> <87iqkjt4t3.fsf@phiwumbda.org> <87d4art0u0.fsf@phiwumbda.org> <87tz3yniog.fsf@phiwumbda.org> <%vONl.26977$0S.20110@newsfe22.iad> said: > No. Here is the source of your confusion. All Shoenfield is doing is > using the two laws above to *define* the notion of a theorem of system > F. Roughly, a theorem is either an axiom or something that you can > prove by means of the rules from previously established theorems. > You are thus confusing the definition of theorem, which requires > that you restrict the rules of F to theorems if you want to generate > *further theorems*, with the (imaginary) stipulation that you must > restrict the rules of F to theorems generally in all contexts. No I'm not. Alas, I fear you are. Be that as it may: > ...You've pointed out one bad but correctable example. You still have > *not* refuted my key point in this thread about Axy[x+y=0] *not* being > a theorem of the lone axiom system T = {Axy[x=y]}, where L(T) has zero > nonlogical symbols, using the rules of inference, and using the > definitions of theorem proof in FOL. Well, I should hope I didn't refute that. It is obviously true that Axy[x+y=0] is not a theorem of T if the language L(T) of T is stipulated to contain no non-logical symbols. For on that assumption (and assuming + is a non-logical symbol) Axy[x+y=0] is not even a sentence of L(T). Surely no one disagreed with you on *that* point. === Subject: Re: Cantor's argument is erroneous > said: > No. Here is the source of your confusion. All Shoenfield is doing is > using the two laws above to *define* the notion of a theorem of system > F. Roughly, a theorem is either an axiom or something that you can > prove by means of the rules from previously established theorems. > You are thus confusing the definition of theorem, which requires > that you restrict the rules of F to theorems if you want to generate > *further theorems*, with the (imaginary) stipulation that you must > restrict the rules of F to theorems generally in all contexts. > No I'm not. Alas, I fear you are. Be that as it may: > ...You've pointed out one bad but correctable example. You still have > *not* refuted my key point in this thread about Axy[x+y=0] *not* being > a theorem of the lone axiom system T = {Axy[x=y]}, where L(T) has zero > nonlogical symbols, using the rules of inference, and using the > definitions of theorem proof in FOL. Well, I should hope I didn't refute that. It is obviously true that > Axy[x+y=0] is not a theorem of T if the language L(T) of T is > stipulated to contain no non-logical symbols. For on that assumption > (and assuming + is a non-logical symbol) Axy[x+y=0] is not even a > sentence of L(T). Surely no one disagreed with you on *that* point. Well then there's no disagreement between you and me on that point, which is: using the FOL syntactical proof machinery which includes rules of inference. *** But I'm not sure if it's true that no one disagreed with me - at least at the beginning of this (sub) thread. -- To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician. (Shoenfield, Mathematical Logic) === Subject: Re: Cantor's argument is erroneous > Well then there's no disagreement between you and me on that point, which > is: > Axy[x+y=0], using the FOL syntactical proof machinery which includes > rules of inference. Note the subtle change here. All of a sudden, Nam has dropped all > mention of the language for the formal system. You're either too ignorant or borderline not telling the truth. Far from dropping it, I've mentioned numerous times L(T = {Axy[x=y]}) as *the language* in which *the formal system T* is written. I have to go now but I'll be back. Meantime, why don't you stop something you yourself would accuse a crank do: uttering nonsense repeatedly! Let's not forget that when I suggested the obvious proof works when > our formal system has the appropriate language and the single axiom, > Nam claimed > the following: This is where your confusion is (between informal reasoning and > formal proof). Hint: you'd *need some axioms* to make '+' a binary > function, as opposed to just a binary relation. He seems to be working to make his claim appear retroactively more > plausible. Nam, one more time: Do you agree that we can specify a language > L = {0,+} and a theory T in L by T = {(Ax)(Ay)(x=y)}? And do you > agree that, with these specifications, T |- (Ax)(Ay)x+y=0? Because we've all agreed that when L = {} and we regard T as a theory > in L, then T does not prove that formula. Let's just see what you say > about the above case. If you agree (contrary to your earliest > claims), then surely we can let the matter rest. > -- To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician. (Shoenfield, Mathematical Logic) === Subject: Re: Cantor's argument is erroneous <%vONl.26977$0S.20110@newsfe22.iad> <871vqu33uz.fsf@phiwumbda.org> posting-account=yKimjgoAAACk5WwPVD4l9HmbpoR6Hmy4 Presto/2.1.1,gzip(gfe),gzip(gfe) > Well then there's no disagreement between you and me on that point, which > is: > Axy[x+y=0], using the FOL syntactical proof machinery which includes > rules of inference. Note the subtle change here. All of a sudden, Nam has dropped all > mention of the language for the formal system. You're either too ignorant or borderline not telling the truth. Far > from dropping it, I've mentioned numerous times L(T = {Axy[x=y]}) > as *the language* in which *the formal system T* is written. I have to go now but I'll be back. Meantime, why don't you stop something > you yourself would accuse a crank do: uttering nonsense repeatedly! What does a formal system consist of? Is there only one formal system for a given set of axioms? As far as I can see, most people (and every textbook cited) says that we specify a language and a set of axioms, and combined they make a theory/formal system. Nam says we specify a set of axioms, which implies a (unique, minimal, signature) language, and that the specified set of axioms along with the implied language makes a theory/ formal system. Is this close to what you're saying? If so... I have literally no idea where you're getting this idea from, or what it all means. Why this restriction? What is wrong about considering more expanded languages? === Subject: Re: Cantor's argument is erroneous <%vONl.26977$0S.20110@newsfe22.iad> <871vqu33uz.fsf@phiwumbda.org> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Well then there's no disagreement between you and me on that point, which > is: Axy[x+y=0], using the FOL syntactical proof machinery which includes > rules of inference. Note the subtle change here. All of a sudden, Nam has dropped all > mention of the language for the formal system. Oh, yeah. He's done a lot of lateral shifting of position in this thread. It appears pretty clear to me that it's conscious and deliberate, which is to say dishonest. For example, in many cases when specific instances of it have been noted and he's been asked for clarification, he simply doesn't respond. Wiggly, that one. Marshall === Subject: Re: Cantor's argument is erroneous <%vONl.26977$0S.20110@newsfe22.iad> posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse Trident/4.0; MathPlayer 2.10d; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > But I'm not sure if it's true that no one disagreed with me > - at least at the beginning of this (sub) thread. You have just been hacked by the sci.gang: you had said some quite interesting things in your first couple of posts... -LV === Subject: Re: Cantor's argument is erroneous <878wllxa2z.fsf@phiwumbda.org> <87ab5yo00x.fsf@phiwumbda.org> <87tz46cflg.fsf@phiwumbda.org> <87r5z7thw5.fsf@phiwumbda.org> <87iqkjt4t3.fsf@phiwumbda.org> <87d4art0u0.fsf@phiwumbda.org> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Also you (MoeBlee) should have realized that as far as FOL= *proof in a theory* > is concerned, *rules of inference don't pertain to just any formula*! I never said they do. Indeed, every deduction is a deduction in a language. Every formula in the deduction must be formula of the language. If the deduction is in a theory, then every formula must be in the language of the theory. Nothing I've ever said has contradicted that. Again, I stated a language that has '+' and '0' of certain arities, then I stated the theory to be the set of all sentences in said language that are entailed by Axy x=y. And Axy x+y=0 is a formula in the language of the theory and Axy x+y=0 is entailed by Axy x=y. That is, every model (for the language) that makes Axy x=y true is a model (for the language) that makes Axy x+y=0 true. Moreover, there is a derivation, using only formulas of the language, of Axy x+y=0 from Axy x=y. And again, I'm not talking about YOUR situation in which '+' and '0' are not in the language. It's that simple. > Specifically, rules of inference pertain *only* to formulas that are > *theorems*. But since theorems are defined from axioms, it isn't much > an exaggeration to say rules of inference pertains to *axioms*. [And that's > not just (in your words) ordinary convention but it's the very heart of the > definition of proofs - in FOL=]. What in the world? Rules of inference pertain to formulas in the language. A formula doesn't have to be a theorem for a rule to apply to it. P and P->Q might not be theorems, but the rule of modus ponens still applies to them. Maybe what you have in mind is the fact that in a proof each line is either an axiom (assumption) or the result of a rule applied to previous lines, so every line is a theorem of the axioms (or assumptions). That's fine. Except it doesn't apply to NATURAL DEDUCTION, where, for example we may put P on a line as an assumption, then reason to get Q on a line, then conclude P->Q. MoeBlee === Subject: Re: Cantor's argument is erroneous > It's my right as a poster that I stay confined within the context > of syntactical/symbolical proof in axiom-set systems of FOL=, when > talking about valid application of rules of inference. I've already agreed - from the beginning - that if '+' and '0' are not in the language, then Axy x+y=0 is not derivable in such a system. But I also described ANOTHER, DIFFERENT context, which is one where '+' and '0' ARE in the language of the system. And in that context, as I desribed it, Axy x+y=0 IS derivable. Yet you keep denying that. The language has '+' and '0' at appropriate arities; the rules are natural deduction; and the sole non-logical axiom is Axy x=y. So, Axy x +y=0 is then derivable in this system. That you have some OTHER system in which '+' and '0' are not symbols is irrelevent to the fact that, in the system I just described, Axy x+y=0 is derivable. It's that simple. > Any thing else like proof-in-a-language or equational logic > is irrelevant (1) Proof in a language is relevant to the context *I* described but that you argued about. (2) Aside from that, you claimed 'proof in a language' is not a notion that is not found in mathematical logic. But I showed you that it is. I don't see why you can't just admit that you spoke incorrectly on that matter. > If by ordinary context of mathematical logic you mean the context > of FOL= syntactical proofs *in formal systems*, then I already gave > examples explaining what you said below is invalid. No you haven't. There is no example' that can possibly change the fact that I showed a proof such that each formula is in the language I defined and only natural deduction was used on formulas in that language and the proof is of Axy x+y=0 from the sole assumption Axy x=y. > the rules of inference generalize over formulas of > the language - and it is my very point that where '+' and '0' are in > the language, the rules of inference allow instantiations to formulas > that have those symbols EVEN THOUGH those symbols do not occur in any > of the non-logical axioms for some particular axiomatization of some > particular theory. Again, as far as FOL= syntactical proofs *in a formal system* T is concerned, > Rules of inference can *not* allow such generalization (instantiation, > expansion, or the like) without committing to the error of over-generalization, > from specific hypothesis/information to general conclusion, as I gave in one > example before. There is no over-generalization in any step I took. I simply applied universal generalization from one formula in the language to another formula in the language. ASK ANY LOGICIAN, Nam. MoeBlee === Subject: Re: Cantor's argument is erroneous > It's my right as a poster that I stay confined within the context > of syntactical/symbolical proof in axiom-set systems of FOL=, when > talking about valid application of rules of inference. I've already agreed - from the beginning - that if '+' and '0' are not > in the language, then Axy x+y=0 is not derivable in such a system. OK, so if you agreed with me ,in the context I've specified, Axy[x+y=0] is not a theorem of T = {Axy[x=y]} then there's no argument between us then. As per your context below where you'd claim Axy[x+y=0] is derivable, I have to go over what you said and reflect more on that. [It's a fact that right now I'm trying to bring some closure on the issue in my context. But I also described ANOTHER, DIFFERENT context, which is one where > '+' and '0' ARE in the language of the system. And in that context, as > I desribed it, Axy x+y=0 IS derivable. Yet you keep denying that. The language has '+' and '0' at appropriate arities; the rules are > natural deduction; and the sole non-logical axiom is Axy x=y. So, Axy x > +y=0 is then derivable in this system. That you have some OTHER system > in which '+' and '0' are not symbols is irrelevent to the fact that, > in the system I just described, Axy x+y=0 is derivable. It's that simple. > Any thing else like proof-in-a-language or equational logic > is irrelevant (1) Proof in a language is relevant to the context *I* described but > that you argued about. (2) Aside from that, you claimed 'proof in a > language' is not a notion that is not found in mathematical logic. But > I showed you that it is. I don't see why you can't just admit that you > spoke incorrectly on that matter. > If by ordinary context of mathematical logic you mean the context > of FOL= syntactical proofs *in formal systems*, then I already gave > examples explaining what you said below is invalid. No you haven't. There is no example' that can possibly change the > fact that I showed a proof such that each formula is in the language I > defined and only natural deduction was used on formulas in that > language and the proof is of Axy x+y=0 from the sole assumption Axy > x=y. the rules of inference generalize over formulas of > the language - and it is my very point that where '+' and '0' are in > the language, the rules of inference allow instantiations to formulas > that have those symbols EVEN THOUGH those symbols do not occur in any > of the non-logical axioms for some particular axiomatization of some > particular theory. > Again, as far as FOL= syntactical proofs *in a formal system* T is concerned, > Rules of inference can *not* allow such generalization (instantiation, > expansion, or the like) without committing to the error of over-generalization, > from specific hypothesis/information to general conclusion, as I gave in one > example before. There is no over-generalization in any step I took. I simply applied > universal generalization from one formula in the language to another > formula in the language. ASK ANY LOGICIAN, Nam. Again we're basing our arguments in 2 different context, so there's nothing for us to debate here. (At least not yet). -- To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician. (Shoenfield, Mathematical Logic) === Subject: Re: Cantor's argument is erroneous <87ab5yo00x.fsf@phiwumbda.org> <87tz46cflg.fsf@phiwumbda.org> <87r5z7thw5.fsf@phiwumbda.org> <87iqkjt4t3.fsf@phiwumbda.org> <87d4art0u0.fsf@phiwumbda.org> posting-account=yKimjgoAAACk5WwPVD4l9HmbpoR6Hmy4 Presto/2.1.1,gzip(gfe),gzip(gfe) > It's my right as a poster that I stay confined within the context > of syntactical/symbolical proof in axiom-set systems of FOL=, when > talking about valid application of rules of inference. I've already agreed - from the beginning - that if '+' and '0' are not > in the language, then Axy x+y=0 is not derivable in such a system. OK, so if you agreed with me ,in the context I've specified, Axy[x+y=0] > is not a theorem of T = {Axy[x=y]} then there's no argument between us > then. The context you've specifiied being that the language has no non- logical symbols not used in the axioms? Nobody has disagreed with that. What about the context where '+' and '0' are in the language? Do you agree that it is a theorem in that context? === Subject: Re: Cantor's argument is erroneous <87ab5yo00x.fsf@phiwumbda.org> <87tz46cflg.fsf@phiwumbda.org> <87r5z7thw5.fsf@phiwumbda.org> <87iqkjt4t3.fsf@phiwumbda.org> <87d4art0u0.fsf@phiwumbda.org> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > you claimed 'proof in a > language' is not a notion that is not found in mathematical logic. Typo. I meant you claimed it is not a notion found in mathematical logic. MoeBlee === Subject: Re: Point charges inside a sphere I find the results truly beautiful. The fact that even though the > charge > density on the sphere (or circle) continues to climb as new charges > are > added - there is never a time when it's better to place one of the > charges internally is very nice. > But, as I noted earlier in the thread, that's _not_ true if we're > dealing > with a circular disk. > To clarify: > I'm not a physicist; I assume that the corresponding mathematical > problem > is minimization of the sum of reciprocals of distances between pairs of > points. Given 12 or more points on a circular disk, minimizing that sum > requires that some of the points lie _in the interior_ of the disk. > David W. Cantrell > I've been trying to understand things in terms of potential theory > and harmonic functions. In two dimensions, I think U(x, y) = log( sqrt(x^2 + y^2) ) satisfies Laplace's equation ( del^2 /del x^2 + del^2 /del y^2 ) U == 0 , > except at (0, 0). So for example with 12 point-charges in the unit disk, > D = {(x,y): x^2+y^2 <= 1}, with position vectors > u_1, u_2, ... u_12, one asks to maximize sum_{1<=i < j <=12} ( log ( || u_i - u_j ||_2 ) ), (***) so ||.|| is the Euclidean norm. Then I'm wondering if all 12-point configurations that > maximize the function (***) will have > u_1, u_2, ... on the unit circle, or if some can be in the interior of D, D being a closed > set for R^2 in the usual sense. Cf. Maximum Principle, > http://en.wikipedia.org/wiki/Harmonic_function David > [cut] > How can a 2D laplacian be applicable in this situation? The original problem, whether in 2-D or 3-D, was, (quoting David Cantrell): << I assume that the corresponding mathematical problem > is minimization of the sum of reciprocals of distances between > pairs of points. I changed the function to maximize for 2-D to: sum_{1<=i < j <=12} ( log ( || u_i - u_j ||_2 ) ), (***) The 2D laplacian is used to to state that U as defined below > is harmonic in R^2 {(0, 0)}. > Ah. Gotcha. Sorry I was focusing on the physics too much. Yes defining the problem as a purely two-dimensional potential (which means we are obviously not talking about physical charge) and asking the question you are asking seems reasonable. My guess would be yes, if the potential were defined as you have it, then all stable configurations would have to exist on the circumference of the disk, and no stable points could exist within the circumference. As you say the reason comes down to the maximum principal for harmonic functions, which is a result of the mean value theorem. The reason the mean value theorem applies can possibly be seen in many ways and from many view points - Green's identity - complex analysis - dirichlet problems - potential theory. So if U is harmonic then U equals its average over the circumference of the disc. if U were then to attain a maximum at a point within the disk, points surrounding it must necessarily be less than U at this point. The average in a small neighbourhood surrounding this point, must then also be less (via the mean value theorem) than U at this point, including U at this point. Which is a contradiction. So if U did have a maximum within the disk it must be constant. However, U can not be a constant throughout the disk if we are talking about a finite set of point charges obeying that particular potential, and so the only stable configurations must lie upon the circumference. (yes/no/maybe)? === Subject: Re: Point charges inside a sphere > I find the results truly beautiful. The fact that even though the > charge > density on the sphere (or circle) continues to climb as new charges > are > added - there is never a time when it's better to place one of the > charges internally is very nice. > But, as I noted earlier in the thread, that's _not_ true if we're > dealing > with a circular disk. To clarify: > I'm not a physicist; I assume that the corresponding mathematical > problem > is minimization of the sum of reciprocals of distances between pairs of > points. Given 12 or more points on a circular disk, minimizing that sum > requires that some of the points lie _in the interior_ of the disk. David W. Cantrell > I've been trying to understand things in terms of potential theory > and harmonic functions. > In two dimensions, I think U(x, y) = log( sqrt(x^2 + y^2) ) satisfies > Laplace's equation ( del^2 /del x^2 + del^2 /del y^2 ) U == 0 , > except at (0, 0). > So for example with 12 point-charges in the unit disk, > D = {(x,y): x^2+y^2 <= 1}, with position vectors > u_1, u_2, ... u_12, one asks to maximize > sum_{1<=i < j <=12} ( log ( || u_i - u_j ||_2 ) ), (***) > so ||.|| is the Euclidean norm. > Then I'm wondering if all 12-point configurations that > maximize the function (***) will have > u_1, u_2, ... on the unit circle, > or if some can be in the interior of D, D being a closed > set for R^2 in the usual sense. > Cf. Maximum Principle, > http://en.wikipedia.org/wiki/Harmonic_function > David > [cut] How can a 2D laplacian be applicable in this situation? > The original problem, whether in 2-D or 3-D, was, > (quoting David Cantrell): > << I assume that the corresponding mathematical problem > is minimization of the sum of reciprocals of distances between > pairs of points. > I changed the function to maximize for 2-D to: > sum_{1<=i < j <=12} ( log ( || u_i - u_j ||_2 ) ), (***) > The 2D laplacian is used to to state that U as defined below > is harmonic in R^2 {(0, 0)}. Ah. Gotcha. Sorry I was focusing on the physics too much. Yes defining the > problem as a purely two-dimensional potential (which means we are obviously > not talking about physical charge) and asking the question you are asking > seems reasonable. My guess would be yes, if the potential were defined as you have it, then > all stable configurations would have to exist on the circumference of the > disk, and no stable points could exist within the circumference. As you say > the reason comes down to the maximum principal for harmonic functions, which > is a result of the mean value theorem. The reason the mean value theorem > applies can possibly be seen in many ways and from many view points - > Green's identity - complex analysis - dirichlet problems - potential theory. So if U is harmonic then U equals its average over the circumference of the > disc. if U were then to attain a maximum at a point within the disk, points > surrounding it must necessarily be less than U at this point. The average > in a small neighbourhood surrounding this point, must then also be less (via > the mean value theorem) than U at this point, including U at this point. > Which is a contradiction. So if U did have a maximum within the disk it > must be constant. However, U can not be a constant throughout the disk if we are talking about > a finite set of point charges obeying that particular potential, and so > the only stable configurations must lie upon the circumference. (yes/no/maybe)? Yes. I've been trying to think about the details, or how I would try to prove it. If we have m>=2 points, confined say to the unit disk D, then we can freeze the positions of m-1 of them, and consider the domain D. Each function - log( || x - u_j ||_2 ) is harmonic except at u_j, where there is a singularity. The u_j, 1<=j<= m-1 are the frozen positions of the first m-1 charges. The part of the potential due to the relative positions of the u_j relative to one another is constant, since they are considered frozen,1<=j<= m-1. We need only look at the A sum (or a linear combination of) harmonic functions is harmonic. With respect to the superposition of m-1 harmonic functions, from the point of view of charge #m, the domain is more properly D{u_1, ... u_m} . To use the Maximum Principle as in Axler, Bourdon and Ramey , Result 1.8, page 7 here: we can let Omega = { y in D, || y - x_0 || < || u_j - x_0 || , j in {1, ... m-1 } }. Here, x_0 is a (presumed) position in the interior of the closed set D, the u_j are the frozen positions of the first m-1 charges, and the condition for y in Omega amounts to d(y, x_0) < d(u_j, x_0), j=1, ... m-1. Another condition on x_0, apart from x_0 being in int(D), is x_0 =/= u_j, 1<=j<= m-1 . x_0 = u_j means infinite potential energy in the - log( || x - u_j ||_2 ) term, and we are trying to minimize it. So x_0 is any point in the interior of D, but not one of the u_j, 1<=j<= m-1. Then Omega, (which depends on x_0, and the u_j), is an open disk in D, and using y as a variable ranging over Omega, as y approaches the (or one of) u_j closest to x_0, - log( || y - u_j || ) grows without bound. So: sum_{1<= j <= m-1} ( - log ( || y - u_j || ) ) is harmonic and unbounded in Omega. This harmonic function is not constant in Omega, so by the Maximum Principle (1.8 in A.B.R.), it has no local maximum in Omega. D, wlog, we can assume that one of them is #m, and that #m is in int(D). Then, reasoning as above, we can move #m a bit and lower the part of the potential involving #m and one of the u_j, 1<=j< m, considered frozen. To establish that there are minimal (total) potential configurations, it seems to me one could apply the compactness of D^m to a function of the type minimum( K, sum_{1<=i < j <=m} ( - log( ||u_i - u_j|| ) ) ) where K is a sufficiently large positive constant. For the classical Thomson problem on the sphere S^2 lying in R^3, I thought this was interesting: Table 2: The symmetry group G of the configuration of n points which minimizes the energy of the Thomson problem, for 2 .b2 n .b2 32. from: http://arxiv.org/abs/math-ph/0303071 (by Atiyah and Sutcliffe) David Bernier === Subject: Re: Singular n-cube We are studying singular homology theory using a > Massey's A basic course in algebraic topology. The book uses a singular cubic homology groups rather > than using singular simplicial homology groups. We tried to find references for singular cubic > homology groups, but most books use the singular > simplicial homology groups. We are having a hard time understanding basic concepts > of a singular n-cubic. link: > dq=Faces+of+a+singular+n-cube&hl=en A singular n-cube is defined as > A singular n-cube in a topological space X is a > continuous map T:I^n --> X (n>=0). A singular n-1 cubes are defined as > A_i T, B_i T:I^(n-1) --> X by the formulas > A_i T(x_1,.....,x_(n-1) ) = T(x_1,...,x_(i-1),0, > x_i,...,x_(n-1)), B_i T(x_1,....,x_n-1)) = T(x_1,...,x_(i-1), 1, > x_i,...x_(n-1) The boundary of n-cube T is the sum of i=1 to n of > (-1)^i[A_i T - B_i T] (The above link includes the above definitions). For instance, how do we get the boundary of a > singular 2-cube if the space X is a solid unit cube. The book has only an example of one point space X, > but I could not figure out how to get the boundary of > a singular 2-cube. Best wishes, > Don and [Martin Musatov] Psst...[http://coding.derkeiler.com/pdf/Archive/General/comp.theory/2009-04/ m sg00122.pdf] === Subject: Re: need to have an answer here-- pi a constant in Eucl yet variable in Elliptic; 1/2 point-sphere concept #486 new book 2nd edition: New True Mathematics > I find myself wanting an answer and sometimes the > best way of > achieving that > goal is to talk it too death. By going over and over > it, it naturally > squeezes out. So I have the situation that Elliptic geometry and > Hyperbolic > are Curved geometries because the positives and > negatives > are separate and alone. I have Eucl as flat plane > because > the positive number is 1/2 the point-sphere and the > negative > the other 1/2 point-sphere. In this book I never defined point-sphere and should > do so > now. I mean by point-sphere that a point in geometry > can > be considered a tiniest of spheres, an > infinitesimal-sphere. > A sphere so small that it is a sphere yet it has no > circumference > no diameter, no radius. We define or axiomatize the > point-sphere and we build the AP-Coordinate System > out of > point spheres. Now a regular sphere like the globe of > Earth > we can define the points north of the Equator as all > positive > points and the points in the southern hemisphere as > all > negative points. And that is sort of what is desired > in the > case of AP-Reals Coordinate System in that every > point > except 0, but every other point such as > 1d000...00000, has > both positive +1d000...0000 and negative > (-)1d000...0000 > situated at that exact same point where the positive > number > occupies 1/2 point-sphere and the negative number > occupies > the other 1/2 point-sphere. Now the beauty of the AP-Reals Coordinate System is > that > there are no negative quadrants. That the graph of > any function > is simply the positive function. It is sort of like a > absolute-value- > function. > Since the negative AP-Reals is the same point as the > positive. So the Identity function Y= X is simply a > diagonal > in the 1st quadrant only. In fact all graphs are just > the > 1st quadrant only and this AP-Reals-Coordinate System > throw > out the window the 2nd, 3rd and 4th quadrants. Now let me get back to pi a constant in Eucl. The > reason that > Eucl is flat plane, is because, well, the positives > have a negative > at each point which, like in physics the positive > charge and > negative charge together flattens out or has the > tendency to > neutralize the desire of points to bend or curve. But > in Elliptic > Geometry, all points are positive only and thus the > geometry > is bent and curved. Ditto for Hyperbolic only all > negative numbers. Now we can get at this question of pi a constant in > Eucl. If you > have a highly bent and curved space as Elliptic, your > pi value > can vary, and vary between 2 yet less than > 3.14159.... In > Hyperbolic, you cannot construct a circle since its > center is no > center, just as it is impossible to construct a > square in Elliptic. Now why is Eucl have pi as a constant? The answer is > quite > clearly due to the positives next to negatives does > not allow > for any extra curvature that Elliptic allows. So if > you have a > circle, the numbers cannot contribute to a less curve > or greater > curve and so the curve is a constant in Eucl. Alright, that explains why pi is a constant in Eucl > but does not > explain why pi is not algebraic in Eucl? Why pi is > imaginary, > just as sqrt(-1) is imaginary in Eucl. For an answer to that, I have not sorted it out in my > head yet. > I believe the answer has to do with the fact that in > the > development of the AP-Reals then the > AP-Reals-Rationals, > and finally to the AP-Reals-Rationals-Irrationals, > that pi is > an AP-Reals-Rational-Irrational that cannot go any > further. What I had called not fully grown number. The > Irrationals > in AP-Reals is the last and final stop, just as there > are no > speeds beyond the speed of light in physics or just > as there > are no temperatures below absolute zero in physics. Archimedes Plutonium > www.iw.net/~a_plutonium > whole entire Universe is just one big atom > where dots of the electron-dot-cloud are galaxies That is certainly one .. to ... a ...:http://coding.derkeiler.com/pdf/Archive/General/comp.theory/2009-04/m sg00122.pdf === Subject: Musatov strikes again (in the past) http://coding.derkeiler.com/pdf/Archive/General/comp.theory/2009-04/msg00122 .pdf === Subject: Re: Musatov strikes again (in the past) > http://coding.derkeiler.com/pdf/Archive/General/comp.theory/2009-04/msg00122 . pdf That leaves at least five unsolved Millennium Problems. < http://www.claymath.org/millennium/ > David Bernier === Subject: solutions manual for Finiancial Accounting 7e by Horngren posting-account=ldzOOQoAAAALmEmlIRVd1wCu2DOhImIB CLR 2.0.50727),gzip(gfe),gzip(gfe) solutions manual and Test Bank Solutions Manuals and Test Bank in Electronic (PDF)Format! Just contact with , solutionsservice (at) hotmail.com (my email address,solutionsservice@hotmail.com ), these are parts of our solutions, if the solution you want isnÁøt on the list, please email to http://getsolutions.spaces.live.com is my blog. solution manuall for fundamentals of thermodynamics, 7th edition,sonntag,borgnak, test bank for managerial accounting 12th Edition authors Garrison Noreen Instructor's Manual Of Fundamental Methods Of Mathematical Economics Chiang & Wainwright 2005 Mc Graw Hill Fundamentals of Advanced Accounting 3rd Edition by Joe B. Hoyle, test bank for Operations Management 10e William J. Stevenson solutions manual for Financial and Mangerial Accounting 2e by Horngren test bank for Financial and Mangerial Accounting 2e by Horngren test bank for Finiancial Accounting 7e by Horngren solutions manual for Finiancial Accounting 7e by Horngren solutions manual for Intermediata Accounting 13e Kieso test bank for Intermediata Accounting 13e Kieso TB solutions manual for Fundamentals of financial management 12e Brigham SM test bank for Fundamentals of financial management 12e Brigham TB Solution manuall for Fundamentals of Engineering Thermodynamics 6th Edition by Michael Moran and Howard Shapiro A Computer System Architecture 3rd Edition by Morris Mano Solution Manual Complete Assignment of All Chapters A FIRST COURSE IN DIFFERENTIAL EQUATIONS WITH MODELING APPLICATION 7TH EDITION BY DENNIS G. ZILL 2500 Solved Problems in Fluid Mechanics and Hydraulics (Schaum's Solved Problems) by Jack B. Evett, Cheng Liu A Course in Game Theory Osborne and Rubinstein A First Course In Probability Solution Manual,Ross 6th A First Course in Abstract Algebra 7th by Fraleigh A First Course in Differential Equations with Modeling Applications (7th ed.) and Zill & ; Cullen s Diferential Equations with Boundary- Value Problems (5th ed.) zill A First Course in Probability: SOLUTIONS MANUAL (7th Edition) by sheldon ross A First Course in String Theory chapter 1 to 16 A First Course in the Finite Element Method, 4th Edition Daryl L. Logan A Friendly Introduction to Number Theory 3rd by Silverman A Guide to Physics Problems, Part 1 - Mechanics, Relativity, and Electrodynamics A Guide to Physics Problems, Part 2 - Thermodynamics, Statistical Physics, and Quantum Mechanics A Practical Introduction to Data Structures and Algorithm analysis 2nd edition by Clifford A. Shaffer A Quantum Approach to Condensed Matter Physics Solutions by philip l. Taylor Absolute Java, 3rd Ed by W. Savitch instructor manual and test bank Accounting Concepts and Applications (9th Ed.) by W. Steve Accounting 7e by horngren solution manual Accounting 7e by horngren TB accounting 7e by horngren TB (test generator File) Accounting 8th edition by horngren test bank and solution manual Accounting Information Systems - james hall 6ed sm Accounting Information Systems - james hall 6ed tb Accounting Information Systems 10E Romney solution manual Accounting Information Systems 10E Romney test bank Accounting Information Systems 11E Romney solution manual Accounting Information Systems 11E Romney test bank Accounting Information Systems 7E Edition Ulric J. Gelinas, Richard B. Dull instructor manual Accounting Information Systems 7E Edition Ulric J. Gelinas, Richard B. Dull solution manual Accounting Information Systems 7E Edition Ulric J. Gelinas, Richard B. Dull test bank Accounting Information Systems, 9E George H. Bodnar William S. Hopwood solution manual Accounting Information Systems, 9E George H. Bodnar William S. Hopwood test bank Accounting Principles 8E by Kieso SM chapter 1 to 10 Accounting Principles 8E by Kieso SM chapter 11 to 26 Accounting Principles, Edition 8E, Weygandt, Kieso, Kimmel (Test Bank) volume 1 Accounting Principles, Edition 8E, Weygandt, Kieso, Kimmel (Test Bank) volume 2 Accounting Text and Cases 12e by Anthony IM Accounting what number means 8e by Marshall Adaptive Control 2E. by Karl J. Astrom solution manual Adaptive Filter Theory, 4th edition S. Haykin Advance corporate finance 1e by Ogden Instructor manual and test bank Advanced accounting 10E by Flyd Beams (SM+IM+TB) Advanced Accounting 10th edition by Fischer (SolutionsManual) Advanced Accounting 10th edition by Fischer (test bank) Advanced Accounting 9e by Beams solution manual Advanced Accounting 9E Hoyle,Schaefer,Doupnik Solution Manual Advanced Accounting 9E Hoyle,Schaefer,Doupnik Test Bank Advanced Accounting 9th edition by Fischer (SolutionsManual) Advanced Accounting 9th edition by Fischer (test bank) ADVANCED CORPORATE FINANCE Policies and Strategies by Joseph P. Ogden,Frank C. Jen,Philip F. O£Àonnor ( solution manual) ADVANCED CORPORATE FINANCE Policies and Strategies by Joseph P. Ogden,Frank C. Jen,Philip F. O£Àonnor (test bank) Advanced Digital Design with the Verilog HDL Michael D. Ciletti selected solutions Advanced Dynamics by Donald T. Greenwood Advanced Engineering Mathematics 3rd Edition by Dennis G Zill and Michael R Cullen Advanced Engineering Mathematics by Erwin Kreyszig 8ed solutions manual Advanced Engineering Mathematics Dennis G Zill 2nd Solution Advanced Engineering Mathematics, 6th Edition Peter V. O'Neil - University of Alabama, Birmingham Advanced Engineering Mathematics, 9th Edition By Erwin Kreyszig Advanced Financial Accounting, 6th edition, by Baker, Lembke, and King solution manual Advanced Macroeconomics 1996 romer Advanced Macroeconomics, Solutions Manual 1996 Romer Advanced Modern Engineering Mathematics, 3rd Edt by Glyn James solution manual Aerodynamics for Engineers 5th ed. solution manual John J. Bertin Russell M. Cummings Algebra by Thomas W. Hungerford Published by Springer Algebra, Pure and Applied by Aigli Papantonopoulou An Introduction to Abstract Algebra with Notes to the Future Teacher by Nicodemi, Sutherland, and Towsley An Introduction to Economic Dynamics An Introduction to Economic Dynamics by Ronald Shone An Introduction to Mass and Heat Transfer Principles of Analysis and Design Middleman An Introduction to Mathematical Statistics and Its Application (4th Edition) by Richard J. Larsen An Introduction to Modern Astrophysics (2nd Ed., Bradley W. Carroll & Dale A. Ostlie) An Introduction to Numerical Analysis by Endre Suli An Introduction to Numerical Analysis by Endre S£Ài, David F. Mayers An Introduction to Ordinary Differential Equations James C. Robinson Publisher: Cambridge University Press An Introduction To The Finite Element Method, 3rd Edition by J. N. Reddy An introduction to the mathematics of financial derivatives Neftci solution manual Analysis and Design of Analog Integrated Circuits (4th Edition) Gray, Hurst, Lewis and Meyer analysis design of analog IC design Analytical Mechanics: Solutions Manual 7ed Grant R. Fowles, George L. Cassiday Anderson J.D. Fundamentals of aerodynamics, 2nd edition - problems and solutions Andrew Tanenbaum Structured Computer Organization Solutions Manual antenna balanis solution manual antenna balanis solution manual 2nd edition ANTENNAS FOR ALL APPLICATIONS, THIRD EDITION Antennas for all Applications 3rd Ed. by Kraus & Marhefka Anton calculus book+ solution manual + test bank 8th edition Anton, Bivens, Davis Calculus Early Transcendentals Combined, 8th Edition instructor solution manual Anton, Bivens, Davis Calculus Early Transcendentals Combined, 8th Edition instructor solution manual Anton, Bivens, Davis Calculus Multivariable, 8th Edition Applied Fluid Mechanics 6th Ed. by Robert L. Mott Applied Mechanics for Engineering Technology 8e Keith M Walker Applied Numerical Analysis 7Ed - Curtis F. Gerald, Patrick O. Wheatley - Solutions manual Applied Partial Differential Equations David Logan Applied Quantum Mechanics by A. F. J. Levi Applied Statistics and Probability for Engineers 3rd.Ed edition student manual Applied Statistics and Probability for Engineers by Douglas C. Montgomery 3rd edition complete Applied Statistics and Probability for Engineers, 4th Edition Montgomery, Runger Applied Strength of Materials (4th Edition) SOLUTION MANUAL by Robert L. Mott Artificial Intelligence A Modern Approach 2e Stuart Russell Peter Norvig Audit and Assurance service An Integrated Approach 11e TB auditing and assuance services by messier test bank 6th edition Auditing and Assurance Services An Intergrated Approach and ACL Software, 12e by Alvin Arens Randal J. Elder, ark Beasley solution manual Auditing and Assurance Services An Intergrated Approach and ACL Software, 12e by Alvin Arens Randal J. Elder, ark Beasley test bank Auditing Cases, 3E Mark S. Beasley solution manual Auditing Cases: An Interactive Learning Approach, 4/E Mark S BeasleyFrank A. BucklessSteven M GloverDouglas F Prawitt Bank Management & Financial Services, 7/e By Peter S. Rose, Sylvia C. Hudgins (IM+SM) Bank management 7e by peter s. rose TB Bank management 7e by Rose ( instructor manual ) Basic Electrical Engineering By Nagrath, D P Kothari, Nagrath D P Kothari I J Nagrath, I J Nagrath Published by Tata 2002 Basic Engineering Circuit Analysis, 8th Edition by J. David Irwin, R. Mark Nelms Basic Engineering Circuit Analysis, 9th Edition Irwin, Nelms Basic Technical Mathematics with Calculus 8e Allyn J. Washington Biological Science and CW+ Grade Tracker Access Card, 2/E test bank .bok file Biology Concepts and Connections 6e Neil A. Campbell Jane B. Reece Martha R. Taylor Eric J. Simon Jean L. Dickey test bank Biology with MasteringBiolog, 8E Neil A. Campbell Jane B. Reece Biomaterials - The Intersection of Biology and Materials Science (Temenoff & Mikos) Bioprocess Engineering Principles - Solutions Manual (Original) by pauline m. Doran Book keeping and Accounting 3e Joel J. Learnef sm + book Borgnakke, Sonntag Fundamentals of Thermodynamics, 7th Edition Brief History of Western Civilization 5e vol 1 im and tb Brief History of Western Civilization, A: The Unfinished Legacy, Volume 2 im and tb BUSINESS STATISTICS A Decision Making Approach 7e SM Business Communication Essentials 3rd edition bovee and thill test bank Business Data Networks and Telecommunications, 7/E Raymond R. Panko test bank Business Law by Cheeseman 6E (IM) Business Law by Cheeseman 6E (TB) Business Law Today: Comprehensive 8th edition Roger LeRoy Miller, Gaylord A. Jentz instructor manual Business Law Today: Comprehensive 8th edition Roger LeRoy Miller, Gaylord A. Jentz test bank Business Statistics (A Decision Making Approach), Groebner, Shannon, Fry, Smith, 7 sm and tb Business Statistics 4e by Leonard J. Kazmier book + sm Business Statistics Decision Making and Student CD Package test items and solution manual, 7E test bank and sm Butterworth Heinemann - Coulson And Richardson - Chemical Engineering Vol I (Solutions Manual V Edition) c ++ how to program deitel 6th edition solution manual and test bank C++ How to Program 3rd edition by deitel Calculus A Complete Course 6th by R.A. Adams calculus by gilbert strang calculus by leithold solution manual Calculus Early transcendentals 5Th Ed - Complete Instructor 's Solutions Manual by James Stewart 0534393217 CALCULUS early transcendentals 7th edition Anton Bivens Davis Calculus Early Transcendentals Single Variable, 8th Edition Howard Anton, Irl Bivens, Stephen Davis Calculus of Variations Solution Manual Russak Calculus Single Variable 4ed chapter 1 to 11 Hughes-Hallett, Gleason, McCallum, et al. Calculus Third Editon By Strauss, Bradley and Smith not complete Calculus With Analytic Geometry (6th) By Bruce E. Edwards, Ron Larson, Robert Hostetler Calculus With Analytic Geometry (7th) By Bruce E. Edwards, Ron Larson, Robert Hostetler student manual Calculus, Early Transcendentals, 7E by C. Henry Edwards ,David E. Penney Callister Fundamentals of Materials Science and Engineering An Integrated Approach, 2nd Edition Capital Budgeting and Long-Term Financing Decisions Neil Seitz, Mitch Ellison 4th Edition instructor manual Carey, Study guide and solution manual for organic chemistry Chapra Applied Numerical Methods With Matlab For Engieers Solutions Manual 1st edition nearly same with 2nd edition Chemical and Engineering Thermodynamics- 3rd Edition- Solutions Manual Chemical Engineering Design, Fourth Edition: Chemical Engineering Volume 6 (Coulson & Richardson's Chemical Engineering) Chemical Engineering Solutions manual for Volumes( 2 and 3) 3 edition Backurst J. R., Harker J.H. & Richardson J. F chemical Engineering: Solutions for Volumes 2 and 3 by coulson 2002-12-11 Chemical Reaction Engineering, 3rd Edition Levenspiel Chemical, Biochemical, and Engineering Thermodynamics, 4th Edition Sandler Chemistry: The Central Science (Hardcover, 2005) Author: Bruce E. Bursten, H. Eugene Lemay Jr., Lemay test bank 10th Classical Dynamics A Contemporary Approach by Jorge V. Jose, Eugene J. Saletan T. Thornton, Jerry B. Marion Classical Electrodynamics - 2nd Ed. John David Jackson byKasper van Wijk Classical Electrodynamics 3rd edition by Jackson Classical Mechanics (2nd Edition) by Herbert Goldstein Classical Mechanics by R. Douglas Gregory Classical Mechanics, 2ed Partial Solutions Manual by Safko Close, Frederick, Newell Modeling and Analysis of Dynamic Systems, 3rd Edition Cmos analog circuit design 2nd edition homework solutions by allen holberg CMOS Analog Circuit Design, 2ed Solutions by Phillip E. Allen, Douglas R. Holberg CMOS VLSI Design 3rd edition David Harris H E Weste College Accounting (Chapters 1-25), 10E by Jeffrey Slater sm and tb College Mathematics for Business, Economics, Life Sciences & Social Sciences (11th Edition) (Hardcover) by Raymond A. Barnett (Author), Michael R. Ziegler (Author), Karl E. Byleen (Author) Collins Mechanical Design of Machine Elements and Machines A Failure Prevention Perspective Communication Networks (2nd Edition) leon Communication Networks Fundamental concepts & key Architectures By Leon Garcia Widjaja not complete 3 4 5 6 7 8 10 Communication Networks Fundamental concepts & key Architectures Alberto Leon-Garcia Communication Systems (4th edt) by Simon Haykin Communication Systems 4Ed - A Bruce Carlson Solutions Manual communication systems engineering by proakis Communication Systems Engineering Proakis J (2002) Solutions Manual 2nd edition Compensation Management in a Knowledge-Based World, 10E Richard I Henderson instructor manual Complex Variables with Applications (Pie) by A.David Wunsch Computational Techniques for Fluid Dynamics: A Solutions Manual By Karkenahalli Srinivas, Clive A. J. Fletcher Computer Architecture A Quantitative Approach, 4th Edition, 2006 by John L. Hennessy, David A. Patterson Computer Architecture: Pipelined and Parallel Processor Design (Solutions Manual) by Michael J. Flynn selected solutions computer networking a top down approach 3rd edition solution manual by James F.Kurose, Keith W. Ross Computer Networking: A Top-Down Approach, 4/E solution manual and lab solutions Computer Networking: A Top-Down Approach, 5/E solution manual computer networks Andrew S. Tanenbaum 4th edition Computer Networks Systems Approach 3ed by davie peterson solutions manual Computer Networks: A Systems Approach 2nd edition Peterson and Davie£À Computer Organization and Architecture: Designing for Performance, 7/E William Stallings Computer Organization and Design, Revised Printing, 3rd Edition Solutions Manual By David A. Patterson, John L. Hennessy, Computer Organization and Design: The Hardware/Software Interface, 3rd Edition by David A. Patterson, John L. Hennessy Concepts In Federal Taxation 2007 (14thEd) - Murphy Solutions Manual Concepts of Genetics, 9e by Klug, Cummings, Spencer & Palladino test generator Concepts of Programming Languages, 8/E Robert W. Sebesta, University of Colorado, Colorado Springs Construction Surveying and layout 2nd edition by wesley g. Crawford Construction Surveying and layout 3rd edition by wesley g. Crawford Consumer Behavior, 8/E Michael R. Solomon test bank contemporary engineering economy by chan s. park 4th edition Contemporary Financial Management R. Charles Moyer, James R. McGuigan, William J. Kretlow 10th edition instructor manual Contemporary Financial Management R. Charles Moyer, James R. McGuigan, William J. Kretlow 10th edition test bank Contemporary Financial Management, 11th Edition R. Charles Moyer James R. McGuigan William J. Kretlow instructor manual Contemporary Financial Management, 11th Edition R. Charles Moyer James R. McGuigan William J. Kretlow test bank Control Systems Engineering by Nise 4ed£À? Corporate Computer and Network Security Raymond Panko Corporate Finance By Stephen A. Ross 6 edition Corporate Finance By Stephen A. Ross 8th edition corporate finance 1e by berk sm corporate finance 1e by berk tb Corporate Finance, 8e Stephen A. Ross Randolph W. Westerfield Jeffrey Jaffe instructor manual and solution manual Corporate Finance-7th Edition by Stephen A. Ross , Randolph W. Westerfield , Jeffrey Jaffe cost Accounting 12e by Horngren Test Bank cost accounting 12e by Charles T. Horngren Srikant Datar George Foster Madhav Rajan Chris Ittner solution manual cost accounting 13e by Charles T. Horngren Srikant Datar George Foster Madhav Rajan Chris Ittner solution manual cost accounting 13e by Charles T. Horngren Srikant Datar George Foster Madhav Rajan Chris Ittner test bank Cost Accounting: Foundations and Evolutions 7E By Kinney solution manual Cost Accounting; Foundations and Evolutions, Edition 7, Kinney, Raiborn Cost Management A Strategic Emphasis, 4e Blocher Cost ManagementMeasuring Monitoring and Motivating Performance by Eldenburg Wolcott SM TB Cryptography & network security 4e william stallings Daniel Shanks - Solved And Unsolved Problems In Number Theory (2Nd Ed), 1978 Data and Computer Communications William Stallings 8th edition William Stallings Data and Computer Communications, 7th Edition by William Stallings Data Communications and Networking fourth edition by Behrouz A.Forouzan odd numbered solutions Data structure and Problem Solving using Java 3rd Mark Allen Weiss, Data Structures and Algorithm Analysis in C++, 3/E Data Structures with Java by John R. Hubbard Anita Huray University of Richmond Database Concepts, 3E david kroenke, david auer tb and im DATABASE MANAGEMENT SYSTEMS 3rd Edition by Ramakrishnan, Gehrke, Derstad, Seliko, Zhu- Solution Manual only odd solutions Database Processing Fundamentals, Design, and Implementation, 10E David Kroenke test bank Database System Concepts, Fifth Edition by Avi Silberschatz ,Henry F. Korth solutions to exercises Database Systems: An Application Oriented Approach, Compete Version, 2/ E Michael Kifer Arthur Bernstein Philip M. Lewis Databases systems: An Application-Oriented Approach 2nd edition Michael Kifer, Arthur Bernstein, Philip M. Lewis test bank + solution manual David Kroenke's Database Processing: Fundamentals, Design and Implementation (10th Edition) test bank Derivatives Markets 2nd edition by Yufeng Guo Solution Manual Derivatives Markets 2nd by Rober L. McDonald solution manual Derivatives Markets 2nd by Rober L. McDonald test bank Design and Analysis of Experiments Solutions Manual 6th edition Design and Analysis of Experiments, 6th Edition Montgomery complete all chapters Design of Analog CMOS Integrated Circuits McGraw Hill Solutions Manual Design of Fluid Thermal Systems, 2nd Edition William S. Janna design of machinary by norton 3rd edition Design with Operational Amplifiers and Analog Integrated Circuits, 3rd edt. by Franco Device Electronics for Integrated Circuits 3Edition Muller Kamins Device Electronics for Integrated Circuits Solutions Manual 3ed DIGITAL DESIGN FOURTH EDITION by M. MORRIS MANO DIGITAL SIGNAL PROCESSING: Signals, Systems, and Filters Andreas Antoniou Differential Equations & Linear Algebra, 2nd ed., Farlow Differential Equations & Linear Algebra, edition 2, by Edwards Penny differential equations 5th edition by zill classic fifth edition Differential Equations And Boundary Value Problems C. Henry Edwards - David E. Penney 2nd edition Differential Equations and Boundary Value Problems Computing and Modeling, 4E C. Henry Edwards David E. Penney Differential Equations and Linear Algebra, 3e by Stephen W. Goode and Scott A. Annin instructor manual Digital & Analog Communication Systems - Leon Couch (7th ed) (ISBN 0131424920) Digital Communications, 4th edition, 2000-08 book+solution by John Proakis Digital Communications: Fundamentals And Applications (2nd Edition)- Bernard Sklar Digital Design (3rd Edition) by M. Morris Mano Digital Electronics with VHDL (Quartus II Version) By William Kleitz Digital Fundamentals (10th Edition) floyd Digital Integrated Circuits by Rabaey 2nd edition solution manuel chapter 3,5,6,10 Digital image processing - Gonzalez 2Ed- Solutions Manual (209p) Digital Signal Processing - A Modern Introduction, 1st Edition Cengage learning Ashok Ambardar Digital Signal Processing - Proakis & Manolakis - Solutions Manual 3ed Digital Signal Processing (2nd Ed.) (Mitra) Solution Manual? Digital Signal Processing A Computer-Based Approach 1st ed Solutions Manual mitra Digital Signal Processing by Thomas J. Cavicchi - solution manuel Digital Signal Processing Principles, Algorithms and Applications (International Edition) by John Proakis ,Dimitris Manolakis Digital Signal Processing Using Matlab- Solution Manual Vinay K Ingle Proakis 2nd edition Discrete and Combinatorial Mathematics 5e (Solutions Manual Only) by Ralph P. Grimaldi Discrete Mathematics (5th Edition) By Dossey, Otto, Spence, Vanden Eynden Discrete Mathematics (5th Edition) by John A. Dossey , Albert D. Otto, Lawrence E. Spence ,Charles Vanden Eynden Discrete Mathematics, 5e John A. Dossey Albert D. Otto Lawrence E. Spence Charles Vanden Eynden Discrete-Event System Simulation 3rd edition by Jerry Banks John S. Carson II Barry L. Nelson David M. Nicol Discrete-Time Signal Processing 2nd Edition, 1999-02 by oppenheim Distributed Systems, Concepts and Design (Exercise Solutions) - G. Coulouris, J. Dollimore and T. Kindberg Doets & Eijck - The Haskell Road To Logic, Math and Programming - Solutions to Exercises Dorf, Svoboda Introduction to Electric Circuits, 7th Edition DSP First: A Multimedia Approach-Mclellan, Schafer & Yoder Solution Manual Dynamics of Mechanical Systems Solutions Manual (Horwood Engineering Science Series) by C. T. F. Ross Econometric Analysis Solutions Manual to the 6th Edition By William H. Greene Econometric Analysis, 5th edition william h. Greene Econometrics - [Instructor Solution Manual] The Econometrics of Financial Markets john y. campbell, andrew w. Lo Economics for Managers by Paul Farnham, 2008 custom edition sm + tb Economists Solution Manual (Blume, 1994) Effective Writing 8e May & May instructor manual Electric Circuits, Nilsson Riedel , 7th edition Electric Circuits,Nilsson Riedel , 8th edition Electric Machinery and Power System Fundamentals Electric Machinery and Power System Fundamentals Stephen J. Chapman first edition Electric Machinery by A. E. Charles Kingsley, Jr.Fitzgerald 6th edition electric machinery fundamentals 4th edition stephen j chapman Electric Machines By D. P. Kothari, I. J. Nagrath Electrical Engineering Principles and Applications 4th Allan R. Hambley Electrical Engineering: Principles and Applications 3ed Allan R. Hambley Electrical Machines Drives and Power Systems 6th edition by Theodore Wildi Electrical Machines, Drives and Power Systems 6th edition ISBN 0131969188 Electrical Power and controls Skvarenina 2nd Electromagnetics for Engineers by Fawwaz T. Ulaby Electronic Circuit Analysis and Design 2nd edt. by Donald A. Neamen - solution manuel Electronic Devices (Electron Flow Version), 8e floyd Electronic Devices and Circuit Theory 8th Ed Instructors Resource Manual with Text Solutions, Lab Solutions, and Test Item File Electronic Devices and Circuit Theory, 10e Boylestad & Nashelsky Electronic Devices and Circuit Theory, 9e Boylestad & Nashelsky Electronic Physics Strabman Electronics Fundamentals Circuits Devices and Applications by Thomas Floyd 7th edition Electronics, 2nd ed. by Allan R. Hambley Elementary Algebra with Applications, 3rd Edition Author: Terry H. Wesner Harry L. Nustad Elementary Differential Equations and Boundary Value Problems , 8th Edition Elementary Differential Equations And Boundary Value Problems, 7Th Ed - Boyce And Diprima Student Solutions Manual, Charles W Haines Ode Architect Companion Elementary Differential Equations with Boundary Value Problems, 6E Henry Edwards avid Penney Elementary Differential Equations With Boundary Value Problems, 4E,Edwards, Penney Elementary Linear Algebra with Applications 9 edition by Howard Anton, Chris Rorres Instructor Solutions Manual and Instructor Testbank Elementary Linear Algebra with Applications, 9/E Bernard Kolman David Hill Elementary Mechanics & Thermodynamics [2000 by Professor Jhon W. Norbury Elementary Number Theory (5th Edition) by Kenneth H. Rosen Elementary Principles of Chemical Processes Solutions Manual 3 ed By Richard M.Felder Elementary Principles of Chemical Processes Solutions Manual 3 ed By Richard M.Felder chapters 2 to 14 Elementary Statistics by Mario F. Triola, 10th Elementary Statistics With Multimedia Study Guide, 10/E solution manual Elementary Statistics With Multimedia Study Guide, 10/E test bank Elements of Chemical Reaction Engineering By H Fogler, 3rd ed Elements of Electromagnetics by Sadiku 2nd Ed. Elements of Electromagnetics, 3rd Ed., Matthew N.O. Sadiku Elements of Electromagnetics, 4th Ed., Matthew N.O. Sadiku homework and midterm problems Elements Of Information Theory - Solution Manual by thomas m. cover and joy a. Thomas E-Marketing, 4e Judy Strauss, Adel El-Ansary, Raymond Frost sm E-Marketing, 4e Judy Strauss, Adel El-Ansary, Raymond Frost tb E-Marketing, 5e Judy Strauss, Adel El-Ansary, Raymond Frost sm E-Marketing, 5e Judy Strauss, Adel El-Ansary, Raymond Frost tb Embedded Microcomputer Systems: Real Time Interfacing, 2nd Edition Jonathan W. Valvano Energy Management 5 ed 2005-12 Klaus-Dieter E. Pawlik Engineering - Materials Science, Milton Ohring Solutions Manual Engineering and Chemical Thermodynamics [Solution Manual] by Milo D. Koretsky Engineering and Chemical Thermodynamics Milo D. Koretsky Engineering biomechanics statics by Beatriz Guevarez, Joshua R?os, Nayka Rivera, Sharon V?zquez and Melvia Villegas Engineering Circuit Analysis 6Ed - Hayt Solutions Manual.pdf Engineering Circuit Analysis 7Ed William Hart Hayt Jack E. Kemmerly Solutions Manual contains chapters 2,3,4,5,7,9,10,11 Engineering Economy - Leland Blank & Anthony Tarquin 6th Edition selected solutions ( student solution) Engineering Economy 14e Sullivan solution manual Engineering Electromagnetics -Hayt (2001) Engineering Electromagnetics -Hayt (2001).rar Engineering Electromagnetics Nathan Ida 2nd edition Engineering Electromagnetics, 7th Ed., Hayt, Buck Engineering Fluid Mechanics, 7th Edition - Student Solutions Manual by Clayton T. Crowe, Donald F. Elger, John A. Roberson Engineering Fluid Mechanics, 9th Edition Crowe, Elger, Roberson, Williams Engineering Mathematics, 4th edt. by John Bird - solution manual Engineering Mechanics - Statics (11th ) by R.C.HIBBELER Engineering mechanics statics 12th edition by hibbeler Engineering Mechanics Dynamics (11th Edition) by Russell C. Hibbeler Engineering Mechanics Dynamics 3rd edition solution manual Hibbeler R.C. updated fixed 09-2006 Engineering Mechanics Dynamics, 4E Anthony M. Bedford and Wallace Fowler Engineering Mechanics Dynamics, 5E Anthony M. Bedford and Wallace Fowler Engineering Mechanics Dynamics, SI 6th Edition Meriam, Kraige Engineering Mechanics Statics 11th Edition By R.C.Hibbeler Engineering Mechanics Statics, 4E Anthony M. Bedford and Wallace Fowler Engineering Mechanics Statics, 5E Anthony M. Bedford and Wallace Fowler Engineering Mechanics, Dynamics 5E - Solutions manual By J. L. Meriam, L. G. Kraige, chapter 1-8 Engineering Mechanics, statics 5th edition Solutions manual By J. L. Meriam, L. G. Kraige Engineering Mechanics, statics 6th edition Solutions manual By J. L. Meriam, L. G. Kraige Engineering Mechanics: Dynamics 2 Ed. by Riley and Sturges contains chapters 13,14,15,16,17 chapters Engineering Mechanics: Statics: Solutions Manual (10th edition) by R.C. Hibbeler Engineering Problem Solving with C, 3E Delores M. Etter Engineering Problem Solving with Matlab 2nd edition by etter sm-tb- quiz Engineering Statistics, 4th Edition Montgomery, Runger, Hubele Engineering Vibration, 3e Daniel J. Inman Entrepreneurial Finance Chris Leach, Ronald W. Melicher 2nd edition instructor manual Entrepreneurial Finance Chris Leach, Ronald W. Melicher 2nd edition test bank Entrepreneurial Finance, 3rd Edition Chris Leach Ronald W. Melicher test bank Entrepreneurial Finance, 3rd Edition Chris Leach Ronald W. Melicher instructor manual Equilibrium and Non-Equilibrium Statistical Thermodynamics By Michel Le Bellac Essentials of Accounting for Governmental and Not for Profit Organizations 9e by Paul A. Copley Essentials of Entrepreneurship and Small Business Management 5e tb and im Essentials of Fluid Mechanics: Fundamentals and Applications, 1st Ed., by Cengel & Cimbala ISBN-13 9780073301129 Essentials of Investments 7th edition Zvi Bodie Alex Kane, Alan marcus test bank Essentials of Investments Zvi Bodie, Alex Kane, Alan J. Marcus Essentials of Managerial Finance 14e Brigham TB Essentials of Managerial Finance 13th Edition Scott Besley, Eugene F. Brigham instructor manual Essentials of Managerial Finance 13th Edition Scott Besley, Eugene F. Brigham test bank Essentials of Managerial Finance, 14th EditionScott Besley Eugene F. Brigham instructor manual Essentials of Managerial Finance, 14th EditionScott Besley Eugene F. Brigham spreadsheet problem solutions Essentials of Managerial Finance, 14th EditionScott Besley Eugene F. Brigham test bank Essentials of Organizational Behavior 9e Stephen P. Robbins Tim Judge Essentials of Statistics, by Triola, 3rd edition sm Essentials of Statistics, by Triola, 3rd edition tb Essentials of Strategic Management 4E DAVID HUNGER THOMAS L. WHEELEN im with tb Ethical Theory and Business, 8/E Tom L. Beauchamp Norman Bowie Denis Arnold Federal Taxation 2008 Corporations Partnerships Estates and Trusts 21E Anderson Pope Kramer test bank Feedback Control of Dynamic Systems 4th edition Franklin - Solutions Manual Feedback Control of Dynamic Systems, 5/E franklin Festo didactic Process Control System Field and Wave Electromagnetics, 2nd edition, Cheng Finance Management Test Bank brigham 11 test bank Financial & managerial Accounting 13E By william Haka bettner Financial accounting an introduction to concepts, methods, and uses by clyde Stickney and Roman Weil solutions manual 11th edition financial Accounting 4e by John Wild Financial Accounting 6e by kieso solution manual Financial Accounting 6e by kieso test bank Financial Accounting 6e Harrison Horngren Financial Accounting 6th edition by Harrison Solution Manual financial accounting 6th edition harrison test bank financial accounting 7th edition harrison solution manual financial accounting 7th edition harrison test bank Financial Accounting; Tools for Business Decision Making, 4th Edition, Kimmel.Weygandt.Kieso instructor manual Financial Accounting; Tools for Business Decision Making, 4th Edition, Kimmel.Weygandt.Kieso solution manual Financial Accounting; Tools for Business Decision Making, 4th Edition, Kimmel.Weygandt.Kieso test bank Financial Analysis with Microsoft?Excel 4th Edition Timothy R. Mayes, Todd M. Shank instructor manual Financial Analysis with Microsoft?Excel 4th Edition Timothy R. Mayes, Todd M. Shank spreadsheet problems Financial Analysis with Microsoft?Excel?2007 Timothy R. Mayes, Todd M. Shank Instructor Spreadsheet Files Financial and managerial accounting 14e by williams haka bettner SM Financial management 2e by jae K shim Financial management Principles and application 10e by By Arthur J. Keown, John D. Martin, John W. Petty, David F. Scott solution manual Financial Management Theory & Practice,Eugene Brigham,12th edition [ Test Bank ] financial management theory and practice 10e by Brigham solution manual Financial Management Theory And Practice 11e by Brigham solution manual Financial management theory and practice 12e by Brigham sm financial management theory and practice 12e by Brigham TB financial Management Theory and Practice, 11e By Eugene F. Brigham test bank Financial Markets And Institution (7thEd) Madura TestBank Financial Reporting Analysis 10th edition TB by gibson Finite Element Method: Volume 1 The Basis 5th edition by O. C. Zienkiewicz, R. L. Taylor Finite Mathematics ?nstructor's Resource Guide and Solutions Manual, 8E Margaret L. Lial Raymond N. Greenwell Nathan P. Ritchey Fluid Mechanics Solutions Manual by yunus a. cengel Fluid Mechanics Fundamentals and Applications by Cengel & Cimbala Fluid Mechanics With Engineering Applications -- Solutions Manual by E. John Finnemore, Joseph B Franzini Fluid Mechanics with Student Resources, 5th edition 2002-12 by Frank M. White Fluid Power with Applications, 7E esposito Foundations of Finance The Logic and Practice of Financial Management Art J Keown John D Martin 6th edition im + tb Foundations of Financial Management, 12e By Stanley B. Block Geoffrey A. Hirt Foundations of Financial Markets and Institutions 4e Fabozzi, Modigliani & Jones instructor manual and test gen Fourier and Laplace Transform - Antwoorden Fractal Geometry Mathematical Foundations and Applications Solutions Fracture mechanics fundamentals and applications 2nd edition northam anderson solution manual framework for Marketing Management, A - Philip Kotler (4th ed) (ISBN 0136026605) framework for Marketing Management, A - Philip Kotler (4th ed) (ISBN test bank Friendly Introduction to Analysis - Witold A.J. Kosmala (2nd ed) fundamental accounting principles 17th edition larson solution manual Fundamental Accounting Principles, 18/e John J. Wild Barbara Chiappetta Kermit D. Larson solution manual and test bank fundamentals accounting principles by larson 18e SM Fundamentals of Actuarial Mathematics?by D. Promislow Fundamentals of Advanced Accounting 1e Fisher taylor chang solution manual Fundamentals of Advanced Accounting 1e Fisher taylor chang test bank fundamentals of advanced accounting 3e by hoyle Fundamentals of Advanced Accounting, 2e Joe B. Hoyle Thomas F. Schaefer Timothy S. Doupnik ( Solution Manual ) Fundamentals of Applied Electromagnetics 5th edition by Fawwaz T. Ulaby solution manual Fundamentals of Biochemistry Life at the Molecular Level, 3rd Edition Fundamentals of Chemical Reaction Engineering - Solutions Manual By Mark E. E. Davis, Robert J. J. Davis Fundamentals of Communication Systems by John G. Proakis ,Masoud Salehi computer solutions Fundamentals of Contemporary Financial Management, 2nd Edition R. Charles Moyer James R. McGuigan Ramesh P. Rao instructor manual Fundamentals of Contemporary Financial Management, 2nd Edition R. Charles Moyer James R. McGuigan Ramesh P. Rao test bank fundamentals of corporate finance 8e by Ross Ross, Westerfield,jordan Fundamentals of Corporate Finance, 4th Edition (Brealey, Myers, Marcus) by Bruce Swenson Fundamentals of Database Systems, 5E Ramez Elmasri,Shamkant B. Navathe Fundamentals of Differential Equations and Boundary Value Problems (4th Ed., Kent B. Nagle, Late, Edward B. Saff & Arthur David Snider) Fundamentals of Digital Logic with Verilog Design by s. Brown z vranesic Fundamentals of Digital Logic with VHDL Design-1st edition by S. Brown, Z. Vranesic Fundamentals of Electric Circuits 2nd by Alexander Sadiku Fundamentals of Electric Circuits 3rd edition by Alexander Sadiku Fundamentals of Electromagnetics for Electrical and Computer Engineering Nannapaneni Narayana Rao Fundamentals of Electromagnetics with Engineering Applications by Stuart M. Wentworth Fundamentals of Electronic Circuit Design by David and Donal Comer Fundamentals of Engineering Electromagnetics--Cheng Fundamentals of engineering thermodynamics moran shapiro Fundamentals of engineering thermodynamics moran shapiro 6th edition Fundamentals of Engineering Thermodynamics: Si Version / 5th Edition by Michael J. Moran Howard N. Shapiro Fundamentals of Financial Management 12th edition instructor manual Fundamentals of Financial Management 12th edition test bank Fundamentals of Financial Management 10th Edition by Eugene Brigham Solution manual Fundamentals of Financial Management 10th Edition by Eugene Brigham test bank Fundamentals of Financial Management 11e by Eugene F. Brigham, Joel F. Houston spreadsheet problems Fundamentals of Financial Management 11e by Eugene F. Brigham, Joel F. Houston test bank and cyberproblems Fundamentals of Financial Management 11e by Brigham Instructor manual Fundamentals of financial management 12e by james c. van horne Fundamentals of Financial Management Eugene F. Brigham, Joel F. Houston 12th edition instructor manual Fundamentals of Financial Management Eugene F. Brigham, Joel F. Houston 12th edition spreadsheets problems and web appendix solutions Fundamentals of Financial Management Eugene F. Brigham, Joel F. Houston 12th edition test bank Fundamentals of Financial Management, 12th Edition (Instructors guide ONLY) by James C. Van Horne, John M Wachowic Fundamentals of Financial Management, Concise Edition 5th Eugene F. Brigham, Joel F. Houston test bank Fundamentals of Financial Management, Concise Edition 5th Eugene F. Brigham, Joel F. Houston spreadsheets problems Fundamentals of Financial Management, Concise Edition 5th Eugene F. Brigham, Joel F. Houston instructor manual Fundamentals of Financial Management, Concise Edition Eugene F. Brigham, Joel F. Houston 6th edition instructor manual Fundamentals of Financial Management, Concise Edition Eugene F. Brigham, Joel F. Houston 6th edition spreadsheet problem solutions Fundamentals of Financial Management, Concise Edition Eugene F. Brigham, Joel F. Houston 6th edition test bank Fundamentals Of Fluid Mechanics 3Rd And 4Th Edition Solutions Manual Fundamentals of Fluid Mechanics, 6th Edition Munson, Young, Okiishi, Huebsch Fundamentals of Heat and Mass Transfer, 5th Edition by Frank P. Incropera Fundamentals of Heat and Mass Transfer, 6th Edition Incropera, DeWitt, Bergman, Lavine Fundamentals of Investing, 10th Edition by Gitman and Joehnk Fundamentals of Investments 3e Gordon J. Alexander William F. Sharpe Fundamentals of Logic Design 5th edition by charles roth Fundamentals of Machine Component Design, 3rd ed - Student Solutions Manual by Juvinall, Marshek Fundamentals of manufacturing 2nd edition by philip d. rufe solutions manual Fundamentals of Momentum, Heat and Mass Transfer, 5th Edition Welty, Wicks, Rorrer, Wilson Fundamentals of Multinational Finance, 2nd edition by Michael H. Moffett test bank and solution manual Fundamentals of Multinational Finance, 3E by Michael H. Moffett (Instructor's Manual) Fundamentals of Multinational Finance, 3E by Michael H. Moffett test bank Fundamentals of Organic Chemistry, 5E, Study Guide and Solutions Manual By T. W. Graham Solomons Fundamentals of Organizational Communication 7e Pamela S. Shockley- Zalabak Fundamentals of physics Halliday Resnick 8th student solution not complete Fundamentals of Physics, 7th Edition - Instructor's SOLUTIONS MANUAL halliday and resnick Fundamentals of Physics, Edition 8, Halliday, Resnick, Walker (Solution Manual) Fundamentals of Probability With stochastic processes 3/e (Solutions Manual ) By Saeed Ghahramani Fundamentals of Quantum Mechanics 0521829526 sols Fundamentals of Quantum Mechanics For Solid State Electronics and Optics by C. L. Tang fundamentals of selling 10e by futrell TB Fundamentals of Semiconductor Devices - Anderson Fundamentals of Signals and systems using web and matlab third edition by Edward W. Kamen, Bonnie S Heck Fundamentals of Solid State Electronics by chih tang sah Fundamentals of Thermal-Fluid Sciences Solution Manual 2ed By Yunus A. Cengel, Robert H. Turner Fundamentals of Thermodynamics [Sonntag-Borgnakke-Van Wylen Solutions Manual volume 1 and volume 2 Fundamentals of Thermodynamics SOLUTION MANUAL 6ed By Richard E. Sonntag, Claus Borgnakke, Gordon J. Van Wylen, Fundamentals of Wireless Communication by Tse and Viswanath Fundementals of differential equations 7E & Fundementals of differential equations and boundary value problems 5E Nagle , Saff , Snider even problems Fundementals of engineering economics 2E Chan S. Park Further Mathematics for Economic Analysis General Chemistry, 8th Edition - Solution Manual by Ralph H. Petrucci William S. Harwood; Geoffrey Herring General Chemistry: Principles and Modern Application & Basic Media Pack, 9/E Ralph H Petrucci test bank Geology for Engineers & Environmental Scientists by Alan Kehew 3rd edition Gilat MATLAB An Introduction with Applications, 3rd Edition Government and Nonprofit Accounting Theory & Practice - 8th edition by Freeman, Shoulders, Allison, Patton, Smith sm Government and Nonprofit Accounting Theory & Practice - 8th edition by Freeman, Shoulders, Allison, Patton, Smith tb Government and Nonprofit Accounting Theory & Practice - 9th edition by Freeman, Shoulders, Allison, Patton, Smith instructor manual Government and Nonprofit Accounting Theory & Practice - 9th edition by Freeman, Shoulders, Allison, Patton, Smith tb Government and Nonprofit Accounting Theory & Practice - 9th edition by Freeman, Shoulders, Allison, Patton, Smith sm Government and Not-for-Profit Accounting: Concepts & Practices (4th edition) by Granof and Wardlow sm Government and Not-for-Profit Accounting: Concepts & Practices (4th edition) by Granof and Wardlow tb Gravity An Introduction to Einstein's General Relativity by hartle Gravity An Introduction to Einstein's General Relativity James B. Hartle Haberman Applied Partial Differential Equations 4e Instructor's Manual Harcourt mathematics 12 Advanced Functions and Introductory Calculus - Solutions Manual by Brian / Nelson Harcourt Mathematics 12 Geometry and Discrete Mathematics Solutions Manual By McGraw-Hill Heat and Mass Transfer 3e SM Yunus A. Cengel Heat Transfer: A Practical Approach. Solution Manual ONLY by cengel 2nd edition Heat Transfer-Fundamentals of Heat and Mass Transfer-Incropera & Dewitt Solution Manual Heating, Ventilating and Air Conditioning Analysis and Design, 6th Edition McQuiston, Parker, Spitler High-Speed Digital System Design A Handbook of Interconnect Theory and Design Practicesbby Stephen H. Hall Human Biology by Colleen Belk Virginia Borden-Maier test bank Human Culture Highlights of Cultural Anthropology By Melvin R Ember, Carol R. Ember Human resources management 10e Gary dessler (IM+TB) Hydraulics in Civil and Environmental Engineering By Andrew Chadwick Hydrology and Floodplain Analysis (4th Ed., Philip Bedient, Wayne Huber & Baxter Vieux) Hydrology and Floodplain Analysis, 4e Philip B. Bedient Wayne C. Huber Baxter E. Vieux IBM WebSphere RFID Handbook A Solution Guide by IBM Redbooks In Experiments with Economic Principles, Instructor's Manual By Bergstrom Industrial Safety and Health Management 5e C. Ray Asfahl all resources Instructor Manual to A Course in Modern Mathematical Physics By S. M. SZE 2nd edition Instructor Manual to An Introduction to Thermodynamics and Statistical Mechanics 2ed By Keith Stowe Instructor Manual to Introduction to Solid State Physics Eighth Edition By Charles Kittel Instructor Manual to Introductory Quantum Optics By Christopher Gerry and Peter Knight Instructor Manual to Quantum Physics 3rd ediiton by Gasiorowicz, S. - Instructor Manual to Quantum Physics Third Edition by Stephen Gasiorowicz Instructor Manual to SEMICONDUCTOR DEVICES Physics and Technology Second Edition By S.M.Sze Instructor Manual to Special Relativity by Patricia M. Schwarz and John H. Schwarz Instructor Solutions Manual for Building Java Programs Stuart Reges,Martin Stepp Instructor Solutions Manual to accompany Boyce Elementary Differential Equations and Boundary Value Problem 8ed by Charles W. Haines, William E. Boyce INSTRUCTOR£À SOLUTIONS MANUAL Basic Technical Mathematics with Calculus SI Version John R. Martin Eighth Canadian Edition Instructor's Manual for Solving ODEs with MATLAB By L. F. Shampine, I. Gladwell, S. Thompson Instructor's Manual for Sipser's Introduction to the Theory of Computation by Ching Law Instructor's Manual for Solving ODEs with MATLAB By L. F. Shampine, I. Gladwell, S. Thompson Instructor's Manual Im Experiments with Economic Principles By Bergstrom Instructor's Manual Of Fundamental Methods Of Mathematical Economics Chiang & Wainwright 2005 Mc Graw Hill Instructor's Manual(Information Technology Project Management 3Rd Edition) by Kathy Schwalbe Instructor's Manual: Im Experiments with Economic Principles By Bergstrom INSTRUCTOR'S SOLUTIONS MANUAL (to accompany Elementary Statistics Ninth Edition) by milton loyer Instructor's solution manual ISBN 0534382150 A Transition to Advanced Mathematics solution manual by douglas smith 5th edition Instructor's Solutions for: Design of Analog CMOS Integrated Circuits by razavi Instructors Solutions Manual for Differential Equations with Boundary Value Problems, 2/E by john polking Instructor's Solutions Manual for Serway and Jewett's Physics for Scientists and Engineers Sixth Edition Volume One by Ralph V. McGrew Instructor's Solutions Manual for Serway and Jewett's Physics for Scientists and Engineers Sixth Edition Volume two by Ralph V. McGrew Intermediate Accounting 10e by Nikolai sm Intermediate Accounting 11e by Kieso Intermediate Accounting 12e by Kieso Intermediate accounting 12th Updated by Kieso Solution manual Intermediate accounting 12th Updated by Kieso test bank Intermediate Accounting 2e by Baruch Englard Intermediate Accounting 3e by J. David Spiceland Intermediate Accounting 4e revised by J. David Spiceland solution manual Intermediate accounting by Spiceland 4e Solution manual Intermediate Accounting, Update, 12th Edition international student solution manual Intermediate Algebra, 8th Edition By Margaret L. Lial, John Hornsby, Terry McGinnis International Accounting 1e by Doupnik solution manual International Accounting 6e Frederick D. Choi Gary K. Meek International Business The Challenges of Globalization, 4E John J. Wild,Kenneth L. Wild tb and im International Economics, 7e Husted Melvin test bank and solution manual International Financial Management 9th Edition jeff madura instructor manual International Financial Management 9th Edition jeff madura test bank International Financial Management Geert Bekaert Robert J. Hodrick test bank and sol manual International Management Managing Across Borders and Cultures 6e test bank and instructor manual Introduction Fluid Mechanics, 6Th Edition Solution by fox CHAPTER 1-8 Introduction To Analysis (3rd) Wade Solution Manual Introduction to Chemical Engineering Thermodynamics 7th edition (solution manual) By J.M. Smith, Hendrick C Van Ness Introduction to Corporate Finance William L. Megginson, Scott B. Smart 1st edition instructor manual Introduction to Corporate Finance William L. Megginson, Scott B. Smart 1st edition test bank Introduction to Corporate Finance, 2nd Edition William L. Megginson Scott B. Smart test bank Introduction to Corporate Finance, 2nd Edition William L. Megginson Scott B. Smart solution manual Introduction To Electric Circuits 6th Ed [Solutions Manual] By R. C. Dorf and J. A. Svoboda Introduction to Electrodynamics (Third Edition) by David J. Griffiths Introduction to Engineering Experimentation 2e wheeler and ganji Introduction to Engineering Thermodynamics, Edition 2, Sonntag, Borgnakke (Solution Manual) Introduction to Environmental Engineering and Science, 3E Gilbert M. Masters Wendell P. Ela, Introduction to Financial Accounting, 9E Charles T. Horngren Gary L. Sundem John A. Elliott Donna Philbrick test bank + solution manual Introduction to Fluid Mechanics (Fox, 5th ed) Solutions Manual Introduction to Fluid Mechanics, Edition 7, Fox, Pritchard, McDonald (Solutions Manual) Introduction to Fourier Optics Third Edition Problem Solutions by Joseph W. Goodman Introduction to Government and Non-for-Profit Accounting 6th edition by martin ives sm Introduction to Government and Non-for-Profit Accounting 6th edition test bank by martin ives Introduction To Graph Theory, 2nd ed. Douglas B. West Introduction to Heat Transfer - 013391061X Solution's Manual By peyman pourmoghaddam , Vedat S. Arpaci not complete Introduction to Heat Transfer 4th Edition SOLUTION MANUAL By Frank P. Incropera, David P. DeWitt Introduction to Heat Transfer, 5th Edition Incropera, DeWitt, Bergman, Lavine Introduction to Java Programming, Comprehensive Version, 7/E Y. Daniel Liang Introduction To Linear Algebra 3Ed - Gilbert Strang Solutions Manual Introduction To Management Accounting horngren 14e TB Introduction to management sciences 9e by Taylor solution manaul Introduction to Managerial Accounting 2nd ed Brewer test bank Introduction to Mathematical Statistics 6th Robert V. Hogg, allen t. Craig Introduction to Operations and Supply Chain Management 2e Cecil Bozarth Introduction to Operations and Supply Chain Management 2e Cecil Bozarth Robert B. Handfield test bank and sol manual Introduction to Ordinary Differential Equations by James C. Robinson Introduction to Probability by Dimitri P. Bertsekas - solution isbn: 1-886529-40-X Introduction to Quantum Mechanics (Second Edition) - Solutions Manual By David J. Griffiths Introduction to Statistical Quality Control, 6th Edition Montgomery Introduction to VLSI Circuits and Systems (2001 draft) - John P Uyemura - Solutions Manual Introduction To Wireless Systems - P M Shankar - Solutions Manual Introductory Circuit Analysis, 11E Robert L. Boylestad Introductory Econometrics A Modern Approach 2ed Jeffrey Wooldridge Introductory Econometrics A Modern Approach, 3Ed (with Economic Applications Online, Econometrics Data Sets with Solutions Manual Web Site Printed Access Card) by Jeffrey Wooldridge Introductory Linear Algebra An Applied First Course, 8e by Kolman Hill Introductory Mathematical Analysis for Business, Economics and the Life and Social Sciences, 12/E Ernest F. Haeussler Richard S. Paul Richard J. Wood test bank and sol manual Introductory Quantum Optics by Knight and Gerry Investment analysis and portfolio management 8e by Reily Brown Investment Analysis and Portfolio Management- Solutions Manual, 7th Edition by Frank K. et al. Reily Investment analysisnvestment analysis and management 9e by charles p. jones & management, 9e By jones some chapters missings sm IR: The New World of International Relations, 7e Michael G. Roskin, Lycoming College Nicholas O. Berry instructor manual and test bank instructor s manual with powerpoints to accompany pic microcontroller and embedded systems by muhammed ali mazidi rolin d mckinlay instructor s resource manual to accompany electronic devices,6th edition and electronic devices electron flow version 4th edition thomas floyd intermediate Accounting 12 E Kieso (TB) intermediate accounting 5e spiceland test bank and solution manual intermediate Accounting 12e by Keiso sm intermediate accounting 4th edition spiceland test bank introduction to algorithms 2nd edition instructors manual McGraw-Hill by thomas h. Cormen introduction to Environmental Engineering and Science (2nd Edition) (Hardcover) by Gilbert M. Masters introduction to linear algebra (5th, johnson) introduction to management accounting 14e Charles T. Horngren, Gary L. Sundem, William O. Stratton, Jeff Schatzberg, Dave Burgstahler solution manual introduction to mechatronics and measurement systems 2nd edition David G. Alciatore and Michael B. Histand introduction to Operations Research - 7th by Frederick Hillier, Gerald Lieberman introduction to probability Charles M. Grinstead and J. Laurie Snell odd solutions investment Analysis & Portfolio Management, 7th edition by Reilly and Brown Java, an Introduction to problem solving and programming , fifth Ed. by W. Savith and F. Carrano john E. Freund's Mathematical Statistics with Applications, 7th edition, by Miller and Miller Journey into Mathematics: An Introduction to Proofs (with solution manual) by Joseph J. Rotman Kc's Problems and Solutions for Microelectronic Circuits by k.c smith 4th edition Kinematics, Dynamics, and Design of Machinery by K. J. Waldron (Author), G. L. Kinzel (Author) Kinetics of Catalytic Reactions--Solutions Manual by: M. Albert Vannice Labor Relations, 12E Arthur A Sloane instructor manual laboratory manual to accompany Introductory Circuit Analysis 11e boylestad Lagrangian and Hamiltonian Mechanics Solutions to the Exercises by M. G. Calkin laser fundamentals 2nd edition by william t. Silfvast lectures on corporate finance 2e by Peter Bossaerts Lectures on Corporate Finance, Second Edition by Peter Bossaerts and Bernt Arne ??degaard legal environmental A critical thinking approach by KUBASKE, BRENNAN and BROWNE 5th edition test bank ?m legal environmental: A critical thinking approach by KUBASKE, BRENNAN and BROWNE 5th edition test bank Linear algebra and it s applications 3rd edition by david c. Lay linear algebra Juan de Burgos Solutions Manual spanish Linear algebra with applications 3e Otto Bretcher - Solutions Manual Linear Algebra with Applications 6 edition by Leon Linear Algebra with Applications, 7E by Steven J. Leon Linear Algebra, 4E Stephen H. Friedberg Arnold J. Insel Lawrence E. Spence Linear circuit analysis by R. A. DeCarlo and P. Lin - solution manuel 2nd edition linear systems and signals by lathi solutions manual blue covered Living Religions, 7E Mary Pat Fisher test bank Logic And Computer Design Fundamentals (4thEd) - Mano - SolutionsManual Lu, Likos Unsaturated Soil Mechanics Machine Design, An Integrated Approach, 3rd edition, by Robert L. Norton Macroeconomics (8thEd) - Froyen - Solutions Manual Macroeconomics 6e Andrew B. Abel, Ben S. Bernanke, Dean Croushore Macroeconomics, 4E Olivier Blanchard instructor manual Macroeconomics, 4E Olivier Blanchard test bank Macroeconomics, 5E Olivier Blanchard instructor manual and test bank Management 5th Edition Chuck Williams instructor manual Management 9E Stephen P. Robbins Mary Coulter test bank and instructor manual management accounting 5e Anthony A. Atkinson Robert S. Kaplan Ella Mae Matsumura S. Mark Young instructor manual test bank and solution manual Management accounting 5E atkinson solution manual Management accounting 5E atkinson test bank Management Control Systems Performance Measurement, Evaluation and Incentives 2e Merchant & Van der Stede Management Information Systems; Managing the Digital Firm, Edition 10, Laudon test bank and solution manual Management Robbins Coulter 9th edition (test bank ) Management10E Stephen P. Robbins Mary Coulter Management10E Stephen P. Robbins Mary Coulter test bank Managerial Accounting 12e By Garrison Noreen ( Solution Manual ) Managerial Accounting 12e By Garrison Noreen (Test Bank) Managerial Accounting Bamber, L. S., K. W. Braun, and W. T. Harrison, Jr. 2008 Managerial Accounting international edition Garrison11 ( Solution Manual) Managerial Accounting, 11th Edition by Ray H Garrison, Eric Noreen, Peter C. Brewer Managerial Accounting: An Introduction to Concepts, Methods and Uses 10th Edition Michael W. Maher, Clyde P. Stickney, Roman L. Weil instructor manual Managerial Accounting: An Introduction to Concepts, Methods and Uses 10th Edition Michael W. Maher, Clyde P. Stickney, Roman L. Weil solution manual Managerial Accounting: An Introduction to Concepts, Methods and Uses 10th Edition Michael W. Maher, Clyde P. Stickney, Roman L. Weil test bank Managerial Economics Applications, Strategies, and Tactics 11th Edition James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris instructor manual Managerial Economics Applications, Strategies, and Tactics 11th Edition James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris test bank Managerial Finance, Gitman,Lawrence 12e [SM] Managing Human Resources by Luis R. Gomez-Mejia; David B. Balkin; Robert L. Cardy. 5th Edition im and tb Manufacturing Engineering and Technology 5E by Kalpakjian and Schimid Marketing Kotler Armstrong 11th edition (Test bank ) Marketing Management (13th Edition) by Philip Kotler, Kevin Keller im Marketing Management (13th Edition) by Philip Kotler, Kevin Keller tb Materials and Processes in Manufacturing, 9 edition, [Solution manual] by E. Paul DeGarmo, Solutions Manual by: Barney E. Klamecki Materials Science and Engineering An Introduction 6E By William D. Callister materials science and engineering,callister 7th edition Mathematical Methods for Physics and Engineering A Comprehensive Guide 3rd edition by riley Mathematical Methods in the Physical Sciences, 3rd Edition Mary L. Boas Mathematical Models in Biology An Introduction by Elizabeth S. Allman john a. Rhodes Mathematical Models in Biology Sols by Elizabeth S. Allman, John A. Rhodes Mathematics for Economists Solution Manual - Simon and Blume (ver 2) Mathematics with Applications (9th Edition) Margaret L. Lial McgrawHill - William H. Hayt, John A. Buck - Engineering Electromagnetics, 6ed Solutions Manual Mechanical Behavior of Materials, 3E Norman E Dowling Mechanical Measurements 6th Edition by Thomas G. Beckwith, Roy D. Marangoni, John H. Lienhard V Mechanical Vibrations - Singiresu Rao - Solutions Manual 3rd edition chapters missing 6 9 12 ( all links same ) 35 mb Mechanics of Fluids by Merle C. Potter Mechanics of Fluids Solutions Manual 8ed By John Ward-Smith Mechanics of Materials [solutions manual] hibbeler 7th edition Mechanics of Materials An Integrated Learning System Philpot Mechanics of Materials Sixth Edition by R.C.Hibbeler Mechanics Of Materials Solution Manual (3Rd Ed , By Beer) nearly same 4th edition just numbered different Mechanics of Materials, 2nd Edition Craig Mechanics of Materials, 5th Edition by James M. Gere Mechanics of Materials, 6th Edition by James M. Gere (missing small portion, section 8.5) Mechanics of Materials, 7th Edition James M. Gere - Stanford University (Professor Emeritus) mechanics of solids by carl t. t. ross, d.sc. Microeconomic Analysis Solution Manual - Varian 3rd edition Microeconomic Analysis Third Edition by Hal R. Varian Microeconomic Theory Solutions Manual for Mas-Colell Microeconomics 7e Robert Pindyck Daniel Rubinfeld Microeconomics Theory and Applications with Calculus by perloff insrructor manual Microeconomics Theory and Applications with Calculus by perloff test bank Microelectronic Circuit Design 2nd and 3rd edition by R. Jaeger selected solutions answers not solution microelectronic circuits 5th ed. By Adel S. Sedra, Kenneth C. Smith microelectronics 3rd edition by jeager Microelectronics Circuit Analysis Desing 3rd Edition by Donald A. Neamen Microelectronics I & II 1st edition by by Dr. Wen-Ching Chang Microelectronics-Digital and Analog Circuits and Systems by Millman Microprocessors and Interfacing,Revised Second Edition Douglas V Hall Microwave And Rf Design Of Wireless Systems - Solution Manual (D M Pozar) Microwave Engineering 3e - David M Pozar Microwave Transistor Amplifiers Analysis and Design, 2nd Edition (Solutions Manual) by Guillermo Gonzalez Miller and Freund's Probability and Statistics for Engineers 7th edition by Richard A. Johnson Modern Advanced Accounting , 10th Edition, by Larsen, (publisher McGraw Hill) Modern Advanced Accounting 10th edition Test Bank LARSEN Modern Auditing Assurance Services 8e Boynton sm and tb modern control engineering by katsuhiko ogata 3rd edition ISBN: 0132273071 modern control engineering by katsuhiko ogata 4th edition Modern Control System 11th edition by richard c dorf, robert H bishop Modern Control System 9th by richard c dorf, robert H bishop Modern Database management 9e by Jeffrey A. Hoffer im Modern Database management 9e by Jeffrey A. Hoffer TB Modern Database Management, Ninth Edition Jeffrey A. Hoffer, University of Dayton Mary Prescott Heikki Topi Modern Digital and Analog Communications Systems - B P Lathi Solutions Manual 3rd edition Modern Digital Electronics,3E R P JAIN modern electronic communication 8th edition by gary miller and beasley Modern Electronic Communication, Beasly and Miller, 9th edition instructor manual Modern Electronic Communication, Beasly and Miller, 9th edition test bank Modern elementary statistics 12 ed freund e john test bank Modern Elementary Statistics 12e John E. Freund Benjamin M. Perles solution manual Modern Elementary Statistics 12e John E. Freund Benjamin M. Perles test bank modern Operating Systems 3rd edition - Andrew S. Tanenbaum Modern Organic Synthesis: An Introduction By Michael H. Nantz, Hasan Palandoken, George S. Zweifel Modern Physics, 2/E Randy Harris Modern Quantum Mechanics - J. J. Sakurai - Solution Modern Systems Analysis and Design, 5E Jeffrey A. Hoffer Joey F George Joseph S Valacich test bank Money, Banking, & Financial Markets 2nd edition Roger LeRoy Miller David D. VanHouse tb Multinational Business Finance 11E by David K. Eiteman SM Multinational Business Finance 11E by David K. Eiteman tb Multivariable Calculus: Student Solutions Manual 4th edition by James Stewart (Author) Nanoengineering of Structural Functional and Smart Materials Nanoengineering of Structural, Functional and Smart Materials Network flows theory, algorithms, and applications by Ravindra K. Ahuja (Author), Thomas L. Magnanti (Author), James B. Orlin (Author) odd solutions Network Security Essentials: Applications and Standards, 3/e stallings Nonlinear programming 2nd edition solution manual by dimitri bertsekas Numerical Methods For Engineers Solutions Manual by chapra 4th edition Numerical Methods for Engineers,5E,Steven C Chapra Numerical Solution of Partial Differential Equations: An Introduction 2ed By K. W. Morton, D. F. Mayers Operating System Concepts (6Th Ed)-Instructor'S Manual (A Silberschatz) Operating System Concepts, 7th Edtion, Instructor's Manual by A Silberschatz Operating systems Internals and Design principles 4th by william stallings Operating Systems Internals and Design principles( 5th Ed,William Stallings) Operating systems internals and design principles William STALLINGS 5.edition test bank Operating Systems: Internals and Design Principles, 6/e stallings solution manual and test bank Operations Management 9e heizer test bank Operations Management 5th edition Slack Nigel Instructors Manual Operations Management 8e by Jay Heizer Barry Render sm Operations Management 8e by Jay Heizer Barry Render tb Operations Management 9th by Jay Heizer Barry Render sm Operations Management 9th by Jay Heizer Barry Render tb Operations Management for MBAs, 3rd Edition Meredith, Shafer tb+im Operations Management Reid, Sanders An Integrated Approach 2nd edition Operations Management Reid, Sanders An Integrated Approach 2nd edition test bank Operations Management, 10e William J. Stevenson instructor manual with solutions to supplementary problems Operations Management, 10e William J. Stevenson test bank Operations Management, 9e William J. Stevenson instructor manual with solutions to supplementary problems Operations Management, William J Stevenson, 9e [TB] operations managenment processes and value chains 8e by Lee J. Krajewski solution manual operations managenment processes and value chains 8e by Lee J. Krajewski test bank Operations Research An Introduction, 8E Hamdy A. Taha Optimal Control Theory An Introduction By Donald E. Kirk Options, Futures and Other Derivatives, 4th Edition (Solutions Manual) by John Hull, John C. Hull Options, Futures and Other Derivatives, Fifth Edition (Solutions Manual) by John Hull, John C. Hull Options, Futures, and Other Derivatives 7E by JOHN C HULL TB Organic Chemistry by Robert C. Atkins, Francis A Carey, Robert Atkins, Francis Carey Organic Chemistry 2nd ed [Student SOLUTIONS MANUAL and Study Gde] - J. Hornback, B. Murugaverl (Thomson, 2006) WW Organic Chemistry and CW+ GradeTracker Access Card Package, 6/E Leroy G. Wade, JR., Whitman College test bank Organic Chemistry, 5/E Paula Y. Bruice test bank Organizational Behavior Stephen P Robbins 12th edition (test bank) Organizational Behavior, by Stephen P. and Timothy A. 13th Edition Robbins Judge im and tb Organizational Behavior: Managing People and Organizations, 9th Edition Ricky W. Griffin Gregory Moorhead instructor manual Organizational Behavior: Managing People and Organizations, 9th Edition Ricky W. Griffin Gregory Moorhead test bank organizational behaviour 12e by Robbins ( instructor manual ) organizational behaviour 12e by Robbins ( Test bank ) Organizational Theory, Design and Change, 5/E Gareth R. Jones Organizational Theory, Design and Change, 5E Gareth R. Jones Instructor's Manual and test bank Organizational Theory, Design and Change, 5E Gareth R. Jones im and tb Papathomas T.V. - Solutions Manual to Accompany Millman Integrated Electronics Analog and Digital Circuits and Systems (McGraw-Hill) Partial Differential Equations and Boundary Value Problems with Fourier Series (2nd Edition) (Student Solutions Manual) by asmar Pattern Recognition and Machine Learning (Solution Manual) - Bishop Personal Finance Turning Money into Wealth 5e Arthur J. Keown Personal Finance Turning Money into Wealth and Student Workbook, 4E keown instructor manual and test bank Physical Chemistry (Instructor's Solutions Manual) Peter Atkins & Julio de Paula 7ed Physics by Resnick Halliday Krane, 5th Ed. Vol 2 physics concepts and connections combined edition solutions manual by igor nowikow brian heimbecker christopher t. Howes Physics for Scientist & Engineers with Modern Physics - A strategic Approach by Randall D. Knight chapter 1-42 Physics for Scientist & Engineers with Modern Physics - A strategic Approach chapter by Randall D. Knight 1-35 Physics for Scientists and Engineers 5th edition by Serway And Jewett Physics for Scientists and Engineers Extended Version, 5th edition, Physics for Scientists and Engineers with Modern Physics (3rd Edition) Physics for Scientists and Engineers with Modern Physics (3rd Edition) by giancoli Physics For Scientists Engineers With Modern Physics (4thEd) - Giancoli - Solutions Manual Physics of the Solar Corona: An Introduction with Problems and Solutions by Markus Aschwanden Physics Principles with Applications Instructor's Solutions Manual (Giancoli, Volume 1and 2) 6th edition power analysis and design by glover , sarma 3rd ediiton Power Electronics, Converters, Applications, and Design By Ned Mohan, Tore M. Undeland, William P. Robbins Power System Analysis and Design Glover and Sarma 4e Thomson Learning Glover J. Duncan, Sarma Mulkutla .S. Power System Analysis Hadi Saadat 2nd Edition Power System Analysis Solution Manual John Grainger, William D. Stevenson Practical Financial Management 5th Edition William R. Lasher instructor manual Practical Financial Management 5th Edition William R. Lasher test bank Practical Financial Management William R. Lasher 4th edition instructor manual Practical Financial Management William R. Lasher 4th edition test bank Prentice Hall - Solutions Manual; Communication Systems Engineering Prentice Hall's Federal Taxation 2008 Individual , 21th Edition By Thomas R. Pope, Kenneth instructor guide Prentice Hall's Federal Taxation 2009 Comprehensive, 22nd Edition By Thomas R. Pope, Kenneth E. Anderson, John L. Kramer solution manual Prentice Hall's Federal Taxation 2009 Comprehensive, 22nd Edition By Thomas R. Pope, Kenneth E. Anderson, John L. Kramer test bank Prentice Hall's Federal Taxation 2009 Individual , 22th Edition By Thomas R. Pope, Kenneth instructor guide Prentice Hall's Federal Taxation 2009: Corporations, 22/E Kenneth E. Anderson Thomas R. Pope John L. Kramer instructor guide Prentice Hall's Federal Taxation 2009: Corporations, 22/E Kenneth E. Anderson Thomas R. Pope John L. Kramer tb +sm Prentice Hall's Federal Taxation 2009: Individuals, 22/E Kenneth E. Anderson Thomas R. Pope John L. Kramer tb +sm Principles and Applications of Electrical Engineering giorigo rizzoni 4th edition Principles and Applications of Electrical Engineering giorigo rizzoni 5th edition Principles Geotechnical Engineering Braja Das 6th ed solution manual Principles of Auditing 15e by Whittington TB Principles of Communication: Systems, Modulation and Noise (5th Ed) by R. E. Ziemer, W. H. Tranter, solution manual Principles of Communications, 6th Edition Ziemer, Tranter Principles of corporate finance 7e Principles of corporate finance 7e by brealy mayers Principles Of Corporate Finance 8E By Brealey Myers Allen Principles of corporate finance 9e by brealy mayers allen (SM+TB) Principles of Electric Circuits Conventional Current Version 8e by floyd Principles of Electric Circuits Conventional Current Version by Floyd 8th edition principles of electric circuits- electron flow version Floyd 8th edition Principles of Electronic Materials and Devices, Solutions Manual ONLY Safa O. Kasap 2nd edition principles of managerial finance 10e by gitman Lawrence principles of managerial finance 11e by gitman Lawrence Principles of managerial finance 11e by gitman Lawrence solution manual Principles of Managerial Finance 11e by Gitman Lawrence test bank Principles of managerial finance 12e by gitman Lawrence test bank Principles of Managerial Finance Brief plus 5e sm Principles of Managerial Finance Brief plus 5e tb Principles of Managerial Finance Brief plus My Finance Lab Student Access Kit, 5E Lawrence J. Gitman principles of marketing 11e by Kotler ( instructor manual ) principles of marketing 11e by Kotler ( test bank ) principles of marketing 12e by Kotler TB Principles of Microeconomics, 9/e Case, Fair & Oster instructor manual Principles of Microeconomics, 9/e Case, Fair & Oster test bank Principles of Microeconomics, 9e Case, Fair & Oster instructor manual ad test bank Principles of Neurocomputing for Science and Engineering, 1st Edition Probability & Statistics for Engineers & Scientists, 8th Edition: Instructors Solution Manual ONLY by Sharon Myers , Keying Ye, Walpole Probability and Statistical Inference 7th edition Hogg Tanis solution Probability and Statistics for Engineering and the Sciences [Solutions Manual] 6th edition by Jay L. Devore Probability and Statistics for Engineers and Scientists Manual HAYLER Solutions Manual Probability, Random Variables, and Stochastic Processes [Only Solutions Manual] by Athanasios Papoulis ,S.Unnikrishna Pillai 4th edition Probability, Statistics, and Random Processes For Electrical Engineering - Alberto Leon-Garcia (3rd ed) (ISBN 0131471228) Probability, Statistics, and Random Processes for Engineers, 1st Edition Richard H. Williams - University of New Mexico Problem Solving and Programming Concepts, 8/E Maureen Sprankle Jim Hubbard Problem Solving with C++ The Object of Programming, 5E Walter Savitch tb and im Problems and Solutions on Electromagnetism by zhao shu ping - you jun han Process Dynamics and Control, 2nd Edition Seborg, Edgar, Mellichamp PROCESS SYSTEMS ANALYSIS AND CONTROL - DONALD R. COUGHANOWR Solution Manual Process Systems Analysis and Control by Donald R Coughanowr Programming the World Wide Web, 4E Robert W. Sebesta Quantitative Analysis for management TENTH EDITION by Barry Render, Ralph M. Stair, Jr, Michael E.Hanna solution manual Quantitative Analysis for management TENTH EDITION by Barry Render, Ralph M. Stair, Jr, Michael E.Hanna test bank Quantum Field Theory (draft version) & Instructor's Manual by Mark Srednicki book + solution manual Radiation Protection in the Health Sciences: With Problem Solutions Manual by: Marilyn E. Noz, Gerald Q. Maguire Sargen Recursive Methods in Economic Dynamics By Claudio Irigoyen ,Esteban Reinforced Concrete: Mechanics and Design by James MacGregor and James Wight, either 5th edition Research in Education 10th edition by Best, J., & Kahn, J instructor manual and test bank Retail Management A Strategic Approach, 10E Barry Berman, Joel R. Evans tb and sm RF circuit Design Theory and Application by Ludwig bretchko solution manuel Roy D. Yates and David J. Goodman, Probability and Stochastic Processes - A Friendly Introduction for Electrical and Computer Engineers, 2nd edition, Satellite Communications By Timothy Pratt, Charles W. Bostian Schaums Mathematical Handbook of Formulas and Tables Schaum's Outline of Discrete Mathematics, 2nd edition, 1997 by Seymor Lipschutz, Marc Lipson Schaum's Outline of Discrete Mathematics, 3rd Ed. (Schaum's Outlines) by Seymour Lipschutz, Marc Lipson Schaum's Outline of Probability, 2nd Edition by Seymour Lipschutz Schaum's Outline of Theory and Problems of Linear Algebra, 2nd Edition (Schaum's Outlines) by Seymour Lipschutz Schaum's Outline of Theory and Problems of Managerial Accounting. [1998.ISBN0070580413] book + sm Selected Answers-Basic Engineering Circuit Analysis-7th Ed. by J. David Irwin Semiconductor Device Fundamentals 1st edition by Robert F Semiconductor Physics and Devices: Basic Principles 3rd edition by neamen separation Process Principles, 2nd Ed by Seader, Henley Shigley's Mechanical Engineering Design 7th Ed - Solution Manual By by Richard Budynas, J.Charles Mischke Shigley's Mechanical Engineering Design 8th Ed - Solution Manual By by Richard Budynas, J. Keith Nisbett Signal , System and transforms fourth edition BY charles L. philips , John M Parr, Eve A. Riskin Signal , System and transforms fourth edition BY charles L. philips , John M Parr, Eve A. Riskin international edition Signal Processing and Linear Systems by lathi Signal Processing First-Mclellan, Schafer & Yoder Solution Manual chapter 3-12 Signals and Systems - Analysis using Transform methods and MATLAB M. Signals and Systems 2nd edition simon Haykin Solutions Manual Signals and Systems, Prentice-Hall Oppenheim, Willsky, Young 2nd edition Silicon VLSI Technology: Fundamentals, Practice, and Modeling James D. Plummer Michael D. Deal Peter B. Griffin sociology John J. Macionis 12th edition test bank and instructor manual Soil Mechanics Concepts and Applications By William Powrie Soil Mechanics Solutions Manual (2nd Edition) By William Powrie soils and foundations 7th by cheng liu and jack b evett Solid State Electronic Device by Ben Streetman Solid State Electronic Devices, 6E Ben Streetman Sanjay Banerjee Solid State Physics - Solutions Manual by Ashcroft & Mermin some chapters missings SOLUTIONS MANUAL AND WORKBOOK to accompany Quantitative Methods for Management Solution Automatic Control Systems 8Ed by Kuo and Golnaraghi Solution to Skill - Assessment Exercises to Accompany Control Systems Engineering 3rd edt. by Norman S. Nise Solution To Two-Dimensional Incompressible Navier-Stokes Equations- Maciej Matyka Solutions & Supplements Introductory Chemical Engineering Thermodynamics by J.R. Elliott & C. T. Lira selected solutions Solutions Irodov's Prob. Gen. Physics Volume 1 by Abhay K. Singh Solutions Irodov's Prob. Gen. Physics Volume 2 by Abhay K. Singh South-Western Federal Taxation 2009 (Individual), Edition 32, Hoffman, Smith, Wills (Test Bank) South-Western Federal Taxation 2009 Corporations - William Hoffman (Solutions Manual) (32nd ed) (ISBN 0324660219) South-Western Federal Taxation 2009 Corporations - William Hoffman (test bank) (32nd ed) (ISBN 0324660219) South-Western Federal Taxation 2009: Comprehensive - William Hoffman (Solutions Manual) (32nd ed) (ISBN 0324660529) South-Western Federal Taxation 2009: Comprehensive - William Hoffman (Test Bank) (32nd ed) (ISBN 0324660529 South-Western Federal Taxation Individual Income Taxes, 32nd by William Hoffman, Jr. James Smith Eugene WillisEdition sm 2009 STATISTICS FOR THE SCIENCES by Martin Buntinas & Gerald M. Funk, Statics & Mechanics of Materials SI 2nd edition Russell Hibbeler Jul Statics and Mechanics of Materials Anthony M. Bedford Kenneth M. Liechti Wallace Fowler Statistical Digital Signal Processing and Modeling - SOLUTIONS MANUAL By Monson H. Hayes Statistical Inference - Casella & Berger 2nd edition statistical Quality Design and Control, 2/E Richard E. DeVor, Tsong- how Chang, John W. Sutherland Statistics 4e by Murray R Spiegel, Larry J. Stephens book + sm Statistics for Business and Economics and Student CD (6th Edition) (Hardcover) by Paul Newbold (Author), William L. Carlson (Author), Betty Thorne (Author Statistics for Engineering and the Sciences, 5e by William Mendenhall and Terry Sincich Statistics for Engineers and Scientists by William Navidi Steel Design, 4th Edition solution Manual William T. Segui Steel Structures Design and Behavior, 5th Edition by Charles G. Salmon Strategic Brand Management 3e keller Strategic management 11e by Fred R. David ( instructor manual ) Strategic management 11e by Fred R. David ( test bank ) Strategic Management and Business Policy, 11e Thomas L. Wheelen David L. Hunger instructor manual with test items Strategic Management and Competitive Advantage: Concepts and Cases, 2/ E instructor manual with test items Jay Barney William S Hesterly Strength of Materials 4th Ed. by Ferdinand L. Singer & Andrew Pytel Strength of Materials 4th Ed. by Singer & Pytel Structural Analysis by Hibbeler 5th edition Structural Analysis by Hibbeler 7th edition Structured computer organization 5E Andrew S. Tanenbaum Student Solution Manual for Mathematical Methods for Physics and Engineering Third Edition by K. F. Riley, M. P. Hobson not complete Student Solutions Manual and Study Guide to accompany Fundamentals of Fluid Mechanics, 5th Edition by Bruce R. Munson Donald F. Young, Theodore H. Okiishi Study Guide with Solutions Manual for McMurry's Organic Chemistry, 7th Ed. by John E. McMurry Fawcett Lisa M. Ellram Jeffrey A. Ogden Survey of Accounting, Fourth Edition Carl S. Warren System Dynamics 3rd Ed By Katsuhiko Ogata System Dynamics and Response, 1st Edition S. Graham Kelly Techniques of Problem Solving by luiz fernandez TestGen for Physics for Scientists and Engineers - A Strategic Approach 2E knight The Analysis and Design of Linear Circuits Laplace Early, 4th Edition Thomas, Rosa The Art of Electronics. Thomas C. Hayes, Paul Horowitz The Economic Way of Thinking 11e Peter J. Boettke Peter J. Boettke David L. Prychitko test bank and solution manual The Economics of Financial Markets By Roy E. Bailey The Economics of Financial Markets by Roy E. Bailey The Economics of Money,Banking,and Financial Market 8th edition by Frederic S. Mishkin instructor manual and test bank The Legal Environment of Business, 5th edition Kubasek The Science and Engineering of Materials by Donald R. Askeland Frank Haddleton 4th edition The Spacetime Frontier Science and Society in the 21st Century by Stewart Swain Theory & Design for Mechanical Measurements 4th edition by Richard S. Figliola & Donald E. Beasley Thermodynamics an engineering approach sixth edition ( SI units ) : solutions manual by Yunus A. Cengel, Michael A. Boles Thermodynamics Of TurboMachinery 5th edition Thermodynamics: An Engineering Approach by: Yunus A. Cengel 5th edition thomas calculus 10th edition instructor solution manual volume 1and 2 Thomas Calculus 11th edition Thomas Calculus updated 10E by finney, weir giordano Thomas, Rosa The Analysis and Design of Linear Circuits Laplace Early, 4th Edition Transport Phenomena - 2nd edition by Bird, Stewart and Lightfoot Solution Manual Ulaby Applied Electromagnetics Undergraduate Econometrics Solutions Manual - Hill, Judge and Griffiths Understanding Corporate Annual Reports, 6e William R. Pasewark, Texas Tech University Understanding Financial Statements 8e Lyn M. Fraser Aileen Ormiston test bank and sol manual Understanding Financial Statements 8e Lyn M. Fraser sm and tb Unit Operations of Chemical Engineering, 6th Edition, Solutions Manual by Warren McCabe, Julian Smith, Peter Harriott Unit Operations of Chemical Engineering, 7th Edition, Solutions Manual by: Warren McCabe, Julian Smith, Peter Harriott University Calculus, Part Two Multivariable, Chap 9-14 Joel Hass Maurice D. Weir sm University Calculus, Part Two Multivariable, Chap 9-14 Joel Hass Maurice D. Weir test generator University physics 11th edition solution manual by Young and Freedman University Physics with Modern Physics (12th Edition) by Hugh D. Young, Roger A. Freedman Unsaturated Soil Mechanics by Ning Lu and William J. Likos Using Econometrics A Practical Guide, 5th edition by Studenmund Vector Calculus (3rd Ed., Susan J. Colley) R. Johnston Jr., E. R. Eisenberg, & G. H. Staab Vector Mechanics for Engineers STATICS, 7th Edition by F. P. Beer, E. R. Johnston Jr., E. R. Eisenberg, & G. H. Staab Vector Mechanics for Engineers; Dynamics 8th edition Beer Johnston Vector Mechanics for Engineers; Statics 8th edition Beer Johnston solution manual Wankat & Oreovicz - Teaching Engineering water and wastewater technology mark j. hammer mark J. Hammer 6th edition Water Supply and Pollution Control, 8E Warren Viessman, Jr. Mark J. Hammer Elizabeth M. Perez Paul A. Chadik WHO-DUN-IT Fifth Edition Shari L. DeMarco wiley applied corporate finance 2nd ed Wiley Chemical And Engineering Thermodynamics 3Ed Solutions Manual Wireless Communication and Networks second edition William Stallings solutions manual Wireless Communications: Principles and Practice, 2nd edition theodore rappaport solutions manual === Subject: test bank for Intermediata Accounting 13e Kieso TB posting-account=ldzOOQoAAAALmEmlIRVd1wCu2DOhImIB CLR 2.0.50727),gzip(gfe),gzip(gfe) solutions manual and Test Bank Solutions Manuals and Test Bank in Electronic (PDF)Format! Just contact with , solutionsservice (at) hotmail.com (my email address,solutionsservice@hotmail.com ), these are parts of our solutions, if the solution you want isnÁøt on the list, please email to http://getsolutions.spaces.live.com is my blog. solution manuall for fundamentals of thermodynamics, 7th edition,sonntag,borgnak, test bank for managerial accounting 12th Edition authors Garrison Noreen Instructor's Manual Of Fundamental Methods Of Mathematical Economics Chiang & Wainwright 2005 Mc Graw Hill Fundamentals of Advanced Accounting 3rd Edition by Joe B. Hoyle, McGraw-Hill test bank for Operations Management 10e William J. Stevenson solutions manual for Financial and Mangerial Accounting 2e by Horngren test bank for Financial and Mangerial Accounting 2e by Horngren test bank for Finiancial Accounting 7e by Horngren solutions manual for Finiancial Accounting 7e by Horngren solutions manual for Intermediata Accounting 13e Kieso test bank for Intermediata Accounting 13e Kieso TB solutions manual for Fundamentals of financial management 12e Brigham SM test bank for Fundamentals of financial management 12e Brigham TB Solution manuall for Fundamentals of Engineering Thermodynamics 6th Edition by Michael Moran and Howard Shapiro A Computer System Architecture 3rd Edition by Morris Mano Solution Manual Complete Assignment of All Chapters A FIRST COURSE IN DIFFERENTIAL EQUATIONS WITH MODELING APPLICATION 7TH EDITION BY DENNIS G. ZILL 2500 Solved Problems in Fluid Mechanics and Hydraulics (Schaum's Solved Problems) by Jack B. Evett, Cheng Liu A Course in Game Theory Osborne and Rubinstein A First Course In Probability Solution Manual,Ross 6th A First Course in Abstract Algebra 7th by Fraleigh A First Course in Differential Equations with Modeling Applications (7th ed.) and Zill & ; Cullen s Diferential Equations with Boundary-Value Problems (5th ed.) zill A First Course in Probability: SOLUTIONS MANUAL (7th Edition) by sheldon ross A First Course in String Theory chapter 1 to 16 A First Course in the Finite Element Method, 4th Edition Daryl L. Logan A Friendly Introduction to Number Theory 3rd by Silverman A Guide to Physics Problems, Part 1 - Mechanics, Relativity, and Electrodynamics A Guide to Physics Problems, Part 2 - Thermodynamics, Statistical Physics, and Quantum Mechanics A Practical Introduction to Data Structures and Algorithm analysis 2nd edition by Clifford A. Shaffer A Quantum Approach to Condensed Matter Physics Solutions by philip l. Taylor Absolute Java, 3rd Ed by W. Savitch instructor manual and test bank Accounting Concepts and Applications (9th Ed.) by W. Steve Accounting 7e by horngren solution manual Accounting 7e by horngren TB accounting 7e by horngren TB (test generator File) Accounting 8th edition by horngren test bank and solution manual Accounting Information Systems - james hall 6ed sm Accounting Information Systems - james hall 6ed tb Accounting Information Systems 10E Romney solution manual Accounting Information Systems 10E Romney test bank Accounting Information Systems 11E Romney solution manual Accounting Information Systems 11E Romney test bank Accounting Information Systems 7E Edition Ulric J. Gelinas, Richard B. Dull instructor manual Accounting Information Systems 7E Edition Ulric J. Gelinas, Richard B. Dull solution manual Accounting Information Systems 7E Edition Ulric J. Gelinas, Richard B. Dull test bank Accounting Information Systems, 9E George H. Bodnar William S. Hopwood solution manual Accounting Information Systems, 9E George H. Bodnar William S. Hopwood test bank Accounting Principles 8E by Kieso SM chapter 1 to 10 Accounting Principles 8E by Kieso SM chapter 11 to 26 Accounting Principles, Edition 8E, Weygandt, Kieso, Kimmel (Test Bank) volume 1 Accounting Principles, Edition 8E, Weygandt, Kieso, Kimmel (Test Bank) volume 2 Accounting Text and Cases 12e by Anthony IM Accounting what number means 8e by Marshall Adaptive Control 2E. by Karl J. Astrom solution manual Adaptive Filter Theory, 4th edition S. Haykin Advance corporate finance 1e by Ogden Instructor manual and test bank Advanced accounting 10E by Flyd Beams (SM+IM+TB) Advanced Accounting 10th edition by Fischer (SolutionsManual) Advanced Accounting 10th edition by Fischer (test bank) Advanced Accounting 9e by Beams solution manual Advanced Accounting 9E Hoyle,Schaefer,Doupnik Solution Manual Advanced Accounting 9E Hoyle,Schaefer,Doupnik Test Bank Advanced Accounting 9th edition by Fischer (SolutionsManual) Advanced Accounting 9th edition by Fischer (test bank) ADVANCED CORPORATE FINANCE Policies and Strategies by Joseph P. Ogden,Frank C. Jen,Philip F. O£Àonnor ( solution manual) ADVANCED CORPORATE FINANCE Policies and Strategies by Joseph P. Ogden,Frank C. Jen,Philip F. O£Àonnor (test bank) Advanced Digital Design with the Verilog HDL Michael D. Ciletti selected solutions Advanced Dynamics by Donald T. Greenwood Advanced Engineering Mathematics 3rd Edition by Dennis G Zill and Michael R Cullen Advanced Engineering Mathematics by Erwin Kreyszig 8ed solutions manual Advanced Engineering Mathematics Dennis G Zill 2nd Solution Advanced Engineering Mathematics, 6th Edition Peter V. O'Neil - University of Alabama, Birmingham Advanced Engineering Mathematics, 9th Edition By Erwin Kreyszig Advanced Financial Accounting, 6th edition, by Baker, Lembke, and King solution manual Advanced Macroeconomics 1996 romer Advanced Macroeconomics, Solutions Manual 1996 Romer Advanced Modern Engineering Mathematics, 3rd Edt by Glyn James solution manual Aerodynamics for Engineers 5th ed. solution manual John J. Bertin Russell M. Cummings Algebra by Thomas W. Hungerford Published by Springer Algebra, Pure and Applied by Aigli Papantonopoulou An Introduction to Abstract Algebra with Notes to the Future Teacher by Nicodemi, Sutherland, and Towsley An Introduction to Economic Dynamics An Introduction to Economic Dynamics by Ronald Shone An Introduction to Mass and Heat Transfer Principles of Analysis and Design Middleman An Introduction to Mathematical Statistics and Its Application (4th Edition) by Richard J. Larsen An Introduction to Modern Astrophysics (2nd Ed., Bradley W. Carroll & Dale A. Ostlie) An Introduction to Numerical Analysis by Endre Suli An Introduction to Numerical Analysis by Endre S£Ài, David F. Mayers An Introduction to Ordinary Differential Equations James C. Robinson Publisher: Cambridge University Press An Introduction To The Finite Element Method, 3rd Edition by J. N. Reddy An introduction to the mathematics of financial derivatives Neftci solution manual Analysis and Design of Analog Integrated Circuits (4th Edition) Gray, Hurst, Lewis and Meyer analysis design of analog IC design Analytical Mechanics: Solutions Manual 7ed Grant R. Fowles, George L. Cassiday Anderson J.D. Fundamentals of aerodynamics, 2nd edition - problems and solutions Andrew Tanenbaum Structured Computer Organization Solutions Manual antenna balanis solution manual antenna balanis solution manual 2nd edition ANTENNAS FOR ALL APPLICATIONS, THIRD EDITION Antennas for all Applications 3rd Ed. by Kraus & Marhefka Anton calculus book+ solution manual + test bank 8th edition Anton, Bivens, Davis Calculus Early Transcendentals Combined, 8th Edition instructor solution manual Anton, Bivens, Davis Calculus Early Transcendentals Combined, 8th Edition instructor solution manual Anton, Bivens, Davis Calculus Multivariable, 8th Edition Applied Fluid Mechanics 6th Ed. by Robert L. Mott Applied Mechanics for Engineering Technology 8e Keith M Walker Applied Numerical Analysis 7Ed - Curtis F. Gerald, Patrick O. Wheatley - Solutions manual Applied Partial Differential Equations David Logan Applied Quantum Mechanics by A. F. J. Levi Applied Statistics and Probability for Engineers 3rd.Ed edition student manual Applied Statistics and Probability for Engineers by Douglas C. Montgomery 3rd edition complete Applied Statistics and Probability for Engineers, 4th Edition Montgomery, Runger Applied Strength of Materials (4th Edition) SOLUTION MANUAL by Robert L. Mott Artificial Intelligence A Modern Approach 2e Stuart Russell Peter Norvig Audit and Assurance service An Integrated Approach 11e TB auditing and assuance services by messier test bank 6th edition Auditing and Assurance Services An Intergrated Approach and ACL Software, 12e by Alvin Arens Randal J. Elder, ark Beasley solution manual Auditing and Assurance Services An Intergrated Approach and ACL Software, 12e by Alvin Arens Randal J. Elder, ark Beasley test bank Auditing Cases, 3E Mark S. Beasley solution manual Auditing Cases: An Interactive Learning Approach, 4/E Mark S BeasleyFrank A. BucklessSteven M GloverDouglas F Prawitt Bank Management & Financial Services, 7/e By Peter S. Rose, Sylvia C. Hudgins (IM+SM) Bank management 7e by peter s. rose TB Bank management 7e by Rose ( instructor manual ) Basic Electrical Engineering By Nagrath, D P Kothari, Nagrath D P Kothari I J Nagrath, I J Nagrath Published by Tata 2002 Basic Engineering Circuit Analysis, 8th Edition by J. David Irwin, R. Mark Nelms Basic Engineering Circuit Analysis, 9th Edition Irwin, Nelms Basic Technical Mathematics with Calculus 8e Allyn J. Washington Biological Science and CW+ Grade Tracker Access Card, 2/E test bank .bok file Biology Concepts and Connections 6e Neil A. Campbell Jane B. Reece Martha R. Taylor Eric J. Simon Jean L. Dickey test bank Biology with MasteringBiolog, 8E Neil A. Campbell Jane B. Reece Biomaterials - The Intersection of Biology and Materials Science (Temenoff & Mikos) Bioprocess Engineering Principles - Solutions Manual (Original) by pauline m. Doran Book keeping and Accounting 3e Joel J. Learnef sm + book Borgnakke, Sonntag Fundamentals of Thermodynamics, 7th Edition Brief History of Western Civilization 5e vol 1 im and tb Brief History of Western Civilization, A: The Unfinished Legacy, Volume 2 im and tb BUSINESS STATISTICS A Decision Making Approach 7e SM Business Communication Essentials 3rd edition bovee and thill test bank Business Data Networks and Telecommunications, 7/E Raymond R. Panko test bank Business Law by Cheeseman 6E (IM) Business Law by Cheeseman 6E (TB) Business Law Today: Comprehensive 8th edition Roger LeRoy Miller, Gaylord A. Jentz instructor manual Business Law Today: Comprehensive 8th edition Roger LeRoy Miller, Gaylord A. Jentz test bank Business Statistics (A Decision Making Approach), Groebner, Shannon, Fry, Smith, 7 sm and tb Business Statistics 4e by Leonard J. Kazmier book + sm Business Statistics Decision Making and Student CD Package test items and solution manual, 7E test bank and sm Butterworth Heinemann - Coulson And Richardson - Chemical Engineering Vol I (Solutions Manual V Edition) c ++ how to program deitel 6th edition solution manual and test bank C++ How to Program 3rd edition by deitel Calculus A Complete Course 6th by R.A. Adams calculus by gilbert strang calculus by leithold solution manual Calculus Early transcendentals 5Th Ed - Complete Instructor 's Solutions Manual by James Stewart 0534393217 CALCULUS early transcendentals 7th edition Anton Bivens Davis Calculus Early Transcendentals Single Variable, 8th Edition Howard Anton, Irl Bivens, Stephen Davis Calculus of Variations Solution Manual Russak Calculus Single Variable 4ed chapter 1 to 11 Hughes-Hallett, Gleason, McCallum, et al. Calculus Third Editon By Strauss, Bradley and Smith not complete Calculus With Analytic Geometry (6th) By Bruce E. Edwards, Ron Larson, Robert Hostetler Calculus With Analytic Geometry (7th) By Bruce E. Edwards, Ron Larson, Robert Hostetler student manual Calculus, Early Transcendentals, 7E by C. Henry Edwards ,David E. Penney Callister Fundamentals of Materials Science and Engineering An Integrated Approach, 2nd Edition Capital Budgeting and Long-Term Financing Decisions Neil Seitz, Mitch Ellison 4th Edition instructor manual Carey, Study guide and solution manual for organic chemistry Chapra Applied Numerical Methods With Matlab For Engieers Solutions Manual 1st edition nearly same with 2nd edition Chemical and Engineering Thermodynamics- 3rd Edition- Solutions Manual Chemical Engineering Design, Fourth Edition: Chemical Engineering Volume 6 (Coulson & Richardson's Chemical Engineering) Chemical Engineering Solutions manual for Volumes( 2 and 3) 3 edition Backurst J. R., Harker J.H. & Richardson J. F chemical Engineering: Solutions for Volumes 2 and 3 by coulson 2002-12-11 Chemical Reaction Engineering, 3rd Edition Levenspiel Chemical, Biochemical, and Engineering Thermodynamics, 4th Edition Sandler Chemistry: The Central Science (Hardcover, 2005) Author: Bruce E. Bursten, H. Eugene Lemay Jr., Lemay test bank 10th Classical Dynamics A Contemporary Approach by Jorge V. Jose, Eugene J. Saletan T. Thornton, Jerry B. Marion Classical Electrodynamics - 2nd Ed. John David Jackson byKasper van Wijk Classical Electrodynamics 3rd edition by Jackson Classical Mechanics (2nd Edition) by Herbert Goldstein Classical Mechanics by R. Douglas Gregory Classical Mechanics, 2ed Partial Solutions Manual by Safko Close, Frederick, Newell Modeling and Analysis of Dynamic Systems, 3rd Edition Cmos analog circuit design 2nd edition homework solutions by allen holberg CMOS Analog Circuit Design, 2ed Solutions by Phillip E. Allen, Douglas R. Holberg CMOS VLSI Design 3rd edition David Harris H E Weste College Accounting (Chapters 1-25), 10E by Jeffrey Slater sm and tb College Mathematics for Business, Economics, Life Sciences & Social Sciences (11th Edition) (Hardcover) by Raymond A. Barnett (Author), Michael R. Ziegler (Author), Karl E. Byleen (Author) Collins Mechanical Design of Machine Elements and Machines A Failure Prevention Perspective Communication Networks (2nd Edition) leon Communication Networks Fundamental concepts & key Architectures By Leon Garcia Widjaja not complete 3 4 5 6 7 8 10 Communication Networks Fundamental concepts & key Architectures Alberto Leon-Garcia Communication Systems (4th edt) by Simon Haykin Communication Systems 4Ed - A Bruce Carlson Solutions Manual communication systems engineering by proakis Communication Systems Engineering Proakis J (2002) Solutions Manual 2nd edition Compensation Management in a Knowledge-Based World, 10E Richard I Henderson instructor manual Complex Variables with Applications (Pie) by A.David Wunsch Computational Techniques for Fluid Dynamics: A Solutions Manual By Karkenahalli Srinivas, Clive A. J. Fletcher Computer Architecture A Quantitative Approach, 4th Edition, 2006 by John L. Hennessy, David A. Patterson Computer Architecture: Pipelined and Parallel Processor Design (Solutions Manual) by Michael J. Flynn selected solutions computer networking a top down approach 3rd edition solution manual by James F.Kurose, Keith W. Ross Computer Networking: A Top-Down Approach, 4/E solution manual and lab solutions Computer Networking: A Top-Down Approach, 5/E solution manual computer networks Andrew S. Tanenbaum 4th edition Computer Networks Systems Approach 3ed by davie peterson solutions manual Computer Networks: A Systems Approach 2nd edition Peterson and Davie£À Computer Organization and Architecture: Designing for Performance, 7/E William Stallings Computer Organization and Design, Revised Printing, 3rd Edition Solutions Manual By David A. Patterson, John L. Hennessy, Computer Organization and Design: The Hardware/Software Interface, 3rd Edition by David A. Patterson, John L. Hennessy Concepts In Federal Taxation 2007 (14thEd) - Murphy Solutions Manual Concepts of Genetics, 9e by Klug, Cummings, Spencer & Palladino test generator Concepts of Programming Languages, 8/E Robert W. Sebesta, University of Colorado, Colorado Springs Construction Surveying and layout 2nd edition by wesley g. Crawford Construction Surveying and layout 3rd edition by wesley g. Crawford Consumer Behavior, 8/E Michael R. Solomon test bank contemporary engineering economy by chan s. park 4th edition Contemporary Financial Management R. Charles Moyer, James R. McGuigan, William J. Kretlow 10th edition instructor manual Contemporary Financial Management R. Charles Moyer, James R. McGuigan, William J. Kretlow 10th edition test bank Contemporary Financial Management, 11th Edition R. Charles Moyer James R. McGuigan William J. Kretlow instructor manual Contemporary Financial Management, 11th Edition R. Charles Moyer James R. McGuigan William J. Kretlow test bank Control Systems Engineering by Nise 4ed£À? Corporate Computer and Network Security Raymond Panko Corporate Finance By Stephen A. Ross 6 edition Corporate Finance By Stephen A. Ross 8th edition corporate finance 1e by berk sm corporate finance 1e by berk tb Corporate Finance, 8e Stephen A. Ross Randolph W. Westerfield Jeffrey Jaffe instructor manual and solution manual Corporate Finance-7th Edition by Stephen A. Ross , Randolph W. Westerfield , Jeffrey Jaffe cost Accounting 12e by Horngren Test Bank cost accounting 12e by Charles T. Horngren Srikant Datar George Foster Madhav Rajan Chris Ittner solution manual cost accounting 13e by Charles T. Horngren Srikant Datar George Foster Madhav Rajan Chris Ittner solution manual cost accounting 13e by Charles T. Horngren Srikant Datar George Foster Madhav Rajan Chris Ittner test bank Cost Accounting: Foundations and Evolutions 7E By Kinney solution manual Cost Accounting; Foundations and Evolutions, Edition 7, Kinney, Raiborn Cost Management A Strategic Emphasis, 4e Blocher Cost ManagementMeasuring Monitoring and Motivating Performance by Eldenburg Wolcott SM TB Cryptography & network security 4e william stallings Daniel Shanks - Solved And Unsolved Problems In Number Theory (2Nd Ed), 1978 Data and Computer Communications William Stallings 8th edition William Stallings Data and Computer Communications, 7th Edition by William Stallings Data Communications and Networking fourth edition by Behrouz A.Forouzan odd numbered solutions Data structure and Problem Solving using Java 3rd Mark Allen Weiss, Data Structures and Algorithm Analysis in C++, 3/E Data Structures with Java by John R. Hubbard Anita Huray University of Richmond Database Concepts, 3E david kroenke, david auer tb and im DATABASE MANAGEMENT SYSTEMS 3rd Edition by Ramakrishnan, Gehrke, Derstad, Seliko, Zhu- Solution Manual only odd solutions Database Processing Fundamentals, Design, and Implementation, 10E David Kroenke test bank Database System Concepts, Fifth Edition by Avi Silberschatz ,Henry F. Korth solutions to exercises Database Systems: An Application Oriented Approach, Compete Version, 2/ E Michael Kifer Arthur Bernstein Philip M. Lewis Databases systems: An Application-Oriented Approach 2nd edition Michael Kifer, Arthur Bernstein, Philip M. Lewis test bank + solution manual David Kroenke's Database Processing: Fundamentals, Design and Implementation (10th Edition) test bank Derivatives Markets 2nd edition by Yufeng Guo Solution Manual Derivatives Markets 2nd by Rober L. McDonald solution manual Derivatives Markets 2nd by Rober L. McDonald test bank Design and Analysis of Experiments Solutions Manual 6th edition Design and Analysis of Experiments, 6th Edition Montgomery complete all chapters Design of Analog CMOS Integrated Circuits McGraw Hill Solutions Manual Design of Fluid Thermal Systems, 2nd Edition William S. Janna design of machinary by norton 3rd edition Design with Operational Amplifiers and Analog Integrated Circuits, 3rd edt. by Franco Device Electronics for Integrated Circuits 3Edition Muller Kamins Device Electronics for Integrated Circuits Solutions Manual 3ed DIGITAL DESIGN FOURTH EDITION by M. MORRIS MANO DIGITAL SIGNAL PROCESSING: Signals, Systems, and Filters Andreas Antoniou Differential Equations & Linear Algebra, 2nd ed., Farlow Differential Equations & Linear Algebra, edition 2, by Edwards Penny differential equations 5th edition by zill classic fifth edition Differential Equations And Boundary Value Problems C. Henry Edwards - David E. Penney 2nd edition Differential Equations and Boundary Value Problems Computing and Modeling, 4E C. Henry Edwards David E. Penney Differential Equations and Linear Algebra, 3e by Stephen W. Goode and Scott A. Annin instructor manual Digital & Analog Communication Systems - Leon Couch (7th ed) (ISBN 0131424920) Digital Communications, 4th edition, 2000-08 book+solution by John Proakis Digital Communications: Fundamentals And Applications (2nd Edition)- Bernard Sklar Digital Design (3rd Edition) by M. Morris Mano Digital Electronics with VHDL (Quartus II Version) By William Kleitz Digital Fundamentals (10th Edition) floyd Digital Integrated Circuits by Rabaey 2nd edition solution manuel chapter 3,5,6,10 Digital image processing - Gonzalez 2Ed- Solutions Manual (209p) Digital Signal Processing - A Modern Introduction, 1st Edition Cengage learning Ashok Ambardar Digital Signal Processing - Proakis & Manolakis - Solutions Manual 3ed Digital Signal Processing (2nd Ed.) (Mitra) Solution Manual? Digital Signal Processing A Computer-Based Approach 1st ed Solutions Manual mitra Digital Signal Processing by Thomas J. Cavicchi - solution manuel Digital Signal Processing Principles, Algorithms and Applications (International Edition) by John Proakis ,Dimitris Manolakis Digital Signal Processing Using Matlab- Solution Manual Vinay K Ingle Proakis 2nd edition Discrete and Combinatorial Mathematics 5e (Solutions Manual Only) by Ralph P. Grimaldi Discrete Mathematics (5th Edition) By Dossey, Otto, Spence, Vanden Eynden Discrete Mathematics (5th Edition) by John A. Dossey , Albert D. Otto, Lawrence E. Spence ,Charles Vanden Eynden Discrete Mathematics, 5e John A. Dossey Albert D. Otto Lawrence E. Spence Charles Vanden Eynden Discrete-Event System Simulation 3rd edition by Jerry Banks John S. Carson II Barry L. Nelson David M. Nicol Discrete-Time Signal Processing 2nd Edition, 1999-02 by oppenheim Distributed Systems, Concepts and Design (Exercise Solutions) - G. Coulouris, J. Dollimore and T. Kindberg Doets & Eijck - The Haskell Road To Logic, Math and Programming - Solutions to Exercises Dorf, Svoboda Introduction to Electric Circuits, 7th Edition DSP First: A Multimedia Approach-Mclellan, Schafer & Yoder Solution Manual Dynamics of Mechanical Systems Solutions Manual (Horwood Engineering Science Series) by C. T. F. Ross Econometric Analysis Solutions Manual to the 6th Edition By William H. Greene Econometric Analysis, 5th edition william h. Greene Econometrics - [Instructor Solution Manual] The Econometrics of Financial Markets john y. campbell, andrew w. Lo Economics for Managers by Paul Farnham, 2008 custom edition sm + tb Economists Solution Manual (Blume, 1994) Effective Writing 8e May & May instructor manual Electric Circuits, Nilsson Riedel , 7th edition Electric Circuits,Nilsson Riedel , 8th edition Electric Machinery and Power System Fundamentals Electric Machinery and Power System Fundamentals Stephen J. Chapman first edition Electric Machinery by A. E. Charles Kingsley, Jr.Fitzgerald 6th edition electric machinery fundamentals 4th edition stephen j chapman Electric Machines By D. P. Kothari, I. J. Nagrath Electrical Engineering Principles and Applications 4th Allan R. Hambley Electrical Engineering: Principles and Applications 3ed Allan R. Hambley Electrical Machines Drives and Power Systems 6th edition by Theodore Wildi Electrical Machines, Drives and Power Systems 6th edition ISBN 0131969188 Electrical Power and controls Skvarenina 2nd Electromagnetics for Engineers by Fawwaz T. Ulaby Electronic Circuit Analysis and Design 2nd edt. by Donald A. Neamen - solution manuel Electronic Devices (Electron Flow Version), 8e floyd Electronic Devices and Circuit Theory 8th Ed Instructors Resource Manual with Text Solutions, Lab Solutions, and Test Item File Electronic Devices and Circuit Theory, 10e Boylestad & Nashelsky Electronic Devices and Circuit Theory, 9e Boylestad & Nashelsky Electronic Physics Strabman Electronics Fundamentals Circuits Devices and Applications by Thomas Floyd 7th edition Electronics, 2nd ed. by Allan R. Hambley Elementary Algebra with Applications, 3rd Edition Author: Terry H. Wesner Harry L. Nustad Elementary Differential Equations and Boundary Value Problems , 8th Edition Elementary Differential Equations And Boundary Value Problems, 7Th Ed - Boyce And Diprima Student Solutions Manual, Charles W Haines Ode Architect Companion Elementary Differential Equations with Boundary Value Problems, 6E Henry Edwards avid Penney Elementary Differential Equations With Boundary Value Problems, 4E,Edwards, Penney Elementary Linear Algebra with Applications 9 edition by Howard Anton, Chris Rorres Instructor Solutions Manual and Instructor Testbank Elementary Linear Algebra with Applications, 9/E Bernard Kolman David Hill Elementary Mechanics & Thermodynamics [2000 by Professor Jhon W. Norbury Elementary Number Theory (5th Edition) by Kenneth H. Rosen Elementary Principles of Chemical Processes Solutions Manual 3 ed By Richard M.Felder Elementary Principles of Chemical Processes Solutions Manual 3 ed By Richard M.Felder chapters 2 to 14 Elementary Statistics by Mario F. Triola, 10th Elementary Statistics With Multimedia Study Guide, 10/E solution manual Elementary Statistics With Multimedia Study Guide, 10/E test bank Elements of Chemical Reaction Engineering By H Fogler, 3rd ed Elements of Electromagnetics by Sadiku 2nd Ed. Elements of Electromagnetics, 3rd Ed., Matthew N.O. Sadiku Elements of Electromagnetics, 4th Ed., Matthew N.O. Sadiku homework and midterm problems Elements Of Information Theory - Solution Manual by thomas m. cover and joy a. Thomas E-Marketing, 4e Judy Strauss, Adel El-Ansary, Raymond Frost sm E-Marketing, 4e Judy Strauss, Adel El-Ansary, Raymond Frost tb E-Marketing, 5e Judy Strauss, Adel El-Ansary, Raymond Frost sm E-Marketing, 5e Judy Strauss, Adel El-Ansary, Raymond Frost tb Embedded Microcomputer Systems: Real Time Interfacing, 2nd Edition Jonathan W. Valvano Energy Management 5 ed 2005-12 Klaus-Dieter E. Pawlik Engineering - Materials Science, Milton Ohring Solutions Manual Engineering and Chemical Thermodynamics [Solution Manual] by Milo D. Koretsky Engineering and Chemical Thermodynamics Milo D. Koretsky Engineering biomechanics statics by Beatriz Guevarez, Joshua R?os, Nayka Rivera, Sharon V?zquez and Melvia Villegas Engineering Circuit Analysis 6Ed - Hayt Solutions Manual.pdf Engineering Circuit Analysis 7Ed William Hart Hayt Jack E. Kemmerly Solutions Manual contains chapters 2,3,4,5,7,9,10,11 Engineering Economy - Leland Blank & Anthony Tarquin 6th Edition selected solutions ( student solution) Engineering Economy 14e Sullivan solution manual Engineering Electromagnetics -Hayt (2001) Engineering Electromagnetics -Hayt (2001).rar Engineering Electromagnetics Nathan Ida 2nd edition Engineering Electromagnetics, 7th Ed., Hayt, Buck Engineering Fluid Mechanics, 7th Edition - Student Solutions Manual by Clayton T. Crowe, Donald F. Elger, John A. Roberson Engineering Fluid Mechanics, 9th Edition Crowe, Elger, Roberson, Williams Engineering Mathematics, 4th edt. by John Bird - solution manual Engineering Mechanics - Statics (11th ) by R.C.HIBBELER Engineering mechanics statics 12th edition by hibbeler Engineering Mechanics Dynamics (11th Edition) by Russell C. Hibbeler Engineering Mechanics Dynamics 3rd edition solution manual Hibbeler R.C. updated fixed 09-2006 Engineering Mechanics Dynamics, 4E Anthony M. Bedford and Wallace Fowler Engineering Mechanics Dynamics, 5E Anthony M. Bedford and Wallace Fowler Engineering Mechanics Dynamics, SI 6th Edition Meriam, Kraige Engineering Mechanics Statics 11th Edition By R.C.Hibbeler Engineering Mechanics Statics, 4E Anthony M. Bedford and Wallace Fowler Engineering Mechanics Statics, 5E Anthony M. Bedford and Wallace Fowler Engineering Mechanics, Dynamics 5E - Solutions manual By J. L. Meriam, L. G. Kraige, chapter 1-8 Engineering Mechanics, statics 5th edition Solutions manual By J. L. Meriam, L. G. Kraige Engineering Mechanics, statics 6th edition Solutions manual By J. L. Meriam, L. G. Kraige Engineering Mechanics: Dynamics 2 Ed. by Riley and Sturges contains chapters 13,14,15,16,17 chapters Engineering Mechanics: Statics: Solutions Manual (10th edition) by R.C. Hibbeler Engineering Problem Solving with C, 3E Delores M. Etter Engineering Problem Solving with Matlab 2nd edition by etter sm-tb- quiz Engineering Statistics, 4th Edition Montgomery, Runger, Hubele Engineering Vibration, 3e Daniel J. Inman Entrepreneurial Finance Chris Leach, Ronald W. Melicher 2nd edition instructor manual Entrepreneurial Finance Chris Leach, Ronald W. Melicher 2nd edition test bank Entrepreneurial Finance, 3rd Edition Chris Leach Ronald W. Melicher test bank Entrepreneurial Finance, 3rd Edition Chris Leach Ronald W. Melicher instructor manual Equilibrium and Non-Equilibrium Statistical Thermodynamics By Michel Le Bellac Essentials of Accounting for Governmental and Not for Profit Organizations 9e by Paul A. Copley Essentials of Entrepreneurship and Small Business Management 5e tb and im Essentials of Fluid Mechanics: Fundamentals and Applications, 1st Ed., by Cengel & Cimbala ISBN-13 9780073301129 Essentials of Investments 7th edition Zvi Bodie Alex Kane, Alan marcus test bank Essentials of Investments Zvi Bodie, Alex Kane, Alan J. Marcus Essentials of Managerial Finance 14e Brigham TB Essentials of Managerial Finance 13th Edition Scott Besley, Eugene F. Brigham instructor manual Essentials of Managerial Finance 13th Edition Scott Besley, Eugene F. Brigham test bank Essentials of Managerial Finance, 14th EditionScott Besley Eugene F. Brigham instructor manual Essentials of Managerial Finance, 14th EditionScott Besley Eugene F. Brigham spreadsheet problem solutions Essentials of Managerial Finance, 14th EditionScott Besley Eugene F. Brigham test bank Essentials of Organizational Behavior 9e Stephen P. Robbins Tim Judge Essentials of Statistics, by Triola, 3rd edition sm Essentials of Statistics, by Triola, 3rd edition tb Essentials of Strategic Management 4E DAVID HUNGER THOMAS L. WHEELEN im with tb Ethical Theory and Business, 8/E Tom L. Beauchamp Norman Bowie Denis Arnold Federal Taxation 2008 Corporations Partnerships Estates and Trusts 21E Anderson Pope Kramer test bank Feedback Control of Dynamic Systems 4th edition Franklin - Solutions Manual Feedback Control of Dynamic Systems, 5/E franklin Festo didactic Process Control System Field and Wave Electromagnetics, 2nd edition, Cheng Finance Management Test Bank brigham 11 test bank Financial & managerial Accounting 13E By william Haka bettner Financial accounting an introduction to concepts, methods, and uses by clyde Stickney and Roman Weil solutions manual 11th edition financial Accounting 4e by John Wild Financial Accounting 6e by kieso solution manual Financial Accounting 6e by kieso test bank Financial Accounting 6e Harrison Horngren Financial Accounting 6th edition by Harrison Solution Manual financial accounting 6th edition harrison test bank financial accounting 7th edition harrison solution manual financial accounting 7th edition harrison test bank Financial Accounting; Tools for Business Decision Making, 4th Edition, Kimmel.Weygandt.Kieso instructor manual Financial Accounting; Tools for Business Decision Making, 4th Edition, Kimmel.Weygandt.Kieso solution manual Financial Accounting; Tools for Business Decision Making, 4th Edition, Kimmel.Weygandt.Kieso test bank Financial Analysis with Microsoft?Excel 4th Edition Timothy R. Mayes, Todd M. Shank instructor manual Financial Analysis with Microsoft?Excel 4th Edition Timothy R. Mayes, Todd M. Shank spreadsheet problems Financial Analysis with Microsoft?Excel?2007 Timothy R. Mayes, Todd M. Shank Instructor Spreadsheet Files Financial and managerial accounting 14e by williams haka bettner SM Financial management 2e by jae K shim Financial management Principles and application 10e by By Arthur J. Keown, John D. Martin, John W. Petty, David F. Scott solution manual Financial Management Theory & Practice,Eugene Brigham,12th edition [ Test Bank ] financial management theory and practice 10e by Brigham solution manual Financial Management Theory And Practice 11e by Brigham solution manual Financial management theory and practice 12e by Brigham sm financial management theory and practice 12e by Brigham TB financial Management Theory and Practice, 11e By Eugene F. Brigham test bank Financial Markets And Institution (7thEd) Madura TestBank Financial Reporting Analysis 10th edition TB by gibson Finite Element Method: Volume 1 The Basis 5th edition by O. C. Zienkiewicz, R. L. Taylor Finite Mathematics ?nstructor's Resource Guide and Solutions Manual, 8E Margaret L. Lial Raymond N. Greenwell Nathan P. Ritchey Fluid Mechanics Solutions Manual by yunus a. cengel Fluid Mechanics Fundamentals and Applications by Cengel & Cimbala Fluid Mechanics With Engineering Applications -- Solutions Manual by E. John Finnemore, Joseph B Franzini Fluid Mechanics with Student Resources, 5th edition 2002-12 by Frank M. White Fluid Power with Applications, 7E esposito Foundations of Finance The Logic and Practice of Financial Management Art J Keown John D Martin 6th edition im + tb Foundations of Financial Management, 12e By Stanley B. Block Geoffrey A. Hirt Foundations of Financial Markets and Institutions 4e Fabozzi, Modigliani & Jones instructor manual and test gen Fourier and Laplace Transform - Antwoorden Fractal Geometry Mathematical Foundations and Applications Solutions Fracture mechanics fundamentals and applications 2nd edition northam anderson solution manual framework for Marketing Management, A - Philip Kotler (4th ed) (ISBN 0136026605) framework for Marketing Management, A - Philip Kotler (4th ed) (ISBN test bank Friendly Introduction to Analysis - Witold A.J. Kosmala (2nd ed) fundamental accounting principles 17th edition larson solution manual Fundamental Accounting Principles, 18/e John J. Wild Barbara Chiappetta Kermit D. Larson solution manual and test bank fundamentals accounting principles by larson 18e SM Fundamentals of Actuarial Mathematics?by D. Promislow Fundamentals of Advanced Accounting 1e Fisher taylor chang solution manual Fundamentals of Advanced Accounting 1e Fisher taylor chang test bank fundamentals of advanced accounting 3e by hoyle Fundamentals of Advanced Accounting, 2e Joe B. Hoyle Thomas F. Schaefer Timothy S. Doupnik ( Solution Manual ) Fundamentals of Applied Electromagnetics 5th edition by Fawwaz T. Ulaby solution manual Fundamentals of Biochemistry Life at the Molecular Level, 3rd Edition Fundamentals of Chemical Reaction Engineering - Solutions Manual By Mark E. E. Davis, Robert J. J. Davis Fundamentals of Communication Systems by John G. Proakis ,Masoud Salehi computer solutions Fundamentals of Contemporary Financial Management, 2nd Edition R. Charles Moyer James R. McGuigan Ramesh P. Rao instructor manual Fundamentals of Contemporary Financial Management, 2nd Edition R. Charles Moyer James R. McGuigan Ramesh P. Rao test bank fundamentals of corporate finance 8e by Ross Ross, Westerfield,jordan Fundamentals of Corporate Finance, 4th Edition (Brealey, Myers, Marcus) by Bruce Swenson Fundamentals of Database Systems, 5E Ramez Elmasri,Shamkant B. Navathe Fundamentals of Differential Equations and Boundary Value Problems (4th Ed., Kent B. Nagle, Late, Edward B. Saff & Arthur David Snider) Fundamentals of Digital Logic with Verilog Design by s. Brown z vranesic Fundamentals of Digital Logic with VHDL Design-1st edition by S. Brown, Z. Vranesic Fundamentals of Electric Circuits 2nd by Alexander Sadiku Fundamentals of Electric Circuits 3rd edition by Alexander Sadiku Fundamentals of Electromagnetics for Electrical and Computer Engineering Nannapaneni Narayana Rao Fundamentals of Electromagnetics with Engineering Applications by Stuart M. Wentworth Fundamentals of Electronic Circuit Design by David and Donal Comer Fundamentals of Engineering Electromagnetics--Cheng Fundamentals of engineering thermodynamics moran shapiro Fundamentals of engineering thermodynamics moran shapiro 6th edition Fundamentals of Engineering Thermodynamics: Si Version / 5th Edition by Michael J. Moran Howard N. Shapiro Fundamentals of Financial Management 12th edition instructor manual Fundamentals of Financial Management 12th edition test bank Fundamentals of Financial Management 10th Edition by Eugene Brigham Solution manual Fundamentals of Financial Management 10th Edition by Eugene Brigham test bank Fundamentals of Financial Management 11e by Eugene F. Brigham, Joel F. Houston spreadsheet problems Fundamentals of Financial Management 11e by Eugene F. Brigham, Joel F. Houston test bank and cyberproblems Fundamentals of Financial Management 11e by Brigham Instructor manual Fundamentals of financial management 12e by james c. van horne Fundamentals of Financial Management Eugene F. Brigham, Joel F. Houston 12th edition instructor manual Fundamentals of Financial Management Eugene F. Brigham, Joel F. Houston 12th edition spreadsheets problems and web appendix solutions Fundamentals of Financial Management Eugene F. Brigham, Joel F. Houston 12th edition test bank Fundamentals of Financial Management, 12th Edition (Instructors guide ONLY) by James C. Van Horne, John M Wachowic Fundamentals of Financial Management, Concise Edition 5th Eugene F. Brigham, Joel F. Houston test bank Fundamentals of Financial Management, Concise Edition 5th Eugene F. Brigham, Joel F. Houston spreadsheets problems Fundamentals of Financial Management, Concise Edition 5th Eugene F. Brigham, Joel F. Houston instructor manual Fundamentals of Financial Management, Concise Edition Eugene F. Brigham, Joel F. Houston 6th edition instructor manual Fundamentals of Financial Management, Concise Edition Eugene F. Brigham, Joel F. Houston 6th edition spreadsheet problem solutions Fundamentals of Financial Management, Concise Edition Eugene F. Brigham, Joel F. Houston 6th edition test bank Fundamentals Of Fluid Mechanics 3Rd And 4Th Edition Solutions Manual Fundamentals of Fluid Mechanics, 6th Edition Munson, Young, Okiishi, Huebsch Fundamentals of Heat and Mass Transfer, 5th Edition by Frank P. Incropera Fundamentals of Heat and Mass Transfer, 6th Edition Incropera, DeWitt, Bergman, Lavine Fundamentals of Investing, 10th Edition by Gitman and Joehnk Fundamentals of Investments 3e Gordon J. Alexander William F. Sharpe Fundamentals of Logic Design 5th edition by charles roth Fundamentals of Machine Component Design, 3rd ed - Student Solutions Manual by Juvinall, Marshek Fundamentals of manufacturing 2nd edition by philip d. rufe solutions manual Fundamentals of Momentum, Heat and Mass Transfer, 5th Edition Welty, Wicks, Rorrer, Wilson Fundamentals of Multinational Finance, 2nd edition by Michael H. Moffett test bank and solution manual Fundamentals of Multinational Finance, 3E by Michael H. Moffett (Instructor's Manual) Fundamentals of Multinational Finance, 3E by Michael H. Moffett test bank Fundamentals of Organic Chemistry, 5E, Study Guide and Solutions Manual By T. W. Graham Solomons Fundamentals of Organizational Communication 7e Pamela S. Shockley- Zalabak Fundamentals of physics Halliday Resnick 8th student solution not complete Fundamentals of Physics, 7th Edition - Instructor's SOLUTIONS MANUAL halliday and resnick Fundamentals of Physics, Edition 8, Halliday, Resnick, Walker (Solution Manual) Fundamentals of Probability With stochastic processes 3/e (Solutions Manual ) By Saeed Ghahramani Fundamentals of Quantum Mechanics 0521829526 sols Fundamentals of Quantum Mechanics For Solid State Electronics and Optics by C. L. Tang fundamentals of selling 10e by futrell TB Fundamentals of Semiconductor Devices - Anderson Fundamentals of Signals and systems using web and matlab third edition by Edward W. Kamen, Bonnie S Heck Fundamentals of Solid State Electronics by chih tang sah Fundamentals of Thermal-Fluid Sciences Solution Manual 2ed By Yunus A. Cengel, Robert H. Turner Fundamentals of Thermodynamics [Sonntag-Borgnakke-Van Wylen Solutions Manual volume 1 and volume 2 Fundamentals of Thermodynamics SOLUTION MANUAL 6ed By Richard E. Sonntag, Claus Borgnakke, Gordon J. Van Wylen, Fundamentals of Wireless Communication by Tse and Viswanath Fundementals of differential equations 7E & Fundementals of differential equations and boundary value problems 5E Nagle , Saff , Snider even problems Fundementals of engineering economics 2E Chan S. Park Further Mathematics for Economic Analysis General Chemistry, 8th Edition - Solution Manual by Ralph H. Petrucci William S. Harwood; Geoffrey Herring General Chemistry: Principles and Modern Application & Basic Media Pack, 9/E Ralph H Petrucci test bank Geology for Engineers & Environmental Scientists by Alan Kehew 3rd edition Gilat MATLAB An Introduction with Applications, 3rd Edition Government and Nonprofit Accounting Theory & Practice - 8th edition by Freeman, Shoulders, Allison, Patton, Smith sm Government and Nonprofit Accounting Theory & Practice - 8th edition by Freeman, Shoulders, Allison, Patton, Smith tb Government and Nonprofit Accounting Theory & Practice - 9th edition by Freeman, Shoulders, Allison, Patton, Smith instructor manual Government and Nonprofit Accounting Theory & Practice - 9th edition by Freeman, Shoulders, Allison, Patton, Smith tb Government and Nonprofit Accounting Theory & Practice - 9th edition by Freeman, Shoulders, Allison, Patton, Smith sm Government and Not-for-Profit Accounting: Concepts & Practices (4th edition) by Granof and Wardlow sm Government and Not-for-Profit Accounting: Concepts & Practices (4th edition) by Granof and Wardlow tb Gravity An Introduction to Einstein's General Relativity by hartle Gravity An Introduction to Einstein's General Relativity James B. Hartle Haberman Applied Partial Differential Equations 4e Instructor's Manual Harcourt mathematics 12 Advanced Functions and Introductory Calculus - Solutions Manual by Brian / Nelson Harcourt Mathematics 12 Geometry and Discrete Mathematics Solutions Manual By McGraw-Hill Heat and Mass Transfer 3e SM Yunus A. Cengel Heat Transfer: A Practical Approach. Solution Manual ONLY by cengel 2nd edition Heat Transfer-Fundamentals of Heat and Mass Transfer-Incropera & Dewitt Solution Manual Heating, Ventilating and Air Conditioning Analysis and Design, 6th Edition McQuiston, Parker, Spitler High-Speed Digital System Design A Handbook of Interconnect Theory and Design Practicesbby Stephen H. Hall Human Biology by Colleen Belk Virginia Borden-Maier test bank Human Culture Highlights of Cultural Anthropology By Melvin R Ember, Carol R. Ember Human resources management 10e Gary dessler (IM+TB) Hydraulics in Civil and Environmental Engineering By Andrew Chadwick Hydrology and Floodplain Analysis (4th Ed., Philip Bedient, Wayne Huber & Baxter Vieux) Hydrology and Floodplain Analysis, 4e Philip B. Bedient Wayne C. Huber Baxter E. Vieux IBM WebSphere RFID Handbook A Solution Guide by IBM Redbooks In Experiments with Economic Principles, Instructor's Manual By Bergstrom Industrial Safety and Health Management 5e C. Ray Asfahl all resources Instructor Manual to A Course in Modern Mathematical Physics By S. M. SZE 2nd edition Instructor Manual to An Introduction to Thermodynamics and Statistical Mechanics 2ed By Keith Stowe Instructor Manual to Introduction to Solid State Physics Eighth Edition By Charles Kittel Instructor Manual to Introductory Quantum Optics By Christopher Gerry and Peter Knight Instructor Manual to Quantum Physics 3rd ediiton by Gasiorowicz, S. - Instructor Manual to Quantum Physics Third Edition by Stephen Gasiorowicz Instructor Manual to SEMICONDUCTOR DEVICES Physics and Technology Second Edition By S.M.Sze Instructor Manual to Special Relativity by Patricia M. Schwarz and John H. Schwarz Instructor Solutions Manual for Building Java Programs Stuart Reges,Martin Stepp Instructor Solutions Manual to accompany Boyce Elementary Differential Equations and Boundary Value Problem 8ed by Charles W. Haines, William E. Boyce INSTRUCTOR£À SOLUTIONS MANUAL Basic Technical Mathematics with Calculus SI Version John R. Martin Eighth Canadian Edition Instructor's Manual for Solving ODEs with MATLAB By L. F. Shampine, I. Gladwell, S. Thompson Instructor's Manual for Sipser's Introduction to the Theory of Computation by Ching Law Instructor's Manual for Solving ODEs with MATLAB By L. F. Shampine, I. Gladwell, S. Thompson Instructor's Manual Im Experiments with Economic Principles By Bergstrom Instructor's Manual Of Fundamental Methods Of Mathematical Economics Chiang & Wainwright 2005 Mc Graw Hill Instructor's Manual(Information Technology Project Management 3Rd Edition) by Kathy Schwalbe Instructor's Manual: Im Experiments with Economic Principles By Bergstrom INSTRUCTOR'S SOLUTIONS MANUAL (to accompany Elementary Statistics Ninth Edition) by milton loyer Instructor's solution manual ISBN 0534382150 A Transition to Advanced Mathematics solution manual by douglas smith 5th edition Instructor's Solutions for: Design of Analog CMOS Integrated Circuits by razavi Instructors Solutions Manual for Differential Equations with Boundary Value Problems, 2/E by john polking Instructor's Solutions Manual for Serway and Jewett's Physics for Scientists and Engineers Sixth Edition Volume One by Ralph V. McGrew Instructor's Solutions Manual for Serway and Jewett's Physics for Scientists and Engineers Sixth Edition Volume two by Ralph V. McGrew Intermediate Accounting 10e by Nikolai sm Intermediate Accounting 11e by Kieso Intermediate Accounting 12e by Kieso Intermediate accounting 12th Updated by Kieso Solution manual Intermediate accounting 12th Updated by Kieso test bank Intermediate Accounting 2e by Baruch Englard Intermediate Accounting 3e by J. David Spiceland Intermediate Accounting 4e revised by J. David Spiceland solution manual Intermediate accounting by Spiceland 4e Solution manual Intermediate Accounting, Update, 12th Edition international student solution manual Intermediate Algebra, 8th Edition By Margaret L. Lial, John Hornsby, Terry McGinnis International Accounting 1e by Doupnik solution manual International Accounting 6e Frederick D. Choi Gary K. Meek International Business The Challenges of Globalization, 4E John J. Wild,Kenneth L. Wild tb and im International Economics, 7e Husted Melvin test bank and solution manual International Financial Management 9th Edition jeff madura instructor manual International Financial Management 9th Edition jeff madura test bank International Financial Management Geert Bekaert Robert J. Hodrick test bank and sol manual International Management Managing Across Borders and Cultures 6e test bank and instructor manual Introduction Fluid Mechanics, 6Th Edition Solution by fox CHAPTER 1-8 Introduction To Analysis (3rd) Wade Solution Manual Introduction to Chemical Engineering Thermodynamics 7th edition (solution manual) By J.M. Smith, Hendrick C Van Ness Introduction to Corporate Finance William L. Megginson, Scott B. Smart 1st edition instructor manual Introduction to Corporate Finance William L. Megginson, Scott B. Smart 1st edition test bank Introduction to Corporate Finance, 2nd Edition William L. Megginson Scott B. Smart test bank Introduction to Corporate Finance, 2nd Edition William L. Megginson Scott B. Smart solution manual Introduction To Electric Circuits 6th Ed [Solutions Manual] By R. C. Dorf and J. A. Svoboda Introduction to Electrodynamics (Third Edition) by David J. Griffiths Introduction to Engineering Experimentation 2e wheeler and ganji Introduction to Engineering Thermodynamics, Edition 2, Sonntag, Borgnakke (Solution Manual) Introduction to Environmental Engineering and Science, 3E Gilbert M. Masters Wendell P. Ela, Introduction to Financial Accounting, 9E Charles T. Horngren Gary L. Sundem John A. Elliott Donna Philbrick test bank + solution manual Introduction to Fluid Mechanics (Fox, 5th ed) Solutions Manual Introduction to Fluid Mechanics, Edition 7, Fox, Pritchard, McDonald (Solutions Manual) Introduction to Fourier Optics Third Edition Problem Solutions by Joseph W. Goodman Introduction to Government and Non-for-Profit Accounting 6th edition by martin ives sm Introduction to Government and Non-for-Profit Accounting 6th edition test bank by martin ives Introduction To Graph Theory, 2nd ed. Douglas B. West Introduction to Heat Transfer - 013391061X Solution's Manual By peyman pourmoghaddam , Vedat S. Arpaci not complete Introduction to Heat Transfer 4th Edition SOLUTION MANUAL By Frank P. Incropera, David P. DeWitt Introduction to Heat Transfer, 5th Edition Incropera, DeWitt, Bergman, Lavine Introduction to Java Programming, Comprehensive Version, 7/E Y. Daniel Liang Introduction To Linear Algebra 3Ed - Gilbert Strang Solutions Manual Introduction To Management Accounting horngren 14e TB Introduction to management sciences 9e by Taylor solution manaul Introduction to Managerial Accounting 2nd ed Brewer test bank Introduction to Mathematical Statistics 6th Robert V. Hogg, allen t. Craig Introduction to Operations and Supply Chain Management 2e Cecil Bozarth Introduction to Operations and Supply Chain Management 2e Cecil Bozarth Robert B. Handfield test bank and sol manual Introduction to Ordinary Differential Equations by James C. Robinson Introduction to Probability by Dimitri P. Bertsekas - solution isbn: 1-886529-40-X Introduction to Quantum Mechanics (Second Edition) - Solutions Manual By David J. Griffiths Introduction to Statistical Quality Control, 6th Edition Montgomery Introduction to VLSI Circuits and Systems (2001 draft) - John P Uyemura - Solutions Manual Introduction To Wireless Systems - P M Shankar - Solutions Manual Introductory Circuit Analysis, 11E Robert L. Boylestad Introductory Econometrics A Modern Approach 2ed Jeffrey Wooldridge Introductory Econometrics A Modern Approach, 3Ed (with Economic Applications Online, Econometrics Data Sets with Solutions Manual Web Site Printed Access Card) by Jeffrey Wooldridge Introductory Linear Algebra An Applied First Course, 8e by Kolman Hill Introductory Mathematical Analysis for Business, Economics and the Life and Social Sciences, 12/E Ernest F. Haeussler Richard S. Paul Richard J. Wood test bank and sol manual Introductory Quantum Optics by Knight and Gerry Investment analysis and portfolio management 8e by Reily Brown Investment Analysis and Portfolio Management- Solutions Manual, 7th Edition by Frank K. et al. Reily Investment analysisnvestment analysis and management 9e by charles p. jones & management, 9e By jones some chapters missings sm IR: The New World of International Relations, 7e Michael G. Roskin, Lycoming College Nicholas O. Berry instructor manual and test bank instructor s manual with powerpoints to accompany pic microcontroller and embedded systems by muhammed ali mazidi rolin d mckinlay instructor s resource manual to accompany electronic devices,6th edition and electronic devices electron flow version 4th edition thomas floyd intermediate Accounting 12 E Kieso (TB) intermediate accounting 5e spiceland test bank and solution manual intermediate Accounting 12e by Keiso sm intermediate accounting 4th edition spiceland test bank introduction to algorithms 2nd edition instructors manual McGraw-Hill by thomas h. Cormen introduction to Environmental Engineering and Science (2nd Edition) (Hardcover) by Gilbert M. Masters introduction to linear algebra (5th, johnson) introduction to management accounting 14e Charles T. Horngren, Gary L. Sundem, William O. Stratton, Jeff Schatzberg, Dave Burgstahler solution manual introduction to mechatronics and measurement systems 2nd edition David G. Alciatore and Michael B. Histand introduction to Operations Research - 7th by Frederick Hillier, Gerald Lieberman introduction to probability Charles M. Grinstead and J. Laurie Snell odd solutions investment Analysis & Portfolio Management, 7th edition by Reilly and Brown Java, an Introduction to problem solving and programming , fifth Ed. by W. Savith and F. Carrano john E. Freund's Mathematical Statistics with Applications, 7th edition, by Miller and Miller Journey into Mathematics: An Introduction to Proofs (with solution manual) by Joseph J. Rotman Kc's Problems and Solutions for Microelectronic Circuits by k.c smith 4th edition Kinematics, Dynamics, and Design of Machinery by K. J. Waldron (Author), G. L. Kinzel (Author) Kinetics of Catalytic Reactions--Solutions Manual by: M. Albert Vannice Labor Relations, 12E Arthur A Sloane instructor manual laboratory manual to accompany Introductory Circuit Analysis 11e boylestad Lagrangian and Hamiltonian Mechanics Solutions to the Exercises by M. G. Calkin laser fundamentals 2nd edition by william t. Silfvast lectures on corporate finance 2e by Peter Bossaerts Lectures on Corporate Finance, Second Edition by Peter Bossaerts and Bernt Arne ??degaard legal environmental A critical thinking approach by KUBASKE, BRENNAN and BROWNE 5th edition test bank ?m legal environmental: A critical thinking approach by KUBASKE, BRENNAN and BROWNE 5th edition test bank Linear algebra and it s applications 3rd edition by david c. Lay linear algebra Juan de Burgos Solutions Manual spanish Linear algebra with applications 3e Otto Bretcher - Solutions Manual Linear Algebra with Applications 6 edition by Leon Linear Algebra with Applications, 7E by Steven J. Leon Linear Algebra, 4E Stephen H. Friedberg Arnold J. Insel Lawrence E. Spence Linear circuit analysis by R. A. DeCarlo and P. Lin - solution manuel 2nd edition linear systems and signals by lathi solutions manual blue covered Living Religions, 7E Mary Pat Fisher test bank Logic And Computer Design Fundamentals (4thEd) - Mano - SolutionsManual Lu, Likos Unsaturated Soil Mechanics Machine Design, An Integrated Approach, 3rd edition, by Robert L. Norton Macroeconomics (8thEd) - Froyen - Solutions Manual Macroeconomics 6e Andrew B. Abel, Ben S. Bernanke, Dean Croushore Macroeconomics, 4E Olivier Blanchard instructor manual Macroeconomics, 4E Olivier Blanchard test bank Macroeconomics, 5E Olivier Blanchard instructor manual and test bank Management 5th Edition Chuck Williams instructor manual Management 9E Stephen P. Robbins Mary Coulter test bank and instructor manual management accounting 5e Anthony A. Atkinson Robert S. Kaplan Ella Mae Matsumura S. Mark Young instructor manual test bank and solution manual Management accounting 5E atkinson solution manual Management accounting 5E atkinson test bank Management Control Systems Performance Measurement, Evaluation and Incentives 2e Merchant & Van der Stede Management Information Systems; Managing the Digital Firm, Edition 10, Laudon test bank and solution manual Management Robbins Coulter 9th edition (test bank ) Management10E Stephen P. Robbins Mary Coulter Management10E Stephen P. Robbins Mary Coulter test bank Managerial Accounting 12e By Garrison Noreen ( Solution Manual ) Managerial Accounting 12e By Garrison Noreen (Test Bank) Managerial Accounting Bamber, L. S., K. W. Braun, and W. T. Harrison, Jr. 2008 Managerial Accounting international edition Garrison11 ( Solution Manual) Managerial Accounting, 11th Edition by Ray H Garrison, Eric Noreen, Peter C. Brewer Managerial Accounting: An Introduction to Concepts, Methods and Uses 10th Edition Michael W. Maher, Clyde P. Stickney, Roman L. Weil instructor manual Managerial Accounting: An Introduction to Concepts, Methods and Uses 10th Edition Michael W. Maher, Clyde P. Stickney, Roman L. Weil solution manual Managerial Accounting: An Introduction to Concepts, Methods and Uses 10th Edition Michael W. Maher, Clyde P. Stickney, Roman L. Weil test bank Managerial Economics Applications, Strategies, and Tactics 11th Edition James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris instructor manual Managerial Economics Applications, Strategies, and Tactics 11th Edition James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris test bank Managerial Finance, Gitman,Lawrence 12e [SM] Managing Human Resources by Luis R. Gomez-Mejia; David B. Balkin; Robert L. Cardy. 5th Edition im and tb Manufacturing Engineering and Technology 5E by Kalpakjian and Schimid Marketing Kotler Armstrong 11th edition (Test bank ) Marketing Management (13th Edition) by Philip Kotler, Kevin Keller im Marketing Management (13th Edition) by Philip Kotler, Kevin Keller tb Materials and Processes in Manufacturing, 9 edition, [Solution manual] by E. Paul DeGarmo, Solutions Manual by: Barney E. Klamecki Materials Science and Engineering An Introduction 6E By William D. Callister materials science and engineering,callister 7th edition Mathematical Methods for Physics and Engineering A Comprehensive Guide 3rd edition by riley Mathematical Methods in the Physical Sciences, 3rd Edition Mary L. Boas Mathematical Models in Biology An Introduction by Elizabeth S. Allman john a. Rhodes Mathematical Models in Biology Sols by Elizabeth S. Allman, John A. Rhodes Mathematics for Economists Solution Manual - Simon and Blume (ver 2) Mathematics with Applications (9th Edition) Margaret L. Lial McgrawHill - William H. Hayt, John A. Buck - Engineering Electromagnetics, 6ed Solutions Manual Mechanical Behavior of Materials, 3E Norman E Dowling Mechanical Measurements 6th Edition by Thomas G. Beckwith, Roy D. Marangoni, John H. Lienhard V Mechanical Vibrations - Singiresu Rao - Solutions Manual 3rd edition chapters missing 6 9 12 ( all links same ) 35 mb Mechanics of Fluids by Merle C. Potter Mechanics of Fluids Solutions Manual 8ed By John Ward-Smith Mechanics of Materials [solutions manual] hibbeler 7th edition Mechanics of Materials An Integrated Learning System Philpot Mechanics of Materials Sixth Edition by R.C.Hibbeler Mechanics Of Materials Solution Manual (3Rd Ed , By Beer) nearly same 4th edition just numbered different Mechanics of Materials, 2nd Edition Craig Mechanics of Materials, 5th Edition by James M. Gere Mechanics of Materials, 6th Edition by James M. Gere (missing small portion, section 8.5) Mechanics of Materials, 7th Edition James M. Gere - Stanford University (Professor Emeritus) mechanics of solids by carl t. t. ross, d.sc. Microeconomic Analysis Solution Manual - Varian 3rd edition Microeconomic Analysis Third Edition by Hal R. Varian Microeconomic Theory Solutions Manual for Mas-Colell Microeconomics 7e Robert Pindyck Daniel Rubinfeld Microeconomics Theory and Applications with Calculus by perloff insrructor manual Microeconomics Theory and Applications with Calculus by perloff test bank Microelectronic Circuit Design 2nd and 3rd edition by R. Jaeger selected solutions answers not solution microelectronic circuits 5th ed. By Adel S. Sedra, Kenneth C. Smith microelectronics 3rd edition by jeager Microelectronics Circuit Analysis Desing 3rd Edition by Donald A. Neamen Microelectronics I & II 1st edition by by Dr. Wen-Ching Chang Microelectronics-Digital and Analog Circuits and Systems by Millman Microprocessors and Interfacing,Revised Second Edition Douglas V Hall Microwave And Rf Design Of Wireless Systems - Solution Manual (D M Pozar) Microwave Engineering 3e - David M Pozar Microwave Transistor Amplifiers Analysis and Design, 2nd Edition (Solutions Manual) by Guillermo Gonzalez Miller and Freund's Probability and Statistics for Engineers 7th edition by Richard A. Johnson Modern Advanced Accounting , 10th Edition, by Larsen, (publisher McGraw Hill) Modern Advanced Accounting 10th edition Test Bank LARSEN Modern Auditing Assurance Services 8e Boynton sm and tb modern control engineering by katsuhiko ogata 3rd edition ISBN: 0132273071 modern control engineering by katsuhiko ogata 4th edition Modern Control System 11th edition by richard c dorf, robert H bishop Modern Control System 9th by richard c dorf, robert H bishop Modern Database management 9e by Jeffrey A. Hoffer im Modern Database management 9e by Jeffrey A. Hoffer TB Modern Database Management, Ninth Edition Jeffrey A. Hoffer, University of Dayton Mary Prescott Heikki Topi Modern Digital and Analog Communications Systems - B P Lathi Solutions Manual 3rd edition Modern Digital Electronics,3E R P JAIN modern electronic communication 8th edition by gary miller and beasley Modern Electronic Communication, Beasly and Miller, 9th edition instructor manual Modern Electronic Communication, Beasly and Miller, 9th edition test bank Modern elementary statistics 12 ed freund e john test bank Modern Elementary Statistics 12e John E. Freund Benjamin M. Perles solution manual Modern Elementary Statistics 12e John E. Freund Benjamin M. Perles test bank modern Operating Systems 3rd edition - Andrew S. Tanenbaum Modern Organic Synthesis: An Introduction By Michael H. Nantz, Hasan Palandoken, George S. Zweifel Modern Physics, 2/E Randy Harris Modern Quantum Mechanics - J. J. Sakurai - Solution Modern Systems Analysis and Design, 5E Jeffrey A. Hoffer Joey F George Joseph S Valacich test bank Money, Banking, & Financial Markets 2nd edition Roger LeRoy Miller David D. VanHouse tb Multinational Business Finance 11E by David K. Eiteman SM Multinational Business Finance 11E by David K. Eiteman tb Multivariable Calculus: Student Solutions Manual 4th edition by James Stewart (Author) Nanoengineering of Structural Functional and Smart Materials Nanoengineering of Structural, Functional and Smart Materials Network flows theory, algorithms, and applications by Ravindra K. Ahuja (Author), Thomas L. Magnanti (Author), James B. Orlin (Author) odd solutions Network Security Essentials: Applications and Standards, 3/e stallings Nonlinear programming 2nd edition solution manual by dimitri bertsekas Numerical Methods For Engineers Solutions Manual by chapra 4th edition Numerical Methods for Engineers,5E,Steven C Chapra Numerical Solution of Partial Differential Equations: An Introduction 2ed By K. W. Morton, D. F. Mayers Operating System Concepts (6Th Ed)-Instructor'S Manual (A Silberschatz) Operating System Concepts, 7th Edtion, Instructor's Manual by A Silberschatz Operating systems Internals and Design principles 4th by william stallings Operating Systems Internals and Design principles( 5th Ed,William Stallings) Operating systems internals and design principles William STALLINGS 5.edition test bank Operating Systems: Internals and Design Principles, 6/e stallings solution manual and test bank Operations Management 9e heizer test bank Operations Management 5th edition Slack Nigel Instructors Manual Operations Management 8e by Jay Heizer Barry Render sm Operations Management 8e by Jay Heizer Barry Render tb Operations Management 9th by Jay Heizer Barry Render sm Operations Management 9th by Jay Heizer Barry Render tb Operations Management for MBAs, 3rd Edition Meredith, Shafer tb+im Operations Management Reid, Sanders An Integrated Approach 2nd edition Operations Management Reid, Sanders An Integrated Approach 2nd edition test bank Operations Management, 10e William J. Stevenson instructor manual with solutions to supplementary problems Operations Management, 10e William J. Stevenson test bank Operations Management, 9e William J. Stevenson instructor manual with solutions to supplementary problems Operations Management, William J Stevenson, 9e [TB] operations managenment processes and value chains 8e by Lee J. Krajewski solution manual operations managenment processes and value chains 8e by Lee J. Krajewski test bank Operations Research An Introduction, 8E Hamdy A. Taha Optimal Control Theory An Introduction By Donald E. Kirk Options, Futures and Other Derivatives, 4th Edition (Solutions Manual) by John Hull, John C. Hull Options, Futures and Other Derivatives, Fifth Edition (Solutions Manual) by John Hull, John C. Hull Options, Futures, and Other Derivatives 7E by JOHN C HULL TB Organic Chemistry by Robert C. Atkins, Francis A Carey, Robert Atkins, Francis Carey Organic Chemistry 2nd ed [Student SOLUTIONS MANUAL and Study Gde] - J. Hornback, B. Murugaverl (Thomson, 2006) WW Organic Chemistry and CW+ GradeTracker Access Card Package, 6/E Leroy G. Wade, JR., Whitman College test bank Organic Chemistry, 5/E Paula Y. Bruice test bank Organizational Behavior Stephen P Robbins 12th edition (test bank) Organizational Behavior, by Stephen P. and Timothy A. 13th Edition Robbins Judge im and tb Organizational Behavior: Managing People and Organizations, 9th Edition Ricky W. Griffin Gregory Moorhead instructor manual Organizational Behavior: Managing People and Organizations, 9th Edition Ricky W. Griffin Gregory Moorhead test bank organizational behaviour 12e by Robbins ( instructor manual ) organizational behaviour 12e by Robbins ( Test bank ) Organizational Theory, Design and Change, 5/E Gareth R. Jones Organizational Theory, Design and Change, 5E Gareth R. Jones Instructor's Manual and test bank Organizational Theory, Design and Change, 5E Gareth R. Jones im and tb Papathomas T.V. - Solutions Manual to Accompany Millman Integrated Electronics Analog and Digital Circuits and Systems (McGraw-Hill) Partial Differential Equations and Boundary Value Problems with Fourier Series (2nd Edition) (Student Solutions Manual) by asmar Pattern Recognition and Machine Learning (Solution Manual) - Bishop Personal Finance Turning Money into Wealth 5e Arthur J. Keown Personal Finance Turning Money into Wealth and Student Workbook, 4E keown instructor manual and test bank Physical Chemistry (Instructor's Solutions Manual) Peter Atkins & Julio de Paula 7ed Physics by Resnick Halliday Krane, 5th Ed. Vol 2 physics concepts and connections combined edition solutions manual by igor nowikow brian heimbecker christopher t. Howes Physics for Scientist & Engineers with Modern Physics - A strategic Approach by Randall D. Knight chapter 1-42 Physics for Scientist & Engineers with Modern Physics - A strategic Approach chapter by Randall D. Knight 1-35 Physics for Scientists and Engineers 5th edition by Serway And Jewett Physics for Scientists and Engineers Extended Version, 5th edition, Physics for Scientists and Engineers with Modern Physics (3rd Edition) Physics for Scientists and Engineers with Modern Physics (3rd Edition) by giancoli Physics For Scientists Engineers With Modern Physics (4thEd) - Giancoli - Solutions Manual Physics of the Solar Corona: An Introduction with Problems and Solutions by Markus Aschwanden Physics Principles with Applications Instructor's Solutions Manual (Giancoli, Volume 1and 2) 6th edition power analysis and design by glover , sarma 3rd ediiton Power Electronics, Converters, Applications, and Design By Ned Mohan, Tore M. Undeland, William P. Robbins Power System Analysis and Design Glover and Sarma 4e Thomson Learning Glover J. Duncan, Sarma Mulkutla .S. Power System Analysis Hadi Saadat 2nd Edition Power System Analysis Solution Manual John Grainger, William D. Stevenson Practical Financial Management 5th Edition William R. Lasher instructor manual Practical Financial Management 5th Edition William R. Lasher test bank Practical Financial Management William R. Lasher 4th edition instructor manual Practical Financial Management William R. Lasher 4th edition test bank Prentice Hall - Solutions Manual; Communication Systems Engineering Prentice Hall's Federal Taxation 2008 Individual , 21th Edition By Thomas R. Pope, Kenneth instructor guide Prentice Hall's Federal Taxation 2009 Comprehensive, 22nd Edition By Thomas R. Pope, Kenneth E. Anderson, John L. Kramer solution manual Prentice Hall's Federal Taxation 2009 Comprehensive, 22nd Edition By Thomas R. Pope, Kenneth E. Anderson, John L. Kramer test bank Prentice Hall's Federal Taxation 2009 Individual , 22th Edition By Thomas R. Pope, Kenneth instructor guide Prentice Hall's Federal Taxation 2009: Corporations, 22/E Kenneth E. Anderson Thomas R. Pope John L. Kramer instructor guide Prentice Hall's Federal Taxation 2009: Corporations, 22/E Kenneth E. Anderson Thomas R. Pope John L. Kramer tb +sm Prentice Hall's Federal Taxation 2009: Individuals, 22/E Kenneth E. Anderson Thomas R. Pope John L. Kramer tb +sm Principles and Applications of Electrical Engineering giorigo rizzoni 4th edition Principles and Applications of Electrical Engineering giorigo rizzoni 5th edition Principles Geotechnical Engineering Braja Das 6th ed solution manual Principles of Auditing 15e by Whittington TB Principles of Communication: Systems, Modulation and Noise (5th Ed) by R. E. Ziemer, W. H. Tranter, solution manual Principles of Communications, 6th Edition Ziemer, Tranter Principles of corporate finance 7e Principles of corporate finance 7e by brealy mayers Principles Of Corporate Finance 8E By Brealey Myers Allen Principles of corporate finance 9e by brealy mayers allen (SM+TB) Principles of Electric Circuits Conventional Current Version 8e by floyd Principles of Electric Circuits Conventional Current Version by Floyd 8th edition principles of electric circuits- electron flow version Floyd 8th edition Principles of Electronic Materials and Devices, Solutions Manual ONLY Safa O. Kasap 2nd edition principles of managerial finance 10e by gitman Lawrence principles of managerial finance 11e by gitman Lawrence Principles of managerial finance 11e by gitman Lawrence solution manual Principles of Managerial Finance 11e by Gitman Lawrence test bank Principles of managerial finance 12e by gitman Lawrence test bank Principles of Managerial Finance Brief plus 5e sm Principles of Managerial Finance Brief plus 5e tb Principles of Managerial Finance Brief plus My Finance Lab Student Access Kit, 5E Lawrence J. Gitman principles of marketing 11e by Kotler ( instructor manual ) principles of marketing 11e by Kotler ( test bank ) principles of marketing 12e by Kotler TB Principles of Microeconomics, 9/e Case, Fair & Oster instructor manual Principles of Microeconomics, 9/e Case, Fair & Oster test bank Principles of Microeconomics, 9e Case, Fair & Oster instructor manual ad test bank Principles of Neurocomputing for Science and Engineering, 1st Edition Probability & Statistics for Engineers & Scientists, 8th Edition: Instructors Solution Manual ONLY by Sharon Myers , Keying Ye, Walpole Probability and Statistical Inference 7th edition Hogg Tanis solution Probability and Statistics for Engineering and the Sciences [Solutions Manual] 6th edition by Jay L. Devore Probability and Statistics for Engineers and Scientists Manual HAYLER Solutions Manual Probability, Random Variables, and Stochastic Processes [Only Solutions Manual] by Athanasios Papoulis ,S.Unnikrishna Pillai 4th edition Probability, Statistics, and Random Processes For Electrical Engineering - Alberto Leon-Garcia (3rd ed) (ISBN 0131471228) Probability, Statistics, and Random Processes for Engineers, 1st Edition Richard H. Williams - University of New Mexico Problem Solving and Programming Concepts, 8/E Maureen Sprankle Jim Hubbard Problem Solving with C++ The Object of Programming, 5E Walter Savitch tb and im Problems and Solutions on Electromagnetism by zhao shu ping - you jun han Process Dynamics and Control, 2nd Edition Seborg, Edgar, Mellichamp PROCESS SYSTEMS ANALYSIS AND CONTROL - DONALD R. COUGHANOWR Solution Manual Process Systems Analysis and Control by Donald R Coughanowr Programming the World Wide Web, 4E Robert W. Sebesta Quantitative Analysis for management TENTH EDITION by Barry Render, Ralph M. Stair, Jr, Michael E.Hanna solution manual Quantitative Analysis for management TENTH EDITION by Barry Render, Ralph M. Stair, Jr, Michael E.Hanna test bank Quantum Field Theory (draft version) & Instructor's Manual by Mark Srednicki book + solution manual Radiation Protection in the Health Sciences: With Problem Solutions Manual by: Marilyn E. Noz, Gerald Q. Maguire Sargen Recursive Methods in Economic Dynamics By Claudio Irigoyen ,Esteban Reinforced Concrete: Mechanics and Design by James MacGregor and James Wight, either 5th edition Research in Education 10th edition by Best, J., & Kahn, J instructor manual and test bank Retail Management A Strategic Approach, 10E Barry Berman, Joel R. Evans tb and sm RF circuit Design Theory and Application by Ludwig bretchko solution manuel Roy D. Yates and David J. Goodman, Probability and Stochastic Processes - A Friendly Introduction for Electrical and Computer Engineers, 2nd edition, Satellite Communications By Timothy Pratt, Charles W. Bostian Schaums Mathematical Handbook of Formulas and Tables Schaum's Outline of Discrete Mathematics, 2nd edition, 1997 by Seymor Lipschutz, Marc Lipson Schaum's Outline of Discrete Mathematics, 3rd Ed. (Schaum's Outlines) by Seymour Lipschutz, Marc Lipson Schaum's Outline of Probability, 2nd Edition by Seymour Lipschutz Schaum's Outline of Theory and Problems of Linear Algebra, 2nd Edition (Schaum's Outlines) by Seymour Lipschutz Schaum's Outline of Theory and Problems of Managerial Accounting. [1998.ISBN0070580413] book + sm Selected Answers-Basic Engineering Circuit Analysis-7th Ed. by J. David Irwin Semiconductor Device Fundamentals 1st edition by Robert F Semiconductor Physics and Devices: Basic Principles 3rd edition by neamen separation Process Principles, 2nd Ed by Seader, Henley Shigley's Mechanical Engineering Design 7th Ed - Solution Manual By by Richard Budynas, J.Charles Mischke Shigley's Mechanical Engineering Design 8th Ed - Solution Manual By by Richard Budynas, J. Keith Nisbett Signal , System and transforms fourth edition BY charles L. philips , John M Parr, Eve A. Riskin Signal , System and transforms fourth edition BY charles L. philips , John M Parr, Eve A. Riskin international edition Signal Processing and Linear Systems by lathi Signal Processing First-Mclellan, Schafer & Yoder Solution Manual chapter 3-12 Signals and Systems - Analysis using Transform methods and MATLAB M. Signals and Systems 2nd edition simon Haykin Solutions Manual Signals and Systems, Prentice-Hall Oppenheim, Willsky, Young 2nd edition Silicon VLSI Technology: Fundamentals, Practice, and Modeling James D. Plummer Michael D. Deal Peter B. Griffin sociology John J. Macionis 12th edition test bank and instructor manual Soil Mechanics Concepts and Applications By William Powrie Soil Mechanics Solutions Manual (2nd Edition) By William Powrie soils and foundations 7th by cheng liu and jack b evett Solid State Electronic Device by Ben Streetman Solid State Electronic Devices, 6E Ben Streetman Sanjay Banerjee Solid State Physics - Solutions Manual by Ashcroft & Mermin some chapters missings SOLUTIONS MANUAL AND WORKBOOK to accompany Quantitative Methods for Management Solution Automatic Control Systems 8Ed by Kuo and Golnaraghi Solution to Skill - Assessment Exercises to Accompany Control Systems Engineering 3rd edt. by Norman S. Nise Solution To Two-Dimensional Incompressible Navier-Stokes Equations- Maciej Matyka Solutions & Supplements Introductory Chemical Engineering Thermodynamics by J.R. Elliott & C. T. Lira selected solutions Solutions Irodov's Prob. Gen. Physics Volume 1 by Abhay K. Singh Solutions Irodov's Prob. Gen. Physics Volume 2 by Abhay K. Singh South-Western Federal Taxation 2009 (Individual), Edition 32, Hoffman, Smith, Wills (Test Bank) South-Western Federal Taxation 2009 Corporations - William Hoffman (Solutions Manual) (32nd ed) (ISBN 0324660219) South-Western Federal Taxation 2009 Corporations - William Hoffman (test bank) (32nd ed) (ISBN 0324660219) South-Western Federal Taxation 2009: Comprehensive - William Hoffman (Solutions Manual) (32nd ed) (ISBN 0324660529) South-Western Federal Taxation 2009: Comprehensive - William Hoffman (Test Bank) (32nd ed) (ISBN 0324660529 South-Western Federal Taxation Individual Income Taxes, 32nd by William Hoffman, Jr. James Smith Eugene WillisEdition sm 2009 STATISTICS FOR THE SCIENCES by Martin Buntinas & Gerald M. Funk, Statics & Mechanics of Materials SI 2nd edition Russell Hibbeler Jul Statics and Mechanics of Materials Anthony M. Bedford Kenneth M. Liechti Wallace Fowler Statistical Digital Signal Processing and Modeling - SOLUTIONS MANUAL By Monson H. Hayes Statistical Inference - Casella & Berger 2nd edition statistical Quality Design and Control, 2/E Richard E. DeVor, Tsong- how Chang, John W. Sutherland Statistics 4e by Murray R Spiegel, Larry J. Stephens book + sm Statistics for Business and Economics and Student CD (6th Edition) (Hardcover) by Paul Newbold (Author), William L. Carlson (Author), Betty Thorne (Author Statistics for Engineering and the Sciences, 5e by William Mendenhall and Terry Sincich Statistics for Engineers and Scientists by William Navidi Steel Design, 4th Edition solution Manual William T. Segui Steel Structures Design and Behavior, 5th Edition by Charles G. Salmon Strategic Brand Management 3e keller Strategic management 11e by Fred R. David ( instructor manual ) Strategic management 11e by Fred R. David ( test bank ) Strategic Management and Business Policy, 11e Thomas L. Wheelen David L. Hunger instructor manual with test items Strategic Management and Competitive Advantage: Concepts and Cases, 2/ E instructor manual with test items Jay Barney William S Hesterly Strength of Materials 4th Ed. by Ferdinand L. Singer & Andrew Pytel Strength of Materials 4th Ed. by Singer & Pytel Structural Analysis by Hibbeler 5th edition Structural Analysis by Hibbeler 7th edition Structured computer organization 5E Andrew S. Tanenbaum Student Solution Manual for Mathematical Methods for Physics and Engineering Third Edition by K. F. Riley, M. P. Hobson not complete Student Solutions Manual and Study Guide to accompany Fundamentals of Fluid Mechanics, 5th Edition by Bruce R. Munson Donald F. Young, Theodore H. Okiishi Study Guide with Solutions Manual for McMurry's Organic Chemistry, 7th Ed. by John E. McMurry Fawcett Lisa M. Ellram Jeffrey A. Ogden Survey of Accounting, Fourth Edition Carl S. Warren System Dynamics 3rd Ed By Katsuhiko Ogata System Dynamics and Response, 1st Edition S. Graham Kelly Techniques of Problem Solving by luiz fernandez TestGen for Physics for Scientists and Engineers - A Strategic Approach 2E knight The Analysis and Design of Linear Circuits Laplace Early, 4th Edition Thomas, Rosa The Art of Electronics. Thomas C. Hayes, Paul Horowitz The Economic Way of Thinking 11e Peter J. Boettke Peter J. Boettke David L. Prychitko test bank and solution manual The Economics of Financial Markets By Roy E. Bailey The Economics of Financial Markets by Roy E. Bailey The Economics of Money,Banking,and Financial Market 8th edition by Frederic S. Mishkin instructor manual and test bank The Legal Environment of Business, 5th edition Kubasek The Science and Engineering of Materials by Donald R. Askeland Frank Haddleton 4th edition The Spacetime Frontier Science and Society in the 21st Century by Stewart Swain Theory & Design for Mechanical Measurements 4th edition by Richard S. Figliola & Donald E. Beasley Thermodynamics an engineering approach sixth edition ( SI units ) : solutions manual by Yunus A. Cengel, Michael A. Boles Thermodynamics Of TurboMachinery 5th edition Thermodynamics: An Engineering Approach by: Yunus A. Cengel 5th edition thomas calculus 10th edition instructor solution manual volume 1and 2 Thomas Calculus 11th edition Thomas Calculus updated 10E by finney, weir giordano Thomas, Rosa The Analysis and Design of Linear Circuits Laplace Early, 4th Edition Transport Phenomena - 2nd edition by Bird, Stewart and Lightfoot Solution Manual Ulaby Applied Electromagnetics Undergraduate Econometrics Solutions Manual - Hill, Judge and Griffiths Understanding Corporate Annual Reports, 6e William R. Pasewark, Texas Tech University Understanding Financial Statements 8e Lyn M. Fraser Aileen Ormiston test bank and sol manual Understanding Financial Statements 8e Lyn M. Fraser sm and tb Unit Operations of Chemical Engineering, 6th Edition, Solutions Manual by Warren McCabe, Julian Smith, Peter Harriott Unit Operations of Chemical Engineering, 7th Edition, Solutions Manual by: Warren McCabe, Julian Smith, Peter Harriott University Calculus, Part Two Multivariable, Chap 9-14 Joel Hass Maurice D. Weir sm University Calculus, Part Two Multivariable, Chap 9-14 Joel Hass Maurice D. Weir test generator University physics 11th edition solution manual by Young and Freedman University Physics with Modern Physics (12th Edition) by Hugh D. Young, Roger A. Freedman Unsaturated Soil Mechanics by Ning Lu and William J. Likos Using Econometrics A Practical Guide, 5th edition by Studenmund Vector Calculus (3rd Ed., Susan J. Colley) R. Johnston Jr., E. R. Eisenberg, & G. H. Staab Vector Mechanics for Engineers STATICS, 7th Edition by F. P. Beer, E. R. Johnston Jr., E. R. Eisenberg, & G. H. Staab Vector Mechanics for Engineers; Dynamics 8th edition Beer Johnston Vector Mechanics for Engineers; Statics 8th edition Beer Johnston solution manual Wankat & Oreovicz - Teaching Engineering water and wastewater technology mark j. hammer mark J. Hammer 6th edition Water Supply and Pollution Control, 8E Warren Viessman, Jr. Mark J. Hammer Elizabeth M. Perez Paul A. Chadik WHO-DUN-IT Fifth Edition Shari L. DeMarco wiley applied corporate finance 2nd ed Wiley Chemical And Engineering Thermodynamics 3Ed Solutions Manual Wireless Communication and Networks second edition William Stallings solutions manual Wireless Communications: Principles and Practice, 2nd edition theodore rappaport solutions manual === Subject: Could high technology develop in an aquatic species? posting-account=7DfHzwkAAABuiYa0I5v96WrKfef1HlA7 Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) Every once in awhile you see a science fiction story in which the aliens were supposed to have evolved in an aquatic environment. Nevertheless, they have high technology comparable to ours. Further, one of the many factors that the drake equation (an attempt at mathematically estimating the number of technological civilizations in our galaxy) http://en.wikipedia.org/wiki/Drake equation ignores is the fact that if a world has liquid water it may very well be entirely covered with water. There is some speculation among geophysicists that had the earth's Moon not formed, the earth itself would likely be a Waterworld. My question is, how much technology would have developed if an intelligent species had had to try to develop its technology while submerged in water? Could in intelligence species of creature with manipulating appendages have ever gotten from the kind of resources you see lying around on the ocean floor to the kind of technology necessary to build a spacecraft? Fire under water is certainly a no go until the development of fairly exotic high Energy Materials such as magnesium. How you can get such materials without having fire in the first place is unclear to me. Fire or at least intense heat is obviously necessary to the production of many chemicals. It is possible that an intelligent species could use natural sources of heat such as volcanic vents, but the danger associated with geologic heat and its relative inconvenience of location would have been a great hindrance. Metallurgy and glasswork both highly important to the construction of the tools of chemistry and engineering would be extraordinarily hampered. Gold would be about the only metal available at low technology levels without fire. Gold is too soft for most applications. Additionally, chemistry in general would be greatly hindered by the fact that water is such a good solvent. It would be nearly impossible to isolate chemicals, concentrate them, and make them pure. Most of our early chemistry was done in water but if you open a beaker of the water soluble chemical underwater your chemical just floats away. It is also nearly impossible to perform important chemical procedures such as distillation and crystallization while you are in an aquatic environment. It is possible that some isolation of chemicals could be carried out by manipulating them in lipid phases. In other words we could have worked with our chemicals by dissolving them in globs of fat that we could manipulate underwater. This of course would have the downside of being useless for the chemistry of ionic compounds. Engineering and physics would suffer also. The high friction environment under water would hinder all sorts of activities. Even flint napping, one of our ancestors' first tool making activities, would be difficult under water. Our understanding of buoyancy and fluid dynamics would probably have come earlier and be much better, but kinematics generally would suffer. Our basic concepts of mass, acceleration and inertia would be hindered by the effects of buoyancy and viscosity. Soft early technologies such as weaving would be ok, provide the aquatic environment provided strong enough materials and the physics of static equilibria would be largely unaffected because the viscosity of water would not be a factor; still one could argue that buoyancy and the motive power of water currents would make controlled experiments in statics more difficult. Water, presumably saltwater, is a relatively good conductor of both heat and electricity. This would have posed serious problems in the development of both electrodynamics and thermodynamics. Electro dynamics, might suffer less since many aquatic organisms seem to have evolved electrical senses naturally. If an intelligent species could perceive electrical currents with their senses directly it is likely that the behavior of electricity would be relatively easy for them to grasp intuitively. Substances like gold would have interesting effects on the electric fields they perceived. Natural rocks containing magnetite would still work under water and have interesting properties allowing them to learn about magnetism. Certain living organisms would be likely to evolve electrical attacks and defenses and could provide a ready if difficult to control source of electrical power. Nevertheless, electricity would be difficult to control, because any exposed surface would provide a route for current to flow through the surrounding water, and insulators would be hard to come by. Thermodynamics would be even more difficult because heat would leak away into the water quickly and as mentioned before there are no easy to control sources of heat available under water. Experiments involving the ideal gas laws would be very difficult to conduct not only because temperature would be difficult to control and manipulate, but pressure underwater changes dramatically depending upon your vertical location. Vacuums are even harder to generate at the pressures found under water. I'm unsure what effect aquatic life would have on optics and acoustics. Light filtering through the water would certainly presents some interest in learning opportunities as would the interesting optics associated with bubbles and similar gas and liquid interfaces. This is assuming of course that the visual systems of intelligent life were well developed (questionable in an aquatic environment) and those of the filtered rippling light that comes from the surface. The poor state of material science would make tools such as telescopes and microscopes impossible, and the natural turbidity of open water would make telescopes nearly useless. If telescopes were invented, presumably they could be used with some great difficulty at the ocean's surface to develop the science of astronomy, but one has to wonder if the culture that never lived under the open skies and saw the wonders of the constellations, the solar, lunar, and planetary cycles would even be interested or see the relevance of what was above the waves. Acoustics on the other hand might be a very lively theoretical science especially if the aquatic intelligence had natural sonar. Nevertheless, the technology of acoustics would probably come to an early standstill because of the lack of materials to build tools and the lack of electronic devices to generate record and analyze sound. Acoustical science might also be hampered by the fact that the underwater environment tends to be quite noisy. Wave mechanics might well benefit from the natural oscillations of water that would be apparent. Nevertheless, this natural way of motion would be very difficult to isolate and control, therefore in experimental science built around wave mechanics would not develop well. Further, mechanical oscillators such as springs and pendulums do not work well under water because of viscosity. Biology would be hampered too. Antiseptic procedures would be nearly more easily than air, but it is more difficult to filter. There are probably some naturally occurring antibiotics and antiseptics but like most water soluble chemicals they would be difficult to purify and concentrate. Herbal medicine would work too though ingredients would be harder to preserve, purify, standardize and dose. Biology is highly dependent upon chemistry and chemical procedures and would be hobbled by the difficulties in chemistry described above. Saltwater itself is toxic when introduced into sensitive structures like the brain and other organs. Bacterial and viral cultures would be difficult to isolate and grow if they were not species that were already adapted to the ambient conditions of the aquatic environment. That is, gut bacteria, would not grow well once taken from the enclosed environment of the gut because the exposure to open water would interfere with their normal chemistry. Generally, I see little problem with the descriptive aspects of biology; disciplines such as taxonomy should not suffer and anatomy in its primitive form would be fine until more sophisticated tools such as scalpels and microscopes became necessary. If the intelligences had sonar or electrical senses this would great aid their understanding of anatomy and physiology. I see no problems for math except to the extent that it depends on materials and tools that might be hard to make and preserve in an aquatic environment? No calculators and would there be a good substitute for pencil and paper? I suppose also that the poor development of sciences like astronomy and kinematics would make math seem less attractive. A natural ability to see into objects through the use of sonar might have interesting effects on geometry. My overall conclusion is this. All the major branches of science would suffer if they had to develop in an aquatic environment. Chemistry in particular would be crippled. Since chemistry is foundational to materials science and since materials are essential for the development of the other science is all technological development would be held back. I see the development of the technology advanced enough for space flight by an aquatic civilization as extremely unlikely, verging on impossible, particularly on a world with no land masses. One unrelated question. Why isn't there a groups called sci.bio and what is the main general biology discussion group? === Subject: Re: Could high technology develop in an aquatic species? >My question is, how much technology would have developed if an >intelligent species had had to try to develop its technology while >submerged in water? Interesting question. Man didn't really have time to advance until we stopped being hunter gatherers and started farming. === Subject: Re: Could high technology develop in an aquatic species? >My question is, how much technology would have developed if an >intelligent species had had to try to develop its technology while >submerged in water? Interesting question. Man didn't really have time to advance until we stopped being hunter > gatherers and started farming. According to Alice Roberts, homo sapiens left Africa 70,000 years ago. If halfway round the Earth is 10,000 miles (in each direction), that's 7 miles a year. I could manage that, even with a gimpy ankle. http://tinyurl.com/ooaktl Looks like the mythical Garden of Eden was Yemen. 70,000 years, 25 years per generation, ~3000 generations. Technology begins with fire. Control that and you have easy-to- chew cooked meat and metals; along came the Bronze Age followed by the Iron Age. With metals you have improved hunting ability, the iron bodkin, the sword and the spear (and later the rifle) as well as the plough. Pregnant women can do the gathering, hunting is arduous. Dolphins and whales may be intelligent species but they have no fire or hands to manipulate it. They are the shape they are because they don't live in trees as we once did. The chimpanzee is a tool user, using sticks to collect termites. He also has an opposable big toe, very handy for tree climbing. http://assets.espn.go.com/photo/2007/1112/pg2_a_chimp_300.jpg http://www.ugandagorillasafaritours.com/images/chimpanzees.jpg http://tinyurl.com/ou83fw It takes more than intelligence and free time to have a technology. === Subject: Re: Could high technology develop in an aquatic species? > Every once in awhile you see a science fiction story in which the > aliens were supposed to have evolved in an aquatic environment. > Nevertheless, they have high technology comparable to ours. Further, > one of the many factors that the drake equation (an attempt at > mathematically estimating the number of technological civilizations in > our galaxy) http://en.wikipedia.org/wiki/Drake_equation ignores is the fact that if a world has liquid water it may very well > be entirely covered with water. There is some speculation among > geophysicists that had the earth's Moon not formed, the earth itself > would likely be a Waterworld. My question is, how much technology would have developed if an > intelligent species had had to try to develop its technology while > submerged in water? Could in intelligence species of creature with > manipulating appendages have ever gotten from the kind of resources > you see lying around on the ocean floor to the kind of technology > necessary to build a spacecraft? > There is no data to answer that question. Speculation about an intelligent being within a water world have no basis in experience, but obviously, the water world has a surface. If that surface is enveloped by an atmosphere, intelligent beings could master that environment and establish themselves in there. If some kind of combustion is possible in that atmosphere they could develop airborne objects like hot air balloons and other similar devices. Depending on the resources they have, building some kind of propellant is not very far away. If the water world is enveloped by an ice sheet, and then by vacuum like in Europa here in the solar system, the situation is more difficult since they would have to first master living in the ice, and tunneling their way into the vacuum. Then, they would need to find some kind of propellant to lift off the surface of the ice. If we extrapolate from our situation, in only 6 000 years we have gone from intelligent animals to space faring. We have explored the whole surface of the planet where we were born. I suppose any intelligent being would need a similar time in a similar sized planet. Once the boundaries are known, intelligence finds the resource necessary to find a propellant to build the first chemical rockets, either in the atmosphere or in vacuum if enveloped with an ice sheet. > Fire under water is certainly a no go until the development of fairly > exotic high Energy Materials such as magnesium. How you can get such > materials without having fire in the first place is unclear to me. Of course it is unclear TO YOU. You never were in the situation of a fish, no wonder you do not know how to do science in there. What is important is the motivation to build a rational explanation of the world. Chemistry belongs to that explanation, and an intelligent being will find out about chemistry sooner or later... > Fire or at least intense heat is obviously necessary to the production > of many chemicals. It is possible that an intelligent species could > use natural sources of heat such as volcanic vents, but the danger > associated with geologic heat and its relative inconvenience of > location would have been a great hindrance. All water worlds have a surface, and probably some of them an atmosphere. Why do you exclude such possibilities? > Metallurgy and glasswork both highly important to the construction of > the tools of chemistry and engineering would be extraordinarily > hampered. Gold would be about the only metal available at low > technology levels without fire. Gold is too soft for most > applications. > Ice can be strong enough at low temperatures... In Titan, ice is as solid as rock... In low temperature water worlds with methan as the universal solvent, many compounds are very hard that here exist only in liquid form... > Additionally, chemistry in general would be greatly hindered by the > fact that water is such a good solvent. It would be nearly impossible > to isolate chemicals, concentrate them, and make them pure. That could be done in the surface! > Most of > our early chemistry was done in water but if you open a beaker of the > water soluble chemical underwater your chemical just floats away. It > is also nearly impossible to perform important chemical procedures > such as distillation and crystallization while you are in an aquatic > environment. It is possible that some isolation of chemicals could be > carried out by manipulating them in lipid phases. In other words we > could have worked with our chemicals by dissolving them in globs of > fat that we could manipulate underwater. This of course would have > the downside of being useless for the chemistry of ionic compounds. > The basic problem with your arguments is that you are an homo sapiens extrapolating from his own history. What you do not know is that in other environments, process completely unknown to homo sapiens could be common place! When we explore Europa and study the life forms living there we will be able to see the solutions those beings developed and get out of this catastrophic anthropomorphism that pervades all scientific thinking today. -- jacob navia jacob at jacob point remcomp point fr logiciels/informatique http://www.cs.virginia.edu/~lcc-win32 === Subject: Re: Could high technology develop in an aquatic species? <4a094c05$0$12637$ba4acef3@news.orange.fr> posting-account=7DfHzwkAAABuiYa0I5v96WrKfef1HlA7 Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Every once in awhile you see a science fiction story in which the > aliens were supposed to have evolved in an aquatic environment. > Nevertheless, they have high technology comparable to ours. Further, > one of the many factors that the drake equation (an attempt at > mathematically estimating the number of technological civilizations in > our galaxy) http://en.wikipedia.org/wiki/Drake equation ignores is the fact that if a world has liquid water it may very well > be entirely covered with water. There is some speculation among > geophysicists that had the earth's Moon not formed, the earth itself > would likely be a Waterworld. My question is, how much technology would have developed if an > intelligent species had had to try to develop its technology while > submerged in water? Could in intelligence species of creature with > manipulating appendages have ever gotten from the kind of resources > you see lying around on the ocean floor to the kind of technology > necessary to build a spacecraft? There is no data to answer that question. Speculation about an > intelligent being within a water world have no basis in experience, > but obviously, the water world has a surface. If that surface is enveloped by an atmosphere, intelligent beings could > master that environment and establish themselves in there. If some kind > of combustion is possible in that atmosphere they could develop airborne > objects like hot air balloons and other similar devices. Depending on the resources they have, building some kind of propellant > is not very far away. If the water world is enveloped by an ice sheet, and then by vacuum like > in Europa here in the solar system, the situation is more difficult > since they would have to first master living in the ice, and tunneling > their way into the vacuum. Then, they would need to find some kind > of propellant to lift off the surface of the ice. If we extrapolate from our situation, in only 6 000 years we have gone > from intelligent animals to space faring. We have explored the whole > surface of the planet where we were born. I suppose any intelligent > being would need a similar time in a similar sized planet. Once the boundaries are known, intelligence finds the resource necessary > to find a propellant to build the first chemical rockets, either in the > atmosphere or in vacuum if enveloped with an ice sheet. Fire under water is certainly a no go until the development of fairly > exotic high Energy Materials such as magnesium. How you can get such > materials without having fire in the first place is unclear to me. Of course it is unclear TO YOU. You never were in the situation of a > fish, no wonder you do not know how to do science in there. What is important is the motivation to build a rational explanation of > the world. Chemistry belongs to that explanation, and an intelligent > being will find out about chemistry sooner or later... Fire or at least intense heat is obviously necessary to the production > of many chemicals. It is possible that an intelligent species could > use natural sources of heat such as volcanic vents, but the danger > associated with geologic heat and its relative inconvenience of > location would have been a great hindrance. All water worlds have a surface, and probably some of them an > atmosphere. Why do you exclude such possibilities? Metallurgy and glasswork both highly important to the construction of > the tools of chemistry and engineering would be extraordinarily > hampered. Gold would be about the only metal available at low > technology levels without fire. Gold is too soft for most > applications. Ice can be strong enough at low temperatures... In Titan, ice is as > solid as rock... In low temperature water worlds with methan as the > universal solvent, many compounds are very hard that here exist only in > liquid form... Additionally, chemistry in general would be greatly hindered by the > fact that water is such a good solvent. It would be nearly impossible > to isolate chemicals, concentrate them, and make them pure. That could be done in the surface! Most of > our early chemistry was done in water but if you open a beaker of the > water soluble chemical underwater your chemical just floats away. It > is also nearly impossible to perform important chemical procedures > such as distillation and crystallization while you are in an aquatic > environment. It is possible that some isolation of chemicals could be > carried out by manipulating them in lipid phases. In other words we > could have worked with our chemicals by dissolving them in globs of > fat that we could manipulate underwater. This of course would have > the downside of being useless for the chemistry of ionic compounds. The basic problem with your arguments is that you are an homo sapiens > extrapolating from his own history. What you do not know is that in > other environments, process completely unknown to homo sapiens > could be common place! When we explore Europa and study the life forms living there we will > be able to see the solutions those beings developed and get out of > this catastrophic anthropomorphism that pervades all scientific thinking > today. -- > jacob navia > jacob at jacob point remcomp point fr > logiciels/informatiquehttp://www.cs.virginia.edu/~lcc-win32 Yes, it is quite possible that my anthropocentric perspective is causing me to miss some obvious alternative pathways to high technology. I know I am not doing anything more than speculation, but I introduced this thread just for the fun of speculation, not to assert a definitive argument. I want us to use our imaginations to see if we can break out of that at least a little, or come up with some convincing genral arguments as to why we couldn't. I hope you see that I have made some attempt to anticipate advantages an aquatic species might have. I agree that working at the surface would be one way to try to overcome the limitations ofthe aquatic environment and I mentioned it in the context of astronomy. Given the action of waves in the open ocean and the lack of materials that a prechemistry society would have, I see setting up a lab at the surface as very, very difficult and with virtually no motivating factors to drive our aliens in that direction. They would need to treat the surface as we treat a glove box. But how do you build a glove box without glass and rubber and other materials you can make some sort of seal with? How would they deal with waves? Would they even even come to understand the concept of drying out materials, a necessary step to combustion. Though, I think we may disagree about somethings, I think your points are good ones and I am glad you replied. We do agree on the idea that as homesapiens we do have many biases that aliens or even other terrestrial species do not have. === Subject: Re: Could high technology develop in an aquatic species? posting-account=-d2gtwoAAACIgCVyog_mrAWs-fwWRSD9 CLR 1.1.4322; InfoPath.1; .NET CLR 2.0.50727; MS-RTC LM 8),gzip(gfe),gzip(gfe) > Every once in awhile you see a science fiction story in which the > aliens were supposed to have evolved in an aquatic environment. > Nevertheless, they have high technology comparable to ours. Further, > one of the many factors that the drake equation (an attempt at > mathematically estimating the number of technological civilizations in > our galaxy) http://en.wikipedia.org/wiki/Drake equation ignores is the fact that if a world has liquid water it may very well > be entirely covered with water. There is some speculation among > geophysicists that had the earth's Moon not formed, the earth itself > would likely be a Waterworld. My question is, how much technology would have developed if an > intelligent species had had to try to develop its technology while > submerged in water? Could in intelligence species of creature with > manipulating appendages have ever gotten from the kind of resources > you see lying around on the ocean floor to the kind of technology > necessary to build a spacecraft? Fire under water is certainly a no go until the development of fairly > exotic high Energy Materials such as magnesium. How you can get such > materials without having fire in the first place is unclear to me. > Fire or at least intense heat is obviously necessary to the production > of many chemicals. It is possible that an intelligent species could > use natural sources of heat such as volcanic vents, but the danger > associated with geologic heat and its relative inconvenience of > location would have been a great hindrance. > Metallurgy and glasswork both highly important to the construction of > the tools of chemistry and engineering would be extraordinarily > hampered. Gold would be about the only metal available at low > technology levels without fire. Gold is too soft for most > applications. Additionally, chemistry in general would be greatly hindered by the > fact that water is such a good solvent. It would be nearly impossible > to isolate chemicals, concentrate them, and make them pure. Most of > our early chemistry was done in water but if you open a beaker of the > water soluble chemical underwater your chemical just floats away. It > is also nearly impossible to perform important chemical procedures > such as distillation and crystallization while you are in an aquatic > environment. It is possible that some isolation of chemicals could be > carried out by manipulating them in lipid phases. In other words we > could have worked with our chemicals by dissolving them in globs of > fat that we could manipulate underwater. This of course would have > the downside of being useless for the chemistry of ionic compounds. Engineering and physics would suffer also. The high friction > environment under water would hinder all sorts of activities. Even > flint napping, one of our ancestors' first tool making activities, > would be difficult under water. Our understanding of buoyancy and > fluid dynamics would probably have come earlier and be much better, > but kinematics generally would suffer. Our basic concepts of mass, > acceleration and inertia would be hindered by the effects of buoyancy > and viscosity. Soft early technologies such as weaving would be ok, > provide the aquatic environment provided strong enough materials and > the physics of static equilibria would be largely unaffected because > the viscosity of water would not be a factor; still one could argue > that buoyancy and the motive power of water currents would make > controlled experiments in statics more difficult. Water, presumably saltwater, is a relatively good conductor of both > heat and electricity. This would have posed serious problems in the > development of both electrodynamics and thermodynamics. Electro > dynamics, might suffer less since many aquatic organisms seem to have > evolved electrical senses naturally. If an intelligent species could > perceive electrical currents with their senses directly it is likely > that the behavior of electricity would be relatively easy for them to > grasp intuitively. Substances like gold would have interesting > effects on the electric fields they perceived. Natural rocks > containing magnetite would still work under water and have interesting > properties allowing them to learn about magnetism. Certain living > organisms would be likely to evolve electrical attacks and defenses > and could provide a ready if difficult to control source of electrical > power. Nevertheless, electricity would be difficult to control, > because any exposed surface would provide a route for current to flow > through the surrounding water, and insulators would be hard to come > by. Thermodynamics would be even more difficult because heat would > leak away into the water quickly and as mentioned before there are no > easy to control sources of heat available under water. Experiments > involving the ideal gas laws would be very difficult to conduct not > only because temperature would be difficult to control and manipulate, > but pressure underwater changes dramatically depending upon your > vertical location. Vacuums are even harder to generate at the > pressures found under water. I'm unsure what effect aquatic life would have on optics and > acoustics. Light filtering through the water would certainly presents > some interest in learning opportunities as would the interesting > optics associated with bubbles and similar gas and liquid interfaces. > This is assuming of course that the visual systems of intelligent life > were well developed (questionable in an aquatic environment) and those > of the filtered rippling light that comes from the surface. The poor > state of material science would make tools such as telescopes and > microscopes impossible, and the natural turbidity of open water would > make telescopes nearly useless. If telescopes were invented, > presumably they could be used with some great difficulty at the > ocean's surface to develop the science of astronomy, but one has to > wonder if the culture that never lived under the open skies and saw > the wonders of the constellations, the solar, lunar, and planetary > cycles would even be interested or see the relevance of what was above > the waves. Acoustics on the other hand might be a very lively > theoretical science especially if the aquatic intelligence had natural > sonar. Nevertheless, the technology of acoustics would probably come > to an early standstill because of the lack of materials to build tools > and the lack of electronic devices to generate record and analyze > sound. Acoustical science might also be hampered by the fact that the > underwater environment tends to be quite noisy. Wave mechanics might well benefit from the natural oscillations of > water that would be apparent. Nevertheless, this natural way of > motion would be very difficult to isolate and control, therefore in > experimental science built around wave mechanics would not develop > well. Further, mechanical oscillators such as springs and pendulums > do not work well under water because of viscosity. Biology would be hampered too. Antiseptic procedures would be nearly > more easily than air, but it is more difficult to filter. There are > probably some naturally occurring antibiotics and antiseptics but like > most water soluble chemicals they would be difficult to purify and > concentrate. Herbal medicine would work too though ingredients would > be harder to preserve, purify, standardize and dose. Biology is highly > dependent upon chemistry and chemical procedures and would be hobbled > by the difficulties in chemistry described above. Saltwater itself is > toxic when introduced into sensitive structures like the brain and > other organs. Bacterial and viral cultures would be difficult to > isolate and grow if they were not species that were already adapted to > the ambient conditions of the aquatic environment. That is, gut > bacteria, would not grow well once taken from the enclosed environment > of the gut because the exposure to open water would interfere with > their normal chemistry. Generally, I see little problem with the > descriptive aspects of biology; disciplines such as taxonomy should > not suffer and anatomy in its primitive form would be fine until more > sophisticated tools such as scalpels and microscopes became necessary. > If the intelligences had sonar or electrical senses this would great > aid their understanding of anatomy and physiology. I see no problems for math except to the extent that it depends on > materials and tools that might be hard to make and preserve in an > aquatic environment? No calculators and would there be a good > substitute for pencil and paper? I suppose also that the poor > development of sciences like astronomy and kinematics would make math > seem less attractive. A natural ability to see into objects through > the use of sonar might have interesting effects on geometry. My overall conclusion is this. All the major branches of science > would suffer if they had to develop in an aquatic environment. > Chemistry in particular would be crippled. Since chemistry is > foundational to materials science and since materials are essential > for the development of the other science is all technological > development would be held back. I see the development of the > technology advanced enough for space flight by an aquatic civilization > as extremely unlikely, verging on impossible, particularly on a world > with no land masses. One unrelated question. Why isn't there a groups called sci.bio and > what is the main general biology discussion group? here's another one; would warm-blooded animals have evolved at all in an underwater environment, or would the heat loss of immersion proved too much to overcome all at once, without an intermediate spell on land, as in the evolution of dolphins and whales? and if warmblooded animals didn't evolve, would big brains evolve, or are they too much of an energy sink for a coldblooded animal? === Subject: Re: Could high technology develop in an aquatic species? posting-account=7DfHzwkAAABuiYa0I5v96WrKfef1HlA7 Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) Every once in awhile you see a science fiction story in which the > aliens were supposed to have evolved in an aquatic environment. > Nevertheless, they have high technology comparable to ours. Further, > one of the many factors that the drake equation (an attempt at > mathematically estimating the number of technological civilizations in > our galaxy) http://en.wikipedia.org/wiki/Drake equation ignores is the fact that if a world has liquid water it may very well > be entirely covered with water. There is some speculation among > geophysicists that had the earth's Moon not formed, the earth itself > would likely be a Waterworld. My question is, how much technology would have developed if an > intelligent species had had to try to develop its technology while > submerged in water? Could in intelligence species of creature with > manipulating appendages have ever gotten from the kind of resources > you see lying around on the ocean floor to the kind of technology > necessary to build a spacecraft? Fire under water is certainly a no go until the development of fairly > exotic high Energy Materials such as magnesium. How you can get such > materials without having fire in the first place is unclear to me. > Fire or at least intense heat is obviously necessary to the production > of many chemicals. It is possible that an intelligent species could > use natural sources of heat such as volcanic vents, but the danger > associated with geologic heat and its relative inconvenience of > location would have been a great hindrance. > Metallurgy and glasswork both highly important to the construction of > the tools of chemistry and engineering would be extraordinarily > hampered. Gold would be about the only metal available at low > technology levels without fire. Gold is too soft for most > applications. Additionally, chemistry in general would be greatly hindered by the > fact that water is such a good solvent. It would be nearly impossible > to isolate chemicals, concentrate them, and make them pure. Most of > our early chemistry was done in water but if you open a beaker of the > water soluble chemical underwater your chemical just floats away. It > is also nearly impossible to perform important chemical procedures > such as distillation and crystallization while you are in an aquatic > environment. It is possible that some isolation of chemicals could be > carried out by manipulating them in lipid phases. In other words we > could have worked with our chemicals by dissolving them in globs of > fat that we could manipulate underwater. This of course would have > the downside of being useless for the chemistry of ionic compounds. Engineering and physics would suffer also. The high friction > environment under water would hinder all sorts of activities. Even > flint napping, one of our ancestors' first tool making activities, > would be difficult under water. Our understanding of buoyancy and > fluid dynamics would probably have come earlier and be much better, > but kinematics generally would suffer. Our basic concepts of mass, > acceleration and inertia would be hindered by the effects of buoyancy > and viscosity. Soft early technologies such as weaving would be ok, > provide the aquatic environment provided strong enough materials and > the physics of static equilibria would be largely unaffected because > the viscosity of water would not be a factor; still one could argue > that buoyancy and the motive power of water currents would make > controlled experiments in statics more difficult. Water, presumably saltwater, is a relatively good conductor of both > heat and electricity. This would have posed serious problems in the > development of both electrodynamics and thermodynamics. Electro > dynamics, might suffer less since many aquatic organisms seem to have > evolved electrical senses naturally. If an intelligent species could > perceive electrical currents with their senses directly it is likely > that the behavior of electricity would be relatively easy for them to > grasp intuitively. Substances like gold would have interesting > effects on the electric fields they perceived. Natural rocks > containing magnetite would still work under water and have interesting > properties allowing them to learn about magnetism. Certain living > organisms would be likely to evolve electrical attacks and defenses > and could provide a ready if difficult to control source of electrical > power. Nevertheless, electricity would be difficult to control, > because any exposed surface would provide a route for current to flow > through the surrounding water, and insulators would be hard to come > by. Thermodynamics would be even more difficult because heat would > leak away into the water quickly and as mentioned before there are no > easy to control sources of heat available under water. Experiments > involving the ideal gas laws would be very difficult to conduct not > only because temperature would be difficult to control and manipulate, > but pressure underwater changes dramatically depending upon your > vertical location. Vacuums are even harder to generate at the > pressures found under water. I'm unsure what effect aquatic life would have on optics and > acoustics. Light filtering through the water would certainly presents > some interest in learning opportunities as would the interesting > optics associated with bubbles and similar gas and liquid interfaces. > This is assuming of course that the visual systems of intelligent life > were well developed (questionable in an aquatic environment) and those > of the filtered rippling light that comes from the surface. The poor > state of material science would make tools such as telescopes and > microscopes impossible, and the natural turbidity of open water would > make telescopes nearly useless. If telescopes were invented, > presumably they could be used with some great difficulty at the > ocean's surface to develop the science of astronomy, but one has to > wonder if the culture that never lived under the open skies and saw > the wonders of the constellations, the solar, lunar, and planetary > cycles would even be interested or see the relevance of what was above > the waves. Acoustics on the other hand might be a very lively > theoretical science especially if the aquatic intelligence had natural > sonar. Nevertheless, the technology of acoustics would probably come > to an early standstill because of the lack of materials to build tools > and the lack of electronic devices to generate record and analyze > sound. Acoustical science might also be hampered by the fact that the > underwater environment tends to be quite noisy. Wave mechanics might well benefit from the natural oscillations of > water that would be apparent. Nevertheless, this natural way of > motion would be very difficult to isolate and control, therefore in > experimental science built around wave mechanics would not develop > well. Further, mechanical oscillators such as springs and pendulums > do not work well under water because of viscosity. Biology would be hampered too. Antiseptic procedures would be nearly > more easily than air, but it is more difficult to filter. There are > probably some naturally occurring antibiotics and antiseptics but like > most water soluble chemicals they would be difficult to purify and > concentrate. Herbal medicine would work too though ingredients would > be harder to preserve, purify, standardize and dose. Biology is highly > dependent upon chemistry and chemical procedures and would be hobbled > by the difficulties in chemistry described above. Saltwater itself is > toxic when introduced into sensitive structures like the brain and > other organs. Bacterial and viral cultures would be difficult to > isolate and grow if they were not species that were already adapted to > the ambient conditions of the aquatic environment. That is, gut > bacteria, would not grow well once taken from the enclosed environment > of the gut because the exposure to open water would interfere with > their normal chemistry. Generally, I see little problem with the > descriptive aspects of biology; disciplines such as taxonomy should > not suffer and anatomy in its primitive form would be fine until more > sophisticated tools such as scalpels and microscopes became necessary. > If the intelligences had sonar or electrical senses this would great > aid their understanding of anatomy and physiology. I see no problems for math except to the extent that it depends on > materials and tools that might be hard to make and preserve in an > aquatic environment? No calculators and would there be a good > substitute for pencil and paper? I suppose also that the poor > development of sciences like astronomy and kinematics would make math > seem less attractive. A natural ability to see into objects through > the use of sonar might have interesting effects on geometry. My overall conclusion is this. All the major branches of science > would suffer if they had to develop in an aquatic environment. > Chemistry in particular would be crippled. Since chemistry is > foundational to materials science and since materials are essential > for the development of the other science is all technological > development would be held back. I see the development of the > technology advanced enough for space flight by an aquatic civilization > as extremely unlikely, verging on impossible, particularly on a world > with no land masses. One unrelated question. Why isn't there a groups called sci.bio and > what is the main general biology discussion group? here's another one; would warm-blooded animals have evolved at all > in an underwater environment, or would the heat loss of immersion > proved too much to overcome all at once, without an intermediate spell > on land, as in the evolution of dolphins and whales? > and if warmblooded animals didn't evolve, would big brains evolve, or > are they too much of an energy sink for a coldblooded animal? That is another concern. Warmbloodness allows for more efficient energy metabolism (good for brains) and also more rapid evolution since biochemistry does not have to select for rarer enzymes that are stable at a range pf temperatures, just ones that are stable at one temperature. Nevertheless, limited warmbloodness has evolved in fish for example Great Whites warm their digestive tract after a big meal. Presumable to aid in digestion and because after a big meal they can afford the extra energy. === Subject: Re: Could high technology develop in an aquatic species? posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse Trident/4.0; MathPlayer 2.10d; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > I see the development of the > technology advanced enough for space flight by an aquatic civilization > as extremely unlikely, verging on impossible, particularly on a world > with no land masses. You are equating the level of development of a civilisation to its technological level, but this is properly a limit of our dominant culture's point of view (untimately, the quest for the biggest weapon). A civilisation not so biased to externalised technology might, for instance, have developed psychic powers rather than denying and banning their existence. Or they could be simply living a balanced and sane life, in peace and harmony, rather than systematically exploiting each other: doesn't this sound already like an *advanced* level of civilisation compared to our barbarism? -LV === Subject: Re: Could high technology develop in an aquatic species? posting-account=7DfHzwkAAABuiYa0I5v96WrKfef1HlA7 Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) I see the development of the > technology advanced enough for space flight by an aquatic civilization > as extremely unlikely, verging on impossible, particularly on a world > with no land masses. You are equating the level of development of a civilisation to its > technological level, No I am not. I have no problem with the idea that a low tech society can have a very complex and rich civilization though I think that technology does contribute to all aspects of civilization. but this is properly a limit of our dominant > culture's point of view (untimately, the quest for the biggest > weapon). A civilisation not so biased to externalised technology > might, for instance, have developed psychic powers rather than denying > and banning their existence. Well, if such things exists, yes it is possible they could develop them, but we have been working on them for quite a while and nobody has yet been able to come-up with convincing reproducible evidence. I am not saying they don't exist but I have no reason to believe they exist and I think they are beyond the scope of a reasonable conversation on this topic. Or they could be simply living a > balanced and sane life, in peace and harmony, rather than > systematically exploiting each other: doesn't this sound already like > an *advanced* level of civilisation compared to our barbarism? > Yeah, yeah, they may be the nicest sanest people in the universe, but I am talking about spacefaring aliens, that is the premise of the question. As far as we know dolphins might have high culture and be incredibly sane and wise (except around fishing nets). That still does not make them good candidates for the next terrestrial power to venture into space. I think you missed the point of my post and somehow assumed it was an argument that I am saying technology = good civilization. That is obviously not true. > -LV === Subject: On the Limits of cycles of the Collatz function AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) On the Limits of cycles of the Collatz function paper... Hence, from the Grand Conclusion of solving the Collatz conjecture, I come to a very modest conclusion... (Damn!!! if it only were the right proof...;) ) Roupam Ghosh roupam ghosh@vsnl.net Abstract In this paper we will show that if for the Collatz function T(x), If T^n(x) ends in a cycle containing only one odd integer then, the possible cycles are (-1,-2) (1,4,2) Statement of the problem: A problem posed by L.Collatz in 1937, also called the 3x+1 mapping, 3x +1 problem, Hasse's algorithm, Kakutani's problem, Syracuse algorithm, Syracuse problem, Thwaites conjecture, and Ulam's problem. Let x be an natural number. Then the Collatz conjecture states, given that T(x) = 3x+1 when x is odd T(x) = x/2 when x is even if we start iterating from any natural number x, T(T(..T(x)...)), then the value will always reach 1 in a finite number of steps. We reduce the problem to G(x) = (3x + 1)/2^k where k is the highest power of 2 that divides 3x+1 for only odd integers x Then the Collatz Problem becomes: for all positive odd integers x element of Z there exists d element of N such that G^d(x) = 1 where G^d(x) = G(G(G(....{d times}...(G(x))... ))) Here's an outline for the steps of the proof: -> We define the limit lim G^n(x) as n -> infinity -> We prove that if the limit lim G^n(x) as n -> infinity exists for starting values of x = x0, x1, ... xn and if those limits are p0, p1, ... pn Then p0, p1, ... pn are the solutions of the equation G(x) = x -> We find all possible solutions of G(x) = x, and hence show that all non-cyclic trajectories of G end into -1 or -1 . Definition of limit: Consider the function f : Z -> Z where Z denotes the set of integers Then we define the limit lim f^n(x0) = L as n -> infinity for every d > 0, d element of N there exists S element Z such that | f^n(x0) - L | < d whenever n > S x0, n element of Z Note: This is a limit taken on the discontinous function f : Z -> Z Hence, do not confuse this limit with that of limits in calculus Lemma 1: If the limit lim G^n(x) as n -> infinity exists for starting values of x = x0, x1, ... xn and if those limits are p0, p1, ... pn Then p0, p1, ... pn are the solutions of the equation G(x) = x Proof: Given a function, G : Z -> Z Consider lim G^n(x0) = p0 as n -> infinity Then, by our definition of limit, there exists S such that whenever n > S, then | G^n(x0) - p0 | < d for each d > 0 Consider d = 1 Then there exists S such that whenever n > S | G^n(x0) - p0 | < 1 But | G^n(x0) - p0 | is a non-negative integer Hence, | G^n(x0) - p0 | = 0 Hence, G^n(x0) = p0, whenever n > S Consider, m > S, and m+1 > S, m element of Z Then we have G^m(x0) = p0 G^(m+1)(x0) = p0 But G^(m+1)(x0) = G(G^m(x0)) Hence, p0 = G(p0) Hence proved. Note: This is similar to fixed point iteration Solutions of G(x) = x Now we try to find all possible solutions of G(x) = x Consider the equation, G(x) = x Incase, of the Collatz problem G(x) = (3x + 1)/2^k we have, the equation as, x = (3x + 1)/2^k where k is the highest power of 2 that divides 3x+1 Simplifying we get, x = 1/(2^k - 3) Keeping in mind that x is an odd integer and k is a positive integer Then we get, possible solutions as k=1 x = -1 k=2 x = 1 Hence, G(x) = x, has possible integer solutions as -1, 1 Note: Also no integer value of k exists for which x = infinity Can we say from this that divergent trajectories do not exist??? Hence for all odd integers x if G^n(x) ends in a limit with increasing n, the possible values of the limits are -1, 1 But since G(x) is a reduced form of the Collatz function, we can say, if for the Collatz function T(x), T^n(x) ends in a cycle containing only one odd integer then, the possible cycles are (-1,-2) (1,4,2) Q.E.D === Subject: Re: On the Limits of cycles of the Collatz function posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Definition of limit: > ? ? ? ? Consider the function f : Z -> Z > ? ? ? ? where Z denotes the set of integers > ? ? ? ? ? ? ? ? Then we define the limit > ? ? ? ? ? ? ? ? ? ? ? ? lim f^n(x0) = L as n -> infinity > ? ? ? ? ? ? ? ? for every d > 0, d element of N ?there exists S element Z such > that > ? ? ? ? ? ? ? ? | f^n(x0) - L | < d whenever n > S > ? ? ? ? ? ? ? ? x0, n element of Z > ? ? ? ? Note: This is a limit taken on the discontinous function f : Z -> Z > ? ? ? ? ? ? ? ? Hence, do not confuse this limit with that of limits in calculus Hence lim f^n(x0) = L implies especially that there exists S in Z such that f^n(x0) = L whenerver n > S. This is just convergence in the usual sense, why not confuse it with limits of calculus when it's just th same? === Subject: Re: On the Limits of cycles of the Collatz function CLR 2.0.50727; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) Definition of limit: > ? ? ? ? Consider the function f : Z -> Z > ? ? ? ? where Z denotes the set of integers > ? ? ? ? ? ? ? ? Then we define the limit > ? ? ? ? ? ? ? ? ? ? ? ? lim f^n(x0) = L as n -> infinity > ? ? ? ? ? ? ? ? for every d > 0, d element of N ?there exists S element Z such > that > ? ? ? ? ? ? ? ? | f^n(x0) - L | < d whenever n > S > ? ? ? ? ? ? ? ? x0, n element of Z > ? ? ? ? Note: This is a limit taken on the discontinous function f : Z -> Z > ? ? ? ? ? ? ? ? Hence, do not confuse this limit with that of limits in calculus Hence ?lim f^n(x0) = L implies especially > that there exists S in Z such that > f^n(x0) = L ?whenerver n > S. This is just convergence in the usual sense, why not > confuse it with limits of calculus when it's just th same? Because the limits on Calculus are on continuous functions Here the limit is on a function that moves in discrete steps. === Subject: Re: On the Limits of cycles of the Collatz function posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Definition of limit: > Consider the function f : Z -> Z > where Z denotes the set of integers > Then we define the limit > lim f^n(x0) = L as n -> infinity > for every d > 0, d element of N there exists S element Z such > that > | f^n(x0) - L | < d whenever n > S > x0, n element of Z > Note: This is a limit taken on the discontinous function f : Z -> Z > Hence, do not confuse this limit with that of limits in calculus Hence lim f^n(x0) = L implies especially > that there exists S in Z such that > f^n(x0) = L whenerver n > S. This is just convergence in the usual sense, why not > confuse it with limits of calculus when it's just th same? Because the limits on Calculus are on continuous functions > Here the limit is on a function that moves in discrete steps. It's still the same definition of convergence of sequences in a metric (here: discrete) space -- which of course boils down to the concept of eventually constant sequences. === Subject: Re: On the Limits of cycles of the Collatz function AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) Definition of limit: > Consider the function f : Z -> Z > where Z denotes the set of integers > Then we define the limit > lim f^n(x0) = L as n -> infinity > for every d > 0, d element of N there exists S element Z such > that > | f^n(x0) - L | < d whenever n > S > x0, n element of Z > Note: This is a limit taken on the discontinous function f : Z -> Z > Hence, do not confuse this limit with that of limits in calculus Hence lim f^n(x0) = L implies especially > that there exists S in Z such that > f^n(x0) = L whenerver n > S. This is just convergence in the usual sense, why not > confuse it with limits of calculus when it's just th same? Because the limits on Calculus are on continuous functions > Here the limit is on a function that moves in discrete steps. It's still the same definition of convergence of > sequences in a metric (here: discrete) space -- which of > course boils down to the concept of eventually constant sequences. I'm confused??? Have you spotted any errors in the proof??? === Subject: Re: P=NP Proof Published at CERN > Who is JSH?--Martin Musatov Oh please don't tell me Harris has moved from being a crank who thinks he has a simple proof of FLT to a crank who thinks he can prove P vs NP? === Subject: Re: K-12 Calculator Woes I have no disdain for applied math; in fact, I do not > believe it exists. I have always seen it this way. The more abstract is the > mathematics that I know, the easiier it is to apply, and > the more powerful the result. For instance when studying point set topology an exercise > presented itself: For 0 < a,b < 1 the intervals ]a, 1] and [0, b[ are a > subbase for the the usual topology on [0, 1]. Knowing this made writing cascaded iterations over ranges > of integers in computer programs simplest and clearest. The insight can be approximated with the Rule. Write the lower bound (b) and upper bound (B) of the > iteration such that B - b is the number of times to iterate > the loop. > for(index = 0; index < N; index++) > ... The above may seem no great insight; yet when writing > cascaded loops where an inner loop depends on the value > of the index of an enclosing loop, complexities disappear. > All is clear. Compare with Fortran programs where the loops are of the form for I = 1, N ... It would help your point if you made it fairly. If I wanted to write a Fortran program to loop from zero (inclusive) > to N (exclusive) I would use for I = 0, N - 1 How many times does that loop execute? It executes N times, but (N - 1) - 0 = N - 1. > If I wanted to compare a C coding convention with an alternate C > coding convention I would write both in C, e.g. for index = 0; index < N; index++ > versus > for index = 0; index <= N-1; index++ The latter iteration does not execute (N-1) - 0 times. Writing iterations so: for index = a; index < b; index++ or for index = a; index >= b; index-- makes coding up cascaded loops and iterating through zero base arrays much clearer. Take some Fortran codes that do linear algebra, eigenvalues and eigenvectors for instance, and translate to C using the protocol that I recommend. Then compare it with the Fortran code. > Of course, the main point that you started to make was that a > knowledge of abstract math is useful to you. That point stands or > falls independently of the above criticism. However... A farmer who has lost a section of fence can divide the distance lost > by the fencepost spacing interval and subtract one without worrying > much about whether the approach he has used to arrive at the number of > fenceposts needed leads to confusion. There's a point where you have to stop computing and start working. That point is when the calculation to be undertaken is perfectly clear in my mind. -- Michael Press The purpose of computing is insight, not numbers. -- R.W. Hamming === Subject: Re: K-12 Calculator Woes [Long list of references deleted.] ................ There are other means as well. Those in the sciences, and > in many other fields as well, misuse mathematics because > they have learned procedures which require assumptions to > be met to be valid. > You continue to miss the point. > Yes, there are other methods, like base 16 for carpentry using > caricatures of tools instead of letters and numerals because studs are > 16 inches on center. If one understands the numbers, integer, fractional, real, > and if need be complex (try handling AC circuits without it), > the representation becomes of little importance. Being able > to make limited use of a particular representation does NOT > convey understanding. No matter how well one does base 10 > arithmetic, this conveys no more than having the mechanical > processes in ones memory banks. There is not even understanding > of the processes. > I beg to differ. No, the understanding of the integers is not there > from learning to calculate in any given base. > The understanding may not be deep or wide, but it is there. Definitely NOT, as has been demonstrated by example. > Not true. There is some, maybe not as much as you and I would like to see, but there is some. Enough for a very large percentage of society to function within societal norms. Bear in mind that at least 25% of our kids --- 40% in my state, never even graduate HS. > I'd like to see it deeper and wider, but there are limits, and you seem > to know no bounds. Mathematics tries to expand the understanding; this is > what research is doing. > Who's understanding? I'll bashfully admit that I probably could not understand most of what you do No one else I know could come close, and I know some intelligent, well educated people. > But research is not necessarily what is needed to get it > in the first place. One can teach rigorous analysis > easily in middle school or high school, but as experience > has shown, teaching it after calculus operations are > learned does not do it. It _is_ taught in HS, and depending on your definition of rigorous, also in middle school. But as I have frequently complained about int his group, the first thing you have to do is get he kids into a math class. Again, until next year, in my state a single HS math class (with an advanced middle school class) is sufficient to get an *academic* college prep diploma here. An adjacent rural county had so little demand for trigonometry that is doesn't even offer it in its one HS, let alone calculus. Algebra II is its most advanced math course, and a friend teaches the one class of it per block What do we do, hogtie parents and frog march kids to class? Larry There is even a branch of > mathematics known as differential algebra, which adds > calculus operations to the algebraic ones; this is > what is used in considering integration in finite terms. > Wasting time on improving speed and accuracy in hand > calculations detracts from, not adds to, understanding. > Starting with counting to develop the concepts, and > having students understand addition and multiplication, > as well as induction arguments, adds to it. > But for those out there in the real world other methods do not exist, > either because they are unaware of them, because their technology cannot > handle them, or they just plain flat out refuse to acknowledge their > existence. If they are unaware of them, how can they know if they can make > use of them? If they refuse to acknowledge their existence, can > they be said to be educated about the numbers? > Why bother? > I'm aware that the Mandarin language exists, but I'm perfectly > functional knowing no more than that. Since you know it exists, if you needed to read something > in Mandarin, you could get it done. But this knowing > facts, rather than concepts. I did not understand > topological concepts on probability measures on metric > spaces until I got rid of the metric, using concepts > from general topology. It became VERY easy. > Why us a base 16 measuring system when the base 10 is perceived to be > better, the technology is designed around that system and they already > know it? Is base 10 better? We are used to base 10; the Sumerians, > and then the Babylonians, used base 60, and it is still > present in our notation. When it comes to computers, base > 2 is the obvious one to use, and all those who designed > the base 2 computers used base 10 for most of their hand > calculations. I have used base 8 and base 16 for hand > calculations myself, and I have not memorized the tables. > Neither have I memorized the base 60 tables, and I have > done some with that base. Computer technology is NOT designed about base 10, and will not > be in the future. It is really base 2, but 16 is used as a > convenient version so that expressions will not be too long. > The human representation provided by that technology. > Just like people are unconcerned whether the signal traveling across the > country for their telephone call is analog, digital, internet routed , > or central office routed, all they care about is the calculator's > display, just like all they care about is hearing the words of the > person on the other end of the line. The internal technology may or may not be important, but > the concepts are. The linguistic (including numerical) > concepts of what is being sent are important, and the > meaning of the computer's answer are as well. If I am > going to send non-integer values of numerical constants > from one computer to another, I want to use a power of 2 > for may base. Also, if constants are included in a > formula, why should we use the current base 10, and > introduce conversion errors, rather than base 16, and > have none of them? > Does any other system work better. Absolutely no!. Talk a bout wasting > time and effort! See the above. There are other places where these come in. > So if you are an electrical engineer, learn complex numbers. > If you are a financial analyst, who cares? > I am more concerned that the financial analyst understand the base 10 > math that governs my money. The financial analyst does not need to understand base 10, > and it may even be confusing. He needs to understand the > meaning of the base 10 expressions. They have NOT. The string 6.43 means 6 plus 43/100. If they > do not understand the decimal fractions are fractions, they do > not understand decimal fractions. > I said the rational representation of fractions. I agree that not enough is done with this, and that > computers have a lot to do with the problem. It is > easy to get computers to run in base 2 arithmetic, > including for real numbers, and even the important > fixed-point arithmetic is difficult on them. The > representation of fractions is, for a machine, much > harder than floats or decimals, and is more complicated > to work with. However, the teaching of fractions leaves even more > to be desired. In can be made both rigorous and > more simple. Of course, anything is simpler if one > has algebraic notation, by which I mean that a > variable is a symbolic expression which can stand > for anything. So we have, for h not 0, a/b = ah/bh, and if the idea of fractions is presented with a > modicum of intelligence, it is obvious that a/b + c/b = (a+c)/b. > I don;t know about your school district, but that is part of my > districts middle school curriculum. Do they use algebraic notation? Do they still ask > students to use the least common denominator? > Yep. A required part of every school curriculum I have heard of. Even a > part of the standard ( not algebra i) 8th grade curriculum. Introduced > in the 6th grade. Too late; algebraic notation belongs with beginning > reading, as it is simple linguistics. And fractions > probably should be introduced by the third grade, and > made obvious. > You suggest that 6 yr olds can handle this. I disagree, and most other s > do, too, because most 6 yr old have not reached the mental stage to > understand representation. They are still in a concrete world. I have never suggested that 6 year olds have the understanding > of integers needed to start on fractions. I do not know how > fast one can do the work, nor do I have any objection to them > learning arithmetic operations to base 10 before doing much, > or even starting, with fractions. Children are capable of understanding abstract ideas, NOT as > abstractions. It is this confusion that too many are making; > an abstract idea may have come about as the result of a > process of abstraction, but it is more than that, and easier > to understand if taught directly. > There is considerable disagreement about this. The disagreement comes because it has not been tried. Even the poor original new math worked, when taught by > people who themselves had the understanding needed. > I doubt if anything has been more tested before being > introduced. > The biggest problem is that people start with differing abilities, and > develop at differing rates. Yes, and teach them accordingly. Placing all children > of a given age in the same class is utterly stupid, and > it should be obvious that it does not give good education > to most. The John Dewey philosophy was not concerned > with this aspect. ............... Whether they know exactly how the black box works is not > a problem; they need to know how the numbers work. Teach > concepts, and these problems might well not even arise. > Exactly my point. By using calcs they do not need to knwo how hte > numbers work. They need to know the meaning of what the black box > produces; they need to know what numbers MEAN, not > how to operate with strings of digits. > Why bother to understand how a tv works? Push certain buttons and good > things happen. Or a pc, microwave, elevator, or cell pone. > Most don;t care until they break, then they go to an expert to fix it, > or even worse, throw the broken one away and get a new one. This is not the problem. They need to understand the concepts > involved in viewing a TV. These are only optical. > No, they aren't. They certainly are. The concern in designing a TV > is that the images presented have the desired effect. > This is just like designing a computer; it is what > answers are presented, not how they are produced. > For multiple precision arithmetic, it often pays to > use procedures which are not the ones taught in school. > The answers are the same, but the procedures are > completely different. The fastest known method is > using the fast Fourier transform over a finite ring. > === Subject: Re: K-12 Calculator Woes The representation of fractions is, for a machine, much > harder than floats or decimals, and is more complicated > to work with. > Why do you believe that is true? Fractions, unless their size is limited, will require at > least two words, and their operations are more complex. > Machines rarely do exact arithmetic, as would be required > with fractions to give the answers desired. Avoiding > computer roundoff would be expensive. I am reasonably familiar with computer architecture, and > I can see no way around it. > Except most computers these days do not deal with memory structures as > small as a word. They are optimized at 32 bits, 64 bits, even 128 bits > for pipelining and reduced instruction set computing (risc). (Jeez, I'm > old. I remember programming using nibbles, or half bytes, for integers) > The older 8 bit chips are disappearing, and those left in the market are > relegated to the dollar store calcs. Now extremely cheap 16 bit longword > chips are about the only low end chips available. The calculations are > not a problem. And the smallest memory chips made today can easily store > the code. Unfortunately, the x86 architecture, in wide use in > computers, still has too many short procedures. And > I would like to do operations involving bit streams > for bit efficiency, but these are horribly expensive > on the present computers. In generating non-uniform > random numbers for simulation, one now uses wasteful > procedures to get speed. > Well, it depends on your definition of waste. With terabyte disk drives at $80, 4 gigabytes of memory at $35, 3.73 GHz dual core processors for $200, a little inefficiency is not a big deal. Gone are he days where resources were a problem. I agree about the x86 --- it is an old architecture that should have been replaced years ago. I much preferred the DEC Alpha VAX architecture and VMS or PDP's and RSTS --- far more flexible and secure. But they were also far more expensive than the x86, and so they went the way of the dinosaurs. > A more significant problem is the increased number of buttons to be put > in a small acreage. The desire for a small format directly competes > with the inclusion of more functions. Some calculators now have as many > as 3 or 4 functions for each key, and even so a large and bulky.Learning > to use them is a challenge. > And, of course, the marketing problem is huge. Most people just don't > want to calculate in fractions, and have learned the algorithm for > computing in integers instead, that is you can actually enter 3/4*5/9 > into the calculator to get a preferred decimal answer. That said, cheap > calculators that do integer fraction arithmetic are available --- my > Home Depot has a few models designed for carpenters, for ex. They > actually round to 16ths, as well as calculating things like pitch and > angles. In most situations, computing with exact arithmetic > would soon make the numerators and denominators so > large that it would be hopeless. And also much is > done using other algebraic and transcendental functions, > the results of which cannot be written as fractions. > Yes, but not on calculators. > One can work with rounded binary answers, and this is > the way most computer functions work. One can even > do these in multiple precision, but only with considerable > difficulty on the current computers. > This is one reason why I miss VMS. A 64 bit octaword floating point was a precise as almost anyone wanted to get. Not everyone, but most. But I still question almost all requests for precision. After all, we went to the moon using computers that are outperfomed by a TI90. > The lack of precision is probably not an issue. It is doubtful that much > precision would be desired in such a calculation --- no one wants to > have to use 1352685233/28451120769 and would rather deal with > 0.04754418089123340268648521865195 > But then why do we need that level of precision? Unless you are > calculating the trajectory for a Jupiter shot, 3 or 4 decimals is fine, > and if you are starting with 5 significant digits then anything beyond 5 > digits is imprecise, anyway. There are many situations in which multiple precision is > needed, and the trajectory for a Jupiter shot is not one of > them. You are not going to be able to adjust the course > of the spacecraft that accurately, and will have to recompute > the trajectory and readjust it. But inverting a matrix > usually loses many bits, and there are other operations, > which one should find ways around. > With all due respect to those out there in cyberspace, very, very little needs that level of precision, despite what some may think. Back in the day when I was a VAX system manager I talked with scientists doing nuclear research at Livermore and Fermilab who were able to do Nobel level work with 32 bit floating points. In the late '80's a friend spent a winter in Antarctica (I was so jealous -- he actually got to visit the south pole!) installing a PDP11 which supported the entire research community for nearly a decade using 16 bit FP's. It replaced an old 8086. We sent satellites to Jupiter with 8 bit FPs. And despite your assertion that in flight corrections made accuracy nearly moot, communication, weight and fuel considerations made mid-course corrections expensive and limited. Everyone (including me) wanted more, but the bean counters ruled, and more often than not it was determined that more wasn't needed. Not always, but usually. One little factoid I heard at DECUS symposium brought that home to me. In 1970, when massive mainframes ruled the world and people ran around with stacks of punch cards and desktop access was rare, there were about 750,000 CPA's in the US. As computers became cheaper and more generally available, the accounting field was targeted as a market where labor could be significantly reduced with the new mechanization --- spreadsheets and financial apps proliferated. By 1990 when Windows 3 came out there were 1.5 million CPAs in the US. What happened was someone found mew ways to slice and dice the numbers, more accurate ways of breaking down the numbers, and different ways of displaying he numbers, so more people wee required to perform the new functions rather than fewer people to maintain the status quo. Beneficial? I'd suggest no considering the current economic problems caused by derivatives and other investments based on no more than a computer program. Larry === Subject: Re: -1 x -1 ? If we can use the expression -1=e^{ipi}, we can show -1*-1=e^{ipi}*e^{ipi}=1 readily: if we rotate once 1 (the vector 01) by pi rad around the origin anticlockwise on the complex plane, we obtain -1. Further likewise if we rotate -1 by pi rad once more, we can obtain -1*-1=1. At least I understand -1*-1=1 in this way. In that way we can obtain i (imaginary unit), if we rotate 1 by pi/2 around the origin anticlockwise. If we cannnot allow to use the expression -1=e^[ipi}, I don't understand -1*-1=1 vividly. === Subject: Re: -1 x -1 ? posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > If we can use the expression -1=e^{ipi}, we can show > -1*-1=e^{ipi}*e^{ipi}=1 readily: if we rotate once 1 (the vector 01) by pi rad around the origin anticlockwise on the complex plane, we obtain > -1. Further likewise if we rotate -1 by pi rad once more, we can obtain -1*-1=1. At least I understand -1*-1=1 in this way. In that way we can obtain i (imaginary unit), if we rotate 1 by pi/2 around the origin anticlockwise. If we cannnot allow to use the expression -1=e^[ipi}, I don't understand -1*-1=1 vividly. Well, (-1)*(-1) = 1 holds in all rings, not just in the field C. And I doubt you can prove a lot about the exponential function without making use of (-1)*(-1) = 1 somewhere ... === Subject: Re: JSH: The Simple Lie Guys, Harris posts _one_ message, no follow ups, and gets up > to twenty replies. Including yours. Not mine: not a reply to James Harris. I'm sure the trolls take that into consideration when they > compute the reply tally. That is the troll's problem, not mine. It is their own moribund lives at stake. Imagine trying to live on what others throw away; to be dependent on others for scraps: beggars. -- Michael Press === Subject: Re: JSH: The Simple Lie Guys, Harris posts _one_ message, no follow ups, and gets up > to twenty replies. Including yours. Not mine: not a reply to James Harris. Yes, it was. > It is not a reply to James Harris, just as this is not. -- Michael Press === Subject: Re: Mental Event & Mental Object: Is there a Difference? posting-account=g9P5YAkAAACbxBDAyQp7cMVe0-Yk0MYD SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) On May 11, 4:57pm, Patricia Aldoraz event. Makes no difference what objects or events we are talking > about. Hint; Ewe clueless useless name calling arrogant bitch, the nature of an entity is included in its identity. > One question about mental events is what is it that has them. People > have them, of course. Your mission, should ewe ever choose to awaken from your Kantian nightmare, is to identify what it is that your mental events are about, BIG hint; if they are not about or if they are not triggered by matter and matter's nature, then they do not matter a gnat's toss. Ewe got that? Hint; An example of a matterless mental event is god and all that other mystical religious mumbojumbo and an another example is that nauseating nasal whining chant of ewe leftist retards, the greater good > Clear the way? Yes, Yes, off and come back when ewe have grown up and learnt at least a little on HOW to think as against the arbitrary meaningless Kantian garbage you regurgitate on here. MG === Subject: Re: Mental Event & Mental Object: Is there a Difference? How very.................. Have ewe off, moron. -- hz === Subject: Re: Mental Event & Mental Object: Is there a Difference? <4A08F5A5.DD27FCF2@gmail.com> posting-account=g9P5YAkAAACbxBDAyQp7cMVe0-Yk0MYD SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) How very.................. Have ewe off, moron. In your dreams retard. MG === Subject: Re: Why does hydrogen congregate at the right amount to form stars? > Masses of hydrogen form into stars under its own gravity. Why isn't > there on occasion too much or too little hydrogen? Why aren't there > as many black holes as stars? Once enough mass has accumulated (heating as it does so) to ignite fusion, radiation pressure and solar wind should tend to drive away much of the remaining gas. It would take rare conditions for a much larger star to form. > Why aren't there trillions of big globules of hydrogen too small to > start fusion? They're called gas giants, and there are a few right in our own solar system. Most other stars likely have some too, and most of the extrasolar planets detected are almost certainly gas giants. As well as those, it wouldn't surprise me if there were more out in the very dark and cold interstellar space where we can't see them. > Final teaser question. If our sun had one extra molecule of > hydrogen would our history be any different? I expect so, probably we wouldn't have developed to have had a history. Then again, I think the same would be true even if the Sun had exactly the same number of hydrogen atoms and the universe was just re-run. - Tim === Subject: Re: Why does hydrogen congregate at the right amount to form stars? > Masses of hydrogen form into stars under its own gravity. > Why isn't there on occasion too much or too little hydrogen? Happens all the time. > Why aren't there as many black holes as stars? Because having that much mass accrete in one place is fairly rare. ************************************************* So basically, hydrogen collapsing under its own gravity just happened to occur at perfectly the right size for star formation and nothing more!! What an absolute piece of luck ?? And you guys don't believe in God. All luck! Herc === Subject: Re: Why does hydrogen congregate at the right amount to form stars? Masses of hydrogen form into stars under its own gravity. > Why isn't there on occasion too much or too little hydrogen? Happens all the time. Why aren't there as many black holes as stars? Because having that much mass accrete in one place is fairly rare. ************************************************* So basically, hydrogen collapsing under its own gravity just happened > to occur at perfectly the right size for star formation and nothing more!! Don't jump to conclusions you can't support. Most of the stuff was far too small for star formation. And those clumps exist today, but they are not luminous. Therefore they are harder to spot. Every now and again, there happens to be quite of bit of mass available and this will eventually collapse into a black hole. But before it does, it will ignite and burn for a while as a star. While it does, the pressure of the ignition will hold off the gravitational collapse for a while. Does this help? ********************************************** OK, but it does seem rather *efficient* that there's so many stars and not many black holes. If I was the intelligent designer that's exactly what I'd be shooting for! What percentage of all hydrogen is contained in stars? Herc === Subject: Re: Why does hydrogen congregate at the right amount to form stars? > Masses of hydrogen form into stars under its own gravity. > Why isn't there on occasion too much or too little hydrogen? Happens all the time. Why aren't there as many black holes as stars? Because having that much mass accrete in one place is fairly rare. ************************************************* So basically, hydrogen collapsing under its own gravity just happened > to occur at perfectly the right size for star formation and nothing more!! Don't jump to conclusions you can't support. Most of the stuff was far too small for star formation. And those > clumps exist today, but they are not luminous. Therefore they are > harder to spot. Every now and again, there happens to be quite of bit > of mass available and this will eventually collapse into a black hole. > But before it does, it will ignite and burn for a while as a star. > While it does, the pressure of the ignition will hold off the > gravitational collapse for a while. Does this help? ********************************************** OK, but it does seem rather *efficient* that there's so many stars > and not many black holes. Keep in mind what I told you before. You do not have an infinite range of mass densities to choose from, you do not have an infinite spectrum of masses to choose from. You have a budget -- you have a certain mass density in the universe and you have a certain time available to let that mass accumulate. There is a top end to that spectrum of masses and beyond that is *physically impossible* to accumulate with those constraints. And in the range of what is physically possible, there is a distribution with a peak toward the low range of masses and a very thin tail up to high masses. > If I was the intelligent designer that's > exactly what I'd be shooting for! What percentage of all hydrogen is contained in stars? all your questions and fuel many more: http://www.amazon.com/Quintessence-Search-Missing-Mass-Universe/dp/B001NPE9O Y ************************************************* I've got 4 other books from Amazon to get through 1st but I'll check it out. Herc === Subject: Re: Number of Triangles from given set of rods posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/2009042523 Ubuntu/9.04 (jaunty) Firefox/3.0.10,gzip(gfe),gzip(gfe) we have set of n sticks of lengths {1,2,...n}, how many triangles > could one assemble, allowing concatenation (but where order of sticks in > a concatenation does not matter). > Do you wish to equate forms like (4,5,6) == (1+3,5,6) == (4,2+3,6), > or count those as three distinct triangles? -- > jiw Yes you are right (4,5,6) == (1+3,5,6) == (4,2+3,6) so it will get > counted only once. Lt n be big enough. > It is possible to select a subset of {1,2,...,n} with *any* sum > between > n*(n+1)/6 and n*(n+1)/2 for the longest side... For a given n, I believe we can form all triangles (a,b,c) with > a<=b<=c, a+b>c, and a+b+c<=n(n+1)/2 with the exception of > 112, 122, 222, 133, 223, 233, 333, 144, 344, 444. This was done > brute force program to find the number of triangles for a given > n. Starting with n=4 (there are none below this), I find > 3, 24, 74, 175, 368, 710, 1275, 2168, 3539, 5574, 8495, 12590, > which is neither polynomial or in Sloane (even if you add back > in the missing triangles)- Hide quoted text - - Show quoted text - Well so at least I was right for first three 3, 24, 74 beyond which I > think I messed up somewhere:) > Can you share your program? Here it is. It seems more natural to count the triangles with > perimeter < n, instead of perimeter < n(n+1)/2. If you count > with < n, the mods are pretty obvious and it is found in Sloane > at A001400. Note that in my previous post of triangles to delete > I missed 111. def prog(n,plev): > #calculates number of triangles formed from rods of size > (1,2,3,...n) > #rods can be added, so we just need (a,b,c) with a<=b<=c, > #a+b+c<=n(n+1)/2, a+b>c, and not 111, 112, 122, 222, 133, 223, > 333, 144, 344, 444 > #this program does not subtract the ones above > rodtot=n*(n+1)/2 > clim=(rodtot-1)/2 > if plev>4: print 'n, clim',n,clim > allcount=0 > for c in range(1,clim+1): > count=0 > for a in range(1,c+1): > bmin=max(a,c-a+1) > bmax=min(c,rodtot-a-c) > if bmax>=bmin: count+=bmax-bmin+1 > if plev>4: print 'c,a,bmax,bmin,count',c,a,bmax,bmin,count > allcount+=count > if plev>4: print 'c,new total grand total',c,count,allcount > return allcount- Hide quoted text - - Show quoted text - I've looked into this once more and found that a few more cases are > unsolveable. > Given k and integers a,b,c (not necessarily making a triangle), can me > find disjoint > subsets A,B,C of {1,2,...,k} such that a=sum A, b=sum B, c=sum C? > Well, we note that D := {1,2...,k}(A U B U C) is also a set and we > can let d=sum D. > So the problem really is: > For which tuples (k; a,b,c,d) with a+b+c+d = k(+1)/2 does there exist > a partition > of {1,2,...,k} into four sets A,B,C,D such that a=sum A etc.? THEOREM: Problem (k; a, b, c, d) is solveable unless it is > (up to permutation of a, b, c, d) one of > (5; 6, 6, 2, 1), > (6; 8, 8, 3, 2), (6; 8, 7, 3, 3), > (7; 10, 10, 4, 4), (7; 10, 8, 8, 2) or (7; 14, 8, 3, 3) > or matches (up to permutation of a, b, c, d) one of the patterns > (* ; 1, 1, *, *), (* ; 2, 2, *, *), (* ; 3, 3, 1, *), (* ; 3, 3, 2, > *), > (* ; 3, 3, 3, *), (* ; 4, 4, 1, *), (* ; 4, 4, 3, *) or (* ; 4, 4, 4, > *). The proof is more technical than ingenious > (readable here:http://www.von-eitzen.de/math/trianglesticks.pdf). > But it shows that triangle a=6, b=c=4 cannot be built if k=5 (although > 6+4+4 < 5*6/2), > nor can a=b=10, c=4 be built if k=7. > Thus there *are* a few more than the small obvious exceptions ... Taking my extended exception list into account it seems that the number of triangles grows as follows (of course these figures are just slightly smaller than rmill's): 1 0 2 0 3 0 4 3 5 20 6 70 7 172 8 366 9 709 10 1274 11 2166 12 3537 13 5573 14 8494 15 12588 16 18227 17 25846 18 35942 19 49124 20 66138 21 87827 22 115132 23 149166 24 191238 25 242800 26 305447 27 381012 28 471602 29 579518 This was calculated using #include int N[15]; void UP(int x) { if (x<15) ++N[x]; } void DN(int x) { if (x<15) --N[x]; } int main() { for (int i=0; i<15; ++i) N[i]=0; for (int k=1; k<30; ++k) { int ct = 0; int abcd = (k*(k+1))/2; for (int a=1; a+a < abcd; ++a) { UP(a); for (int b=a; b+b>=a; --b) { UP(b); for (int c=a-b+1; c<=b; ++c) { int d = abcd-a-b-c; if (d<0) break; UP(c); UP(d); for (int test=0; test<1; + +test) { if (N[1]>=2) break; if (N[2]>=2) break; if (N[3]>=3) break; if (N[3]>=2 && N[1]+N [2]) break; if (N[4]>=3) break; if (N[4]>=2 && N[1]+N [3]) break; if (N[6]==2 && N[1] && N[2]) break; if (N[8]==2 && N[3] && N[2]) break; if (N[3]==2 && N[7] && N[8]) break; if (N[4]==2 && N[10] ==2) break; if (N[8]==2 && N[10] && N[4]) break; if (N[3]==2 && N[8] && N[14]) break; ++ct; } DN(d); DN(c); } DN(b); } DN(a); } printf (%d %dn, k, ct); } } A slight modification of the above program counts all triangles that can be produced by making use of all rods (i.e. with d=0, the way I originally interpreted the problem -- ): 1 0 2 0 3 0 4 2 5 7 6 12 7 16 8 27 9 48 10 70 11 91 12 127 13 184 14 243 15 300 16 385 17 507 18 631 19 752 20 919 21 1141 22 1365 23 1587 24 1875 25 2241 26 2611 27 2977 28 3434 29 3997 I had thought I had overlooked something when I first posted this sequence, but it seems I did already count all exceptions then ... After all, the triangle condition may appear somewhat artificial, so I counted all solutions (up to permutation) and obtained 1 1 2 2 3 5 4 13 5 35 6 93 7 214 8 437 9 815 10 1436 11 2413 12 3886 13 6041 14 9125 15 13436 16 19323 17 27221 18 37670 19 51293 20 68797 21 91025 22 118982 23 153797 24 196721 25 249206 26 312935 27 389761 28 481709 29 591080 It appears that none of the sequences is known. === Subject: Re: Ooops... An Inertial Motion Detector posting-account=XbpThQoAAACtPo22j_KAb5B7K8_euNJy Gecko/20081217 Firefox/2.0.0.20,gzip(gfe),gzip(gfe) you was enough convincent With apologies to Richard Fiztpatrick, the > substance of the argument is not mine. > It is simply the result of getting a few cobwebs > out of the attic and updating some concepts so > they conform with modern experience. << A book based on these lectures notes, entitled > Maxwell's Equations and the Principles of Electromagnetism, > is now available from Infinity Science Press, Hingham, MA. > (ISBN: 978-1-934015-20-9) >http://farside.ph.utexas.edu/teaching/em/em.html ..and also we cannot continue that > experiment : my dog is completely unreliable ( he took a cheese ' > piece and simulated the starting ..bahh )... i invite you on the other > table '' wave model ..ligth..einsteniana'' is not very good. It is just better than any of > the other languages I have tried to write. There are a few colourful variations used > in this forum that don't seem to correlate > with academic papers but I have a sailor friend > that helps me with the translation. ;-) When I become proficient enough with English > to use it as an effective communication tool, > I might try some of the other languages I see > on your very interesting forum. > Sue... truly , a little problem there is .... when you say : the distance between two bodies is growing 400.000 km.s... somebody can think that a body can fly away 400.000 km.s. from anoter-one ..but we can hope that no professor or einstenian supporter can hear that... === Subject: Re: Ooops... An Inertial Motion Detector posting-account=wigfZgkAAACDgITarXffzxJygX81YRSs Gecko/2009042708 Fedora/3.0.10-1.fc9 Firefox/3.0.10,gzip(gfe),gzip(gfe) > you was enough convincent With apologies to Richard Fiztpatrick, the > substance of the argument is not mine. > It is simply the result of getting a few cobwebs > out of the attic and updating some concepts so > they conform with modern experience. << A book based on these lectures notes, entitled > Maxwell's Equations and the Principles of Electromagnetism, > is now available from Infinity Science Press, Hingham, MA. > (ISBN: 978-1-934015-20-9) >http://farside.ph.utexas.edu/teaching/em/em.html ..and also we cannot continue that > experiment : my dog is completely unreliable ( he took a cheese ' > piece and simulated the starting ..bahh )... i invite you on the other > table '' wave model ..ligth..einsteniana'' is not very good. It is just better than any of > the other languages I have tried to write. There are a few colourful variations used > in this forum that don't seem to correlate > with academic papers but I have a sailor friend > that helps me with the translation. ;-) When I become proficient enough with English > to use it as an effective communication tool, > I might try some of the other languages I see > on your very interesting forum. > Sue... truly , a little problem there is .... when you say : the distance > between two bodies is growing 400.000 km.s... somebody can think that > a body can fly away 400.000 km.s. from anoter-one ..but we can hope > that no professor or einstenian supporter can hear that... I was thinking that might be a little public relations problem when I suggested a heavy ions might substitute for doggy treats. used at RHIC are p + p, d + Au, Cu + Cu and Au + Au. The projectiles typically travel at a speed of 99.995% of the speed of light in vacuum. For Au + Au collision, the center-of-mass energy sqrt{s {NN}} is typically 200 GeV (or 100 GeV per nucleus); > http://en.wikipedia.org/wiki/Relativistic Heavy Ion Collider Sue... === Subject: Re: Alan Schwartz Jewish science in full-bloom posting-account=zPldcAoAAAARb0zYNReBWm9VJxAwWiBG Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > The above posting fails to deal with any of the points raised - your > failure to be able to distinguish between libel and slander, the fact > the site to which you repeatedly refer eulogises Irving ( a true > description of whom I have included ) and that I could not defame > Irving because a) what I posted is true > b) that I could not damage his reputation - it was already at possibly > the lowest point possible. You are manifestly incapable of conducting a debate. As a true seeker of wisdom and truth, > I am disappointed to see that made nasty, open loop comments about > the author David Irving and me. Yet again you are not dealing with the substantive issue. You accuse me of slanedering Irving (wjhilst you mean libelling). You now accuse me of posting garbage and nasty comments about both you and Irving. I posted no comments about Irving: I posted an accurate account of a case before Judge Gray in a British court. Irving himself does not even deny the contents of his own diary nor the little song he sung. Are you accusing me of maliciously telling the truth? > As can be seen with a Google search on David Irving, > Irving, like Jimmy Carter, Mel Gibson and many others, > who have stated truths that Jews don't want the public exposed to, > has been a victim of Jewish systematized bigotry. and compare the them with the garbage posted by I take it you can see nothing bigotted in Irving's little rhyme. > No doubt one can find events to slander and demonize anyone, > even people like Abraham the patriarch of the Jewish lineage. > === Subject: Re: Alan Schwartz Jewish science in full-bloom posting-account=zPldcAoAAAARb0zYNReBWm9VJxAwWiBG Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) There is your bigotry Tom. You and Irving have a lot in common Judge: Why Irving had to lose It took Judge Charles Gray 300 strongly-worded pages to set out why he had found David Irving to be a Holocaust denier, an anti-Semite and a racist. * his dismissal of the eyewitnesses en masse as liars or as suffering from a mental problem * his reference to an Association of Auschwitz Survivors and Other Liars or 'Asshols' * the question he asked of Holocaust survivor Mrs Altman, as to how much money she had made from her concentration camp tattoo. The judge said he also accepted the defendants' case that Mr Irving was anti-Semitic. His words are directed against Jews, either individually or collectively, in the sense that they are by turns hostile, critical, offensive and derisory in their references to Semitic people, their characteristics and appearances, he said. His words are directed against Jews, either individually or collectively, in the sense that they are by turns hostile, critical, offensive and derisory in their references to Semitic people, their characteristics and appearances Mr Justice Gray Examples included Irving's claims that the Jews deserve to be disliked; that they brought the Holocaust on themselves and that they generated anti-Semitism by greed and mendacity. The judge agreed that Jews were as open to criticism as anyone else. But it appears to me that Irving has repeatedly crossed the divide between legitimate criticism and prejudiced vilification of the Jewish race and people, he said. 'Baby Ayran' The judge said he concluded that Irving was also a racist. He said evidence came from sample quotations such as the rhyme Mr Irving composed for his daughter. The court heard it ran I am a Baby Aryan, Not Jewish or Sectarian. I have no plans to marry an ape or Rastafarian. He makes surprising and often unfounded assertions about the Nazi regime which tend to exonerate the Nazis for the appalling atrocities which they inflicted on the Jews Mr Justice Gray Mr Justice Gray said: I accept that Irving is not obsessed with race. He has certainly not condoned or excused racist violence or thuggery. But he has on many occasions spoken in terms which are plainly racist. The judge said he also found that Irving associated regularly with extremist and neo-Nazi organisations and individuals. The conclusion I draw...is that Irving is sympathetic towards, and on occasion promotes the views held by, those individuals and organisations. The content of his speeches and interviews often displays a distinctly pro-Nazi and anti-Jewish bias. The picture of Irving which emerges from the evidence of his extra- curricular activities reveals him to be a right-wing pro-Nazi polemicist Mr Justice Gray He makes surprising and often unfounded assertions about the Nazi regime which tend to exonerate the Nazis for the appalling atrocities which they inflicted on the Jews. He is content to mix with neo-fascists and appears to share many of their racist and anti-Semitic prejudices. The picture of Irving which emerges from the evidence of his extra- curricular activities reveals him to be a right-wing pro-Nazi polemicist. In my view the defendants have established that Irving has a political agenda. It is one which, it is legitimate to infer, disposes him, where he deems it necessary, to manipulate the historical record in order to make it conform with his political beliefs. === Subject: Re: Alan Schwartz Jewish science in full-bloom Deleted your post? Wow, pothead, you are even more insane than before. === Mail-To-News-Contact: abuse@dizum.com A Conspiracy that has been MATHEMATICALLY DEMONSTRATED. prob 99.9999999999%. === posting-account=_UsfzgoAAABaL7VsA2KKRHz79j_7uGGD SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) ******************************************************************* ******************************************************************* CLICK HERE TO ENTER: > http://your-guide-online.com/pages/spiderman-games <<< ******************************************************************* ******************************************************************* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . spiderman 2 on line games free spiderman games online downloads spiderman games play spiderman 3 games online spiderman double wire games play spiderman games free downloadable spiderman games spiderman download games spiderman range games online pc games spiderman 3 spiderman games to play free online spiderman video games spiderman games online free spiderman games online free spiderman 3 games spiderman adult games spiderman playing games spiderman 3 xbox games play free spiderman 1 games online spiderman city attack games play spiderman games free spiderman action games play free online spiderman 3 games crazy monkey games spiderman spiderman daredevil online games pc spiderman games download spiderman 2 games spiderman word games spiderman games on the computer download spiderman games spiderman 2 pc games doc ock spiderman 2 games spiderman 3 online games cheats for spiderman games preschool free online spiderman games free spiderman games for pc spiderman games to play spiderman math games playfree spiderman games play games spiderman spiderman 3 games to play now spiderman 2 games online spiderman games to play on line free spiderman games free spiderman 2 games online spiderman games downloads xbox games spiderman spiderman 3 games to play playstation 2 games spiderman spiderman games spiderman games free downloads online spiderman games games spiderman spiderman 3 trailerand free games online play arcade games spiderman spiderman city raid by addicting games spiderman 2 games spiderman cp games spiderman web games downloadable spiderman games spiderman party games free spiderman video games spiderman 2 flash games spiderman computer games free spiderman 3 games gamecube tv games spiderman spiderman java games spiderman 3 flash games spiderman free online games spiderman and official games spiderman paty games spiderman on line games spiderman computer games mario spiderman games venom online games spiderman ultimate spiderman saved games spiderman and friends games spiderman batman free online games spiderman swinging games free batman and spiderman games spiderman toondisney games spiderman free games onlie top five spiderman games disney spiderman and friends games spiderman on-line computer games spiderman arcade games spiderman free games online free online spiderman games in 3d spiderman 3 the games play free spiderman games free online spiderman 3 games preschool spiderman learning games free spiderman 3 games online free on line spiderman games spiderman video games ultimate spiderman games spiderman 3 free games free spiderman flash games keyboarding games spiderman download free spiderman games free spiderman games to play spiderman 3 games games spiderman 2 spiderman 3 pictures and games spiderman games to play for free playstation 1 games spiderman addicting games spiderman computer spiderman games spiderman games for pc spiderman games entertainment at answers spiderman downloadable games spiderman games to play online spiderman 3 games and cards ultamate spiderman games phones spiderman online games spiderman games free download spiderman internet games play online spiderman games spiderman 3 pc games games cheat spiderman 1 spiderman games free spiderman city raid games spiderman the movie games spiderman and venom games arcade spiderman games spiderman free on line games spiderman free computer games play free spiderman games online spiderman lego games ultimate spiderman save games spiderman game cube games video games spiderman hints spiderman games to download for pc spiderman games on ps2 spiderman games that you can play spiderman birthday party games spiderman tv games green gobion spiderman 1 games online spiderman preschool games free spiderman games on line spiderman and friends online games entertainment spiderman online games spiderman them birthday games spiderman games to play now venom spiderman online games play spiderman games online spiderman games for mac spiderman party games and activities spiderman games to play on computer free computer spiderman games spiderman online free games video games spiderman 2 for xbox spiderman interactive games spiderman 3 video games spiderman pc games spiderman pictures and games spiderman online games free online games spiderman free spiderman online games doc ock spiderman games on line free online spiderman games online multiplayer spiderman games online spiderman adventure games spiderman coloring activities games spiderman games download spiderman 3 trailerand free games is there any spiderman 2 games spiderman 3 internet games full version games of spiderman download spiderman free games downloadable spiderman computer games spiderman 3 games online spiderman 2 download games for pc free spiderman games pc play online free spiderman games online spiderman ragdoll physics games spiderman 2 online games spiderman vs venom games internet spiderman games games spiderman 2 hints free spiderman online games free internet games play spiderman spiderman stickman games spiderman games for playstation 2 spiderman sickman games free spiderman computer games spiderman games with letters games spiderman online games spiderman wolverine's revenge gameboy games play spiderman 3 games on-line spiderman games free spiderman games to download spiderman vidio games spiderman games free online free computer games spiderman free online games preschool spiderman spiderman free on-line games download games for pc and spiderman spiderman 3 games on computer spiderman games by sony online spiderman adventure games spiderman games online for free spiderman games com download free spiderman 3 games ps3 spiderman 3 games freezes spiderman birthday games spiderman the games spiderman video games for sony psp free on-line spiderman games play games like spiderman spiderman online games spiderman nes games spiderman 2 ps2 games spiderman xbox games games spiderman party spiderman flash games spiderman dot games ninja spiderman online games spiderman series video games === Subject: Re: Hitting a spec of debris at .97*c might kill you. posting-account=8k9z_QoAAABfJ6zct3wDB2k3FKfRerU2 2.0.50727),gzip(gfe),gzip(gfe) > NoEinstein Folks: PD is a certifiable LUNY who thinks that if he asks an already- > well-answered question often enough, that he will discredit my science > truths. Smart readers among you must surely understand why I consider > PD to be a persona non grata, unworthy of being given the time of > day! NoEinstein > Well, my crankometer pegs when you post NoE. NoEinstein I do not need to use my credentials to peg you as a crack, NoE. LOL! === Subject: Re: Hitting a spec of debris at .97*c might kill you. NoEinstein Remember, john, that you have demonstrated you have no credentials and that you are willingly ignorant of science. Why do you want to look stupid in public? >Folks: PD is a certifiable LUNY who thinks that if he asks an already- >well-answered question often enough, that he will discredit my science >truths. Smart readers among you must surely understand why I consider >PD to be a persona non grata, unworthy of being given the time of >day! NoEinstein >Well, my crankometer pegs when you post NoE. >NoEinstein >I do not need to use my credentials to peg you as a crack, NoE. LOL! === Subject: Re: Hitting a spec of debris at .97*c might kill you. posting-account=8k9z_QoAAABfJ6zct3wDB2k3FKfRerU2 2.0.50727),gzip(gfe),gzip(gfe) > Would some knowledgeable person address the issue of mass energy and > maybe, just maybe suggest that neither the spec of debris nor the craft > is privileged and therefore the effect would be nil? As objects approach > c relative to each other, they each enter into the dimension of the > creation of space in which there are no privileged masses. > Why thank you, NoEinstein: considering the source, that is a compliment. === Subject: Re: Hitting a spec of debris at .97*c might kill you. posting-account=lpj6eQgAAADfAdNPtqtrw3lMgI8RApJ_ SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > Folks: PD is a certifiable LUNY who thinks that if he asks an already- > well-answered question often enough, that he will discredit my science > truths. Smart readers among you must surely understand why I consider > PD to be a persona non grata, unworthy of being given the time of > day! NoEinstein Well, my crankometer pegs when you post NoE. NoEinstein Interesting that you use the number of '+ new posts' to an unmoderated > newsgroup to stand in for credentials. Nothing like setting yourself > a REALLY low bar for accomplishment, eh John? PD xxein: I hope you didn't think that was from me. My name is John also. Fortunately, I have a good handle on the physic even with a shortcoming in math. But I'm bullet proof. Try me. === Subject: Re: inverse of power series with prime power coefficients posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > The power series with prime coefficients P(x) = 1 + 2x + 3x^2 + 5x^3 + 7x^4 + 11x^5 + 13x^6 + .... and its inverse Q(x) = 1 / P(x) = 1 - 2x + x^2 - x^3 + 2x^4 - 3x^5 + 7x^6 - 10x^7 + > 13x^8 - 21x^9 + 26x^10 - 33x^11 + 53x^12 - .... were studied in the mid-1990s, when Backhouse conjectured, and > Flajolet proved, that the limit of the absolute value of the ratio of > consecutive coefficients of Q(x) is equal to B=1.4560749...., > Backhouse's constant. (By comparison, the limit of the ratio of > consecutive primes is equal to 1.) Consider instead the power series with prime power coefficients: PP(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + 7x^5 + 8x^6 + 9x^7 + 11x^8 + > 13x^9 + 16x^10 + 17x^11 + 19x^12 + 23x^13 + 25x^14 + 27x^15 + 29x^16 + > 31x^17 + 32x^18 + 37x^19 + .... I would expect its inverse to be quite similar to the series Q(x) > above. But it appears to be quite different: QQ(x) = 1 - 2x + x^2 - x^5 + 3x^6 - 3x^7 + 2x^9 - x^10 + x^15 - 4x^16 > + 8x^17 - 5x^18 - 13x^19 + 26x^20 - 9x^21 - 20x^22 + 25x^23 + 9x^24 - > 60x^25 + 65x^26 + .... Note how many low terms are zero: x^3, x^4, x^8, and the four > consecutive terms x^11, x^12, x^13, x^14. I wonder if this is pure > coincidence or if it has some significance. Since so many small numbers are prime powers, we see that PP(x) = (1+2x+3x^2+...) + (x^5+x^6+x^7+...) + O(x^8) = 1/(1-x)^2 + x^5/(1-x) + O(x^8) has a quite simple expression for low terms. Thus QQ(x) = (1 - x)^2/(1 + x^5 - x^6) + O(x^8), which hints to a recursion matching your observations. One could take this a bit further noting that many small odd numbers are prime powers, but then the denominator of QQ would become less nice. The really interesting part comes where the law of small numbers dowsn not hold any more. Also, whereas the terms of Q(x) alternate regularly between positive > and negative, there is no apparent pattern to the signs of the terms > of QQ(x). Finally, the terms of QQ(x) appear to grow much more slowly than the > terms of Q(x). Perhaps as the series continues and the prime powers > become sparser compared to the primes, QQ(x) will start to look and > behave more like Q(x). But I am curious: does the limit of the > absolute value of the ratio of consecutive coefficients of QQ(x) > exist? If so, what is it? Geoffrey Caveney === Subject: Re: inverse of power series with prime power coefficients > The power series with prime coefficients P(x) = 1 + 2x + 3x^2 + 5x^3 + 7x^4 + 11x^5 + 13x^6 + .... and its inverse Q(x) = 1 / P(x) = 1 - 2x + x^2 - x^3 + 2x^4 - 3x^5 + 7x^6 - 10x^7 + > 13x^8 - 21x^9 + 26x^10 - 33x^11 + 53x^12 - .... were studied in the mid-1990s, when Backhouse conjectured, and > Flajolet proved, that the limit of the absolute value of the ratio of > consecutive coefficients of Q(x) is equal to B=1.4560749...., > Backhouse's constant. (By comparison, the limit of the ratio of > consecutive primes is equal to 1.) Consider instead the power series with prime power coefficients: PP(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + 7x^5 + 8x^6 + 9x^7 + 11x^8 + > 13x^9 + 16x^10 + 17x^11 + 19x^12 + 23x^13 + 25x^14 + 27x^15 + 29x^16 + > 31x^17 + 32x^18 + 37x^19 + .... I would expect its inverse to be quite similar to the series Q(x) > above. But it appears to be quite different: QQ(x) = 1 - 2x + x^2 - x^5 + 3x^6 - 3x^7 + 2x^9 - x^10 + x^15 - 4x^16 > + 8x^17 - 5x^18 - 13x^19 + 26x^20 - 9x^21 - 20x^22 + 25x^23 + 9x^24 - > 60x^25 + 65x^26 + .... Note how many low terms are zero: x^3, x^4, x^8, and the four > consecutive terms x^11, x^12, x^13, x^14. I wonder if this is pure > coincidence or if it has some significance. There seem to be no other zero coefficients up to the x^300 term. > Also, whereas the terms of Q(x) alternate regularly between positive > and negative, there is no apparent pattern to the signs of the terms > of QQ(x). Finally, the terms of QQ(x) appear to grow much more slowly than the > terms of Q(x). Perhaps as the series continues and the prime powers > become sparser compared to the primes, QQ(x) will start to look and > behave more like Q(x). But I am curious: does the limit of the > absolute value of the ratio of consecutive coefficients of QQ(x) > exist? If so, what is it? Looking at the first 300 or so terms, I see no pattern. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: inverse of power series with prime power coefficients > The power series with prime coefficients P(x) = 1 + 2x + 3x^2 + 5x^3 + 7x^4 + 11x^5 + 13x^6 + .... and its inverse Q(x) = 1 / P(x) = 1 - 2x + x^2 - x^3 + 2x^4 - 3x^5 + 7x^6 - 10x^7 + > 13x^8 - 21x^9 + 26x^10 - 33x^11 + 53x^12 - .... were studied in the mid-1990s, when Backhouse conjectured, and > Flajolet proved, that the limit of the absolute value of the ratio of > consecutive coefficients of Q(x) is equal to B=1.4560749...., > Backhouse's constant. (By comparison, the limit of the ratio of > consecutive primes is equal to 1.) Consider instead the power series with prime power coefficients: PP(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + 7x^5 + 8x^6 + 9x^7 + 11x^8 + > 13x^9 + 16x^10 + 17x^11 + 19x^12 + 23x^13 + 25x^14 + 27x^15 + 29x^16 + > 31x^17 + 32x^18 + 37x^19 + .... I would expect its inverse to be quite similar to the series Q(x) > above. But it appears to be quite different: QQ(x) = 1 - 2x + x^2 - x^5 + 3x^6 - 3x^7 + 2x^9 - x^10 + x^15 - 4x^16 > + 8x^17 - 5x^18 - 13x^19 + 26x^20 - 9x^21 - 20x^22 + 25x^23 + 9x^24 - > 60x^25 + 65x^26 + .... Note how many low terms are zero: x^3, x^4, x^8, and the four > consecutive terms x^11, x^12, x^13, x^14. I wonder if this is pure > coincidence or if it has some significance. Also, whereas the terms of Q(x) alternate regularly between positive > and negative, there is no apparent pattern to the signs of the terms > of QQ(x). Finally, the terms of QQ(x) appear to grow much more slowly than the > terms of Q(x). Perhaps as the series continues and the prime powers > become sparser compared to the primes, QQ(x) will start to look and > behave more like Q(x). But I am curious: does the limit of the > absolute value of the ratio of consecutive coefficients of QQ(x) > exist? If so, what is it? I don't know any of the answers, but here are a couple of suggestions. 1. Have you gone through Flajolet's proof? Maybe there are ideas in it that could work for your problem. 2. Run the coefficients of QQ(x) through the online encyclopedia of integer sequences and see if anything comes out. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: a subset of continuous functions - is it interesting? posting-account=ZUczkQoAAABCznvSjYJPkJbPEzZyPyBU AppleWebKit/525.27.1 (KHTML, like Gecko) Version/3.2.1 Safari/525.27.1,gzip(gfe),gzip(gfe) Okay let me tighten up my claim. I guess in view of the Stone- Wierstrass Thm, what I said previously has to be true: Let X be a compact subset of R^M and C the set of continuous functions from X to R^N. We consider the topology on C defined by the sup norm. I consider a maximal subset M of C that contains all polynomails and has the following property: for any two functions f and g in M, the set of x in X for which f(x) = g(x) is measure 0. The set of polynomials I claim satisfies the property. The set of polynomials is dense in C by the Stone-Wierstrass Theorem, and hence M is also dense in C. > Yes - uniform approximation on compact subsets was what I was > originally imagining, although I am not married to this topology and > would be fine using a weaker topology. On May 11, 11:57am, Mariano Su.87rez-Alvarez > I want to form the following subset of the set C of continuous > functions from R^M to R^N: The largest subset S of continuous functions such that for any two > functions f and g in S, the set of points x in R^M for which f(x)=g(x) > has Lebesgue measure zero. Is it obvious that this is a good definition? I do not see > at once why there is a maximum set with that property or, even, > that there exists maximal sets with that property. -- m > I also can't see any reason to think a > maximum such set exists. I think Zorn's lemma gets us a maximal set, > since a chain (by set inclusion) of such > sets will have a union with the same > property (because any f,g appearing in > the union must already appear in one > of the chained sets). Is a maximal such set necessarily dense > in C(R^M,R^N)? This requires one to > define what topology those continuous > functions have. Some of the latter > comments from the OP suggest he (or > the shark) has in mind uniform > approximation on compact subsets, > but I'll leave that to him to say. > === Subject: Re: a subset of continuous functions - is it interesting? posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Trident/4.0; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Okay let me tighten up my claim. I guess in view of the Stone- > Wierstrass Thm, what I said previously has to be true: Let X be a compact subset of R^M and C the set of continuous functions > from X to R^N. We consider the topology on C defined by the sup norm. I consider amaximalsubset M of C that contains all polynomails and > has the following property: > for any two functions f and g in M, the set of x in X for which f(x) > = g(x) is measure 0. The set of polynomials I claim satisfies the property. The set of > polynomials is dense in C by the Stone-Wierstrass Theorem, and hence M > is also dense in C. > Yes - uniform approximation on compact subsets was what I was > originally imagining, although I am not married to this topology and > would be fine using a weaker topology. On May 11, 11:57am, Mariano Su.87rez-Alvarez > I want to form the following subset of the set C of continuous > functions from R^M to R^N: The largest subset S of continuous functions such that for any two > functions f and g in S, the set of points x in R^M for which f(x)=g(x) > has Lebesgue measure zero. Is it obvious that this is a good definition? I do not see > at once why there is a maximum set with that property or, even, > that there existsmaximalsets with that property. -- m > I also can't see any reason to think a > maximum such set exists. I think Zorn's lemma gets us amaximalset, > since a chain (by set inclusion) of such > sets will have a union with the same > property (because any f,g appearing in > the union must already appear in one > of the chained sets). Is amaximalsuch set necessarily dense > in C(R^M,R^N)? This requires one to > define what topology those continuous > functions have. Some of the latter > comments from the OP suggest he (or > the shark) has in mind uniform > approximation on compact subsets, > but I'll leave that to him to say. > - Show quoted text - Okay, but if you include the polynomials, it's not much of a deduction that we get a dense subset. I've been trying to construct a maximal subset of the desired kind which is not dense. Let's introduce some terminology for your property so the conversation is a little easier to have. Let C([0,1],R) be the continuous real-valued functions on the unit interval, with topology induced by the max norm. We call a subset S of this function space (measurably) irredundant if {x|f(x)=g(x)} has (Lebesgue) measure zero for any distinct f,g in S. By the Zorn's lemma argument sketched before, maximal irredundant subsets S exist, e.g. as an extension of any particular irredundant subset. Standard cardinality arguments show that |S| is at most the size of the continuum, and that this upper bound can be attained. I don't know if we could have such S be countable. I'd like to prove the following: Conjecture: If S is maximally irredundant, then xS = {x*f(x)| f in S} is maximally irredundant. With that in hand I think we could construct a maximally irredundant set which is not dense. However I keep running into difficulties, so perhaps I'm banging blindly into a wall here. === === Subject: Re: Tracking the minimum in a sliding window > More formally, let W be a vector of values of length k. Define > the sequence of ascending mimima, A, as follows: Let A[0] be the minimum value in W and for j>0 let A[j] be the > minimum value in W with index greater than the index of A[j-1]. > (If two locations have the same minimum value take the later > one.) Example: > > W = 5,2,8,6,4,7 > A = 2,4,7 > Let x be the newly added element. Then A can be updated by (a) removing all elements of A less than or equal to x, > (b) appending x to A, and > (c) removing the initial element of A if it is being removed > from the window. Don't you mean greater or equal in a)? Martin -- Quidquid latine scriptum est, altum videtur. === Subject: Re: Tracking the minimum in a sliding window > sliding > window minimum. =3DA0 >Index. >Matt Brenneman > off hand but the URL ishttp://home.tiac.net/~cri/2001/slidingmin.html Hi Richard, The website is just as good ;>) That's more than I can say for the analysis or the code. :-) I will replace it soon. Richard Harter, cri@tiac.net http://home.tiac.net/~cri, http://www.varinoma.com If I do not see as far as others, it is because I stand in the footprints of giants. === Subject: Re: Is relativity I L L O G I C A L? <4A088392.3040406@somewhere.no> posting-account=0iME0QoAAABHLc0kgVai1vIO0M5rRJGg InfoPath.1; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; MS-RTC LM 8; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) (NetCache NetApp/6.0.2P1) On 11 Maj, 15:59, Paul B. Andersen PART TWO - Continuation > I have a circle with a one meter radius. It is sitting with its > center C coincident on the origin of an X-Y Cartesian frame. There is > a light source at the (1,0) coordinate R on the right side of the > circle and another light source on the (-1,0) coordinate L on the left > side of the circle. > A photon is fired simultaneously from R and L from the reference frame > of C, going in the X+ direction. > When photon R1 hits any point on the X coordinate, photon L1 follows 2 > meters behind it, or 2/c seconds behind it. > An observer O moving at a certain velocity observes photon L1 and R1. > For simplicity O may be assumed to be somewhere along the X > coordinate. Observer O is in possession of a meterstick which is at > rest in frame O. This meterstick will be identical to a meterstick at > rest in frame C.* > Observer O measures the distance between L1 and R1. Since L1 and R1 > are both moving at constant velocity, then the distance between them, > as judged by identical metersticks, must remain the same. Otherwise > one photon will be FASTER than the other, which is not allowed by > relativity. > Thus C measured the distance L1-R1 to be 2 meters in the C frame, and > O measures the distance L1-R1 to be 2 meters in the O frame. In fact, > any observer Q will measure the L1-R1 distance to be 2 meters using a > Q meterstick at rest in Q. > Why do you keep repeating this triviality? > This is obviously correct according to Galilean relativity. > It is however wrong according to Lorentzian relativity (SR). > This is the old fallacy: > Galilean relativity MUST apply because my intuition says so. > Only real experiments can decide if the Galilean > or the Lorentz transform apply in the real world. > The Lorentz transform is experimentally confirmed, > while the Galilean transform is experimentally falsified. > Your naive intuition is proven wrong. > No ad hominems can change that fact. > But that won't prevent you from trying, of course. Perenial idiot Paul, I was not using Galilean relativity as you > misunderstood it to be. QED. You must be pretty ignorant if you don't realize that your analysis > is strictly according to the Galilean transform. I was using the very principles of Einteinian > relativity. Another confirmation of your ignorance. So far, PD and Miguel can only derive contradictory > results, which is what Einsteinian relativity always does. SR has but one answer to your scenario. > SR is consistent and cannot produce contradictory answers. > Below you will find the correct answers according to both > the Galilean and the Lorentzian transform. > (put L = 2m) K': O--------------------->x' -> v > K: |--------------------->x Event E1: First photon emitted. > Coordinates in K: x 1 = 0, t 1 = 0 > Coordinates in K': x 1'= 0, t 1'= 0 Event E2: First photon at x = L > Coordinates in K: x 2 = L, t 2 = L/c > coordinates in K' > According to the Galilean transform: > x 2' = x 2 - v*t 2 = L(1-v/c) > t 2' = t 2' = L/c > According to the Lorentz transform: > x 2' = (x 2 - v*t 2)/sqrt(1-v^2/c^2) = L*sqrt((1-v/c)/(1+v/c)) > t 2' = (t 2 - x 2*v/c^2)/sqrt(1-v^2/c^2) = (L/c)*sqrt((1-v/c)/(1+v/c)) Event E3: Second photon emitted > Coordinates in K: x 3 = 0, t 3 = L/c > Coordinates in K': > According to the Galilean transform: > x 3' = x 3 - v*t 3 = -Lv/c > t 3' = t 3' = L/c > According to the Lorentz transform: > x 3' = (x 3 - v*t 3)/sqrt(1-v^2/c^2) = -(Lv/c)/sqrt(1-v^2/c^2) > t 3' = (t 3 - x 3*v/c^2)/sqrt(1-v^2/c^2) = (L/c)/sqrt(1-v^2/c^2) The question is: What is the distance between the two photons? > 'Distance' is the spatial separation between two events with > the same time coordinate. According to the Galilean transform, the events E2 and E3 > are simultaneous in K', so the the distance between > the photons measured in K' is simply: > d' = x 2' - x 3' = L(1-v/c) - (-Lv/c) = L > ----------------------------------------------- > | According to the Galilean transform is | > | the distance between the photons invariant. | > ----------------------------------------------- According to the Lorentz transform, the events E2 and E3 > are not simultaneous in K', so the distance between them > is not x 2' - x 3'. We must find where the photons > are at the same time in K' . > So let's ask: what is the x' coordinate of the first photon > at the time t 3' = (L/c)/sqrt(1-v^2/c^2) when the second > photon is emitted? > We know that the first photon was emitted at t 1' = 0, > and it moves with the speed c in K'. > So x' = c*t 3' = L/sqrt(1-v^2/c^2) > The distance between the photons in K' is thus: > d' = x' - x 3' = L/sqrt(1-v^2/c^2) - (-(Lv/c)/sqrt(1-v^2/c^2)) > d' = L*sqrt((1+v/c)/(1-v/c)) > Note: > The distance between the photons is always the same in K', > the photons move with the same speed. > But: > ---------------------------------------------------- > | According to the Lorentz transform is | > | the distance between the photons frame dependent.| > ---------------------------------------------------- Case closed. > I am not going to quibble about this. > It is correct, and your ad hominems can't change that. -- > Paul http://home.c2i.net/pb andersen/- D.9alj citerad text - - Visa citerad text - Nice try. But you still have not answered the question: What is the distance between L1 and R1 as seen by O? Your pet theory impotent again with such a simple problem... === Subject: Re: Is relativity I L L O G I C A L? posting-account=0iME0QoAAABHLc0kgVai1vIO0M5rRJGg InfoPath.1; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; MS-RTC LM 8; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) (NetCache NetApp/6.0.2P1) > On May 11, 8:12am, Strich-Reply-To-Idiots On 11 Maj, 05:54, Paul B. Andersen center C coincident on the origin of an X-Y Cartesian frame. There is > a light source at the (1,0) coordinate R on the right side of the > circle and another light source on the (-1,0) coordinate L on the left > side of the circle. A photon is fired simultaneously from R and L from the reference frame > of C, going in the X+ direction. When photon R1 hits any point on the X coordinate, photon L1 follows 2 > meters behind it, or 2/c seconds behind it. An observer O moving at a certain velocity observes photon L1 and R1. > For simplicity O may be assumed to be somewhere along the X > coordinate. Observer O is in possession of a meterstick which is at > rest in frame O. This meterstick will be identical to a meterstick at > rest in frame C.* Observer O measures the distance between L1 and R1. Since L1 and R1 > are both moving at constant velocity, then the distance between them, > as judged by identical metersticks, must remain the same. Otherwise > one photon will be FASTER than the other, which is not allowed by > relativity. Thus C measured the distance L1-R1 to be 2 meters in the C frame, and > O measures the distance L1-R1 to be 2 meters in the O frame. In fact, > any observer Q will measure the L1-R1 distance to be 2 meters using a > Q meterstick at rest in Q. Why do you keep repeating this triviality? This is obviously correct according to Galilean relativity. > It is however wrong according to Lorentzian relativity (SR). This is the old fallacy: > Galilean relativity MUST apply because my intuition says so. Only real experiments can decide if the Galilean > or the Lorentz transform apply in the real world. > The Lorentz transform is experimentally confirmed, > while the Galilean transform is experimentally falsified. Your naive intuition is proven wrong. > No ad hominems can change that fact. > But that won't prevent you from trying, of course. -- > Paul http://home.c2i.net/pb andersen/-D.9alj citerad text - - Visa citerad text - Perenial idiot Paul, I was not using Galilean relativity as you > misunderstood it to be. I was using the very principles of Einteinian > relativity. So far, PD and Miguel can only derive contradictory > results, What contradictory results? which is what Einsteinian relativity always does.- D.9alj citerad text - - Visa citerad text -- D.9alj citerad text - - Visa citerad text - You won't get it if you don't solve the problems Schizo. Just sitting there and spouting your usual nonsense isn't gonna work... === Subject: Re: Is relativity I L L O G I C A L? > On 11 Maj, 05:54, Paul B. Andersen PART TWO - Continuation > I have a circle with a one meter radius. It is sitting with its > center C coincident on the origin of an X-Y Cartesian frame. There is > a light source at the (1,0) coordinate R on the right side of the > circle and another light source on the (-1,0) coordinate L on the left > side of the circle. > A photon is fired simultaneously from R and L from the reference frame > of C, going in the X+ direction. > When photon R1 hits any point on the X coordinate, photon L1 follows 2 > meters behind it, or 2/c seconds behind it. > An observer O moving at a certain velocity observes photon L1 and R1. > For simplicity O may be assumed to be somewhere along the X > coordinate. Observer O is in possession of a meterstick which is at > rest in frame O. This meterstick will be identical to a meterstick at > rest in frame C.* > Observer O measures the distance between L1 and R1. Since L1 and R1 > are both moving at constant velocity, then the distance between them, > as judged by identical metersticks, must remain the same. Otherwise > one photon will be FASTER than the other, which is not allowed by > relativity. > Thus C measured the distance L1-R1 to be 2 meters in the C frame, and > O measures the distance L1-R1 to be 2 meters in the O frame. In fact, > any observer Q will measure the L1-R1 distance to be 2 meters using a > Q meterstick at rest in Q. > Why do you keep repeating this triviality? This is obviously correct according to Galilean relativity. > It is however wrong according to Lorentzian relativity (SR). This is the old fallacy: > Galilean relativity MUST apply because my intuition says so. Only real experiments can decide if the Galilean > or the Lorentz transform apply in the real world. > The Lorentz transform is experimentally confirmed, > while the Galilean transform is experimentally falsified. Your naive intuition is proven wrong. > No ad hominems can change that fact. > But that won't prevent you from trying, of course. > Perenial idiot Paul, I was not using Galilean relativity as you > misunderstood it to be. QED. You must be pretty ignorant if you don't realize that your analysis > is strictly according to the Galilean transform. > I was using the very principles of Einteinian > relativity. Another confirmation of your ignorance. > So far, PD and Miguel can only derive contradictory > results, which is what Einsteinian relativity always does. SR has but one answer to your scenario. > SR is consistent and cannot produce contradictory answers. > Below you will find the correct answers according to both > the Galilean and the Lorentzian transform. > (put L = 2m) K': O--------------------->x' -> v > K: |--------------------->x Event E1: First photon emitted. > Coordinates in K: x_1 = 0, t_1 = 0 > Coordinates in K': x_1'= 0, t_1'= 0 Event E2: First photon at x = L > Coordinates in K: x_2 = L, t_2 = L/c > coordinates in K' > According to the Galilean transform: > x_2' = x_2 - v*t_2 = L(1-v/c) > t_2' = t_2' = L/c > According to the Lorentz transform: > x_2' = (x_2 - v*t_2)/sqrt(1-v^2/c^2) = L*sqrt((1-v/c)/(1+v/c)) > t_2' = (t_2 - x_2*v/c^2)/sqrt(1-v^2/c^2) = (L/c)*sqrt((1-v/c)/(1+v/c)) Event E3: Second photon emitted > Coordinates in K: x_3 = 0, t_3 = L/c > Coordinates in K': > According to the Galilean transform: > x_3' = x_3 - v*t_3 = -Lv/c > t_3' = t_3' = L/c > According to the Lorentz transform: > x_3' = (x_3 - v*t_3)/sqrt(1-v^2/c^2) = -(Lv/c)/sqrt(1-v^2/c^2) > t_3' = (t_3 - x_3*v/c^2)/sqrt(1-v^2/c^2) = (L/c)/sqrt(1-v^2/c^2) The question is: What is the distance between the two photons? > 'Distance' is the spatial separation between two events with > the same time coordinate. > According to the Galilean transform, the events E2 and E3 > are simultaneous in K', so the the distance between > the photons measured in K' is simply: > d' = x_2' - x_3' = L(1-v/c) - (-Lv/c) = L > ----------------------------------------------- > | According to the Galilean transform is | > | the distance between the photons invariant. | > ----------------------------------------------- According to the Lorentz transform, the events E2 and E3 > are _not_ simultaneous in K', so the distance between them > is _not_ x_2' - x_3'. We must find where the photons > are _at the same time in K'_. > So let's ask: what is the x' coordinate of the first photon > at the time t_3' = (L/c)/sqrt(1-v^2/c^2) when the second > photon is emitted? > We know that the first photon was emitted at t_1' = 0, > and it moves with the speed c in K'. > So x' = c*t_3' = L/sqrt(1-v^2/c^2) > The distance between the photons in K' is thus: > d' = x' - x_3' = L/sqrt(1-v^2/c^2) - (-(Lv/c)/sqrt(1-v^2/c^2)) > d' = L*sqrt((1+v/c)/(1-v/c)) > Note: > The distance between the photons is always the same in K', > the photons move with the same speed. > But: > ---------------------------------------------------- > | According to the Lorentz transform is | > | the distance between the photons frame dependent.| > ---------------------------------------------------- This was not exactly the same scenario as described by Strich, but it is equivalent and the answer is the same. The important point is that at the time t_2+, we have two photons moving to the right separated by a distance L in K. Since Strich not can be expected to see that the two scenarios are equivalent, I will redo it with simultaneously emitted photons: Event E1: Left photon emitted. Coordinates in K: x_1 = 0, t_1 = 0 Coordinates in K': Galilean transform: x_1' = x_1 - v*t_1 = 0 t_1' = t_1 = 0 Lorentz transform: x_1' = (x_1 - v*t_1)/sqrt(1-v^2/c^2)= 0 t_1' = (t_1 - x_1*v/c^2)/sqrt(1-v^2/c^2)= 0 Event E2: Right photon emitted. Coordinates in K: x_2 = L, t_2 = 0 Coordinates in K': Galilean transform: x_2' = x_2 - v*t_2 = L t_2' = t_2 = 0 Lorentz transform: x_2' = (x_2 - v*t_2)/sqrt(1-v^2/c^2)= L/sqrt(1-v^2/c^2) t_2' = (t_2 - x_2*v/c^2)/sqrt(1-v^2/c^2)= -(Lv/c^2)/sqrt(1-v^2/c^2) According to the Galilean transform, the events E1 and E2 are simultaneous in K', so the the distance between the photons measured in K' is simply: d' = x_2' - x_1' = L - 0 = L ----------------------------------------------- | According to the Galilean transform is | | the distance between the photons invariant. | ----------------------------------------------- According to the Lorentz transform, the events E1 and E2 are _not_ simultaneous in K', so the distance between them is _not_ x_2' - x_1'. We must find where the photons are _at the same time in K'_. So let's ask: what is the x' coordinate of the right photon at the time t_1' when the left photon is emitted at x_1'? We know that the right photon was emitted from x_2' at the time t_2'. So at the time t_1' it must be at the position: x' = x_2' + (t_1'-t_2')*c = L/sqrt(1-v^2/c^2)+(Lv/c)/sqrt(1-v^2/c^2) x' = L*sqrt((1+v/c)/(1-v/c)) The distance between the photons in K' is thus: d' = x' - x_1' = L*sqrt((1+v/c)/(1-v/c)) ---------------------------------------------------- | According to the Lorentz transform is | | the distance between the photons frame dependent.| ---------------------------------------------------- -- Paul http://home.c2i.net/pb_andersen/ === Subject: Re: Dorn Strich, P.D. is at: TheDraperFamily @ gMail .COM posting-account=0iME0QoAAABHLc0kgVai1vIO0M5rRJGg InfoPath.1; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; MS-RTC LM 8; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) (NetCache NetApp/6.0.2P1) > On May 11, 12:08pm, Strich-Reply-To-Idiots On May 11, 8:14am, Strich-Reply-To-Idiots Dorn Strich, P.D. is at: TheDraperFamily @ gMail .COM By the way, unlike Google Groups, NNTP servers > don't block this eMail address -- eMail harvesters already have it. As I have several inboxes for several purposes and with different > levels of outside exposure, this doesn't really matter. But thank you at least for introducing spaces Still no address. This is a legal document PD. You must willingly > provide your contact information, and so far, you have tried to weasel > away. I will happily provide my legal address to your lawyer when he > contacts me at my email address. There is absolutely no legal reason > for me to post my address in a public space. Ask your lawyer about > that, and stop whining. Since Dave has been whining for the past month about how I'm not being > helpful in giving him a mailing address... Eric Gisse > PO Box 750315 > Fairbanks, Alaska 99775 That I'm leaving Fairbanks and never to return in about 6 hours won't > stop the mail from being forwarded to another address I won't provide. > I'll let you know if he actually sends anything. Personally I'm expecting cardboard with crayon marks. Or if I'm really > lucky, copy paper with crayon marks. So little Eric lives in a PO Box. It must get cold in the winter. A PO Box is a valid mailing address, entirely capable of recieving > registered mail. > - Visa citerad text - stupidity. Actually, it is. Ask your lawyer.- D.9alj citerad text - - Visa citerad text - A PO Box is now a legal residence? You are a really small man Eric :-) It's a legal address. Ask your lawyer. Oh, that's right, you lie about > stuff like this. PD- D.9alj citerad text - - Visa citerad text - Nope. It is not a legal RESIDENCE. That's correct. You don't need a residence. You need an ADDRESS, even > if you *want* a residence. You don't you get what you want, you > whiner. You might be given what you need.- D.9alj citerad text - - Visa citerad text - Stop quibbling schizo... Where is your legal residence? Ah yes, a liar and a coward too. === Subject: Re: STRICH fails to refute Gravity Probe A as evidence for Relativity <4A088D08.7030308@somewhere.no> posting-account=0iME0QoAAABHLc0kgVai1vIO0M5rRJGg InfoPath.1; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; MS-RTC LM 8; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) (NetCache NetApp/6.0.2P1) On 11 Maj, 16:39, Paul B. Andersen is the only way to test a scientific theory: > 1. Calculate what the theory predicts will be measured in an experiment. > 2. Perform the experiment, do the measurements. > 3. Compare the predictions of the theory to the measured values. > There are two possible outcomes: > a) The prediction of the theory are not consistent with the measurements > (Not within the error-bars). The theory is falsified (proven false). > b) The predictions of the theory are consistent with the the measurements. > The theory is confirmed (NOT proven - no theory can be proven). What was measured in Gravity probe A was the rate of the on board clock > relative to a clock on the ground during the flight. > The measurements were consistent with the predictions of GR, > and thus gravity probe A confirmed GR. The challenge to Strich was to prove gravity probe A > wrong by argument or evidence. Since it is an irrefutable fact that gravity probe A confirms GR, > this is obviously an impossible task. Strich's attempt to do that impossible task is rather amusing, though. > Or should I say pathetic? :-) TA - DAAAA : # # # # # The Gravity Probe A experiment* tested the equivalence between > inertial mass and gravitational mass. Here are the main points: 1) There is only ONE mass. > 2) Einstein proposed there were TWO types of mass--inertial and > gravitational. > 3) The nitwit then went on to say that the TWO types of mass are > really equal (or that there was only ONE mass). > 4) The GP-A measured that inertial mass and gravitational mass are > equal with an accuracy of 200 parts per million. > 5) What do you expect, if the two are the same, then any measurement > will show them to be the same. > 6) This is like proving A=A to as many significant figures as you care > to test it. Additional points: Mass has always been of ONE type. When we measure the mass of an > atom, we do not say mass-inertial or mass-gravitational. We simply > say mass. for, we do not say Higgs-inertial or Higgs-gravitational. We simply > say Higgs boson. Einstein the schizophrenic thought there were two masses, then said > they were the same, and now tests of the two masses being found to be > equal are used to prove that Einstein is correct in a circular > reasoning. Some refutation, eh? :-) -- > Paul http://home.c2i.net/pb andersen/- D.9alj citerad text - - Visa citerad text - Here is the question this perenial idiot will probve he is a perenial idiot: What was the primary objective of the Gravity Probe A ? What was the primary objective of the Gravity Probe A ? What was the primary objective of the Gravity Probe A ? === Subject: Linda Rosa and Larry Sarner: Quackery, Pseudoscience, and Calculus is Useless posting-account=A2rtMwoAAACiQZRb17VN8BH3JZMlCZXL Gecko/2009021910 Firefox/3.0.7,gzip(gfe),gzip(gfe) Linda Rosa, RN, is Larry Sarner's stepdaughter. They often collaborate on efforts to denigrate the humanitarian efforts of those dedicated to helping orphans. There is no reason why family members cannot collaborate on vital and important work. The names Bach, Bernoulli, and Curie come to mind. Unfortunately, Linda Rosa and Larry Sarner cannot be included. Time and time again, they have demonstrated a complete lack of objectivity, to say nothing of generally poor performance in key areas such as experimental design. It seems Linda Rosa might have done better to use her nursing training for its intended purpose, helping others, instead of promulgating her ñfamily brandsî of pseudoscience, quackery, and hatred. http://larrysarnerpseudoscience.wordpress.com/2009/05/12/linda-rosa-nepotist ic-nurse/ http://medicalcrackpots.wordpress.com/2009/05/09/larry-sarner/ === Subject: Re: A BLACK HOLE MYTH GETS BUSTED: posting-account=0iME0QoAAABHLc0kgVai1vIO0M5rRJGg InfoPath.1; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; MS-RTC LM 8; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) (NetCache NetApp/6.0.2P1) --- AHAHAHAHA... only on the UseNet! ... AHAHAHAHA... ---- > [snip bantering].... > Folks: > PD is 'Dr. No'. Where is James Bond when we need him, > to terminate such a world-threatening parasite [1] as PD? > NoEinstein > I think it's getting close to time to report you, John, for making > thinly veiled threats on the internet, don't you? > AHAHAHAHA.. What's the matter with you, Paul? John adores > you & he cranks himself over you. You have frightened him as if > you had the power of a world-threatening parasite... This made me smile, ahahahanson. Now you > blew it, Paul, because you reacted like a religious fanatic who is > not quite sure about the veracity of his own (physics)-faith... and > you threaten to report & appeal to a Higher Power!... ahahahaha... > Celebrate John's self-inflicted agony... Enjoy that, Paul... ahaha... Well, people who are a danger to themselves or to others need > treatment.- D.9alj citerad text - - Visa citerad text - Paul, your schizophrenia can be a danger to yourself. Why are you not getting treatment? === Subject: Re: A BLACK HOLE MYTH GETS BUSTED: posting-account=8k9z_QoAAABfJ6zct3wDB2k3FKfRerU2 2.0.50727),gzip(gfe),gzip(gfe) > sci.physics. Most likely they would be divided on two sides: (1.) The gullible who embrace Einstein's 'theories' as solid ground for appeasing their own inferiority complexes; and (2.) The independent thinkers who can see the shallowness of Einstein's counterintuitive argumentstantamount to the absurd. As a matter of course, everyone replying on these news groups should be required to state which side they are on so that those with opposite positions need not waste their time reading that which they find offensive. NoEinstein > --- AHAHAHAHA... only on the UseNet! ... AHAHAHAHA... ---- > [snip bantering].... > Folks: > PD is 'Dr. No'. Where is James Bond when we need him, > to terminate such a world-threatening parasite [1] as PD? > NoEinstein > I think it's getting close to time to report you, John, for making > thinly veiled threats on the internet, don't you? > AHAHAHAHA.. What's the matter with you, Paul? John adores > you & he cranks himself over you. You have frightened him as if > you had the power of a world-threatening parasite... Now you > blew it, Paul, because you reacted like a religious fanatic who is > not quite sure about the veracity of his own (physics)-faith... and > you threaten to report & appeal to a Higher Power!... ahahahaha... > Celebrate John's self-inflicted agony... Enjoy that, Paul... ahaha... But listen Paul, stand back & see what these acrid Einstein pro > & con exchanges do reflect in the real world... The theory of > Relativity, pro or con, has never threatened anybody's livelihood > or lifestyle, but it's purveyors appears to threatens a lot of cultural, > ideological and religious values in the triad of Monotheism. > These debates highlight here locally that: ---- ||| Relativity is a tool used by the few to fool the many ||| ---- [1] --|| The problem with stereotypes is that stereotypes are true ||--. > --- AHAHAHAHA... only on the UseNet! ... AHAHAHAHA... ---- Hardly, cuz on the wider field of socio-physics one can see that:: > ::::::::::::::::::::: Nobody is born religious. :::::: > ::::::::::::::::::: Religion is an acquired disease ::::::::::: > :::::::: Religion is a tool used by the few to the many ::::: [1] > ::::::::::::::::::::: Nobody is born anti-Semitic neither. :::::: > :::::::::::: But Jews are ultimately the teachers for it ::::::: [1] > ::::: No Jews -- No Christians || No Jews -- No Muslims :::::: > ::::::::::::::::::::: No Jews || No anti-Semitism. :::::: Maybe on a given future occasion I'll post about this fascinating > subject of Re: A BLACK HOLE MYTH GETS BUSTED, and why > this BLEAK, WHOLE MYTH GETS TRUSTED... NOT...ahaha.... Ha, ha, ha, jerk.- Hide quoted text - - Show quoted text - === Subject: Re: A BLACK HOLE MYTH GETS BUSTED: <5_5Ol.596$wR5.230@nwrddc02.gnilink.net> posting-account=8k9z_QoAAABfJ6zct3wDB2k3FKfRerU2 2.0.50727),gzip(gfe),gzip(gfe) > Hanson: :-) NoEinstein --- AHAHAHAHA... only on the UseNet! ... AHAHAHAHA... ---- > [snip bantering].... > Folks: > PD is 'Dr. No'. Where is James Bond when we need him, > to terminate such a world-threatening parasite [1] as PD? > NoEinstein > I think it's getting close to time to report you, John, for making > thinly veiled threats on the internet, don't you? > AHAHAHAHA.. What's the matter with you, Paul? John adores > you & he cranks himself over you. You have frightened him as if > you had the power of a world-threatening parasite... Now you > blew it, Paul, because you reacted like a religious fanatic who is > not quite sure about the veracity of his own (physics)-faith... and > you threaten to report & appeal to a Higher Power!... ahahahaha... > Celebrate John's self-inflicted agony... Enjoy that, Paul... ahaha... But listen Paul, stand back & see what these acrid Einstein pro > & con exchanges do reflect in the real world... The theory of > Relativity, pro or con, has never threatened anybody's livelihood > or lifestyle, but it's purveyors appears to threatens a lot of cultural, > ideological and religious values in the triad of Monotheism. > These debates highlight here locally that: ---- ||| Relativity is a tool used by the few to fool the many ||| ---- [1] --|| The problem with stereotypes is that stereotypes are true ||--. > --- AHAHAHAHA... only on the UseNet! ... AHAHAHAHA... ---- Hardly, cuz on the wider field of socio-physics one can see that:: > ::::::::::::::::::::: Nobody is born religious. :::::: > ::::::::::::::::::: Religion is an acquired disease ::::::::::: > :::::::: Religion is a tool used by the few to the many ::::: [1] > ::::::::::::::::::::: Nobody is born anti-Semitic neither. :::::: > :::::::::::: But Jews are ultimately the teachers for it ::::::: [1] > ::::: No Jews -- No Christians || No Jews -- No Muslims :::::: > ::::::::::::::::::::: No Jews || No anti-Semitism. :::::: Maybe on a given future occasion I'll post about this fascinating > subject of Re: A BLACK HOLE MYTH GETS BUSTED, and why > this BLEAK, WHOLE MYTH GETS TRUSTED... NOT...ahaha.... === Subject: Re: Answers > I need the answers of the book Electric Circuits 6th Edition By >Nilsson. > How do I receive? Uhhh --- maybe work the problems?? Or is cheating more preferred? === Subject: Re: Levy proof that R is uncountable. > You have a belief that these number systems have these natural > mathematical properties that are immutable: so you're a platonist. This is a curious notion of Platonism. We might well posit that mathematical objects are nothing but figments of our imagination, a consensual hallucination. Even then it makes perfect sense to say that someone imagining a system of objects for which induction does not hold is simply not imagining the naturals, just as someone thinking of a gelatinous blob of yellow space goo is not thinking of Leopold Bloom. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Levy proof that R is uncountable. <27p8v4tivdguvifhf5fk4hsf1p0oqnl87a@4ax.com> <49f46eb5$0$293$7a628cd7@news.club-internet.fr> posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > so that the standard set theorists will consider the theories > at least as much as they do ZFA? > I can't speak for any standard set theorists (because I'm not a set > theorist), but if you want to know why I chose to learn about ZFA then > here's why: I was looking through the CASM reading list in the hope of > finding something interesting to read, and it mentioned Barwise and > Moss's /Vicious Circles/ as being relevant to the set theory course. I > looked at reviews online and it sounded interesting, so I bought a > copy. The reason I choose not to spend my time considering the your > theories based on so-called so-called crank posts is that they > don't sound interesting. Why would they? /Vicious Circles/ proposes > that hypersets may be used to model circular phenomena in a number of > fields. Interesting! I never imagined that hypersets could be useful in the way that, say, classical analysis is useful. On the other hand, I suppose that the hypersets proposed by the so-called cranks, such as having I = {I} and II = {I, II} yet II not equaling the Quine atom, aren't as useful in the same way that the hypersets of ZFA are useful. > Your theories, on the other hand, seem to exist solely for the > purpose of allowing you to pretend that people who talk nonsense are > actually making sense. The gist of Rotwang post, as well as several posts made by others, is as follows: standard theorists will consider alternate theories proposed in books , but not those proposed on Usenet . But unlike the standard theorists, I would much rather discuss an alternate theory here than in a book, and I have a reason other than the cost (since I'm aware that the standard theorists won't consider cost as a valid argument against books). Here on Usenet, I can actually communicate with the inventor of the theory. If I claim something about another person's proposed theory that's wrong, the inventor can correct me. Many of my posts with cranks have proceeded in this manner. On the other hand, I can't communicate with Aczel, the inventor of the theory AFA. I don't know whether Aczel is even alive, and even if he is, it's doubtful that he posts on Usenet or any other forum. If I claim something wrong about AFA, Aczel isn't there to tell me how he intends his theory to work. I obviously rank this interactivity much more highly than the standard theorists. This is the 21st century and the Internet is now a way of life, but the standard theorists are stuck in the 20th century and are dependent on books. In another thread, someone claimed to have a proof of P=NP. In Surely it is not impossible that P=NP will be proved. It is extremely unlikely, but not impossible, that the proof will be announced on sci.math. So Hughes expresses his skepticism about the proof (which is in itself reasonable), but then implies how unlikely that the proof would be on sci.math . And based on other posts I've read in other threads, what's being implied is that a real proof of P=NP (or even of P The Axiom of Extensionality is another key axiom that, if > replaced by its negation or an axiom implying its negation, > will also be ridiculed by standard set theorists. > I doubt that, though given how fundamental extensionality is to most > people's conception of set theory, if the things in your theory don't > satisfy it then you should probably call them something other than > sets. I agree to some extent. I bet that even the cranks who try to propose theories with potential infinity would accept Extensionality as part of their conception of set theory, but they don't realize that having potential infinity would require that they give up Extensionality. The problem is that Aristotle, who first came up with potential infinity, had never heard of Extensionality. Maybe there can be a way to have both potential infinity and Extensionality.... === Subject: Re: Levy proof that R is uncountable. <87fxfa1o1j.fsf@phiwumbda.org> posting-account=U44YcwkAAAAbGXB70Qr7gA3kornmKE4i Gecko/20080922 Ubuntu/7.10 (gutsy) Firefox/2.0.0.17,gzip(gfe),gzip(gfe) The gist of Rotwang post, as well as several posts made by others, > is as follows: standard theorists will consider alternate theories > proposed in _books_, but not those proposed on _Usenet_. You lie. Again. We could possibly test this if we could arrange a comparison of some theories proposed in free books on an a full-priced Usenet... Brian Chandler === Subject: Re: Levy proof that R is uncountable. > Right, how ironic. It is ironic that these days ironic is used mainly to describe things that are not ironic at all. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Levy proof that R is uncountable. posting-account=aLpfCwoAAACh4BOs3HOlQBCoxUpEgyxc Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) so that the standard set theorists will consider the theories > at least as much as they do ZFA? > I can't speak for any standard set theorists (because I'm not a set > theorist), but if you want to know why I chose to learn about ZFA then > here's why: I was looking through the CASM reading list in the hope of > finding something interesting to read, and it mentioned Barwise and > Moss's /Vicious Circles/ as being relevant to the set theory course. I > looked at reviews online and it sounded interesting, so I bought a > copy. The reason I choose not to spend my time considering the your > theories based on so-called so-called crank posts is that they > don't sound interesting. Why would they? /Vicious Circles/ proposes > that hypersets may be used to model circular phenomena in a number of > fields. Interesting! I never imagined that hypersets could be useful in > the way that, say, classical analysis is useful. On the other hand, I suppose that the hypersets proposed by the > so-called cranks, such as having I = {I} and II = {I, II} yet > II not equaling the Quine atom, aren't as useful in the same > way that the hypersets of ZFA are useful. It seems quite conceivable to me that a theory in which there exist distinct sets I and II such that I = {I} and II = {I,II} could be useful. But you won't find such a theory by reading the posts of cranks, because cranks don't propose rigorous theories or give valid proofs. They just spout nonsense. > Your theories, on the other hand, seem to exist solely for the > purpose of allowing you to pretend that people who talk nonsense are > actually making sense. The gist of Rotwang post, as well as several posts made by > others, is as follows: standard theorists will consider alternate > theories proposed in books , but not those proposed on Usenet . Jeez. No, that isn't the gist of my post at all. > But unlike the standard theorists, I would much rather discuss an > alternate theory here than in a book, and I have a reason other > than the cost (since I'm aware that the standard theorists won't > consider cost as a valid argument against books). Here on Usenet, I can actually communicate with the inventor of > the theory. If I claim something about another person's proposed > theory that's wrong, the inventor can correct me. Many of my > posts with cranks have proceeded in this manner. On the other hand, I can't communicate with Aczel, the inventor > of the theory AFA. I don't know whether Aczel is even alive, and > even if he is, it's doubtful that he posts on Usenet or any > other forum. If I claim something wrong about AFA, Aczel isn't > there to tell me how he intends his theory to work. Actually, while reading /Vicious Circles/ I did communicate with one of the authors (the one who is still alive). I found counterexamples to a couple of theorems and also had some questions about proofs that I didn't understand, so I looked up his email address and emailed him. He was kind enough to reply and we exchanged a few further emails. I don't doubt that my communication with him was much more fruitful than your attempts to get useful feedback from the likes of MR or AP. > I obviously rank this interactivity much more highly than the > standard theorists. This is the 21st century and the Internet is > now a way of life, but the standard theorists are stuck in the > 20th century and are dependent on books. You seem to think that usenet exists as an alternative to conventional means of study. How's that working out for you? > In another thread, someone claimed to have a proof of P=NP. In Surely it is not impossible that P=NP will be proved. It is > extremely unlikely, but not impossible, that the proof will be > announced on sci.math. So Hughes expresses his skepticism about the proof (which is > in itself reasonable), but then implies how unlikely that the > proof would be on sci.math . And based on other posts I've > read in other threads, what's being implied is that a real > proof of P=NP (or even of P book , not on sci.math. No, it would most likely be announced in a journal or at a conference. Generally, books are written to teach people about well-established results, not to announce new ones. > Maybe one way to reconcile standard theorists and cranks > would be for the cranks to adopt theories that agree with > their intuitions, but can be found in some book . No, standard theorists are well aware that books get written which are full of crap - hell, even JSH can write a book (and has). The way to reconcile standard theorists and cranks would be for the cranks to adopt theories that make sense. === Subject: Re: Doubly twisted Moebius band > We probably all know how to 'make' a Moebius band: > You take a cylinder, cut it 'open' in a certain way; then > you turn one side 180 degrees and 'glue' the two sides > together again. > What if instead of turning 180 degrees, we turn it 360 degrees? > The result will obviously be homeomorphic to a cylinder. > So, we have two maps from the cylinder C = S^1 x [0,1] to R^3: > one is the 'ordinary' embedding, and the other is this 'doubly > twisted Moebius band' described above. > Now, it seems clear that these two maps are not homotopic to eachother, > at least not 'in an injective way' (is there an established term for this?). > Question 1) What is the best, smoothest, most elegant way to > prove the above, nowadays? (And is it indeed true?) Another, simple way. Let I = [-1,1] and let C = S^1 x I be the cilinder. > Let f_0, f_1 : C --> R^3 be the two inmersions. Let Z be one of the two components of the boundary of C, > and let M = S^1 x {0} be the equator in C. Then the linking > number between f_0(Z) and f_0(M) is different to the linking > number between f_1(Z) and f_1(M). Since the linking number is > invariant under isotopies, f_0 and f_1 are not isotopic. It seems to me that this can also directly be done with the two boundaries themselves (i.e. no need to bring in the equator), right? If one draws the picture, one can easily see that the two boundaries of the cylinder map to two curves that 'interlink'. But more generally bundles will indeed help for this kind of questions, much; i hadn't realized that. One more thing: Suppose we follow an inhabitant who 'travels around' on this doubly-twisted Mobius band. Then on every round trip, the inhabitant will also 'spin around its axis' over 360 degrees, so to say. That smells like some kind of group homomorphism from the H_1 group to something else (or does it rather have to do with some co-cycle in H^2?) To what algebraic construction does this intuition correspond? -- Herman Jurjus === Subject: Re: Doubly twisted Moebius band > We probably all know how to 'make' a Moebius band: > You take a cylinder, cut it 'open' in a certain way; then > you turn one side 180 degrees and 'glue' the two sides > together again. What if instead of turning 180 degrees, we turn it 360 degrees? > The result will obviously be homeomorphic to a cylinder. > So, we have two maps from the cylinder C = S^1 x [0,1] to R^3: > one is the 'ordinary' embedding, and the other is this 'doubly > twisted Moebius band' described above. > Now, it seems clear that these two maps are not homotopic to eachother, > at least not 'in an injective way' (is there an established term for > this?). Question 1) What is the best, smoothest, most elegant way to > prove the above, nowadays? (And is it indeed true?) > Another, simple way. > Let I = [-1,1] and let C = S^1 x I be the cilinder. > Let f_0, f_1 : C --> R^3 be the two inmersions. > Let Z be one of the two components of the boundary of C, > and let M = S^1 x {0} be the equator in C. Then the linking > number between f_0(Z) and f_0(M) is different to the linking > number between f_1(Z) and f_1(M). Since the linking number is > invariant under isotopies, f_0 and f_1 are not isotopic. It seems to me that this can also directly be done with the two > boundaries themselves (i.e. no need to bring in the equator), right? > If one draws the picture, one can easily see that the two boundaries of > the cylinder map to two curves that 'interlink'. But more generally bundles will indeed help for this kind of questions, > much; i hadn't realized that. One more thing: > Suppose we follow an inhabitant who 'travels around' on this > doubly-twisted Mobius band. Then on every round trip, the inhabitant > will also 'spin around its axis' over 360 degrees, so to say. That > smells like some kind of group homomorphism from the H_1 group to > something else (or does it rather have to do with some co-cycle in H^2?) > To what algebraic construction does this intuition correspond? BTW, just after posting i realized that ordinary (co-)homology won't help with this; because for ordinary (co-)homology, cylinder, Mobius band and doubly twisted Mobius band are indistinguishable (they all homotopically contract to a mere circle). -- Herman Jurjus === === Subject: Re: Property of Algebraically closed fields > Question : Suppose K is a field (not necessarily algebraically > closed) big enough to contain two algebraically > closed fields say F1 and F2. So we have: F1 < K , F2< K (as fields) we assume F1 and F2 not to be equal K. Can we assume or do we know that there is another > algebraically closed field F3 that contains both F1 > and F2? In other words.. does it make sense to talk > about the largest algebraically closed subfield of a > field (provided of course that it already contains at > least one algebraically closed field). I don't think so. Let C be the complex numbers, let > x1 and x2 be > transcendental over C, and let L be the algebraic > closure of C(x1,x2), > the field of rational functions in x1 and x2 over C. Now let F1 be an algebraic closure of C(x1) contained > in L, and let F2 > be an algebraic closure of C(x2) contained in L. > Finally, let K be > the compositum of F1 and F2 in L. I don't see why > this field would be > all of L. Do you see a rather simple argument why the compositum is not L? yields rather long-winded arguments. The whole stuff is related to Hilbert's 13th problem. > -- > Arturo Magidin > === Subject: Re: Property of Algebraically closed fields These are the steps of Abhyankar: 1. Define: a.) k be an algebraically closed field with characteristic p b.) M = k(x)^al , N=k(y)^al (^al means the algebraic closure of the resp. fields) c.) L = k(x,y)^al d.) K = MN (i.e. compositum of M and N in L) e.) M1 = k((x))^al (fraction field of power series wrt x) f.) N1 = k((y))^al .. .. g.) L1 = k((x,y))^al .. .. h.) K1 = M1N1 (in L1) 2. Let n be an integer not divisiple by p and consider the polynomial Z^n - g(x) - h(y) in K[Z] with g(x) in k[x] and h in k[y] such that g(0)=/= 0 and h(0)=/=0 In order to show K is not algebraically closed it suffices to show that this does not have a linear factor. F(Z) := Z^n - g(x) - h(y) has no linear factor in K1[Z] for arbitrary g(x) in k[x] What is a non-Cantorian set? A set for which Cantor's theorem fails. Such exotic wonders provably exist in Quine's New Foundations and related theories. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Diagonal wanderings (incongruent by construction) > Plus, there are good reasons for believing that ZFC etc. ARE > internally consistent. For example, there is an intuitively > not-unreasonable model for ZFC etc. (Goedel's constructible > universe). How is the constructible universe more not-unreasonable than the cumulative hierarchy? Does this not-unreasonableness extend also to, say, the hereditarily ordinal definable sets? Do recall the constructible universe presupposes the (wild and unfathomably impredicative) totality of all ordinals. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Diagonal wanderings (incongruent by construction) > Correct conlusion, but INCORRECT reasoning. If PA is inconsistent > but ZFC proves PA consistent, then all we can infer from those mere > facts is that ZFC proves false arithmetical statements, not > necessarily that ZFC is inconsistent. For statements of the form in question (Pi-1) we can conclude that ZFC is inconsistent from the assumption that it proves a false statement of that form. Do recall that all false Pi-1 statements are refutable in ZFC, or, equivalently, that all true Sigma-1 statements are provable in ZFC. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Diagonal wanderings (incongruent by construction) > As I've mentioned before, Cantor's original theory, naive set > theory, was quickly proved inconsistent by Russell. No it wasn't. If by naive set theory you mean the theory containing extensionality and unrestricted comprehension it is a pertinent observation no set theorist ever worked in such a theory. If by naive set theory you simply refer to the mathematical theory Cantor established it is a pertinent observation it is entirely untouched by any paradoxes. > This is the reason Zermelo came up with ZFC. A major motivation was his desire to establish the correctness of his proof of the well-ordering theorem. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Diagonal wanderings (incongruent by construction) <87fxfao0rb.fsf@alatheia.truth.invalid> posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322),gzip(gfe),gzip(gfe) > As I've mentioned before, Cantor's original theory, naive set > theory, was quickly proved inconsistent by Russell. No it wasn't. If by naive set theory you mean the theory containing > extensionality and unrestricted comprehension Why should he? He said Cantor's theory, and that did not extend itself to any of these nonsense notions. Can't you comprehend that? It simply contained the inconsisent notion of a set. Unter einer Menge verstehen wir jede Zusammenfassung von bestimmten wohlunterschiedenen Objekten unserer Anschauung oder unseres Denkens zu einem Ganzen. As we know today this inconsistency had already been well known by Cantor, but had not been recognized as being desatrous by himself. === Subject: Re: Diagonal wanderings (incongruent by construction) > Why should he? He said Cantor's theory, and that did not extend > itself to any of these nonsense notions. The notion that Cantor worked in anything resembling the theory containing extensionality and unrestricted comprehension is indeed non-sense. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Diagonal wanderings (incongruent by construction) <873abanzru.fsf@alatheia.truth.invalid> posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322),gzip(gfe),gzip(gfe) > Why should he? He said Cantor's theory, and that did not extend > itself to any of these nonsense notions. The notion that Cantor worked in anything resembling the theory > containing extensionality and unrestricted comprehension is indeed > non-sense. Yes, because Cantor abhorred axioms. === Subject: Re: Diagonal wanderings (incongruent by construction) > That seems right to me, as long as the instance of axiom schema of > separation used to carve out the contradicting function is not > impredicative (Aatu had a convincing argument about that, but, alas, > I forgot his conclusion). Well, my argument was really just a (standard and unoriginal) observation: inspecting the definition of the subset D in the diagonal argument D = {x | x not in f(x)} we find the defining formula is quantifier free (or contains only bounded quantifiers when spelled out in the language of set theory) and hence certainly predicative. Depending on the audience an elucidation of the notion of predicativity might be in order, but even then a superficial understanding suffices to render the observation immediate. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Diagonal wanderings (incongruent by construction) <4a04aa1e$0$315$b45e6eb0@senator-bedfellow.mit.edu> <87skjao277.fsf@alatheia.truth.invalid> posting-account=S6jUlgkAAAAS0KYO9CfNqTx523v1YxGt Gecko/2008102920 Firefox/3.0.4 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Well, my argument was really just a (standard and unoriginal) > observation: inspecting the definition of the subset D in the diagonal > argument D = {x | x not in f(x)}. we find the defining formula is quantifier free (or contains only > bounded quantifiers when spelled out in the language of set theory) > and hence certainly predicative. This is missing the bound. The whole point is that all these x's are in some pre-known set. You have no letter mentioning the set! That set IS the bound for this definition of D. === Subject: Re: Diagonal wanderings (incongruent by construction) > Well, my argument was really just a (standard and unoriginal) > observation: inspecting the definition of the subset D in the > diagonal argument > D = {x | x not in f(x)}. > we find the defining formula is quantifier free (or contains only > bounded quantifiers when spelled out in the language of set theory) > and hence certainly predicative. This is missing the bound. It was left implicit. > The whole point is that all these x's are in some pre-known set. > You have no letter mentioning the set! That set IS the bound for > this definition of D. Yes, but this is not relevant: all instances of separation are bounded in that sense but not all of them are predicative. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Diagonal wanderings (incongruent by construction) > Why won't you just let this spavined donkey die a natural death? - - - experience shows that such appeals to a poster's better nature are useless. (Torkel Franz.8en) -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Diagonal wanderings (incongruent by construction) > Wolfgang M.9fckenheim is a classic crank. Why do you imagine, as you seem to > do, that there is any point arguing with him? (Torkel Franzen) It is in the nature of Usenet that there will always be people who find it in their heart to debate the cranks, the trolls, the loons. Why do you imagine, as you seem to do, that there is any point posting endless reminders about the pointlessness of this activity? (Aatu Koskensilta) -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Diagonal wanderings (incongruent by construction) > The arguments of Koskensilta and McCullough in the Yet Another > Disproof Of Cantor's Theorem thread (and another thread too) do > seem sufficient to refute that the diagonal argument is > impredicative. Well, we need no substantial argument. Simple inspection shows the instances of separation in the diagonal argument are predicative. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Diagonal wanderings (incongruent by construction) > To summarize: the basic axiom needed is the power set axiom. A few > other axioms play a role, but AC is not needed and there is no > diagonal argument involved. I wonder, idly as always, what might lead anyone to conclude choice was used -- this is very elementary stuff, after all. Perhaps they're thinking, rather, of the proof that omega_1 is regular? -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Diagonal wanderings (incongruent by construction) <6sj505tf0bif530lrt8bp2a9ds9a31k2qq@4ax.com> <87my9ipjvo.fsf@alatheia.truth.invalid> To summarize: the basic axiom needed is the power set axiom. A few > other axioms play a role, but AC is not needed and there is no > diagonal argument involved. > I wonder, idly as always, what might lead anyone to conclude choice > was used -- this is very elementary stuff, after all. Perhaps they're > thinking, rather, of the proof that omega_1 is regular? I'll confess that the Wiener quote was actually in response to a question of mine (back in 1998) about whether aleph_1 could be shown to exist in ZF. I have completely forgotten what I had in mind at the time, but I vaguely recall that a similar question had come up in other threads before then. -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === Subject: Re: Diagonal wanderings (incongruent by construction) ... > You keep misinterpreting on this: as said, I have no particular > objections to defining an effective computation for the anti- > diagonal. That the sequence so defined is not in the list at any index > is what I am questioning: that *conclusion* requires a leap to > infinity that cannot be proven from the algorithmic (effective) > definitions only. IOW, that's a result that is *not* entailed by the > algorithmic (effective) properties of the definitions in question. You keep saying this, and I keep asking you to clarify what *you* mean > when you say that something is not entailed by algorithmic properties, > and you simply repeat your assertions. I have mentioned a leap to infinity that is not entailed by > effectiveness: I have put this in terms of a two opponents game, or > turing machines, or the many ways to compactification. As for now, I'm > afraid I can't be clearer than that. The mention of ciompactification simply does not help, as others have pointed out. The question is whether the antidiagonal is in the given list, not whether it may be in something else in an enlarged structure. You can choice to ask a different question, but ignoring the given question does not make it go away. As for the two person game, yes there is an asymmetry, but again it is there in the problem; one thing is given, another is constructed. The anti-diagonal is every bit as effective as the given list, though. There is no need to compute infinitely many values, neither of the given list nor of the anti-diagonal. Where exactly is this leap to infinity? > You simply ignore my remark that there is a proof of the result > you say cannot be proved. I have commented to Bishop's proof: my point (my take) is that this > simply does not solve our issue, for the reasons just mentioned. If you refuse to say what counts as > proof, then of course you make it very hard for someone to > come up with an argument you might accept -- that could well > be your intention, of course ... No such intention, but I see no need either to define what counts as a > proof: I have raised a specific objection. But not a clear one. I see you snipped the question of whether it is possible to show that a double function over natural numbers can be proved to never output 3. How would you do that, assuming you think it's possible. I can suppose here that there is a leap to infinity because there are infinitely many inputs to take into consideration. > -LV -- Alan Smaill === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse Trident/4.0; MathPlayer 2.10d; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > If you refuse to say what counts as > proof, then of course you make it very hard for someone to > come up with an argument you might accept -- that could well > be your intention, of course ... No such intention, but I see no need either to define what counts as a > proof: I have raised a specific objection. But not a clear one. That does not make you question proper. -LV === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse Trident/4.0; MathPlayer 2.10d; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > The anti-diagonal is every bit as > effective as the given list, though. There is no need to > compute infinitely many values, neither of the given list > nor of the anti-diagonal. Where exactly is this leap > to infinity? If you don't make a leap to infinity, you simply have no ground to conclude that the anti-diagonal is not in the list. Again: *** by induction only (the effective definitions) you simply cannot prove that conclusion *** !! Unless by resorting to axioms and similar forms of begging the question, that is. > I see you snipped the question of whether it is possible to > show that a double function over natural numbers can be proved > to never output 3. Sorry, but... to put it simply: you keep asking the wrong question there. -LV === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=S6jUlgkAAAAS0KYO9CfNqTx523v1YxGt Gecko/2008102920 Firefox/3.0.4 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > If you don't make a leap to infinity, you simply have no ground to > conclude that the anti-diagonal is not in the list. Again: *** by induction only (the effective definitions) > you simply cannot prove that conclusion *** !! OF COURSE you can prove the conclusion by induction! But you DON'T NEED to! The conduction IS ACTUALLY proved by UNIVERSAL GENERALIZATION! > Unless by resorting to axioms and similar forms of begging the > question, that is. There is no question TO beg. There is NO QUESTION that Cantor's theorem is true. You CANNOT attack a proof for begging the question. That the question HAS been begged is what EVERY proof proves! === Subject: Re: Diagonal wanderings (incongruent by construction) ... > You keep misinterpreting on this: as said, I have no particular > objections to defining an effective computation for the anti- > diagonal. That the sequence so defined is not in the list at any index > is what I am questioning: that *conclusion* requires a leap to > infinity that cannot be proven from the algorithmic (effective) > definitions only. IOW, that's a result that is *not* entailed by the > algorithmic (effective) properties of the definitions in question. You keep saying this, and I keep asking you to clarify what *you* mean > when you say that something is not entailed by algorithmic properties, > and you simply repeat your assertions. I have mentioned a leap to infinity that is not entailed by > effectiveness: I have put this in terms of a two opponents game, or > turing machines, or the many ways to compactification. As for now, I'm > afraid I can't be clearer than that. The mention of ciompactification simply does not help, as others have pointed out. The question is whether the antidiagonal is in the given list, not whether it may be in something else in an enlarged structure. You can choice to ask a different question, but ignoring the given question does not make it go away. As for the two person game, yes there is an asymmetry, but again it is there in the problem; one thing is given, another is constructed. The anti-diagonal is every bit as effective as the given list, though. There is no need to compute infinitely many values, neither of the given list nor of the anti-diagonal. Where exactly is this leap to infinity? > You simply ignore my remark that there is a proof of the result > you say cannot be proved. I have commented to Bishop's proof: my point (my take) is that this > simply does not solve our issue, for the reasons just mentioned. If you refuse to say what counts as > proof, then of course you make it very hard for someone to > come up with an argument you might accept -- that could well > be your intention, of course ... No such intention, but I see no need either to define what counts as a > proof: I have raised a specific objection. But not a clear one. I see you snipped the question of whether it is possible to show that a double function over natural numbers can be proved to never output 3. How would you do that, assuming you think it's possible. I can suppose here that there is a leap to infinity because there are infinitely many inputs to take into consideration. > -LV -- Alan Smaill === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > WM is trying to prove ii: if > all natural numbers and all lines exist > then > there exists a line that contains all natural numbers and: there cannot exist a line that contains all natural numbers But I am not trying to prove that, because the proof is complete. I am trying to help you to understand that proof. Remark: You agreed that every line can be reached by induction. No, I agreed that any line can be reached by induction. Let is make clear: You agreed that there is no line that cannot be > reached by induction. Yes Easier: There is no line that cannot be painted red. Yes Easiest: Every line can be painted red. No you want things to be clear. Say : *Each* line can be painted red. > You play word games with the word every So we have You can reach any line You can reach any line. However, reaching a line does > not produce one line that contains all lines _You_ are playing word games with the word every. If there is no line that cannot be painted, then every line can be painted. If every line can be painted, then all lines can be painted. No is the complement of all in all cases in which all exists as a completed entity. > unless less the line you reach is the last line. > It does not matter how many lines you reach. Unless you > reach the last line you do not produce one > line that contains all lines. It does not matter what I reach or not reach. If no line remains unpainted, then all lines are painted. And you agreed that there is no line that cannot be painted red. So if you reach line one in 1/2 second, line > 2 in 1/4 second and so on, after one second > you will have reached all lines, but since > no line was the last line, you have not produced > one line that contains all lines. First let us stay with painting red. After you will have understood this, we can switch to the related case where painting red is replaced by putting all previous stuff into one line. Please confirm: 1) If there is no line that is not red, then all lines are red. 2) If there is no line that cannot become red, then all lines can become red. 3) If there is no line that cannot be painted red, then all lines can be painted red. Unless you can confirm these basics, further discussion is idle. === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > WM has agreed that i: if all natural numbers and all lines exist then there does not exist a line that contains all natural numbers is true. > WM is trying to prove ii: if > all natural numbers and all lines exist > then > there exists a line that contains all natural numbers and: there cannot exist a line that contains all natural numbers Nope, you are trying to show all natural numbers and all lines exist if false by proving ii (since we both agree that i: holds). - William Hughes === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322),gzip(gfe),gzip(gfe) > WM has agreed that i: if > all natural numbers and all lines exist > then > there does not exist a line that contains all natural numbers is true. WM is trying to prove ii: if > all natural numbers and all lines exist > then > there exists a line that contains all natural numbers and: there cannot exist a line that contains all natural numbers Nope, you are trying to show all natural numbers and all lines > exist > if false by proving ii (since we both agree that i: holds). The question is not what we both agree but what is fact. === Subject: Re: Diagonal wanderings (incongruent by construction) <20090506222934.N59609@agora.rdrop.com> posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) every set of natural numbers has even or odd > cardinality. That's incorrect. Only finite sets have even or odd cardinality. MoeBlee There are not much more, are there? Albrecht === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse Trident/4.0; MathPlayer 2.10d; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) Not that part, it's the conclusion that is problematic. We have a > construction of the list (our 'f'), then we define the construction > of > an item ('d') so that it differs in each place for each item in the > list given by 'f'. If we now run this game of 'f' against 'd', there > is no winner, yet Cantor's conclusion is that the anti-diagonal wins. Your comments reflect a fundamental misunderstanding of basic logic > specifically the principle of universal generalization. There is no > game of f against d. That is a figment borne of your misunderstanding > the basic logic. You have no idea what basic logic is. -LV === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322),gzip(gfe),gzip(gfe) > Not that part, it's the conclusion that is problematic. We have a > construction of the list (our 'f'), then we define the construction > of > an item ('d') so that it differs in each place for each item in the > list given by 'f'. If we now run this game of 'f' against 'd', there > is no winner, yet Cantor's conclusion is that the anti-diagonal wins. > Your comments reflect a fundamental misunderstanding of basic logic > specifically the principle of universal generalization. There is no > game of f against d. That is a figment borne of your misunderstanding > the basic logic. You have no idea what basic logic is. You can tell yourself that if you want, but based on what > people have said here that's not the way things look. The theorem says precisely that given a list f of reals > there is a real d which does not appear on the list. But the theorem withholds the fact that, if actual infinity exists, then it is impossible to check whether d appears on the list. The theorem is only valid for all finite initial segments of the list. But as we have heard, the list is more than all its finite initial segments. === Subject: Re: Diagonal wanderings (incongruent by construction) Nntp-Posting-Host: hera.cwi.nl ... > You can tell yourself that if you want, but based on what > people have said here that's not the way things look. > > The theorem says precisely that given a list f of reals > there is a real d which does not appear on the list. > > But the theorem withholds the fact that, if actual infinity exists, > then it is impossible to check whether d appears on the list. The > theorem is only valid for all finite initial segments of the list. The theorem is not about segments, it is about *lines*. And it holds for all lines in final positions on the list, and so for all lines on the list as there are no other lines. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322),gzip(gfe),gzip(gfe) > ... > You can tell yourself that if you want, but based on what > people have said here that's not the way things look. > The theorem says precisely that given a list f of reals > there is a real d which does not appear on the list. > But the theorem withholds the fact that, if actual infinity exists, > then it is impossible to check whether d appears on the list. The > theorem is only valid for all finite initial segments of the list. The theorem is not about segments, it is about *lines*. And it holds for all > lines in final positions on the list, and so for all lines on the list as there > are no other lines. The theorem that any line of the list 1 1, 2 1, 2, 3 ... contains all natural numbers that are contained in its predecessors, also holds for exactly the same reason for all lines of the list, namely as there are no other lines. === Subject: Re: Diagonal wanderings (incongruent by construction) Nntp-Posting-Host: hera.cwi.nl ... > The theorem is not about segments, it is about *lines*. And it holds for > all lines in final positions on the list, and so for all lines on the list > as there are no other lines. > > The theorem that any line of the list > > 1 > 1, 2 > 1, 2, 3 > ... > > contains all natural numbers that are contained in its predecessors, > also holds for exactly the same reason for all lines of the list, > namely as there are no other lines. Yes. What is the relevance? -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322),gzip(gfe),gzip(gfe) > ... > Less than omega is not enough to evade the case that the diagonal > of the list > > 0.0 > 0.1 > 0.11 > 0.111 > ... > is not distinct from every line. > > Pray tell to which line it is equal. > > If all lines exist, then N is in a line. > > By which rule? (And we were talking about the diagonal...) > > By the rule that the diagonal cannot exist without lines. The diagonal > is made from the last digits of the lines. If the diagonal were longer > than every line, then it could not exist. Wrong. No. The diagonal is made from the last digits of the lines. > The diagonal has more digits > than every line. Right. > Hence there must be a line that has more digits than > every line. Or actual infinity is wrong a concept. On what is the hence based? The diagonal is made from the last digits of the lines. But as WM has left this thread (as is customary when it becomes clear that > he cannot win the argument except by circularity), I left the thread because it will get confusing after exploding into pieces beyond the 1001th contribution. === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322),gzip(gfe),gzip(gfe) > ... > The theorem is not about segments, it is about *lines*. And it holds for > all lines in final positions on the list, and so for all lines on the list > as there are no other lines. > The theorem that any line of the list > 1 > 1, 2 > 1, 2, 3 > ... > contains all natural numbers that are contained in its predecessors, > also holds for exactly the same reason for all lines of the list, > namely as there are no other lines. Yes. What is the relevance? If all there are all natural numbers and all ines in the list (and nothing else), then all natural umbers are in one line. This contradicts the assumption. === Subject: Re: Diagonal wanderings (incongruent by construction) Nntp-Posting-Host: hera.cwi.nl ... > The theorem that any line of the list > > 1 > 1, 2 > 1, 2, 3 > ... > > contains all natural numbers that are contained in its predecessors, > also holds for exactly the same reason for all lines of the list, > namely as there are no other lines. > > Yes. What is the relevance? > > If all there are all natural numbers and all ines in the list (and > nothing else), then all natural umbers are in one line. By what logical reasoning do you come to that conclusion? -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=yKimjgoAAACk5WwPVD4l9HmbpoR6Hmy4 Presto/2.1.1,gzip(gfe),gzip(gfe) > ... > The theorem is not about segments, it is about *lines*. And it holds for > all lines in final positions on the list, and so for all lines on the list > as there are no other lines. > The theorem that any line of the list > 1 > 1, 2 > 1, 2, 3 > ... > contains all natural numbers that are contained in its predecessors, > also holds for exactly the same reason for all lines of the list, > namely as there are no other lines. Yes. What is the relevance? If all there are all natural numbers and all ines in the list (and > nothing else), then all natural umbers are in one line. > This contradicts the assumption. I'm not sure how you get the conclusion from the premise there. Perhaps you could clarify? Why does A) All natural numbers and all lines are in [what does this mean?] the list (and nothing else) imply B) All natural numbers are in one line ? === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse Trident/4.0; MathPlayer 2.10d; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > It's you who are pretending my responses do not exist. No, I've answered you point by point. You better take a course in basic reasoning before digging any further into the technicalities. -LV === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=S6jUlgkAAAAS0KYO9CfNqTx523v1YxGt Gecko/2008102920 Firefox/3.0.4 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > You better take a course in basic reasoning before digging any further > into the technicalities. If you think you can keep talking about reasoning and argument INSTEAD OF PROOFS then YOU are the one who needs help with basics, and I don't think a course will suffice to give it to you. === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse Trident/4.0; MathPlayer 2.10d; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > We can show that uncountable sets exist without using the diagonal > argument. > And could you tell which argument we would use instead? This is from a post by Matthew P. Wiener in 1998: On the one hand, P(NxN) existing is straightforward. Therefore, by > comprehension, so is the P(NxN) subset of those relations on N which > are well-ordered. Call it WO. The von Neumann ordinals can be defined--in one fell swoop--as > well-ordered (by membership) transitive sets. (z is transitive means > x e y e z => x e z.) The elementary theory of well-ordered sets is easy to develop without AC. > In particular, one can prove there exists a unique von Neumann ordinal > of the same order type for any given well-ordered set. (For starts, > this is via the Mostowski collapse: given w well-ordered, let mc(w)= > {mc(x):x < w}. By definition by transfinite recursion, this is > well-defined, and rather easy to prove is transitive.) At this point, apply the axiom of replacement to WO, replacing each > well-ordered relation on N with its von Neumann order type ordinal. The resulting set is aleph 1. For some reason, there is a folklore belief that choice is needed here. > I've even seen this claim show up in books on set theory! (end quote from Matthew P. Wiener) To summarize: the basic axiom needed is the power set axiom. A few other > axioms play a role, but AC is not needed and there is no diagonal argument > involved. set axiom is irrelevant to a discussion on the diagonal argument (just as Levy's proof is irrelevant). -LV === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=S6jUlgkAAAAS0KYO9CfNqTx523v1YxGt Gecko/2008102920 Firefox/3.0.4 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > set axiom is irrelevant to a discussion on the diagonal argument You're just STUPID. THERE IS NO argument!! THERE IS *ONLY* a proof! That is THE ONLY thing happening here! And you can't SAY leverage the power set axiom!! THAT IS NOT happening! The only thing the powerset axiom says is that the existing subsets can be thought of as being all&only the elements of a known collection! That CAN'T BE leveraged! That's just the way it is! What makes you think you can DISallow a collection when you already have ALLOWED everything IN the collection?? === Subject: Re: Diagonal wanderings (incongruent by construction) <6sj505tf0bif530lrt8bp2a9ds9a31k2qq@4ax.com> argument. > And could you tell which argument we would use instead? > This is from a post by Matthew P. Wiener in 1998: > [ snip quote ] > To summarize: ?the basic axiom needed is the power set axiom. ?A few other > axioms play a role, but AC is not needed and there is no diagonal argument > involved. > set axiom is irrelevant to a discussion on the diagonal argument (just > as Levy's proof is irrelevant). That is not the current context. You responded to a statement that the reals are uncountable by claiming That ultimately depends on your stance on the diagonal argument. I was pointing out that the existence of uncountable sets does *not* depend on the diagonal argument. In fact, we can also show the reals are uncountable without using a diagonal argument. -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse Trident/4.0; MathPlayer 2.10d; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > No, I agreed that any line can be reached by induction. > You can reach any line. However, reaching a line does > not mean finishing unless the line is the last line. > There is no last line so you can't finish. On the same line of reasoning, you cannot finish the anti-diagonal either. -LV === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=S6jUlgkAAAAS0KYO9CfNqTx523v1YxGt Gecko/2008102920 Firefox/3.0.4 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) No, I agreed that any line can be reached by induction. > You can reach any line. However, reaching a line does > not mean finishing unless the line is the last line. > There is no last line so you can't finish. On the same line of reasoning, you cannot finish the anti-diagonal > either. On that line of reasoning, you cannot finish THE LIST, since you can't get to ITS last line, since it doesn't have one. On that line of reasoning, you can't finish even ONE NUMBER (one subset, one bit-string) on the list, because no bit-string has a last bit. On that line of reasoning, you can't finish the DIAGONAL ( we are NOT talking about the ANTI-diagonal, but just the DIAGONAL), because IT doesn't have a last bit. Whether you can or can't FINISH anything has NOTHING to do with whether it does or doesn't EXIST!! The concept you are not understanding here is THE INFERENCE RULE OF UNIVERSAL GENERALIZATION. This rule applies to EVERYthing in a domain, REGARDLESS of how the domain is ordered and regardless of whether it can be finished or what size it is. === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I No, I agreed that any line can be reached by induction. > You can reach any line. However, reaching a line does > not mean finishing unless the line is the last line. > There is no last line so you can't finish. On the same line of reasoning, you cannot finish the anti-diagonal > either. > You will never have a time of the last step, but if you do the usual trick of squeezing an infinite number of steps into a finite time, you can have a time after all steps are done. The difference is that you can create an anti-diagonal without doing a last step. You cannot create a line containing all other lines without doing a last step. - William Hughes === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322),gzip(gfe),gzip(gfe) > No, I agreed that any line can be reached by induction. > You can reach any line. However, reaching a line does > not mean finishing unless the line is the last line. > There is no last line so you can't finish. On the same line of reasoning, you cannot finish the anti-diagonal > either. You will never have a time of the last step, but if > you do the usual trick of squeezing an infinite number > of steps into a finite time, you can have a time after all > steps are done. The difference is that you can create an > anti-diagonal without doing a last step. You cannot create > a line containing all other lines without doing a last step. Therefore we *prove* using logics (If there are infinitely many, then there are a least two) and mathematical reasoning (if not all numbers are in one line, but if all numbers and all lines are in the list, then there must be at least two lines, A and B, such that for at least two numbers, a and b, (a e A & b !e A) & (a !e B & b e B), if all numbers are there, then they are in one line. Please confirm: 1) If there is no line that is not red, then all lines are red. 2) If there is no line that cannot become red, then all lines can become red. 3) If there is no line that cannot be painted red, then all lines can be painted red. Unless you can confirm these basics, further discussion is idle. === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322),gzip(gfe),gzip(gfe) > No, I agreed that any line can be reached by induction. > You can reach any line. However, reaching a line does > not mean finishing unless the line is the last line. > There is no last line so you can't finish. On the same line of reasoning, you cannot finish the anti-diagonal > either. You will never have a time of the last step, but if > you do the usual trick of squeezing an infinite number > of steps into a finite time, you can have a time after all > steps are done. The difference is that you can create an > anti-diagonal without doing a last step. You cannot create > a line containing all other lines without doing a last step. > Is this because of the same reason that some magicans can transform gold into ivory but not ivory into gold? Remark: The anti-diagonal contains all other lines, at least one element of *all* other lines. === Subject: Re: Diagonal wanderings (incongruent by construction) No, I agreed that any line can be reached by induction. > You can reach any line. However, reaching a line does > not mean finishing unless the line is the last line. > There is no last line so you can't finish. On the same line of reasoning, you cannot finish the anti-diagonal > either. Why do you think this is a problem? Why would you place the bar higher for the anti-diagonal? > -LV -- Alan Smaill === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse Trident/4.0; MathPlayer 2.10d; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) No, I agreed that any line can be reached by induction. > You can reach any line. However, reaching a line does > not mean finishing unless the line is the last line. > There is no last line so you can't finish. On the same line of reasoning, you cannot finish the anti-diagonal > either. Why do you think this is a problem? Why would you place the bar higher for the anti-diagonal? Why everything always upside down? It's *Cantor*'s argument that places the bar higher: my whole point has been objecting to *that* (the leap to infinity; or, the diagonal that is complete while the list is not; etc. etc). -LV === Subject: Re: Diagonal wanderings (incongruent by construction) No, I agreed that any line can be reached by induction. > You can reach any line. However, reaching a line does > not mean finishing unless the line is the last line. > There is no last line so you can't finish. On the same line of reasoning, you cannot finish the anti-diagonal > either. Why do you think this is a problem? Why would you place the bar higher for the anti-diagonal? Why everything always upside down? It's *Cantor*'s argument that > places the bar higher: my whole point has been objecting to *that* > (the leap to infinity; or, the diagonal that is complete while the > list is not; etc. etc). But the argument in the case at hand doess *not* place the bar any higher for the anti-diagonal. It's *you* that insist on leaping to infinity (by introducing talk about compactification, for example). > -LV -- Alan Smaill === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse Trident/4.0; MathPlayer 2.10d; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) No, I agreed that any line can be reached by induction. > You can reach any line. However, reaching a line does > not mean finishing unless the line is the last line. > There is no last line so you can't finish. On the same line of reasoning, you cannot finish the anti-diagonal > either. Why do you think this is a problem? Why would you place the bar higher for the anti-diagonal? Why everything always upside down? It's *Cantor*'s argument that > places the bar higher: my whole point has been objecting to *that* > (the leap to infinity; or, the diagonal that is complete while the > list is not; etc. etc). But the argument in the case at hand doess *not* place the bar any > higher for the anti-diagonal. I just do not see how you can deny such a thing, given that: *** by induction only (the effective definitions) you simply cannot prove that conclusion. *** > It's *you* that insist on leaping to > infinity (by introducing talk about compactification, for example). What is your objection, if any, to my consideration above? Can you perhaps show how to go from the inductive (effective) definition to a conclusion about the whole infinite set? If you can't, Cantor's argument is invalidated: if you can, then I'll be able to use the very same tool to build an effective complete list, and Cantor's argument will simply resolve to the most unhappy possible choice among others. William Highes: I agreed that any line can be reached by induction. You can reach any line. However, reaching a line does not mean finishing unless the line is the last line. There is no last line so you can't finish. Neither you can! -LV === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse Trident/4.0; MathPlayer 2.10d; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) No, I agreed that any line can be reached by induction. > You can reach any line. However, reaching a line does > not mean finishing unless the line is the last line. > There is no last line so you can't finish. On the same line of reasoning, you cannot finish the anti-diagonal > either. Why do you think this is a problem? Why would you place the bar higher for the anti-diagonal? Why everything always upside down? It's *Cantor*'s argument that > places the bar higher: my whole point has been objecting to *that* > (the leap to infinity; or, the diagonal that is complete while the > list is not; etc. etc). But the argument in the case at hand doess *not* place the bar any > higher for the anti-diagonal. I just do not see how you can deny such a thing, given that: *** by induction only (the effective definitions) > you simply cannot prove that conclusion. *** It's *you* that insist on leaping to > infinity (by introducing talk about compactification, for example). What is your objection, if any, to my consideration above? Can you > perhaps show how to go from the inductive (effective) definition to a > conclusion about the whole infinite set? If you can't, Cantor's argument is invalidated: if you can, then I'll > be able to use the very same tool to build an effective complete list, > and Cantor's argument will simply resolve to the most unhappy possible > choice among others. William Highes: Sorry for the misspelling. -LV > I agreed that any line can be reached by induction. > You can reach any line. However, reaching a line does > not mean finishing unless the line is the last line. > There is no last line so you can't finish. Neither you can! === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) No, I agreed that any line can be reached by induction. > You can reach any line. However, reaching a line does > not mean finishing unless the line is the last line. > There is no last line so you can't finish. On the same line of reasoning, you cannot finish the anti-diagonal > either. Exactly that is the point. The usual escape of set theorists: We do not finish, but we define it for all numbers simultaneously. Obviously nonsense. In order to define something for a number that number must be known. That means one has to be able to count up to this number. But even if all is possibly done simultaneously, once and for all, then we can utilize that means too. === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > then they likely > believe that no proof is possible. And if this is really > the case, then this would actually prove the point that > I was making to Mariano -- that the more time passes, > the less likely a proof of ~Con(ZFC) will be found, and > the fewer mathematicians will even search for a proof! > Except that continued research in ZFC itself exposes ZFC to scrutiny > for contradiction. Every theorem proved allows that someone may prove > the negation of that theorem. So far, this discussion/subthread began with Knox, who first mentioned that finding a proof that ZFC is inconsistent was unlikely at best. I agreed with her on that point and tried to quantify how unlikely it was by comparing it to the likelihood that conjectures such as Riemann or Goldbach would someday be proved. Mariano questioned my claim about the relative likelihood of the three proofs (and eventually stated the neutrality of his opinion regarding the relative likelihood of the proofs). I tried to explain why I thought that Riemann and Goldbach were more likely to be proved than ~Con(ZFC), which led to MoeBlee's line of questioning. In between all of this, Feldmann mentioned how implausible it would be for ZFC or PA to be proved inconsistent. So what's the point of all this? What I was trying to do was characterize the standard theorists' opinion of how likely a proof of ~Con(ZFC) would be. Perhaps, instead of trying to compare ~Con(ZFC) to Riemann or Goldbach (which led to dead end questions from Mariano and MoeBlee), let me give a quote from MH Knowles, another so-called crank who once tried to prove ZFC inconsistent: ALL mathematicians believe that it is theoretically possible that set theory [e.g., ZFC] is inconsistent, but NO mathematicians believe it is actually possible that set theory [ZFC] is inconsistent. (emphasis the author's) And so MoeBlee's post gives evidence for the first half of this theoretical possibility that this theory will someday be proved inconsistent. Meanwhile, Knox's post and eventually Feldmann's as well give universe V=L is sufficient to convince her that ZFC will never actually be proved inconsistent, and Feldmann adds how implausible the possibility of an inconsistency proof actually is. And so that one quote from MH Knowles succinctly summarizes the standard set theorist's position on the consistency of ZFC. I repeat the quote for emphasis: ALL mathematicians believe that it is theoretically possible that set theory [e.g., ZFC] is inconsistent, but NO mathematicians believe it is actually possible that set theory [ZFC] is inconsistent. Well, except for one mathematician, that is. Ed Nelson does believe that it's actually possible that PA, and therefore ZFC, is inconsistent. A natural question to wonder is, if Nelson can really prove that PA and ZFC are inconsistent, then why did it take over a century from Zermelo to the inconsistency proof? And the reason I gave earlier in this thread is that the proof might require the operation of tetration or superexponentiation. Relatively few mathematicians have ever heard of this operation, and so they wouldn't have found a proof that requires it. Just as Fermat had never heard of the necessary elliptic curves to prove FLT, so most mathematicians have never heard of tetration required to prove ~Con(PA). Will Nelson be successful? Who knows? But if Nelson fails, I suspect, like most set theorists and the mathematicians from the Knowles quote, that ZFC will never be proved inconsistent. === Subject: Re: Diagonal wanderings (incongruent by construction) > A natural question to wonder is, if Nelson can really prove that PA > and ZFC are inconsistent, then why did it take over a century from > Zermelo to the inconsistency proof? And the reason I gave earlier in > this thread is that the proof might require the operation of > tetration or superexponentiation. Relatively few mathematicians have > ever heard of this operation, and so they wouldn't have found a > proof that requires it. Just as Fermat had never heard of the > necessary elliptic curves to prove FLT, so most mathematicians have > never heard of tetration required to prove ~Con(PA). A wonderfully baffling piece of baseless speculation! In return I offer the suggestion that one day Quine's New Foundations will be proved consistent but omega-inconsistent as a result of vigorous pondering of the mysteries of the epsilon-0-th iterate of the Ackermann function. Earlier you propounded certain profound remarks about extensionality and the axiom of foundation. Here's some grist for your mill: we're naturally wont to regard extensionality and foundation as mathematically inconsequential, purely aesthetic as it were, but, as I recently learned from a talk by Rathjen, Dana Scott has proved a rather surprising theorem, that ZF without extensionality is of the same strength as Zermelo set theory. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Diagonal wanderings (incongruent by construction) <87vdo6pke9.fsf@alatheia.truth.invalid> posting-account=S6jUlgkAAAAS0KYO9CfNqTx523v1YxGt Gecko/2008102920 Firefox/3.0.4 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Here's some grist for your mill: we're > naturally wont to regard extensionality and foundation as > mathematically inconsequential, purely aesthetic as it were Oh, come ON. Extensionality is basic. It's the definitional feature of the enterprise. It's what makes sets sets. === Subject: Re: Diagonal wanderings (incongruent by construction) <87vdo6pke9.fsf@alatheia.truth.invalid> posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322),gzip(gfe),gzip(gfe) > A natural question to wonder is, if Nelson can really prove that PA > and ZFC are inconsistent, then why did it take over a century from > Zermelo to the inconsistency proof? And the reason I gave earlier in > this thread is that the proof might require the operation of > tetration or superexponentiation. Relatively few mathematicians have > ever heard of this operation, and so they wouldn't have found a > proof that requires it. Just as Fermat had never heard of the > necessary elliptic curves to prove FLT, so most mathematicians have > never heard of tetration required to prove ~Con(PA). A wonderfully baffling piece of baseless speculation! In return I > offer the suggestion that one day Quine's New Foundations will be > proved consistent but omega-inconsistent as a result of vigorous > pondering of the mysteries of the epsilon-0-th iterate of the > Ackermann function. Or maybe it will be recognized that omega-consistence is as reasonable a notion as consistent inconsistence and completed incompleteness? Earlier you propounded certain profound remarks about extensionality > and the axiom of foundation. Here's some grist for your mill: we're > naturally wont to regard extensionality and foundation as > mathematically inconsequential, purely aesthetic as it were, but, as I > recently learned from a talk by Rathjen, Dana Scott has proved a > rather surprising theorem, that ZF without extensionality is of the > same strength as Zermelo set theory. Small wonder. Everything of that stuff is of same strength, because: Set theory is wrong (Ludwig Wittgenstein). > Wovon mann nicht sprechen kann, dar.9fber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Will Nelson be successful? Who knows? But if Nelson fails, I > suspect, like most set theorists and the mathematicians from > the Knowles quote, that ZFC will never be proved inconsistent. Wishful blindness? Mathematics deals with mathematical numbers, i.e., numbers that can be defined as individuals. ZF proves that there are mathematical numbers that cannot be defined as individuals, hence are not mathematical numbers. A perfect contradiction. Or see the binary tree and the clueless counterarguments like 0 + 0 + 0 + ... > alep_0 === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) Mathematics deals with mathematical numbers, i.e., numbers that can be > defined as individuals. > ZF proves that there are mathematical numbers that cannot be defined > as individuals, hence are not mathematical numbers. > A perfect contradiction. Contradiction fail again AGAIN! He's a triple-threat. Marshall === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=S6jUlgkAAAAS0KYO9CfNqTx523v1YxGt Gecko/2008102920 Firefox/3.0.4 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > It is (AFAIK) easy enough to come up with a production of all the > infinite binary strings. A production is NOT a LIST. It is in fact NOT possible, not as far as you OR ANYbody knows, to come up with a LIST of ALL the denumerably-wide bit-strings. > The problem remains how to make the list > bullet-proof to diagonalisation. That is not a problem. Every square list has a diagonal; there is simply nothing you can do about that. Every bit-string has a complement -- there is nothing you can do about that either. If the square list exists, its diagonal MUST exist. If a bit-string exists, its complement MUST ALSO exist. Therefore, the anti-diagonal MUST exist. You DO NOT GET to even TALK about games or bulletproof. === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse Trident/4.0; MathPlayer 2.10d; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) It is (AFAIK) easy enough to come up with a production of all the > infinite binary strings. A production is NOT a LIST. > It is in fact NOT possible, not as far as you OR ANYbody knows, to > come up > with a LIST of ALL the denumerably-wide bit-strings. What you are saying is false: there is not standard math only. > The problem remains how to make the list > bullet-proof to diagonalisation. That is not a problem. > Every square list has a diagonal; there is simply nothing you can do > about that. That list can be all but square: you keep speaking nonsense, and that's the only thing I can do nothing about. -LV === Subject: Re: Diagonal wanderings (incongruent by construction) <6sj505tf0bif530lrt8bp2a9ds9a31k2qq@4ax.com> To summarize: ?the basic axiom needed is the power set axiom. ?A few other > axioms play a role, but AC is not needed and there is no diagonal argument > involved. > Do you mesan that power set axiom that guarantees the existence of > *all* elements of the power set? Or do you mean a power set axiom as > it is required to produce a countable power set of omega (countable > from the outside of course)? I'm not sure I even know what it means to guarantee the existence of *all* elements of the power set. There is only one power set axiom (in ZFC). What it guarantees is simply that if X is a set, then there is a set Y such that for every S, if S is a subset of X, then S is an element of Y. The power set of omega is uncountable when viewed from the inside. -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > To summarize: ?the basic axiom needed is the power set axiom. ?A few other > axioms play a role, but AC is not needed and there is no diagonal argument > involved. > Do you mesan that power set axiom that guarantees the existence of > *all* elements of the power set? Or do you mean a power set axiom as > it is required to produce a countable power set of omega (countable > from the outside of course)? I'm not sure I even know what it means to guarantee the existence of > *all* elements of the power set. It means that also such elements exist, that cannot be defined as individuals (in a given fixed language) because the number of definitions is countable. > There is only one power set axiom (in > ZFC). What it guarantees is simply that if X is a set, then there is a > set Y such that for every S, if S is a subset of X, then S is an element > of Y. The question is, however, must someone be able to name / specify the subset S? Or is it enough to believe that are all (uncountably many)? To put it in other words: Your phrase if S is a subset of X: does it cover all possible subsets or only those that can be picked? > The power set of omega is uncountable when viewed from the > inside. The power set of the smallest infinite set, omega or N or a set isomorphic to N, is always uncountable, if it is complete and if Cantor's or Hessenberg's proofs hold. === Subject: Re: Diagonal wanderings (incongruent by construction) <6sj505tf0bif530lrt8bp2a9ds9a31k2qq@4ax.com> axioms play a role, but AC is not needed and there is no diagonal argument > involved. > Do you mesan that power set axiom that guarantees the existence of > *all* elements of the power set? Or do you mean a power set axiom as > it is required to produce a countable power set of omega (countable > from the outside of course)? > I'm not sure I even know what it means to guarantee the existence of > *all* elements of the power set. > It means that also such elements exist, that cannot be defined as > individuals (in a given fixed language) because the number of > definitions is countable. The axiom says nothing about whether elements can be defined as individuals. > There is only one power set axiom (in > ZFC). What it guarantees is simply that if X is a set, then there is a > set Y such that for every S, if S is a subset of X, then S is an element > of Y. > The question is, however, must someone be able to name / specify the > subset S? Or is it enough to believe that are all (uncountably many)? > To put it in other words: Your phrase if S is a subset of X: does it > cover all possible subsets or only those that can be picked? It covers all the subsets that exist, whether they can be picked or not. There is no way to guarantee that the number of subsets of omega is really uncountable, as seen from the outside, because the axiom can only talk about sets that exist, not sets that don't exist. > The power set of omega is uncountable when viewed from the > inside. > The power set of the smallest infinite set, omega or N or a set > isomorphic to N, is always uncountable, if it is complete and if > Cantor's or Hessenberg's proofs hold. -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322),gzip(gfe),gzip(gfe) > I'm not sure I even know what it means to guarantee the existence of > *all* elements of the power set. > It means that also such elements exist, that cannot be defined as > individuals (in a given fixed language) because the number of > definitions is countable. The axiom says nothing about whether elements can be defined as > individuals. It does not because that is self-evident, or at least, it should be. > There is only one power set axiom (in > ZFC). What it guarantees is simply that if X is a set, then there is a > set Y such that for every S, if S is a subset of X, then S is an element > of Y. > The question is, however, must someone be able to name / specify the > subset S? Or is it enough to believe that are all (uncountably many)? > To put it in other words: Your phrase if S is a subset of X: does it > cover all possible subsets or only those that can be picked? It covers all the subsets that exist, whether they can be picked or > not. How do we prove existence of sets that cannot be picked (= defined as individuals)? > There is no way to guarantee that the number of subsets of omega is > really uncountable, as seen from the outside, because the axiom can > only talk about sets that exist, not sets that don't exist. > Is it possible to put internal omega in (an external) bijection with external omega? === Subject: Re: Mathematical theater of the absurd >The actual infinite is not required for the mathematics of the >physical world (Soloman Feferman) That's his mistake. Why do you think Solomon Feferman is mistaken? Before jumping into any rash conclusion it's a good idea to take into account the context of such statements as quoted above. Here you should recall Feferman has been heavily involved in the project to determine what mathematical principles are essential to these and those mathematical results and techniques, in particular in the study of predicativism. It is certainly plausible that most of the mathematical machinery necessary in physics can be formalised so as to be conservative over PA or even PRA. As the saying goes, the need for powerful mathematical principles and abstractions is probably not a logical need, but a more subtle matter. > But what the philosophers say that scientists do has very little > connection with what scientists actually do. We may further observe that often what scientists say that scientists do has very little connection with what scientists actually do. > Falsificationism turns out to be unfalsifiable, so by its own > criteria it ought to be rejected. Why? No one has claimed the falsifiability criterion is a scientific claim or theory, certainly not Popper. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Mathematical theater of the absurd > A Dedekind cut is an infinite set. [NonBreakingSpace]Infinite sets have only a > potential existence. Hence, the length of the diagonal of a unit square exists only potentially. Did I say that right? Usually when we say something, we have a purpose for saying it. What > would be the purpose for saying that? To clarify that the length (which is a number) of the diagonal does not have an actual existence. Is that what you mean? > I can only repeat things I've already said. I don't know why you have > so much trouble understanding what I say. I was dropped on my head as a child. Perhaps that has something to do with it. Or perhaps the problem I have understanding you stems from your inability to express yourself clearly. On the other hand, have you considered the possibility that the problem may be that I *do* understand you? Well, let's not bother any more with why I do or don't understand you, 'kay? It's off-topic really. > The idea is that infinity is merely a useful fiction, or equivalently, > a figure a speech. A figure of speech? Sure, why not? Some people think that integers are just figures of speech, useful fictions. > The only properties that infinity has are useful properties. I would say that if infinity is merely a figure of speech, then the only properties it would have would be those we ascribe to it, whether they be useful or not. > That is, we're trying to understand the world we observe, > and any given observation contains only finite information. Infinity > is useful when it helps us (as a conceptual aid) to understand those > finite observations. When we think of infinity in that way, then we > say, just once, that infinity has only a potential existence, but > then we never ever have to mention the word potential again. Fine by me. I think it might be less contentious to say here that infinity has a hypothetical existence, rather than a potential existence. Hypothesizing entities for the purposes of explaining phenomena is a hallowed tradition in the most hard-headed of the sciences. E.g., quarks. Personally, the word potential doesn't mean much to me in this context; the phrases potential set, potential existence, potential infinity, etc., just don't seem meaningful english to me in this application. -- hz === Subject: Re: Mathematical theater of the absurd <4A090347.F62CC882@gmail.com> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) The idea is that infinity is merely a useful fiction, or equivalently, > a figure a speech. A figure of speech? Sure, why not? Some people think that integers > are just figures of speech, useful fictions. Indeed; I proposed that his thesis works as well with seven in place of infinity. > Fine by me. I think it might be less contentious to say here that > infinity has a hypothetical existence, rather than a potential > existence. Hypothesizing entities for the purposes of explaining > phenomena is a hallowed tradition in the most hard-headed of the > sciences. E.g., quarks. Personally, the word potential doesn't mean much to me in this context; > the phrases potential set, potential existence, potential infinity, > etc., just don't seem meaningful english to me in this application. I might propose that these are meaningful English but not meaningful mathematics. Perhaps a better word choice would be abstract. Marshall === Subject: Re: Mathematical theater of the absurd posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > The actual infinite is not required for the mathematics of the > physical world (Soloman Feferman) > When we seek to understand the real world, we are forced to do > mathematics. It would be a very reasonable thing to do to define > mathematics in terms of what it is we are forced to do when we seek to > understand the real world. > Math is a tool for understanding the real world. It's also a whole lot > more. It's that whole lot more that separates out the mathematicians > from the physical and social scientists, who use math to understand and > model the real world. But if a so-called crank proposed a alternate theory, that theory isn't allowed to be part of that whole lot more. Instead, the standard set theorist would insist that the crank explain the theory's applicability to science and the real world. Unwritten rule #1: only standard theories are allowed to be part of the whole lot more. Crank theories must have an application to the real world in order to be considered. Also, the OP's quote of Feferman sounds very similar to Robinson's comment about infinite sets not existing. Finitists cranks quote Robinson all the time, yet are ridiculed for doing so. Unwritten rule #2: only standard theorists are allowed to quote Feferman's and Robinson's finitist comments. Also, Feferman and Robinson are considered to be on the standard theorist side of the debate despite their finitist quotes. === Subject: Re: Mathematical theater of the absurd > The actual infinite is not required for the > mathematics of the > physical world (Soloman Feferman) > When we seek to understand the real world, we are > forced to do > mathematics. It would be a very reasonable thing to > do to define > mathematics in terms of what it is we are forced to > do when we seek to > understand the real world. That is, we can and > probably should takes > steps to ensure that mathematics stays in touch with > reality. The > scientists use the notion of falsifiability as a > criterion to > guarantee that science stays in touch with > reality--to distinguish > science from philosophy, theology and > pseudoscience--and that notion > could be incorporated into the foundations of > mathematics where it > could serve to distinguish the mathematics of the > physical world from > the mathematics of the metaphysical world; the > mathematics of the > physical world must meet the criterion of being > falsifiable. It would > be very reasonable to divide mathematics into (at > least) two separate > subjects--the mathematics of the physical world, and > the mathematics > of the metaphysical world--for exactly the same > reasons that the > scientists keep science separate from philosophy, > theology and > pseudoscience; the confusion caused by the failure to > distinguish the > two subjects is an obstacle to progress in the > sciences and in > technology. In particular, the general failure of > people to recognize > that the sacred cows of metaphysical mathematics > (e.g. Cantor's theory > of the actual infinite, and Godel's theorem) have > nothing to tell us > about the physical world is a serious obstacle to > progress in the > field of artificial intelligence. > Experience shows that it is virtually impossible to > discuss the ideas > in the foregoing paragraph in these newsgroups; here, > the inmates have > taken over the asylum. However, the interested reader > may want to read > c14ac6?hl=en 5d4633?hl=en 8d67de?hl=en a0e9dc?hl=en 62cb7e?hl=en > We can make excuses for however long we want still the facts stand:http://coding.derkeiler.com/pdf/Archive/General/comp.theory/2009-04/ms g 00122.pdf[P==NP][Martin Musatov] === Subject: Re: Mathematical theater of the absurd I probably agree with you more than I don't. I'm no expert in > this area, but tend to agree with your paraphrasing of Gauss: The notion of a completed infinity doesn't belong in mathematics; > infinity is merely a figure of speech which helps us talk about limits That remark is a couple centuries old and is no longer correct. Nowadays various notions of infinity are employed in many branches of mathematics with great success. For an introduction accessible to a bright layperson see Rudy Rucker's book Infinity and the Mind. --Bill Dubuque === Subject: Re: Mathematical theater of the absurd I probably agree with you more than I don't. I'm no expert in > this area, but tend to agree with your paraphrasing of Gauss: The notion of a completed infinity doesn't belong in mathematics; > infinity is merely a figure of speech which helps us talk about limits That remark is a couple centuries old and is no longer correct. > Nowadays various notions of infinity are employed in many branches > of mathematics with great success. For an introduction accessible > to a bright layperson see Rudy Rucker's book Infinity and the Mind. A lovely book! -- hz === Subject: Re: Mathematical theater of the absurd > For an introduction accessible to a bright layperson see Rudy > Rucker's book Infinity and the Mind. A lovely book! of pointless philosophical posturing and gratuitous formal logic chopping, with a few rather perplexing reflections on the incompleteness theorems. On an unrelated note, your sig-delimiter is broken. There should be a space after the dashes. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Mathematical theater of the absurd > For an introduction accessible to a bright layperson see Rudy > Rucker's book Infinity and the Mind. > A lovely book! of pointless philosophical posturing and gratuitous formal logic > chopping, with a few rather perplexing reflections on the > incompleteness theorems. gratuitous formal logic chopping What does than mean? If you think you could do better then, by all means, please do so. I'm not aware of anything else worth recommending for such an audience. === Subject: Re: Mathematical theater of the absurd posting-account=OWfgwwgAAADQpH2XgMDMe2wuQ7OFPXlE Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) I probably agree with you more than I don't. I'm no expert in > this area, but tend to agree with your paraphrasing of Gauss: The notion of a completed infinity doesn't belong in mathematics; > infinity is merely a figure of speech which helps us talk about limits That remark is a couple centuries old and is no longer correct. > Nowadays various notions of infinity are employed in many branches > of mathematics with great success. For an introduction accessible > to a bright layperson see Rudy Rucker's book Infinity and the Mind. > and an infinite hierarchy of orders of infinity. But I don't see that David's notion of potential existence is all that useful. And while you might be able to contrive quasi-numerical properties for oo, doesn't Gauss (and David here) have a point that, strictly speaking, it isn't necessary? Can you give an simple example in mathematical theory where it is absolutely essential? It seems to me that even statements like Aleph0 + Aleph0 = Aleph0 can be restated in terms of mappings between sets. Dan Download my DC Proof software at http://www.dcproof.com === Subject: Re: length on a matrix posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Trident/4.0; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > simple question: > I did a matrix in this way: > test = [ test1(1:4, :); test2(1:4, :); ]; and after I like to know the number of the rows: length(test); but it returns '2'. > in fact, I use > size(test) > the number correct '8' is returned. Could anybody tell me why? I'm not sure what package you are using, but it suggests that size is giving the number of entries, e.g. 2 rows by 4 columns. You might try varying the sizes of test1 and test2 to confirm this. === Subject: Re: Integration <110520091338346486%anniel@nym.alias.net.invalid> posting-account=uOO_zwkAAAAGD4AJIMJtIuvVjT4qNyEf Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > The task is to evaluate: Int[x*Sin^2(2x) dx] bwtween the limits x=0 and x=PI/4 Now, if I use integration by parts setting u=x and dv/dx = Sin^2(2x) it all goes fine and I get the correct answer of: PI^2/64+1/16 However, what is wrong with the idea of (instead of integration by > parts) making the substitution u = (x/2)-(1/8) then > my intrgral becomes Int[u^2+u/4] what happened to the sine? between the limits u = -1/8 and u = > PI/8 - 1/8 Which, unfortunately, I evaluate to PI^2/64 Is the method sound and therfore I am making a mistake when > evaluating, or is the method not sound. Mitch. End of day and made an pathetic typo, all solved now. === Subject: Surface like helicoid in S^3? Consider the surface of a helicoid in cylindrical coordinates: z = phi , see for example: Now say I'm sitting in the space S^3 whose radius is much larger then my height. I hold the truncated surface of a helicoid in my hand. Does this surface extend to all of S^3 in some natural way? Is there a simple function for this helicoid like z = phi? === Subject: Surface like helicoid in S^3? Consider the surface of a helicoid in cylindrical coordinates: z = phi , see for example: Now say I'm sitting in the space S^3 whose radius is much larger then my height. I hold the truncated surface of a helicoid in my hand. Does this surface extend to all of S^3 in some natural way? Is there a simple function for this helicoid like z = phi? === Subject: RIEMAN HYPOTHESIS Cc: kwendakagiso@gmail.com posting-account=J_HMSwoAAADQSPDIzAYwrLCeksdquADF 1.1.4322; MySpace;),gzip(gfe),gzip(gfe) will some ever solve this hypothesis ,because ithink iam on verge of something great here. === Subject: minimization problem posting-account=eRNt1QoAAADh0Jk-O8FgaNVRuZUmimKq Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) hi..all.. i would like to get some help for this problem.... it is quite difficult for me.. minimize with respect to A for the function: log det [ AXA'+AHA' AXB' ; BXA' BXB'] === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > [...] In the last 15 > years of this group, the only person who tried to get his ideas > accepted in the normal way, in my opinion, was Tom Van Flandern. This Dr. Tom Van Flandern? http://metaresearch.org/media%20and%20links/press/tomvf%20obituary.asp peace === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole The posters in sci.physics who question the value of General Relativity > are not interested in being spoonfed with a rubber-coated spoon, they are interested in getting members of the Relativity Cult > to engage in serious dichotomies about Relativity, and compare the utility of Relativity > which uses rubber clocks and rulers > to speculate about time travel, aging of twins, gravitons, > worm holes, space warps, and things beyond man's capacity > to ever experience in time and space, > like the beginning and end of time, > and the mind of God, to other models of the world such as > the Watson/Crick DNA model which is used every day > to improve health, improve foods, fight crime, reconstruct history, etc. No doubt, one can condition impressionable minds > by spoonfeeding it dogma with a rubber-coated spoon, but rational, intelligent, independent thinkers > are interested in ferreting out TRUTH > by engaging in focused dichotomies, > rather than being spoonfed cult dogma because a mind is a terrible thing to waste. > My, my Potter, you do try to disparage things you have FAILED to understand, such as General Relativity and it's applications! Hey, Potter, GTR has directly contributed to a $30B+ GPS industry, benefiting people all over the world. Aviation, shipping, asset management, survey, mining, agriculture, time dissemination, communications networks... and on and on! Bluster on, Potter, bluster some more! Froth at the mouth! Whatever! === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole posting-account=0iME0QoAAABHLc0kgVai1vIO0M5rRJGg InfoPath.1; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; MS-RTC LM 8; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) (NetCache NetApp/6.0.2P1) On May 11, 5:19am, Strich-Reply-To-Idiots and useful theory without being familiar with the then-current >theories and experiments. That's why the idiots and crackpots around >here are so pathetic: if they truly wanted to make a contribution, >they would be seriously STUDYING the current theories and experimental >record, and trying to extend one or the other. Miguel Rios Miguel, > who are the idiots and crackpots around here? Regarding your comment about > developing a useful theory, how do you think General Relativity > which uses rubber clocks and rulers > and consumes an enormous amount of > time, money, and minds on such issues as time travel, worm holes, gravitons, > space warps, and things beyond man's capacity > to ever experience in time and space like > the beginning and end of the universe > and the mind of God, compares to the Watson/Crick DNA model > that is used every day to improve health care, > food production, fight crime, reconstruct history, etc.? Do you think that the 'idiots and crackpots around here > are members of the Relativity Cult? -- It is quite easy to observe that there are no crackpots trying to deny > those models on biology or chemistry. But somehow, those very same > people do have opinions (sometimes highly opinionated) about > Relativity subjects, most of the time coming from misunderstandings > caused by not having formally studied the subject. In the last 15 > years of this group, the only person who tried to get his ideas > accepted in the normal way, in my opinion, was Tom Van Flandern. For > years, his opinion on the speed of gravity was conducted at a good and > interesting level with people like Tom Roberts and Steve Carlip, and > those arguments even resulted on a few papers being published in good > journals. What helped there, was that Van Flandern (differently from > many around here) had a good background on physics and, while many > here considered him an illustrated crackpot, he had sometimes good > points in his arguments. For the rest of the wackos here is like > trying to rediscover the wheel day after day (read that on arguments > with Winn, Seto, Spaceman, Strich, etc.). Miguel Rios- D.9alj citerad text - - Visa citerad text - If the science was solid, the so-called crackpots would not be > stumping physicists left and right everyday. Stumping questions like: Where are the right handed neutrinos? Is the LIGO negative? Is the GPB negative? These questions make the relativists squirm each time. A science > backed up be solid logic and evidence will not be stumped by the so- > called crackpots. On the other hand, the so-called orthodoxy is throwing all the ad > hominems they can generate at the so-called crackpots because the > latter are clearly winning the intellectual and logical encounters. http://www.youtube.com/watch?v=znowCx y7nU It never stops being relevant. You betcha. Braying is pretty much all we're getting from Strich9.- D.9alj citerad text - - Visa citerad text - Proven liar and schizo with nothing new to add... === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole posting-account=8k9z_QoAAABfJ6zct3wDB2k3FKfRerU2 2.0.50727),gzip(gfe),gzip(gfe) > Folks: The pen is mightier than the sword. But Dougie can't read; reason; nor defer to those who can. NoEinstein > The best cure for a parasite like PD is to interrupt his food source. > Stop replying to the bastard! NoEinstein You are getting increasingly vicious to those who point out your > mistakes. You need to get some anger therapy before you get > yourself into trouble. Folks: My detailed explanation in my last reply to... PNG, PD, IS the >proof! >Detailed explanation, my foot. Where are your *calculations*. Physics >is a *quantitative* science. If you do not do calculations, then you >are not doing physics. PD makes a good point when he points out that Physics a *quantitative* science. >If you do not do calculations, then you are not doing physics. Hopefully, rather than make unsubstantiated claims, >and attack posters who oppose Relativity, the General Relativity Cultists will begin to post calculations >rather than useless references that may or may not >support their positions. >This is a pointless and inefficient exercise, and I am shocked that >you, who rail at the inefficiencies of science, would encourage such >an activity. >ASCII Usenet is a horrible medium for communicating calculations, and >it would serve absolutely no purpose other than to spoonfeed you at >your chosen trough of convenience. >In an efficient mode of communication, there is an optimum whereby >transmitter does some work and receiver does some work, and a system >design where the transmitter does all the work and the receiver does >none is almost guaranteed to be a wasteful method. >The calculations are easily available to you, Potter, and the best >thing a relativity cultist can do to facilitate your education is to >give you directed pointers to places where you can look, rather than >you attempting a raw search yourself. One would think that if the General Relativity Gurus >and Cultists were privy to such powerful, esoteric knowledge, >that they would use their powerful knowledge >to make a few bucks in the free market. The best I can determine is that the >General Relativity Gurus and Cultists >are either phones, bullshippers, brainwashed students, >or on the public dole. -- >Tom Potterhttp://tdp1001.spaces.live.com/http://www.tompotter.us/misc.htmlhttp:. . .Hide quoted text - >- Show quoted text -- Hide quoted text - >- Show quoted text -- Hide quoted text - - Show quoted text - === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole posting-account=8k9z_QoAAABfJ6zct3wDB2k3FKfRerU2 2.0.50727),gzip(gfe),gzip(gfe) > Folks: The pen is mightier than the sword. But Dougie can't read; reason; nor defer to those who can. NoEinstein > of my science research and analysis. Yes, they do. The depth is zero. It would be most helpful if you would reply and give a similar list of your posts... IF you have any. > NoEinstein >Folks: PD is a certifiable LUNY who thinks that if he asks an already- >well-answered question often enough, that he will discredit my science >truths. Smart readers among you must surely understand why I consider >PD to be a persona non grata, unworthy of being given the time of >day! NoEinstein >And you are one who repeatedly redirects the argument away from proof. >Show us the science.- Hide quoted text - - Show quoted text - === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole posting-account=8k9z_QoAAABfJ6zct3wDB2k3FKfRerU2 2.0.50727),gzip(gfe),gzip(gfe) > Folks: The pen is mightier than the sword. But Dougie can't read; reason; nor defer to those who can. NoEinstein > Folks: Parasites like PD should be dealt with in the most drastic > ways. Unfortunately, these news groups only allow the use of words. > NoEinstein So you are advocating violence against those who point out your > mistakes? That is both crazy and illegal. Folks: PD is a certifiable LUNY who thinks that if he asks an already- >well-answered question often enough, that he will discredit my science >truths. Smart readers among you must surely understand why I consider >PD to be a persona non grata, unworthy of being given the time of >day! NoEinstein Folks: The question which PNG, PG, keeps asking was completely >answered in this original post, as well as in several of my replies. >Please provide a link to your post with the calculations. >Amazing how you tend toward violent thoughts when people ask you to >show your calculations. >PD- Hide quoted text - >- Show quoted text -- Hide quoted text - - Show quoted text - === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole At least Ptolemy had a millenia of evidence. Einstein only had a century. Strich admits that there is a century of evidence supporting Einstein's theories! Yea... progress. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole At least Ptolemy had a millenia of evidence. Einstein only had a century. Strich admits that there is a century of evidence supporting Einstein's theories! Yea... progress. === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole posting-account=0iME0QoAAABHLc0kgVai1vIO0M5rRJGg InfoPath.1; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; MS-RTC LM 8; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) (NetCache NetApp/6.0.2P1) At least Ptolemy had a millenia of evidence. Einstein only had a century. Strich admits that there is a century of evidence supporting > Einstein's theories! Yea... progress. That's nothing. Ptolemy had a millenia of evidence. Ptolemy must be more right than Einstein, aside from both being in that same league... === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole > At least Ptolemy had a millenia of evidence. > Einstein only had a century. > Strich admits that there is a century of evidence supporting > Einstein's theories! Yea... progress. That's nothing. Ptolemy had a millenia of evidence. Ptolemy must be > more right than Einstein, aside from both being in that same league... Ya Think, Strich? === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole posting-account=-Yr6sAoAAAArJwgdHLA4MBxm4oIdzdtZ InfoPath.1; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; MS-RTC LM 8; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) (NetCache NetApp/6.0.2P1) At least Ptolemy had a millenia of evidence. > Einstein only had a century. > Strich admits that there is a century of evidence supporting > Einstein's theories! Yea... progress. That's nothing. Ptolemy had a millenia of evidence. Ptolemy must be > more right than Einstein, aside from both being in that same league... Ya Think, Strich? Sammie cannot count. Ptolemy had 10 centuries of evidence behind him. Poor Einstein only had 1 century. The clincher, both were dead wrong... === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole posting-account=lpj6eQgAAADfAdNPtqtrw3lMgI8RApJ_ SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > logical thinking. Interesting. So you think textbooks only serve to muddle thinking, > John? > You find them too hard to follow and still maintain coherent thoughts, > and you want somebody to blame for that? Science should be easy and obvious? PD After enduring long books on difficult science > (like... relativity), naive readers feel that they have been made to > be 'learned', too. But all they now are is: brainwashed FOOLS. > NoEinstein Real scientists do not do, nor report, their research on this forums. >They study the subjects, they make measurements and simulations and, >finally, report their findings in conferences and journals (that is at >least what I do). Talking of real people doing science, and working in real problems, >here are some references to read: 1) 10 General Relativistic Models for Space-time Coordinates and >XXVIIA Reports on Astronomy 2006-2009, pp55-59. >Tou Ni, Michael C.W. Sandford, Christian Veillet, An-Ming Wu, Patricia >Fridelance, Etienne >Samain, George Spalding and Xiaohui Xu, Adv. Space Res. Vol. 32, No. >4) Relativistic Corrections to Lunar Occultations, Costantino >Sigismondi, Tenth Italian-Korean Meeting on Relativistic Astrophysics >Pescara, June 25-30, 2007. >5) Deep-space laser-ranging missions ASTROD and ASTROD I for >astrodynamics and astrometry, W. T. Ni and the ASTROD I ESA COSMIC >Astrometry Proceedings IAU Symposium No. 248, 2007. Miguel Rios I was not Talking of real people doing science, > I was Talking of real people doing REALITY. As can be seen from your reference > 10 General Relativistic Models for Space-time.. > and Ashby's paper that uses 13 Classical Physics hacks > of GPS data to fit it to General Relativity, unlike DNA scientists, computer scientists, electronics scientists, etc. > all of the General Relativity Gurus seem to be guys on the Public Dole, > who have never made a real world contribution to mankind. One would think that if General Relativity Gurus > possessed such powerful, esoteric knowledge, > they would use it to make a few bucks in the free market, rather than being a burden to hardworking taxpayers, > and wasting the minds and time of young folks > on issues such as time travel, worm holes, > the beginning and end of the universe, > and the mind of God. A mind is a terrible thing to waste. -- > Tom Well Mr. potter, regarding the chicken situation, I'm quite sure you > do not need Galileo, Newton, nor Einstein theories to find your way to > your bathroom at night. That is how reality works!!. Science, on the other hand, has always worked differently to that. You > observe Nature and wonder how and why it works like like you see it. > Then you build a model that may, or may not, be useful to predict the > results of observations. Many advances on physics or mathematics take > several years to understand, and then more years to get into real > useful applications of those new advances. One example: in the early > 60's, Dr. Robert Gallager, on his Ph.D. thesis, found a mathematical > method that could be used in coding systems. The method is called Low > Density Parity Check (LDPC). For over 30 years nothing happened with > his development (even him had forgotten it), until somebody found the > method could be applied to the 3G and 4G cellular systems. So now, > almost all cellular systems are using these LDPC codes to improve the > quality of communications. Furthermore, History has shown that nobody has ever developed a new > and useful theory without being familiar with the then-current > theories and experiments. That's why the idiots and crackpots around > here are so pathetic: if they truly wanted to make a contribution, > they would be seriously STUDYING the current theories and experimental > record, and trying to extend one or the other. Miguel Rios- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text - xxein: Newton would have called Einstein an asshole. Is there no reality to be found? === Subject: Re: R E L A T I V I T Y: Fitting a Square Peg into a Round Hole >Furthermore, History has shown that nobody has ever developed a new >and useful theory without being familiar with the then-current >theories and experiments. That's why the idiots and crackpots around >here are so pathetic: if they truly wanted to make a contribution, >they would be seriously STUDYING the current theories and experimental >record, and trying to extend one or the other. >Miguel Rios > Miguel, > who are the idiots and crackpots around here? > Regarding your comment about > developing a useful theory, > how do you think General Relativity > which uses rubber clocks and rulers > and consumes an enormous amount of > time, money, and minds > on such issues as time travel, worm holes, gravitons, > space warps, and things beyond man's capacity > to ever experience in time and space like > the beginning and end of the universe > and the mind of God, > compares to the Watson/Crick DNA model > that is used every day to improve health care, > food production, fight crime, reconstruct history, etc.? > Do you think that the 'idiots and crackpots around here > are members of the Relativity Cult? > -- It is quite easy to observe that there are no crackpots trying to deny > those models on biology or chemistry. Save the Texas education leaders. ;) === Subject: Re: a question about finite morphisms schrieb im Newsbeitrag > in the particular case, when > X=X. If dominant morphism f:X ---> X with finite fibres, then is f > finite? No, counterexample: That the corresponding mapping on Spec's induced by a non-finite field extension. > So now the question is: > Assume that X is an irreducible affine variety and f:X-->X is a > dominant morphism. Assume also that for every x in X the complete > preimage f^{-1}(f(x)) is finite. Is f a finite morphism? The same counterexample applies here. But - as David has already pointed out - it would be helpful to investigate the difference between quasi-finite and finite (affine) morphisms. HTH. Best wishes, J. === Subject: Re: a question about finite morphisms posting-account=FkjBxgoAAAC9Im8gu67TzgR_biUpxjvK in the particular case, when > X=X. If dominant morphism f:X ---> X with finite fibres, then is f > finite? No, counterexample: That the corresponding mapping on Spec's induced by a > non-finite field extension. So now the question is: > Assume that X is an irreducible affine variety and f:X-->X is a > dominant morphism. Assume also that for every x in X the complete > preimage f^{-1}(f(x)) is finite. Is f a finite morphism? The same counterexample applies here. But - as David has already pointed out - it would be helpful to investigate > the difference between quasi-finite and finite (affine) morphisms. HTH. Best wishes, > J. Can you please give an example, instead of general phrases? I realise the difference between quasi-finite and finite morphisms. But I have not seen any example of non-finite morphism f:X--->X with finite fibres. === Subject: Re: a question about finite morphisms > in the particular case, when > X=X. If dominant morphism f:X ---> X with finite fibres, then is f > finite? > No, counterexample: That the corresponding mapping on Spec's induced by a > non-finite field extension. > So now the question is: > Assume that X is an irreducible affine variety and f:X-->X is a > dominant morphism. Assume also that for every x in X the complete > preimage f^{-1}(f(x)) is finite. Is f a finite morphism? > The same counterexample applies here. > But - as David has already pointed out - it would be helpful to investigate > the difference between quasi-finite and finite (affine) morphisms. > HTH. > Best wishes, > J. > Can you please give an example, instead of general phrases? I realise > the difference between quasi-finite and finite morphisms. But I have > not seen any example of non-finite morphism f:X--->X with finite > fibres. I do not have old excecises at hand, but that is Hartshorne, Alg Geom Ex II.3.5 c: Show by example, that a surjective, finite-type, quasi- finite morphism need not be finite. === Subject: Re: a question about finite morphisms > I do not have old excecises at hand, but that is Hartshorne, Alg Geom > Ex II.3.5 c: Show by example, that a surjective, finite-type, quasi- > finite morphism need not be finite. Several people seem to me to be missing the point of the question. The crucial new piece of information given by the OP is that the domain and range of the morphism are the same algebraic variety. Sure surjective finite type quasi-finite morphisms might not be finite (imagine X=U union V and consider the map from U disjoint V to X; this can easily be made to work) but this is not the question. Here's the question: let X be an irreducible affine variety over a field k and say f:X-->X is a k-morphism of varieties such that f is dominant and quasi-finite. Is f finite? So, for example, X can't be a smooth curve, because if Z is its smooth compactification then f:X-->X will extend to a finite map F:Z-->Z and, if Z=X minus a finite set S, then F^{-1}(S) will have at least #S points, so F^{-1}(S)=S so F^{-1}(X)=X and so f is finite. Someone talked about infinite field extensions etc etc but I don't see how this helps either. Maybe there is some crazy irreducible affine variety X and an open immersion X-->X which isn't an isomorphism, but I don't know of one and would love to know of one! Kevin === Subject: Re: a question about finite morphisms posting-account=FkjBxgoAAAC9Im8gu67TzgR_biUpxjvK > I do not have old excecises at hand, but that is Hartshorne, Alg Geom > Ex II.3.5 c: Show by example, that a surjective, finite-type, quasi- > finite morphism need not be finite. Several people seem to me to be missing the point of the question. The > crucial new piece of information given by the OP is that the > domain and range of the morphism are the same algebraic variety. > Sure surjective finite type quasi-finite morphisms might > not be finite (imagine X=U union V and consider the map > from U disjoint V to X; this can easily be made to work) but this > is not the question. Here's the question: let X be an irreducible affine variety > over a field k and say f:X-->X is a k-morphism of varieties > such that f is dominant and quasi-finite. Is f finite? So, for example, X can't be a smooth curve, because if Z is > its smooth compactification then f:X-->X will extend to a finite map F:Z-->Z > and, if Z=X minus a finite set S, then F^{-1}(S) will > have at least #S points, so F^{-1}(S)=S so F^{-1}(X)=X and so f is finite. > Someone talked about infinite field extensions etc etc but I don't > see how this helps either. Maybe there is some crazy irreducible affine variety X and an > open immersion X-->X which isn't an isomorphism, but I don't know of one > and would love to know of one! Kevin Yes, the key point here is that f is a surjective morphism onto the same variety. Kevin, are there any generalization of your arguments for a smooth curve? In particular, if we know that X is a smooth variety, can we deduce that a surjective morphism f:X--->X with finite fibres is finite? === Subject: JMCAD is an program for the modeling and simulation of complex dynamic systems posting-account=WHYd2AoAAADNnE_imJqwx0y7u0BDtN4X New version JMCAD-08.087 released!!! JMCAD is an program for the modeling and simulation of complex dynamic systems. This includes the ability to construct and simulate block diagrams. The visual block diagram interface offers a simple method for constructing, modifying and maintaining complex system models. The simulation engine provides fast and accurate solutions for linear, nonlinear, continuous time, discrete time, time varying and hybrid system designs. With JMCAD, users can quickly develop software or virtual prototypes of systems or processes to demonstrate their behavior prior to building physical prototypes. Project detail and discuss: http://jmcad.sourceforge.net/ http://sourceforge.net/projects/jmcad/ To join this project, please contact the project administrators === Subject: Re: K-12 Calculator Woes [Long list of references deleted.] ................ > There are other means as well. Those in the sciences, and > in many other fields as well, misuse mathematics because > they have learned procedures which require assumptions to > be met to be valid. > You continue to miss the point. > Yes, there are other methods, like base 16 for carpentry using > caricatures of tools instead of letters and numerals because studs are > 16 inches on center. > If one understands the numbers, integer, fractional, real, > and if need be complex (try handling AC circuits without it), > the representation becomes of little importance. Being able > to make limited use of a particular representation does NOT > convey understanding. No matter how well one does base 10 > arithmetic, this conveys no more than having the mechanical > processes in ones memory banks. There is not even understanding > of the processes. >I beg to differ. No, the understanding of the integers is not there from learning to calculate in any given base. >The understanding may not be deep or wide, but it is there. Definitely NOT, as has been demonstrated by example. >I'd like to see it deeper and wider, but there are limits, and you seem >to know no bounds. Mathematics tries to expand the understanding; this is what research is doing. But research is not necessarily what is needed to get it in the first place. One can teach rigorous analysis easily in middle school or high school, but as experience has shown, teaching it after calculus operations are learned does not do it. There is even a branch of mathematics known as differential algebra, which adds calculus operations to the algebraic ones; this is what is used in considering integration in finite terms. Wasting time on improving speed and accuracy in hand calculations detracts from, not adds to, understanding. Starting with counting to develop the concepts, and having students understand addition and multiplication, as well as induction arguments, adds to it. > But for those out there in the real world other methods do not exist, > either because they are unaware of them, because their technology cannot > handle them, or they just plain flat out refuse to acknowledge their > existence. > If they are unaware of them, how can they know if they can make > use of them? If they refuse to acknowledge their existence, can > they be said to be educated about the numbers? >Why bother? >I'm aware that the Mandarin language exists, but I'm perfectly >functional knowing no more than that. Since you know it exists, if you needed to read something in Mandarin, you could get it done. But this knowing facts, rather than concepts. I did not understand topological concepts on probability measures on metric spaces until I got rid of the metric, using concepts from general topology. It became VERY easy. > Why us a base 16 measuring system when the base 10 is perceived to be > better, the technology is designed around that system and they already > know it? Is base 10 better? We are used to base 10; the Sumerians, and then the Babylonians, used base 60, and it is still present in our notation. When it comes to computers, base 2 is the obvious one to use, and all those who designed the base 2 computers used base 10 for most of their hand calculations. I have used base 8 and base 16 for hand calculations myself, and I have not memorized the tables. Neither have I memorized the base 60 tables, and I have done some with that base. > Computer technology is NOT designed about base 10, and will not > be in the future. It is really base 2, but 16 is used as a > convenient version so that expressions will not be too long. >The human representation provided by that technology. >Just like people are unconcerned whether the signal traveling across the >country for their telephone call is analog, digital, internet routed , >or central office routed, all they care about is the calculator's >display, just like all they care about is hearing the words of the >person on the other end of the line. The internal technology may or may not be important, but the concepts are. The linguistic (including numerical) concepts of what is being sent are important, and the meaning of the computer's answer are as well. If I am going to send non-integer values of numerical constants from one computer to another, I want to use a power of 2 for may base. Also, if constants are included in a formula, why should we use the current base 10, and introduce conversion errors, rather than base 16, and have none of them? > Does any other system work better. Absolutely no!. Talk a bout wasting > time and effort! > See the above. There are other places where these come in. >So if you are an electrical engineer, learn complex numbers. >If you are a financial analyst, who cares? >I am more concerned that the financial analyst understand the base 10 >math that governs my money. The financial analyst does not need to understand base 10, and it may even be confusing. He needs to understand the meaning of the base 10 expressions. > They have NOT. The string 6.43 means 6 plus 43/100. If they > do not understand the decimal fractions are fractions, they do > not understand decimal fractions. > I said the rational representation of fractions. > I agree that not enough is done with this, and that > computers have a lot to do with the problem. It is > easy to get computers to run in base 2 arithmetic, > including for real numbers, and even the important > fixed-point arithmetic is difficult on them. The > representation of fractions is, for a machine, much > harder than floats or decimals, and is more complicated > to work with. > However, the teaching of fractions leaves even more > to be desired. In can be made both rigorous and > more simple. Of course, anything is simpler if one > has algebraic notation, by which I mean that a > variable is a symbolic expression which can stand > for anything. So we have, for h not 0, > a/b = ah/bh, > and if the idea of fractions is presented with a > modicum of intelligence, it is obvious that > a/b + c/b = (a+c)/b. > I don;t know about your school district, but that is part of my > districts middle school curriculum. > Do they use algebraic notation? Do they still ask > students to use the least common denominator? >Yep. A required part of every school curriculum I have heard of. Even a >part of the standard ( not algebra i) 8th grade curriculum. Introduced >in the 6th grade. Too late; algebraic notation belongs with beginning reading, as it is simple linguistics. And fractions probably should be introduced by the third grade, and made obvious. > You suggest that 6 yr olds can handle this. I disagree, and most other s > do, too, because most 6 yr old have not reached the mental stage to > understand representation. They are still in a concrete world. > I have never suggested that 6 year olds have the understanding > of integers needed to start on fractions. I do not know how > fast one can do the work, nor do I have any objection to them > learning arithmetic operations to base 10 before doing much, > or even starting, with fractions. > Children are capable of understanding abstract ideas, NOT as > abstractions. It is this confusion that too many are making; > an abstract idea may have come about as the result of a > process of abstraction, but it is more than that, and easier > to understand if taught directly. >There is considerable disagreement about this. The disagreement comes because it has not been tried. Even the poor original new math worked, when taught by people who themselves had the understanding needed. I doubt if anything has been more tested before being introduced. >The biggest problem is that people start with differing abilities, and >develop at differing rates. Yes, and teach them accordingly. Placing all children of a given age in the same class is utterly stupid, and it should be obvious that it does not give good education to most. The John Dewey philosophy was not concerned with this aspect. ............... > Whether they know exactly how the black box works is not > a problem; they need to know how the numbers work. Teach > concepts, and these problems might well not even arise. >Exactly my point. By using calcs they do not need to knwo how hte >numbers work. They need to know the meaning of what the black box produces; they need to know what numbers MEAN, not how to operate with strings of digits. > Why bother to understand how a tv works? Push certain buttons and good > things happen. Or a pc, microwave, elevator, or cell pone. > Most don;t care until they break, then they go to an expert to fix it, > or even worse, throw the broken one away and get a new one. > This is not the problem. They need to understand the concepts > involved in viewing a TV. These are only optical. >No, they aren't. They certainly are. The concern in designing a TV is that the images presented have the desired effect. This is just like designing a computer; it is what answers are presented, not how they are produced. For multiple precision arithmetic, it often pays to use procedures which are not the ones taught in school. The answers are the same, but the procedures are completely different. The fastest known method is using the fast Fourier transform over a finite ring. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: K-12 Calculator Woes [Long list of references deleted.] ................ There are other means as well. Those in the sciences, and > in many other fields as well, misuse mathematics because > they have learned procedures which require assumptions to > be met to be valid. > You continue to miss the point. > Yes, there are other methods, like base 16 for carpentry using > caricatures of tools instead of letters and numerals because studs are > 16 inches on center. If one understands the numbers, integer, fractional, real, > and if need be complex (try handling AC circuits without it), > the representation becomes of little importance. Being able > to make limited use of a particular representation does NOT > convey understanding. No matter how well one does base 10 > arithmetic, this conveys no more than having the mechanical > processes in ones memory banks. There is not even understanding > of the processes. > I beg to differ. No, the understanding of the integers is not there > from learning to calculate in any given base. > The understanding may not be deep or wide, but it is there. Definitely NOT, as has been demonstrated by example. > Not true. There is some, maybe not as much as you and I would like to see, but there is some. Enough for a very large percentage of society to function within societal norms. Bear in mind that at least 25% of our kids --- 40% in my state, never even graduate HS. > I'd like to see it deeper and wider, but there are limits, and you seem > to know no bounds. Mathematics tries to expand the understanding; this is > what research is doing. > Who's understanding? I'll bashfully admit that I probably could not understand most of what you do No one else I know could come close, and I know some intelligent, well educated people. > But research is not necessarily what is needed to get it > in the first place. One can teach rigorous analysis > easily in middle school or high school, but as experience > has shown, teaching it after calculus operations are > learned does not do it. It _is_ taught in HS, and depending on your definition of rigorous, also in middle school. But as I have frequently complained about int his group, the first thing you have to do is get he kids into a math class. Again, until next year, in my state a single HS math class (with an advanced middle school class) is sufficient to get an *academic* college prep diploma here. An adjacent rural county had so little demand for trigonometry that is doesn't even offer it in its one HS, let alone calculus. Algebra II is its most advanced math course, and a friend teaches the one class of it per block What do we do, hogtie parents and frog march kids to class? Larry There is even a branch of > mathematics known as differential algebra, which adds > calculus operations to the algebraic ones; this is > what is used in considering integration in finite terms. > Wasting time on improving speed and accuracy in hand > calculations detracts from, not adds to, understanding. > Starting with counting to develop the concepts, and > having students understand addition and multiplication, > as well as induction arguments, adds to it. > But for those out there in the real world other methods do not exist, > either because they are unaware of them, because their technology cannot > handle them, or they just plain flat out refuse to acknowledge their > existence. If they are unaware of them, how can they know if they can make > use of them? If they refuse to acknowledge their existence, can > they be said to be educated about the numbers? > Why bother? > I'm aware that the Mandarin language exists, but I'm perfectly > functional knowing no more than that. Since you know it exists, if you needed to read something > in Mandarin, you could get it done. But this knowing > facts, rather than concepts. I did not understand > topological concepts on probability measures on metric > spaces until I got rid of the metric, using concepts > from general topology. It became VERY easy. > Why us a base 16 measuring system when the base 10 is perceived to be > better, the technology is designed around that system and they already > know it? Is base 10 better? We are used to base 10; the Sumerians, > and then the Babylonians, used base 60, and it is still > present in our notation. When it comes to computers, base > 2 is the obvious one to use, and all those who designed > the base 2 computers used base 10 for most of their hand > calculations. I have used base 8 and base 16 for hand > calculations myself, and I have not memorized the tables. > Neither have I memorized the base 60 tables, and I have > done some with that base. Computer technology is NOT designed about base 10, and will not > be in the future. It is really base 2, but 16 is used as a > convenient version so that expressions will not be too long. > The human representation provided by that technology. > Just like people are unconcerned whether the signal traveling across the > country for their telephone call is analog, digital, internet routed , > or central office routed, all they care about is the calculator's > display, just like all they care about is hearing the words of the > person on the other end of the line. The internal technology may or may not be important, but > the concepts are. The linguistic (including numerical) > concepts of what is being sent are important, and the > meaning of the computer's answer are as well. If I am > going to send non-integer values of numerical constants > from one computer to another, I want to use a power of 2 > for may base. Also, if constants are included in a > formula, why should we use the current base 10, and > introduce conversion errors, rather than base 16, and > have none of them? > Does any other system work better. Absolutely no!. Talk a bout wasting > time and effort! See the above. There are other places where these come in. > So if you are an electrical engineer, learn complex numbers. > If you are a financial analyst, who cares? > I am more concerned that the financial analyst understand the base 10 > math that governs my money. The financial analyst does not need to understand base 10, > and it may even be confusing. He needs to understand the > meaning of the base 10 expressions. They have NOT. The string 6.43 means 6 plus 43/100. If they > do not understand the decimal fractions are fractions, they do > not understand decimal fractions. > I said the rational representation of fractions. I agree that not enough is done with this, and that > computers have a lot to do with the problem. It is > easy to get computers to run in base 2 arithmetic, > including for real numbers, and even the important > fixed-point arithmetic is difficult on them. The > representation of fractions is, for a machine, much > harder than floats or decimals, and is more complicated > to work with. However, the teaching of fractions leaves even more > to be desired. In can be made both rigorous and > more simple. Of course, anything is simpler if one > has algebraic notation, by which I mean that a > variable is a symbolic expression which can stand > for anything. So we have, for h not 0, a/b = ah/bh, and if the idea of fractions is presented with a > modicum of intelligence, it is obvious that a/b + c/b = (a+c)/b. > I don;t know about your school district, but that is part of my > districts middle school curriculum. Do they use algebraic notation? Do they still ask > students to use the least common denominator? > Yep. A required part of every school curriculum I have heard of. Even a > part of the standard ( not algebra i) 8th grade curriculum. Introduced > in the 6th grade. Too late; algebraic notation belongs with beginning > reading, as it is simple linguistics. And fractions > probably should be introduced by the third grade, and > made obvious. > You suggest that 6 yr olds can handle this. I disagree, and most other s > do, too, because most 6 yr old have not reached the mental stage to > understand representation. They are still in a concrete world. I have never suggested that 6 year olds have the understanding > of integers needed to start on fractions. I do not know how > fast one can do the work, nor do I have any objection to them > learning arithmetic operations to base 10 before doing much, > or even starting, with fractions. Children are capable of understanding abstract ideas, NOT as > abstractions. It is this confusion that too many are making; > an abstract idea may have come about as the result of a > process of abstraction, but it is more than that, and easier > to understand if taught directly. > There is considerable disagreement about this. The disagreement comes because it has not been tried. Even the poor original new math worked, when taught by > people who themselves had the understanding needed. > I doubt if anything has been more tested before being > introduced. > The biggest problem is that people start with differing abilities, and > develop at differing rates. Yes, and teach them accordingly. Placing all children > of a given age in the same class is utterly stupid, and > it should be obvious that it does not give good education > to most. The John Dewey philosophy was not concerned > with this aspect. ............... Whether they know exactly how the black box works is not > a problem; they need to know how the numbers work. Teach > concepts, and these problems might well not even arise. > Exactly my point. By using calcs they do not need to knwo how hte > numbers work. They need to know the meaning of what the black box > produces; they need to know what numbers MEAN, not > how to operate with strings of digits. > Why bother to understand how a tv works? Push certain buttons and good > things happen. Or a pc, microwave, elevator, or cell pone. > Most don;t care until they break, then they go to an expert to fix it, > or even worse, throw the broken one away and get a new one. This is not the problem. They need to understand the concepts > involved in viewing a TV. These are only optical. > No, they aren't. They certainly are. The concern in designing a TV > is that the images presented have the desired effect. > This is just like designing a computer; it is what > answers are presented, not how they are produced. > For multiple precision arithmetic, it often pays to > use procedures which are not the ones taught in school. > The answers are the same, but the procedures are > completely different. The fastest known method is > using the fast Fourier transform over a finite ring. > === Subject: Re: K-12 Calculator Woes > The representation of fractions is, for a machine, much > harder than floats or decimals, and is more complicated > to work with. > Why do you believe that is true? > Fractions, unless their size is limited, will require at > least two words, and their operations are more complex. > Machines rarely do exact arithmetic, as would be required > with fractions to give the answers desired. Avoiding > computer roundoff would be expensive. > I am reasonably familiar with computer architecture, and > I can see no way around it. >Except most computers these days do not deal with memory structures as >small as a word. They are optimized at 32 bits, 64 bits, even 128 bits >for pipelining and reduced instruction set computing (risc). (Jeez, I'm >old. I remember programming using nibbles, or half bytes, for integers) >The older 8 bit chips are disappearing, and those left in the market are >relegated to the dollar store calcs. Now extremely cheap 16 bit longword >chips are about the only low end chips available. The calculations are >not a problem. And the smallest memory chips made today can easily store >the code. Unfortunately, the x86 architecture, in wide use in computers, still has too many short procedures. And I would like to do operations involving bit streams for bit efficiency, but these are horribly expensive on the present computers. In generating non-uniform random numbers for simulation, one now uses wasteful procedures to get speed. >A more significant problem is the increased number of buttons to be put >in a small acreage. The desire for a small format directly competes >with the inclusion of more functions. Some calculators now have as many >as 3 or 4 functions for each key, and even so a large and bulky.Learning >to use them is a challenge. >And, of course, the marketing problem is huge. Most people just don't >want to calculate in fractions, and have learned the algorithm for >computing in integers instead, that is you can actually enter 3/4*5/9 >into the calculator to get a preferred decimal answer. That said, cheap >calculators that do integer fraction arithmetic are available --- my >Home Depot has a few models designed for carpenters, for ex. They >actually round to 16ths, as well as calculating things like pitch and >angles. In most situations, computing with exact arithmetic would soon make the numerators and denominators so large that it would be hopeless. And also much is done using other algebraic and transcendental functions, the results of which cannot be written as fractions. One can work with rounded binary answers, and this is the way most computer functions work. One can even do these in multiple precision, but only with considerable difficulty on the current computers. >The lack of precision is probably not an issue. It is doubtful that much >precision would be desired in such a calculation --- no one wants to >have to use 1352685233/28451120769 and would rather deal with >0.04754418089123340268648521865195 >But then why do we need that level of precision? Unless you are >calculating the trajectory for a Jupiter shot, 3 or 4 decimals is fine, >and if you are starting with 5 significant digits then anything beyond 5 >digits is imprecise, anyway. There are many situations in which multiple precision is needed, and the trajectory for a Jupiter shot is not one of them. You are not going to be able to adjust the course of the spacecraft that accurately, and will have to recompute the trajectory and readjust it. But inverting a matrix usually loses many bits, and there are other operations, which one should find ways around. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: K-12 Calculator Woes The representation of fractions is, for a machine, much > harder than floats or decimals, and is more complicated > to work with. > Why do you believe that is true? Fractions, unless their size is limited, will require at > least two words, and their operations are more complex. > Machines rarely do exact arithmetic, as would be required > with fractions to give the answers desired. Avoiding > computer roundoff would be expensive. I am reasonably familiar with computer architecture, and > I can see no way around it. > Except most computers these days do not deal with memory structures as > small as a word. They are optimized at 32 bits, 64 bits, even 128 bits > for pipelining and reduced instruction set computing (risc). (Jeez, I'm > old. I remember programming using nibbles, or half bytes, for integers) > The older 8 bit chips are disappearing, and those left in the market are > relegated to the dollar store calcs. Now extremely cheap 16 bit longword > chips are about the only low end chips available. The calculations are > not a problem. And the smallest memory chips made today can easily store > the code. Unfortunately, the x86 architecture, in wide use in > computers, still has too many short procedures. And > I would like to do operations involving bit streams > for bit efficiency, but these are horribly expensive > on the present computers. In generating non-uniform > random numbers for simulation, one now uses wasteful > procedures to get speed. > Well, it depends on your definition of waste. With terabyte disk drives at $80, 4 gigabytes of memory at $35, 3.73 GHz dual core processors for $200, a little inefficiency is not a big deal. Gone are he days where resources were a problem. I agree about the x86 --- it is an old architecture that should have been replaced years ago. I much preferred the DEC Alpha VAX architecture and VMS or PDP's and RSTS --- far more flexible and secure. But they were also far more expensive than the x86, and so they went the way of the dinosaurs. > A more significant problem is the increased number of buttons to be put > in a small acreage. The desire for a small format directly competes > with the inclusion of more functions. Some calculators now have as many > as 3 or 4 functions for each key, and even so a large and bulky.Learning > to use them is a challenge. > And, of course, the marketing problem is huge. Most people just don't > want to calculate in fractions, and have learned the algorithm for > computing in integers instead, that is you can actually enter 3/4*5/9 > into the calculator to get a preferred decimal answer. That said, cheap > calculators that do integer fraction arithmetic are available --- my > Home Depot has a few models designed for carpenters, for ex. They > actually round to 16ths, as well as calculating things like pitch and > angles. In most situations, computing with exact arithmetic > would soon make the numerators and denominators so > large that it would be hopeless. And also much is > done using other algebraic and transcendental functions, > the results of which cannot be written as fractions. > Yes, but not on calculators. > One can work with rounded binary answers, and this is > the way most computer functions work. One can even > do these in multiple precision, but only with considerable > difficulty on the current computers. > This is one reason why I miss VMS. A 64 bit octaword floating point was a precise as almost anyone wanted to get. Not everyone, but most. But I still question almost all requests for precision. After all, we went to the moon using computers that are outperfomed by a TI90. > The lack of precision is probably not an issue. It is doubtful that much > precision would be desired in such a calculation --- no one wants to > have to use 1352685233/28451120769 and would rather deal with > 0.04754418089123340268648521865195 > But then why do we need that level of precision? Unless you are > calculating the trajectory for a Jupiter shot, 3 or 4 decimals is fine, > and if you are starting with 5 significant digits then anything beyond 5 > digits is imprecise, anyway. There are many situations in which multiple precision is > needed, and the trajectory for a Jupiter shot is not one of > them. You are not going to be able to adjust the course > of the spacecraft that accurately, and will have to recompute > the trajectory and readjust it. But inverting a matrix > usually loses many bits, and there are other operations, > which one should find ways around. > With all due respect to those out there in cyberspace, very, very little needs that level of precision, despite what some may think. Back in the day when I was a VAX system manager I talked with scientists doing nuclear research at Livermore and Fermilab who were able to do Nobel level work with 32 bit floating points. In the late '80's a friend spent a winter in Antarctica (I was so jealous -- he actually got to visit the south pole!) installing a PDP11 which supported the entire research community for nearly a decade using 16 bit FP's. It replaced an old 8086. We sent satellites to Jupiter with 8 bit FPs. And despite your assertion that in flight corrections made accuracy nearly moot, communication, weight and fuel considerations made mid-course corrections expensive and limited. Everyone (including me) wanted more, but the bean counters ruled, and more often than not it was determined that more wasn't needed. Not always, but usually. One little factoid I heard at DECUS symposium brought that home to me. In 1970, when massive mainframes ruled the world and people ran around with stacks of punch cards and desktop access was rare, there were about 750,000 CPA's in the US. As computers became cheaper and more generally available, the accounting field was targeted as a market where labor could be significantly reduced with the new mechanization --- spreadsheets and financial apps proliferated. By 1990 when Windows 3 came out there were 1.5 million CPAs in the US. What happened was someone found mew ways to slice and dice the numbers, more accurate ways of breaking down the numbers, and different ways of displaying he numbers, so more people wee required to perform the new functions rather than fewer people to maintain the status quo. Beneficial? I'd suggest no considering the current economic problems caused by derivatives and other investments based on no more than a computer program. Larry === Subject: Re: K-12 Calculator Woes >com>, [Long string of citations deleted.} ...................... > <> Understanding numbers is NOT understanding strings of > <> decimal digits. There are two main concepts for the > <> integers, and neither of them has anything to do with > <> the decimal representation. > <> The decimal representation is only ONE way of representing > <> numbers; it is sometimes not even useful. > <> For 99% of he population it is the only way of representing and > <> manipulating numbers. Even rational fractions disappear from th= >eir > <> lexicon for most, unless they user a tool like a tape measure. = >Hec=3D>k, > <> even fractions have virtually disappeared from he stock market. > <> There are other means as well. Those in the sciences, and > <> in many other fields as well, misuse mathematics because > <> they have learned procedures which require assumptions to > <> be met to be valid. > You continue to miss the point. >I think the problem is you two are talking about two >different points. >IMO the purpose of a math class is not to teach maths >but to teach a mode of knowing---of course, the only >way available to do that is to teach some mathematics, >so the teaching of maths in a good math class is unavoidable. > Knowing WHAT? >Knowing mathematical facts. One can learn the concepts without being burdened by a mass of facts. Learning the concepts first reduces the set of facts which need to be committed to memory. One can reconstruct facts if one has the understanding. Also, one can often avoid the need for certain facts if one understands; there are workarounds. The somewhat touted Chisenbop method of multiplication requires little of the multiplication table. >But as soon as we lose sight of the fact that the real purpose >is not to have students know facts about, say, numbers but to >know them in a specific way, chaos (in various manifestations) >follows. > It is NOT important to know facts, other than a few, > but to UNDERSTAND. The present methods of teaching > teach lots of facts, but little or even NO understanding. >Understanding only exists with respect to facts: it is >impossible to understand, what you can do is understand >something. One can understand the basic properties of the integers from a FEW facts, not now taught to most, including many teachers of mathematics in the schools. The classical method of teaching geometry, more than two millennia old, derives the facts from a few principles; it does not start with a mass of facts. The same holds for the number systems; it takes research to derive concepts from facts, and research ability is not that common. It takes only the ability to think in a logical manner to go the other way. >It is, as evidence shows, quite possible to teach facts without >teaching any understanding, but it is impossible to teach >understanding without teaching facts. Here you are wrong. One can teach the grammatical structure of a language without any knowledge of the vocabulary. Memorizing lots of vocabulary does not even help in this.l In a very strong sense, mathematics is pure grammar. One may CONJECTURE from many facts, but in mathematics, this is not accepted as KNOWLEDGE. The Riemann hypothesis has lots of implications in number theory, but these weaker attempts at facts are not accepted. Much effort is spent in mathematics to get the understanding behind many of the facts, and in some cases, understanding has yet to come. Why are there exactly 26 sporadic simple groups? I do not believe that anyone is working on that. The integers are form INFINITE set, and there are several basic concepts about them, which are not all related. One cannot learn an infinite set of facts, but one can understand this, and other, important infinities. > As of this time, the students coming out of a calculus > class may know many facts and methods, but few of them > will have an understanding of derivative or integral. > Concepts are not taught by teaching the words, but by > getting the student to internalize the ideas. >Carrying out the plan implicit in this last sentence of you >results in one of many ways in which education fails. >It surprises me quite deeply to observe how little people >involved in education (specially higher education) fail >to recognize that one cannot teach understanding, for it is >a rather intransmissible thing. One cannot communicate ideas >directly: in order to communicate ideas one must necessarily >go through the process of packaging them in the form of concrete >specific knowledge. Far less concrete than you think. One understood the concept of regular spaces which were not completely regular before Tychonov produced one, and one understood the concept of a regular space on which all continuous real-valued functions were constant before Hewitt produced one. There are other examples like this. Mathematics proceeds by reasoning, not by memorizing facts. >You simply cannot give your students understanding of the >integers, nor of any other subject. Not completely, but far more than the teachers of mathematics seem able to comprehend. >It is my conclusion after having participated in way too >many discussions about math education (at the college level >but, really, mutatis mutandi, the issues are exactly the same >as in any other level) that essentially all disagreements >are due to the opposing parties having a different opinion >on this. Moreover, in the enormous majority of cases none >of the parties recognize that the matter of contention is >not material (*what* subjects to teach?, for example) but >epistemiological---and this results in countless hours/emails >of argumentation for and against the teaching of various >subjects and other trivia, discussion which will never ever >help in getting close to an agreement for it is a discussion on >something rather irrelevant to the disagreement. > No, you do not get it. I would require much of a good > course on real variables before calculus. Solving > problems is not the goal, except for adepts; knowing > the concepts well enough to set up the problem and > communicate with the specialist is primary. Only then > does one become a specialist. >How can you tell if someone knows the concepts unless >they put them into use, assuming it is not something >of the same nature as religious experience, whatever >that may be? You are right about this. One problem now is that the concepts are taught AFTER many facts and procedures, and thus these are not connected. This is why I believe that the concepts come first, so that the understanding can be used in applications. In the fall semester, I convinced a student that what he needed to do was to get his hands dirty, to go back and use the theory to compute the solution of problems. I often apply abstract theory to the understanding, and even the solution, of concrete problems. Sometimes, I apply the concepts of one area to work in another, and get simple solutions to problems. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: K-12 Calculator Woes posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2009042708 Fedora/3.0.10-1.fc10 Firefox/3.0.10,gzip(gfe),gzip(gfe) com>, [Long string of citations deleted.} ...................... > <> Understanding numbers is NOT understanding strings of > <> decimal digits. There are two main concepts for the > <> integers, and neither of them has anything to do with > <> the decimal representation. > <> The decimal representation is only ONE way of representing > <> numbers; it is sometimes not even useful. > <> For 99% of he population it is the only way of representing and > <> manipulating numbers. Even rational fractions disappear from th= >eir > <> lexicon for most, unless they user a tool like a tape measure. = >Hec=3D>k, > <> even fractions have virtually disappeared from he stock market. > <> There are other means as well. Those in the sciences, and > <> in many other fields as well, misuse mathematics because > <> they have learned procedures which require assumptions to > <> be met to be valid. > You continue to miss the point. >I think the problem is you two are talking about two >different points. >IMO the purpose of a math class is not to teach maths >but to teach a mode of knowing---of course, the only >way available to do that is to teach some mathematics, >so the teaching of maths in a good math class is unavoidable. > Knowing WHAT? >Knowing mathematical facts. One can learn the concepts without being burdened > by a mass of facts. Learning the concepts first > reduces the set of facts which need to be committed > to memory. One can reconstruct facts if one has > the understanding. Also, one can often avoid the > need for certain facts if one understands; there > are workarounds. The somewhat touted Chisenbop > method of multiplication requires little of the > multiplication table. >But as soon as we lose sight of the fact that the real purpose >is not to have students know facts about, say, numbers but to >know them in a specific way, chaos (in various manifestations) >follows. > It is NOT important to know facts, other than a few, > but to UNDERSTAND. The present methods of teaching > teach lots of facts, but little or even NO understanding. >Understanding only exists with respect to facts: it is >impossible to understand, what you can do is understand >something. One can understand the basic properties of the integers > from a FEW facts, not now taught to most, including many > teachers of mathematics in the schools. The classical > method of teaching geometry, more than two millennia old, > derives the facts from a few principles; it does not start > with a mass of facts. The same holds for the number systems; it takes research > to derive concepts from facts, and research ability is > not that common. It takes only the ability to think in > a logical manner to go the other way. It is, as evidence shows, quite possible to teach facts without >teaching any understanding, but it is impossible to teach >understanding without teaching facts. Here you are wrong. One can teach the grammatical structure > of a language without any knowledge of the vocabulary. > Memorizing lots of vocabulary does not even help in this.l That appears to be true, but it isn't. Grammar is indisolubly linked to meaning, and dictated by it. This makes the simile of quite restricted utility ;-) Moreover, you cannot do this to a kid who does not speak any language. Which is, more or less, a status similar to our students with respect to maths. > In a very strong sense, mathematics is pure grammar. One > may CONJECTURE from many facts, but in mathematics, this is > not accepted as KNOWLEDGE. The Riemann hypothesis has lots > of implications in number theory, but these weaker attempts > at facts are not accepted. Much effort is spent in > mathematics to get the understanding behind many of the > facts, and in some cases, understanding has yet to come. > Why are there exactly 26 sporadic simple groups? I do not > believe that anyone is working on that. The integers are form INFINITE set, and there are several > basic concepts about them, which are not all related. One > cannot learn an infinite set of facts, but one can > understand this, and other, important infinities. > As of this time, the students coming out of a calculus > class may know many facts and methods, but few of them > will have an understanding of derivative or integral. > Concepts are not taught by teaching the words, but by > getting the student to internalize the ideas. >Carrying out the plan implicit in this last sentence of you >results in one of many ways in which education fails. >It surprises me quite deeply to observe how little people >involved in education (specially higher education) fail >to recognize that one cannot teach understanding, for it is >a rather intransmissible thing. One cannot communicate ideas >directly: in order to communicate ideas one must necessarily >go through the process of packaging them in the form of concrete >specific knowledge. Far less concrete than you think. Of course. Or rather, not really: at the time non-completely regular spaces become an issue, one's notion of concrete has evolved in ways that the very little concrete that you mention is really quite concrete. > One understood the > concept of regular spaces which were not completely regular > before Tychonov produced one, and one understood the > concept of a regular space on which all continuous > real-valued functions were constant before Hewitt produced > one. There are other examples like this. Mathematics > proceeds by reasoning, not by memorizing facts. You simply cannot give your students understanding of the >integers, nor of any other subject. Not completely, but far more than the teachers of > mathematics seem able to comprehend. Well, they do seem to comprehend so little! >It is my conclusion after having participated in way too >many discussions about math education (at the college level >but, really, mutatis mutandi, the issues are exactly the same >as in any other level) that essentially all disagreements >are due to the opposing parties having a different opinion >on this. Moreover, in the enormous majority of cases none >of the parties recognize that the matter of contention is >not material (*what* subjects to teach?, for example) but >epistemiological---and this results in countless hours/emails >of argumentation for and against the teaching of various >subjects and other trivia, discussion which will never ever >help in getting close to an agreement for it is a discussion on >something rather irrelevant to the disagreement. > No, you do not get it. I would require much of a good > course on real variables before calculus. Solving > problems is not the goal, except for adepts; knowing > the concepts well enough to set up the problem and > communicate with the specialist is primary. Only then > does one become a specialist. >How can you tell if someone knows the concepts unless >they put them into use, assuming it is not something >of the same nature as religious experience, whatever >that may be? You are right about this. One problem now is that the > concepts are taught AFTER many facts and procedures, and > thus these are not connected. This is why I believe that > the concepts come first, so that the understanding can be > used in applications. This is where we differ: to me, they come together and reinforce each other ;-) -- m === Subject: Re: K-12 Calculator Woes <4dqdndxEOYJTop7XRVn_vwA@comporium.net> posting-account=nHkyWQoAAAAZj13mfknn7vPxoYn-Mvx3 CLR 1.1.4322; InfoPath.1; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) I have no disdain for applied math; in fact, I do not > believe it exists. I have always seen it this way. The more abstract is the > mathematics that I know, the easiier it is to apply, and > the more powerful the result. For instance when studying point set topology an exercise > presented itself: For 0 < a,b < 1 the intervals ]a, 1] and [0, b[ are a > subbase for the the usual topology on [0, 1]. Knowing this made writing cascaded iterations over ranges > of integers in computer programs simplest and clearest. The insight can be approximated with the Rule. Write the lower bound (b) and upper bound (B) of the > iteration such that B - b is the number of times to iterate > the loop. > for(index = 0; index < N; index++) > ... The above may seem no great insight; yet when writing > cascaded loops where an inner loop depends on the value > of the index of an enclosing loop, complexities disappear. > All is clear. Compare with Fortran programs where the loops are of the form for I = 1, N ... It would help your point if you made it fairly. If I wanted to write a Fortran program to loop from zero (inclusive) to N (exclusive) I would use for I = 0, N - 1 If I wanted to compare a C coding convention with an alternate C coding convention I would write both in C, e.g. for index = 0; index < N; index++ versus for index = 0; index <= N-1; index++ Of course, the main point that you started to make was that a knowledge of abstract math is useful to you. That point stands or falls independently of the above criticism. However... A farmer who has lost a section of fence can divide the distance lost by the fencepost spacing interval and subtract one without worrying much about whether the approach he has used to arrive at the number of fenceposts needed leads to confusion. There's a point where you have to stop computing and start working. === Subject: Re: K-12 Calculator Woes I have no disdain for applied math; in fact, I do not > believe it exists. I have always seen it this way. The more abstract is the > mathematics that I know, the easiier it is to apply, and > the more powerful the result. For instance when studying point set topology an exercise > presented itself: For 0 < a,b < 1 the intervals ]a, 1] and [0, b[ are a > subbase for the the usual topology on [0, 1]. Knowing this made writing cascaded iterations over ranges > of integers in computer programs simplest and clearest. The insight can be approximated with the Rule. Write the lower bound (b) and upper bound (B) of the > iteration such that B - b is the number of times to iterate > the loop. > for(index = 0; index < N; index++) > ... The above may seem no great insight; yet when writing > cascaded loops where an inner loop depends on the value > of the index of an enclosing loop, complexities disappear. > All is clear. Compare with Fortran programs where the loops are of the form for I = 1, N ... It would help your point if you made it fairly. If I wanted to write a Fortran program to loop from zero (inclusive) > to N (exclusive) I would use for I = 0, N - 1 How many times does that loop execute? It executes N times, but (N - 1) - 0 = N - 1. > If I wanted to compare a C coding convention with an alternate C > coding convention I would write both in C, e.g. for index = 0; index < N; index++ > versus > for index = 0; index <= N-1; index++ The latter iteration does not execute (N-1) - 0 times. Writing iterations so: for index = a; index < b; index++ or for index = a; index >= b; index-- makes coding up cascaded loops and iterating through zero base arrays much clearer. Take some Fortran codes that do linear algebra, eigenvalues and eigenvectors for instance, and translate to C using the protocol that I recommend. Then compare it with the Fortran code. > Of course, the main point that you started to make was that a > knowledge of abstract math is useful to you. That point stands or > falls independently of the above criticism. However... A farmer who has lost a section of fence can divide the distance lost > by the fencepost spacing interval and subtract one without worrying > much about whether the approach he has used to arrive at the number of > fenceposts needed leads to confusion. There's a point where you have to stop computing and start working. That point is when the calculation to be undertaken is perfectly clear in my mind. -- Michael Press The purpose of computing is insight, not numbers. -- R.W. Hamming === Subject: Re: K-12 Calculator Woes On 6 May 2009 15:02:53 -0400, hrubin@odds.stat.purdue.edu (Herman >What mathematics do non-mathematicians need? They need >to understand the mathematical concepts which will be >used in their work, and for most this will need an >understanding of the real numbers and limits. They >need to be able to use mathematical notation to >translate their real-world problems into mathematical >representations which then can be solved by mathematical >methods. What they will not be able to do, except in >very simple situations, is carry out the solution. Herman, most non-mathematicians in non-technical fields do NOT use >these concepts at all. The use of technical fields is expanding, and even > those in non-technical fields could make good use > of the concepts to simplify their calculations. Furthermore, if they have not seen the concepts, and > have even moderate difficulty with the manipulations, > they may well not enter a technical field. The world does need people other than mathematicians. -- ... when we came back, late, from the hyacinth garden, Your arms full, and your hair wet, I could not Speak, and my eyes failed... === Subject: Re: Spherical map; Delaunay Triangulation posting-account=PPzH5gkAAAAHEC4yUVc-0eQpq5EOMM1Y Gecko/2008120121 Firefox/3.0.5,gzip(gfe),gzip(gfe) > Oh, for clarification, I have some oldish code called STRIPACK > (spherical triangulation pack), which does a lot of this stuff in > Fortran 90. I've been adapting some of it to C, since I never studied > Fortran, Fortran is uncommon in game programming, and the development > environments for Fortran are expensive. They are? what about g77 and the free FORTRAN compiler that's part of Sun Studio? http://gcc.gnu.org/onlinedocs/gcc-3.4.6/g77/What-is-GNU-Fortran 003f.html http://developers.sun.com/sunstudio/features/whatsin.jsp > Since this requires learning > enough Fortran to translate the code and a lot of work to translate, I > was hoping there might be a similar library in C already. :) Then again, since Fortran writers tend to make highly optimized code, > maybe using STRIPACK as a base would get me the best possible result. > I don't know. === Subject: Re: Apologies: Really easy problem. I have been swatting up using a book I purchased, which has a question > about area: How much will it cost to build a wall round a house that is 25 metres > long and 75 metres wide, What a badly worded question. Is it the house or the wall that is 25m long and 75m wide? If it's the house, how does knowing its size help you to work out the size of the wall? > if the cost of 50 cm is £10? Your options are > A) £4,000, B) 40,000 or C) £400. The answers are meant to be > estimations, What is the answer? I have calculated the answer, but when I checked > it I found I was wrong, but I cannot get the supposed answer. I am > trying to just work out if the book is wrong or not. > D -- ... when we came back, late, from the hyacinth garden, Your arms full, and your hair wet, I could not Speak, and my eyes failed... === Subject: Re: Apologies: Really easy problem. > I have been swatting up using a book I purchased, which has a question > about area: > How much will it cost to build a wall round a house that is 25 metres > long and 75 metres wide, What a badly worded question. Is it the house or the wall that is 25m > long and 75m wide? If it's the house, how does knowing its size help > you to work out the size of the wall? > Its been a long time since I did such problems, but... I always found that when a question contained an ambiguity such that one interpretation enabled the calculation of a solution and the other interpretation did not, it was best to assume the case that enabled the solution. In such a case I would feel grateful that such an easy technique had been provided to resolve the ambiguity. Hence I would hazard a guess they are referring to the wall and not the house. > if the cost of 50 cm is £10? Your options are > A) £4,000, B) 40,000 or C) £400. The answers are meant to be > estimations, > What is the answer? I have calculated the answer, but when I checked > it I found I was wrong, but I cannot get the supposed answer. I am > trying to just work out if the book is wrong or not. > D === Subject: Assumptions behind the OLS regression model? In many statistics textbooks I read the following text: ñA models based on ordinary linear regression equation models Y, the dependent variable, as a normal random variable, whose mean is linear function of the predictors, b0 + b1*X1 + ... , and whose variance is constant. While generalized linear models extend the linear model in two ways. First, assumption of linearity in the parameters is relaxed, by introducing the link function. Second, error distributions other than the normal can be modeled.î My Stat teacher never bothered to explain these things to us. He started the regression lesson with the equation Y = b0 + b1 * X1, and an example based on the Weight and Height relation. He never talked about these assumptions about normality and the variance. As a result for quite some time, I treated this equation was an identity, similar to Assets= Liability + Equity. I have never understood what difference those underlying assumptions make. Can anybody please explain me why these assumptions are required for this model, and what happens to the result of this model if these assumptions are violated? MG. === Subject: Re: JSH: The Simple Lie posting-account=mgs1FwoAAABD3j5T_RLZ06yrgt2dghDu Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) Guys, Harris posts one message, no follow ups, and gets up > to twenty replies. Including yours. Not mine: not a reply to James Harris. Yes, it was. === Subject: Re: JSH: The Simple Lie Guys, Harris posts _one_ message, no follow ups, and gets up > to twenty replies. Including yours. Not mine: not a reply to James Harris. Yes, it was. > It is not a reply to James Harris, just as this is not. -- Michael Press === Subject: Re: JSH: The Simple Lie posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 CLR 1.1.4322; .NET CLR 2.0.50727; InfoPath.1),gzip(gfe),gzip(gfe) Guys, Harris posts one message, no follow ups, and gets up > to twenty replies. Including yours. Not mine: not a reply to James Harris. I'm sure the trolls take that into consideration when they compute the reply tally. -- > Michael Press === Subject: Re: JSH: The Simple Lie Guys, Harris posts _one_ message, no follow ups, and gets up > to twenty replies. Including yours. Not mine: not a reply to James Harris. I'm sure the trolls take that into consideration when they > compute the reply tally. That is the troll's problem, not mine. It is their own moribund lives at stake. Imagine trying to live on what others throw away; to be dependent on others for scraps: beggars. -- Michael Press === Subject: Re: JSH: The Simple Lie As Michael Press has implied, adding energy to a system (e.g. threads begun by JSH) is likely to lead to that system's increase, so I add this with some trepidation, but fishfry deserves recognition for this one. Well done. === Subject: Re: JSH: EMIS has my old paper back up? > On May 9, 10:48=A0pm, Mariano Su=E1rez-Alvarez > > > Actually, the most common source of that particular problem occurs > when, while typing the *.tex source, one inadvertedly types > > instead of ., usually because the immediate prior character requires > the shift key. It happens to me a lot when a period follows an in- > line formula (which ends in $). > > Hmm. That may be the case in an USian keyboard layout, I guess. > > The usual keyboard (US) does not have an opening question mark. I saw your correction. But what Mariano meant was that possibly on a US keyboard the period was paired with the greater than sign. This is not universally true on all keyboards. And I think that Mariano's explamation initial position in the body of a mail message by >. This is due to the original Unix mailbox format where all messages are lumped after each other delivery software. (And I still do use such software...) -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: EMIS has my old paper back up? Nntp-Posting-Host: hera.cwi.nl ... > They trashed a decades worth of papers, which should be disquieting to > people who publish in electronic only journals. > > EMIS saved its archives though, and now it seems saw fit to save my > paper as well. Oh no. EMIS saved only two parts of a single issue of the journal. > That paper historically may be considered one of the biggest in > mathematical history. Probably because the author of it acknowledged that it contained errors before it was published, but refused to retract it. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Math- Organized posting-account=-LsU5QoAAADlmZ8sORNW6vX0Eva6j_04 InfoPath.2; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) http://www.geocities.com/multiplicationfacts/bestlinks.html Links to tutorials, worksheets, interactive practice and tests. Organized by subject: Fractions, Decimals, Percents, Negative Numbers and more. Brett Taylor === Subject: Re: The set of 2x2 matrices with nonnegative integers as entries and determinant 1 <488r05d5pdhgukrstegl74m817hehcalbt@4ax.com> posting-account=-PngCgkAAAD2yUjosqWv1Nf1lkqWP4lp InfoPath.2),gzip(gfe),gzip(gfe) So we're comparing a with c, and b with d, and in that case there are, >of course, theoretically nine possibilities: I) a = c, b = d >II) a = c, b < d >III) a = c, b > d >IV) a < c, b = d >V) a < c, b < d >VI) a < c, b > d >VII) a > c, b = d >VIII) a > c, b < d >IX) a > c, b > d You seem to be saying that, for matrices that have a determinant of 1 >(i.e., are members of M), cases I and VI are impossible, and case >VIII can only arise from the identity matrix. Actually, I've experimented with my problem in the past (including the > comparison of a with c and b with d ), and I think it is > possible to collapse the nine possibilities above into only four: VIII) a > c, b < d > VII/IX) a > c, b >= d > II/V) a <= c, b < d > I/III/IV/VI) a <= c, b >= d So, with regard to matrices in M (where a, b, c, d are integers >= > 0, and a*d - b*c = 1), the object would then be to show that the first > case above applies only to the identity matrix [[1 0] [0 1]], and the > last case is totally impossible. I would appreciate it if someone > would write out a proof of this so that I'd be able to verify it more > clearly, although I may try this weekend going over it again myself. If a>c, b=c+1, d>=b+1, so ad-bc >= (c+1)(b+1)-bc >= b+c+1, sot ad-bc=1 if and only if b=c=1, a=d=1. Try the other one yourself - the argument is similar! Derek Holt. === Subject: scrambling a word posting-account=5KI_ewoAAACPf426M9-P_p-sfT6Q5yxy Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) Let u be a word over some alphabet S. Call the scramble of a word u as the set of all words over S having the same image under the Parikh map. So for example, if u = a^2b^3, then the scramble of u is the set of all words consisting of two a's and three b's. Is this concept studied in formal language theory? Define the scramble of a language L as the union of the scrambles of words in L. Questions: if L is regular, is the scramble of L regular? if L is context-free, is the scramble of L context-free? if L is context-sensitive, is the scramble of L context-sensitive? === Subject: Re: scrambling a word +M[5[U[QT7xFN%^gR=tuJw%TXXR'Fp~W;(T1(739R%m0Yyyv*gkGoPA.$b,D.w:z+<'=-lV T?6 {T?=R^:W5g|E2#EhjKCa+nt:4b}dU7GYB*HBxn&Td$@f%.kl^:7X8rQWd[NTcPu6nkisze/ Q;8 9Z{peQF,w)7UjV$c|RO/mQW/NMgWfr5*$-Z%u46/00mx-,R'fLPe.)^ Let u be a word over some alphabet S. Call the scramble of a word u > as the set of all words over S having the same image under the Parikh > map. So for example, if u = a^2b^3, then the scramble of u is the set > of all words consisting of two a's and three b's. Is this concept studied in formal language theory? IDNK > Define the scramble of a language L as the union of the scrambles of > words in L. Questions: if L is regular, is the scramble of L regular? I can see no way that allowing scrambled substiutes of words could make it irregular > if L is context-free, is the scramble of L context-free? I can see no way that allowing scrambled substiutes of words could make it contecxt sensitive > if L is context-sensitive, is the scramble of L context-sensitive? not necessarily, === Subject: Re: Planar graphs vs. graphs of contact of convex polygons posting-account=JdyD6goAAAAEmGYaeNSr70Vkn3kWnqq6 AppleWebKit/525.27.1 (KHTML, like Gecko) Version/3.2.1 Safari/525.27.1,gzip(gfe),gzip(gfe) I'd like to know whether every planar graph G is [topologically > equivalent to] the graph of contact of a set of non-overlapping convex > polygons. NB: The graph of contact of a finite set S of non-overlapping convex > polygons is formed as follows: > - each polygon is devoted by a vertex v, > - there is an edge e between two vertices v1 and v2 iff the > corresponding polygons share a portion of side of length greater than > zero (no contacts by the corners). Is it a known result or are there known counter-examples? Any pointers welcome! > Yann David Below is a planar graph (for instance A has edges to B, J, E, T, etc.). What does it look like drawn with convex polygons? Note that I am not offering this as a counter example. I am interested in what the process is to convert graph to drawing. Bob H A: B,J,E,T,K,D,S,G,C,I,H,O,N,R B: A,R,N,O,H,F,Q,M,C,J C: A,G,L,D,E,J,B,M,F,I D: A,K,E,C,L,P,G,S E: A,J,C,D,K,T F: B,H,I,C,M,Q G: A,S,D,P,L,C H: A,I,F,B,O I: A,C,F,H J: A,B,C,E K: A,T,E,D L: C,G,P,D M: B,Q,F,C N: A,O,B,R O: A,H,B,N P: D,L,G Q: B,F,M R: A,N,B S: A,D,G T: A,E,K === Subject: Re: Planar graphs vs. graphs of contact of convex polygons posting-account=mMlFNwkAAAB4mlUHsL9lydZIKUPXmAkO Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) I'd like to know whether every planar graph G is [topologically > equivalent to] the graph of contact of a set of non-overlapping convex > polygons. NB: The graph of contact of a finite set S of non-overlapping convex > polygons is formed as follows: > - each polygon is devoted by a vertex v, > - there is an edge e between two vertices v1 and v2 iff the > corresponding polygons share a portion of side of length greater than > zero (no contacts by the corners). Is it a known result or are there known counter-examples? Any pointers welcome! > Yann David I guess my last message didn't go through. Apply Fary's theorem to the > dual graph. You'll need to add the outside faces, but this is no > trouble. I thought about this a little more. This construction works only if > both G and its dual, G*, are simple. If G has multiple edges, there is > no trouble if we allow degenerate polygons: If the two polygons are in > contact at two different points, then by convexity they are in contact > along the edge between them. (So maybe this is disallowed by the > definition of the graph of contact.) The other case is if G has > vertices of degree two, so that G' has multiple edges. To resolve this > case, replace each stretch of such vertices by a single edge, and then > apply Fary's theorem to the dual. Here's an example: G * G' * . G'* . > /| /| . . > / * / | . . > * | * -> * | * -> x....x > * / | / . . > |/ |/ . . > * * . . Now replace the edge in G' with a slighly larger rectangle, preserving > all the contacts except for the two faces on either side of the edge, > and then fill in the inside of the rectangle with as many rectangles > as vertices were deleted in G, making sure that they only touch the > face above and face below (in my diagram): . G* . G * > x..x.x..x /| > . . . . / * > . x.x . -> * | * > . . . . * / > x..x.x..x |/ > . . * This G' is the graph of contact for the originial G. The graphs I'm considering are simple. Are G* and G' the same in your explanation above? Could you be a bit more specific about your core idea (applying Fary's theorem to the dual of the graph)? 1) The contact graph of the dual is not necessarily the initial graph. How do you make that happen? 2) How do you introduce the convexity in that approach? Yann === Subject: Re: Planar graphs vs. graphs of contact of convex polygons posting-account=YUVDEAoAAADy0b0DWFp7F-O3S06gZA3W Gecko/2009042523 Ubuntu/9.04 (jaunty) Firefox/3.0.10,gzip(gfe),gzip(gfe) > I'd like to know whether every planar graph G is [topologically > equivalent to] the graph of contact of a set of non-overlapping convex > polygons. NB: The graph of contact of a finite set S of non-overlapping convex > polygons is formed as follows: > - each polygon is devoted by a vertex v, > - there is an edge e between two vertices v1 and v2 iff the > corresponding polygons share a portion of side of length greater than > zero (no contacts by the corners). Is it a known result or are there known counter-examples? Any pointers welcome! > Yann David I guess my last message didn't go through. Apply Fary's theorem to the > dual graph. You'll need to add the outside faces, but this is no > trouble. I thought about this a little more. This construction works only if > both G and its dual, G*, are simple. If G has multiple edges, there is > no trouble if we allow degenerate polygons: If the two polygons are in > contact at two different points, then by convexity they are in contact > along the edge between them. (So maybe this is disallowed by the > definition of the graph of contact.) The other case is if G has > vertices of degree two, so that G' has multiple edges. To resolve this > case, replace each stretch of such vertices by a single edge, and then > apply Fary's theorem to the dual. Here's an example: G * G' * . G'* . > /| /| . . > / * / | . . > * | * -> * | * -> x....x > * / | / . . > |/ |/ . . > * * . . Now replace the edge in G' with a slighly larger rectangle, preserving > all the contacts except for the two faces on either side of the edge, > and then fill in the inside of the rectangle with as many rectangles > as vertices were deleted in G, making sure that they only touch the > face above and face below (in my diagram): . G* . G * > x..x.x..x /| > . . . . / * > . x.x . -> * | * > . . . . * / > x..x.x..x |/ > . . * This G' is the graph of contact for the originial G. The graphs I'm considering are simple. Are G* and G' the same in your explanation above? No, G' is the graph obtained from G by replacing stretches of vertices of degree 2 with single edges, and G'* is its dual. In my example, G* is not quite the dual of G--I should have used better notation--because it's not even a proper graph, but G is the graph of contact of the polygons in the construction of G*. > Could you be a bit more specific about your core idea (applying Fary's > theorem to the dual of the graph)? Sure. The above was only needed when there were vertices of degree two, so I'll assume that G doesn't have any, and that it doesn't have any multiple edges. In the dual graph G*, the faces correspond to vertices of G, and vice-versa, while the edges remain edges. (There are issues with the outside face and edges, but we'll deal with them later.) Since G* is also planar, we can choose an embedding such that all the edges are straight lines. Then each polygon in G* (i.e., the faces of G*) corresponds to a vertex in the original graph, and two polygons share and edge if and only if there's an edge between the two vertices in the original graph. Thus G is the graph of contact for the faces of G*. The only issues are (1) outside faces and edges, (2) convexity of the polygons, and (3) dealing with multiple edges and vertices of degree two. For (1), there are different ways to do this. Here's one way that seems relatively simple. For each outside vertex v, add another vertex v' (to the outside of the graph) and an edge between v and v'. Then add edges between two of the new vertices if there's an edge between the two correponding vertices in the original graph--i.e., if v is adjacent to u and w (all outside vertices), then create edges between u' and v' and between v' and w'. If there are n outside vertices (and n outside edges), then this will create n new quadrilateral faces. Here's an example, where G is the original graph, G' is the altered graph, and G* is the dual of G' (neglecting the outside face of G'): * /| G * G' / * G* x-----x /| / /| | /| / | / / | | x-x | *--*--* -> *-*--*--*-* -> | | | | | / | / / | x-x | |/ |/ / |/ | * * / x-----x |/ * Adding these quadrilaterals to the graph ensures that the outside vertices of G will correspond to faces in G*. Just to be clear here, the graph of contact of the five faces in G* is exactly G. (Sorry that I'm abusing notation with G* again.) For (2), convexity isn't clear from Fary's Theorem (since the proof involves adding edges until every face is a triangle), but there is a result that if G is three-connected, then it can be embedded such that every face is a convex polygon. (G is three-connected if it remains connected after removing any two vertices.) Unfortunately, that's not enough for our case. I'll try to think about how to resolve this. (I originally thought that you could perturb the positions of the vertices so that every face is convex.) Finally, (3) was addressed in my last message. > 1) The contact graph of the dual is not necessarily the initial graph. > How do you make that happen? > 2) How do you introduce the convexity in that approach? > Yann === Subject: Inverse of a Legendre Polynomial posting-account=ZPMZHQoAAABDJE-oNaVfDOGq3q488giM CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) Hello Assume that F(X) and G(X) are two functions that are inverses of each other. That is, F(G(X)) =X and G(F(X)) =X. Assume that F(X) is a linear combination of Legendre polynomials. Are there any properties of Legendre polynomials that will simplify the calcuation of G(X)? Ideally, G(X) would also be a linear combination of Legendre polynomials. Bob === Subject: Re: Inverse of a Legendre Polynomial posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Hello Assume that F(X) and G(X) are two functions that are inverses of each > other. That is, F(G(X)) =X and G(F(X)) =X. Assume that F(X) is a linear combination of Legendre polynomials. That is: F is an arbitrary univariate polynomial? Are there any properties of Legendre polynomials that will simplify > the calcuation of > G(X)? Ideally, G(X) would also be a linear combination of Legendre > polynomials. That is: G is also a univariate polynomial? Not that in this case deg(FoG) = deg(F)*deg(G), so there are not many options to have FoG = identity > Bob === Subject: math fun -sbk posting-account=qttmhgkAAAChICW4cnappYtBT9kM3JuS .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) Vendetta Theory - 4X /2 f(X)= reduction to 1 where 1 therefore becomes 0 slow motion stabbing quanta - consecutive x1, x2, X3, x2, x1 ^ 9 = set A becomes subset B and C exclusively godbomb - 1 integrated with inf. -> imaginary number Rt. 1 lovemaking theory - 3X E Converted into = Mass * speed of light into reduction of Brute.. Math Fun - by sEung b. Kim === Subject: Logical Necessities and the End of Diablo -by Seung Bum Kim posting-account=qttmhgkAAAChICW4cnappYtBT9kM3JuS .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) Diablo.. the theory that moral deposits are lies is a primitive notion. Lust doesn't produce delusion based on spiritual principle but based on the change of blood pressure as one proceeds to beat the crap outta her.. there is a biological basis for all things.. where math can explain everything as opposed to Pascal's claim that math cannot be confused with the topic of man.. it comes from mythical archetypes of Pascal's primitive mind.. Pascal's Wager that inf. reward through suspension of finite reward is > than finite reward * limited duration. the logical contingencies here are measurable by automatic processing through values that by quotient are comparable and leading to identity of it by the quotient from the origin of componential values..where this process cannot be in constriction of logical contingencies or necessities there must therefore exist a conscious relation to the data..by which it must be interpreted by consciousness since logical schizms are within the data that must be interpreted by that which isn't based on logical contingencies..or factors that are not automatic, or cannot be answered by automatic processes which are in fact logically contingent by necessity since it operates through the matrix factor of all things material..in which that which is higher than the matrix factor becomes non-corporeal.. from which consciousness operates in this transcendental field.. and becomes a phenomenon that is not apart from body but pregnant with life that exists by the blood factor and its open system as a open system it allows for morphic processes..and in which the hemoglobin in the blood with O2 factor leads to a unity with the Earth and its whole objects.. Logical Necessities as Automatic Processes - by Seung Bum Kim === Subject: mathfun2 -sbk posting-account=qttmhgkAAAChICW4cnappYtBT9kM3JuS .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) The torque of the pork * the speed of the deed = the heat of the meat (a classic) The mass of the ass/ by the speed of the deed = the torque of the Pork Things that are imaginary are just concepts since only existance is a Predicate.. it would be great for me to have lots and lots of money but I only have a few $100 from writing books.. And as such things are facts not ideals in Truth.. God / 2 = 0 Blair Witch rooted by 3^M = beer time science * Science = -|pi| f(pi) -> inf. * 3000 love * the torque of the Pork equals sin or irrational numbers rt. by 3 mathFun2 -sbk === Subject: Prime Polynomials for P=NP Proof posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Gecko/20081029 Firefox/2.0.0.18,gzip(gfe),gzip(gfe) 2*2^606 - 1 may be prime. (a = 3) 2*2^606 - 1 is prime! (P = 1, Q = -1) [183 digits] 333333335*2^1188 - 1 may be prime. (a = 3) 333333335*2^1188 - 1 is prime! (P = 5, Q = -1) [367 digits] 2*2^1278 - 1 may be prime. (a = 3) 2*2^1278 - 1 is prime! (P = 1, Q = -1) [386 digits] 2*2^2202 - 1 may be prime. (a = 3) 2*2^2202 - 1 is prime! (P = 1, Q = -1) [664 digits] 2*2^2280 - 1 may be prime. (a = 3) 2*2^2280 - 1 is prime! (P = 12, Q = -1) [687 digits] 2*2^3216 - 1 may be prime. (a = 3) 2*2^3216 - 1 is prime! (P = 5, Q = -1) [969 digits] 2*2^4422 - 1 may be prime. (a = 3) 2*2^4422 - 1 is prime! (P = 1, Q = -1) [1332 digits] 333333335*2^10440 - 1 may be prime. (a = 3) 333333335*2^10440 - 1 is prime! (P = 5, Q = -1) [3152 digits] 333333335*2^16200 - 1 may be prime. (a = 3) 333333335*2^16200 - 1 is prime! (P = 7, Q = -1) [4886 digits] 2*2^18456 - 1 ? Computing power [0.00%] 333333335*2^16200 - 1 is prime! (P = 7, Q = -1) [4886 digits] I assert the polynomial# For example, x2 - 4x + 7 is a polynomial, but x2 - 4/x + 7x3/2 is not, ... A polynomial is a sum of terms. For example, the following is a polynomial: ... en.wikipedia.org/wiki/Polynomial - 89k # Simple Polynomial Multiplication The next step up in complexity is a one-term polynomial times a multi- term polynomial. For example: Simplify -3x(4x2 - x + 10) ... 2*n^p - 1 converge at 333333335*2^10440 - 1. === posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Gecko/20081029 Firefox/2.0.0.18,gzip(gfe),gzip(gfe) The LISTSERV kept thinking I was trying to process a command when posting this, so I apologize if this submission posts twice. This is a method from which to calculate infinite sets of prime numbers, which should not be too taxing computationally. I have not written the program, only outlined this underlying conjecture I would like to put forth to my peers for your confirmation/refinements: TO ANY GIVEN SET OF NUMBERS (Infinite, Countable or Non- Countable (I dis-enjoy the latter phrase, always seemed so defeatist and anti-to understanding: 1) First eliminate all EVEN numbers and multiples of FIVE. 2) Then, with the exception of the first number (one), delete all numbers ending in 1,3,5,7,9 that were multiplied by numbers ending in 1,3,5,7,9. 3) THE REST ARE PRIME. Can you confirm these findings? Martin M. Musatov === > The LISTSERV kept thinking I was trying to process a command when > posting > this, so I apologize if this submission posts twice. This is a method from which to calculate infinite sets of prime > numbers, > which should not be too taxing computationally. I have not written > the > program, only outlined this underlying conjecture I would like to put > forth > to my peers for your confirmation/refinements: TO ANY GIVEN SET OF NUMBERS (Infinite, Countable or Non- > Countable (I > dis-enjoy the latter phrase, always seemed so defeatist and anti-to > understanding: 1) First eliminate all EVEN numbers and multiples of FIVE. > 2) Then, with the exception of the first number (one), delete all > numbers > ending in 1,3,5,7,9 that were multiplied by numbers ending in > 1,3,5,7,9. > 3) THE REST ARE PRIME. Can you confirm these findings? > Martin M. Musatov How is this in any way an improvement on the sieve of Aritosthenes? === Subject: 27 percent of aspiring teachers pass math test posting-account=NgMGSwkAAABJni6NIYF05Yc4jNzXwHf- 1.1.4322),gzip(gfe),gzip(gfe) The grim results of the math tests were also published in the front page of the Boston Globe on 19 May 2009. May 20, 2009 Strength in numbers Math results ID need for harder work If 600 aspiring teachers take a math test and 27 percent pass, how many new and genuinely competent math teachers will be available to elementary schoolchildren this fall? Such a question involves only elementary math skills, but the results emerging from the new math section on the state's licensing exam for teachers show that many would-be teachers are in need of a thorough review of such elementary skills before they enter a classroom. On the one hand, Massachusetts deserves credit for having broken math out as a separate section of the test. Previously, math was incorporated into the general exam, so that teachers could achieve certification without demonstrating specific competence in mathematics. Massachusetts has correctly decided that math merits separate treatment. Indeed, our state's economy is reliant upon strong math skills, whether in the high technology and life sciences sector, financial services, insurance, accounting, manufacturing, or even and perhaps especially education. To be fair, colleges that train teachers need time to ramp up course requirements and instruction to meet the demands of a separate math licensing test. Still, a failure rate of nearly 75 percent on a math test speaks to an unfortunate neglect of math instruction. For too long, elementary, secondary and post-secondary schools have shied from rigorous math requirements for all. But math is not the province of the gifted only. Math literacy is a vital skill for every graduate, including potential teachers. Massachusetts and the nation needs to recapture the sense of urgency that attended the launch of Sputnik by the Soviet Union in 1957, for today's task is not merely to close a gap with one rival, but to better compete with the world for technology and information jobs. The formula is no secret: More course requirements, at high schools and colleges alike, with texts that balance problem-solving and applications. If budgets are tight, schools must give reading, writing and mathematics priority over social sciences, electives and sports. If, by the way, you came up with anything other than 162 for the question above, it's time to hit the books again. === Subject: Re: 27 percent of aspiring teachers pass math test Random comments inline..... Martin > The grim results of the math tests were also published in the front > page of the Boston Globe on 19 May 2009. > May 20, 2009 Strength in numbers > Math results ID need for harder work If 600 aspiring teachers take a math test and 27 percent pass, how > many new and genuinely competent math teachers will be available to > elementary schoolchildren this fall? Hmm... sounds like a trick question..... since when has being genuinely competent been a qualifier?... > Such a question involves only elementary math skills, but the results > emerging from the new math section on the state's licensing exam for > teachers show that many would-be teachers are in need of a thorough > review of such elementary skills before they enter a classroom. The same could probably be said about any subject - not just math... > On the one hand, Massachusetts deserves credit for having broken math > out as a separate section of the test. Previously, math was > incorporated into the general exam, so that teachers could achieve > certification without demonstrating specific competence in > mathematics. Massachusetts has correctly decided that math merits > separate treatment. Indeed, our state's economy is reliant upon strong > math skills, whether in the high technology and life sciences sector, > financial services, insurance, accounting, manufacturing, or even > and perhaps especially education. So... they (the state of Mass..) is actually doing a good thing.... > To be fair, colleges that train teachers need time to ramp up course > requirements and instruction to meet the demands of a separate math > licensing test. Teaching to the test is probably the best way to go.... > Still, a failure rate of nearly 75 percent on a math test speaks to an > unfortunate neglect of math instruction. For too long, elementary, > secondary and post-secondary schools have shied from rigorous math > requirements for all. But math is not the province of the gifted only. > Math literacy is a vital skill for every graduate, including potential > teachers. Didn't these graduates have to take the SAT to get into college? And isn't there a math section for that? > Massachusetts and the nation needs to recapture the sense of > urgency that attended the launch of Sputnik by the Soviet Union in > 1957, for today's task is not merely to close a gap with one rival, > but to better compete with the world for technology and information > jobs. Problem is... there isn't a space race these days, and the only substitute we have for the Cold War is either the war on terrorism or the war on drugs... or maybe the war on 'Climate Change'.... maybe the green movement might inspire a few people to want to learn a little math... > The formula is no secret: More course requirements, at high schools > and colleges alike, with texts that balance problem-solving and > applications. If budgets are tight, schools must give reading, writing > and mathematics priority over social sciences, electives and sports. The problem with maligning electives is that those classes are typically the ones that the students are interested in, and where some of the academic stuff that students learn gets used in practical applications.... Dump all of the non-academic classes and the drop-out rate will most likely get bigger... and you will have to put more demand on including practical applications (hands-on) in academic classes.... Martin > If, by the way, you came up with anything other than 162 for the > question above, it's time to hit the books again. === Subject: Re: Axiomatic introduction to algebra <4A01642F.E25EE00C@tesco.net> posting-account=MuV5xwkAAAAsy5Tle8h2R8TWyzSL60Q4 1.1.4322; .NET CLR 3.0.04506.30; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) On May 6, 6:19am, Frederick Williams structured than what we find here: http://www.themathpage.com/aPreCalc/algebraPre.htm I don't want something as detailed and pedantic as a text on abstract > algebra. There is nothing fundamentally wrong with the above linked > page. It is fine for teaching people the basic skills required for > future mathematical advancement. It doesn't, however, rigorously > (step-by-step) introduce each new form while maintaining the > distinction between precedents and consequences. IOW, I'm seeking a > development of basic algebra ad more geometrico. I am more > interested in formalism than in examples and application. Any > suggestions? Actually, the text mentioned would be much better if it distinguished > (group/ring/field) axioms from theorems (e.g. -(-a)=a is atheorem). > And of course statements like 0/0 = Any number are to be avoided and > required conditions (such as a != 0 in 19, a>=0 in 26) must be > carefully stated. > It looks like that is a formulary of rather than an introduction to > basic algebra. Well, yes. I am not looking for a treatment which discusses rings, > fields, groups, etc. I am interested in a fairly structured > development of basic algebra which would be accessible to a person > learning algebra for the first time. I learned algebra from Birkhoff and Mac Lane A Survey of Modern > Algebra . -- > ... when we came back, late, from the hyacinth garden, > Your arms full, and your hair wet, I could not > Speak, and my eyes failed...- Hide quoted text - - Show quoted text - It looks like a good book. It doesn't, however provide the treatment I was looking for. I kind of like http://mitpress.mit.edu/catalog/author/default.asp?aid=1422 but it doesn't satisfy the criteria either. === Subject: Honeywell Digital Thermostat Honeywell HZ-315 Quick Heat Ceramic Heater Price:$29.99 Image: http://bestdeallocator.info/image.php?id=B0006I9WHS Best deal: http://bestdeallocator.info/index.php?id=B0006I9WHS Honeywell Digital Round Thermostat Price:$81.10 Image: http://bestdeallocator.info/image.php?id=B00099Q91G Best deal: http://bestdeallocator.info/index.php?id=B00099Q91G Honeywell Focus 5100 Digital Manual Heat/Cool Thermostat Price:$69.00 Image: http://bestdeallocator.info/image.php?id=B000E3AVZ2 Best deal: http://bestdeallocator.info/index.php?id=B000E3AVZ2 Honeywell RLV310 Digital Manual Baseboard Heat Thermostat Price:$34.99 Image: http://bestdeallocator.info/image.php?id=B000IMY9O8 Best deal: http://bestdeallocator.info/index.php?id=B000IMY9O8 === Subject: Lens Ski Snowboard Goggles Dragon DX 08 Snowboard Goggles - Coal Frame / Amber Lens Price:$59.95 Image: http://bestdeallocator.info/image.php?id=B001GDHMTC Best deal: http://bestdeallocator.info/index.php?id=B001GDHMTC Anon Figment Men's Snowboard Goggles - Irie Frame - Green Solex Mirror Lens Price:$94.99 Image: http://bestdeallocator.info/image.php?id=B001IXYWLG Best deal: http://bestdeallocator.info/index.php?id=B001IXYWLG Dragon DX Goggle Price:$80.00 Image: http://bestdeallocator.info/image.php?id=B000WCC5PO Best deal: http://bestdeallocator.info/index.php?id=B000WCC5PO Von Zipper Dojo 09 Snowboard Goggles - White Gloss Frame / Fire Chrome Lens Price:$119.99 Image: http://bestdeallocator.info/image.php?id=B001FODRBY Best deal: http://bestdeallocator.info/index.php?id=B001FODRBY Scott 2009 STORM o.t.g. Goggles Silver / Amp Lens Price:$80.00 Image: http://bestdeallocator.info/image.php?id=B001U72398 Best deal: http://bestdeallocator.info/index.php?id=B001U72398 UVEX Downhill II Ski Goggle,Black Frame with Double Gold Lite Lens Price:$34.99 Image: http://bestdeallocator.info/image.php?id=B000YYKIZE Best deal: http://bestdeallocator.info/index.php?id=B000YYKIZE Von Zipper Dojo 09 Snowboard Goggles - Ribbons & Gold w/ Skull Candy / Gold Chrome Lens Price:$179.99 Image: http://bestdeallocator.info/image.php?id=B001IY193Y Best deal: http://bestdeallocator.info/index.php?id=B001IY193Y Dragon ROGUE Weapons Vice / Gold Ionized Lens / Weapons Print Strap Price:$130.00 Image: http://bestdeallocator.info/image.php?id=B0017XLRMO Best deal: http://bestdeallocator.info/index.php?id=B0017XLRMO Dragon DX 08 Snowboard Goggles - Coal Frame / Amber Lens Price:$59.95 Image: http://bestdeallocator.info/image.php?id=B001GDHMTC Best deal: http://bestdeallocator.info/index.php?id=B001GDHMTC Anon Figment Men's Snowboard Goggles - Irie Frame - Green Solex Mirror Lens Price:$94.99 Image: http://bestdeallocator.info/image.php?id=B001IXYWLG Best deal: http://bestdeallocator.info/index.php?id=B001IXYWLG Dragon DX Goggle Price:$80.00 Image: http://bestdeallocator.info/image.php?id=B000WCC5PO Best deal: http://bestdeallocator.info/index.php?id=B000WCC5PO Von Zipper Dojo 09 Snowboard Goggles - White Gloss Frame / Fire Chrome Lens Price:$119.99 Image: http://bestdeallocator.info/image.php?id=B001FODRBY Best deal: http://bestdeallocator.info/index.php?id=B001FODRBY Scott 2009 STORM o.t.g. Goggles Silver / Amp Lens Price:$80.00 Image: http://bestdeallocator.info/image.php?id=B001U72398 Best deal: http://bestdeallocator.info/index.php?id=B001U72398 UVEX Downhill II Ski Goggle,Black Frame with Double Gold Lite Lens Price:$34.99 Image: http://bestdeallocator.info/image.php?id=B000YYKIZE Best deal: http://bestdeallocator.info/index.php?id=B000YYKIZE Von Zipper Dojo 09 Snowboard Goggles - Ribbons & Gold w/ Skull Candy / Gold Chrome Lens Price:$179.99 Image: http://bestdeallocator.info/image.php?id=B001IY193Y Best deal: http://bestdeallocator.info/index.php?id=B001IY193Y Dragon ROGUE Weapons Vice / Gold Ionized Lens / Weapons Print Strap Price:$130.00 Image: http://bestdeallocator.info/image.php?id=B0017XLRMO Best deal: http://bestdeallocator.info/index.php?id=B0017XLRMO === Subject: Re: 27 percent of aspiring teachers pass math test posting-account=NgMGSwkAAABJni6NIYF05Yc4jNzXwHf- 1.1.4322),gzip(gfe),gzip(gfe) > The grim results of the math tests were also published in the front > page of the Boston Globe on 19 May 2009. === Subject: DKNY CHRONOGRAPH BROWN LEATHER MENS WATCH - NY1324 Price:$175.00 Image: http://bestdeallocator.info/image.php?id=B000O8V4OO Best deal: http://bestdeallocator.info/index.php?id=B000O8V4OO DKNY watches are distinctive sleek and modern. === Subject: Skagen Women's Cyrstal Accented Brown Mesh Watch #686XSMM Price:$140.00 Image: http://bestdeallocator.info/image.php?id=B000IZDXA6 Best deal: http://bestdeallocator.info/index.php?id=B000IZDXA6 The watch is very pretty and is so light weight you wouldn't even know you had it on. Crystals don't show up as much as I would like. Brown in style. Ladies brown IP oval case with brown Mother-of-Pearl dial and indicators Made with CRYSTALLIZED - Swarovski Elements. Signature Skagen brown IP mesh band is fully adjustable. Add a uniquely stylish timepiece to your collection with this contemporary, espresso-tone Skagen stainless steel women's watch, which offers a mother-of-pearl dial and Swarovski crystal accents. The oval watch case features small yellow crystals at the top and bottom connectors as well as larger, sparkling yellow crystals embedded into the bezel that act as hour markers. Manufactured in Austria, Swarovski crystals are machine-faceted, optically pure, and made of 32 percent lead, producing a highly refractive quality. The spare dial just features a pair of luminous-tipped, rounded skeleton hands. It's completed by a mesh stainless steel bracelet band that adds an elegant texture. Other features include Japanese quartz movement, a scratch-resistant mineral crystal, and water resistance to 100 meters (330 feet)--offering protection from accidental splashes as well suitable for swimming, snorkeling, and light recreational diving. About Skagen Named after a fishing hamlet in Denmark on the northern tip of the Jutland peninsula, Skagen Designs was founded by Henrik and Charlotte Jorst in 1991 after the two Copenhagen natives moved to New York City. The company subscribes to the principle that beautifully designed, high-quality objects can be created at reasonable prices. Skagen watches represent the technical excellence, refined design and operational simplicity that have created the unique reputation of Danish design. The fishing village of Skagen--the Skaw--lies where the northern-most tip of the Danish peninsula known as Jutland bends East and breaks the surging waters of the Kattegat and Skagerak seas. The spectacular natural beauty of the area is compellingly beautiful, and its white sandy beaches have been visited by artists for centuries. ACCESSORIES: Skagen Women's Gold-Tone Bracelet Watch #572SGXG:http://bestdeallocator.info/index.php?id=B0012ON4N8 Invicta Women's Slim Collection Round Black Leather Watch #5157:http://bestdeallocator.info/index.php?id=B001AQCRHC Skagen Women's Gunmetal Mesh Watch #107SSTTM:http://bestdeallocator.info/index.php?id=B0012OSSDY Invicta Women's Slim Collection Round Black Leather Watch #5158:http://bestdeallocator.info/index.php?id=B001AQCRIQ Skagen Women's Steel Rose-Gold Mesh Watch #496SRR:http://bestdeallocator.info/index.php?id=B000QYDYT4 === Subject: Disney Women's Mickey Mouse Melody Watch #MC0179D Price:$35.00 Image: http://bestdeallocator.info/image.php?id=B0000UIWW8 Best deal: http://bestdeallocator.info/index.php?id=B0000UIWW8 This is a beautiful watch, and a must have for any Mickey Mouse collector. I just wish the button for music was a little easier to push. Overall, I am very happy with this purchase. i love this watch it is great. i have had no promblems with it. my niece loves the song it plays and likes to push the button when she sees me. I was disappointed with my purchase. The watch does not play a melody when you push the button. A waste of my money it would cost more to return than it was worth. I love it!!! I love it!!! I love it!!!. This is one of the best mickey watches that I've added to my collection. I love it!!! Musical Mickey Mouse Watch. Plays Mickey Mouse March. Silver Case. Black Genuine Leather Strap. Blue Sunray Dial with Lapped Mickey. A silver shadow image of Mickey Mouse adorns the midnight blue dial of this stylish Disney watch. Presented in a silvertone case, the dial features three-hand function with silvertone hands and Arabic numerals. A smooth black leather strap with a buckle closure completes the look. For added fun, push the button on the side of the case to play the Mickey Mouse theme song. SIMILAR PRODUCTS: Disney Mickey Mouse Club Collectible Watch, MU2332, Special Packaging, Leather Strap:http://bestdeallocator.info/index.php?id=B000N4QNH2 Disney Mickey Musical Watch MU0668:http://bestdeallocator.info/index.php?id=B000EDNCIK High School Musical 3: Senior Year (Extended Edition):http://bestdeallocator.info/index.php?id=B001NY42KQ Disney Winnie the Pooh Sport Watch #MC2282D:http://bestdeallocator.info/index.php?id=B000RY8PW4 Casio Men's 10-Year Battery Analog Resin Watch #MW600E-2AV:http://bestdeallocator.info/index.php?id=B000GB1RFU ACCESSORIES: Disney Mickey Mouse Rhinestone Charm Watch #MC0229:http://bestdeallocator.info/index.php?id=B0000UIWWS Disney Men's Mickey Mouse Two-Tone Motion Hands Watch #MU0959D:http://bestdeallocator.info/index.php?id=B0000UIX8Q Disney Winnie the Pooh & Friends Embossed Strap Watch #MU1077D:http://bestdeallocator.info/index.php?id=B0002OSNPS Disney Kids' Pink Plaid Glitter Watch #DK0081:http://bestdeallocator.info/index.php?id=B000AEGM58 Disney Kids' Thumper Jelly Strap Watch #MC0469:http://bestdeallocator.info/index.php?id=B000AEGMO4 === Subject: Invicta Men's Slim Collection Square Stainless Steel Mesh Watch #5145 Price:$265.00 Image: http://bestdeallocator.info/image.php?id=B001AQCRAY Best deal: http://bestdeallocator.info/index.php?id=B001AQCRAY Thin is in with this stainless steel men's watch from Invicta's Slim Collection, which is a great choice for everyday casual wear. But with its amazingly svelte 5mm (0.2 inches) profile, it can also be perfectly tucked under the cuff when dress or business wear is called for. Offering Invicta's refined, modern styling, this watch is powered by a precisely timed Swiss-manufactured quartz movement. And while it might be stereotypical, the truth is many of the world's finest and most accurate timing movements are created in Switzerland. This is due to the wealth of knowledge and infrastructure that's been built up in the Swiss watch industry over the centuries, and with many of the Swiss practitioners having the trade and watchmaking traditions passed down over the generations. Matched with a uniquely designed metal mesh bracelet in silver, this watch has a square, silver stainless steel watch case that measures 39mm wide (1.54 inches). Its silvery white dial face features a two-handed movement placed in a slightly recessed center, which is surrounded by a silver tone markers and Arabic numerals. It's water resistant to 30 meters (99 feet), also known as 3 ATM. This means that it can withstand rain and splashes of water--such as car washing and showering--but it shouldn't be worn swimming. Other features include a scratch-resistant mineral crystal, fold-over safety clasp, and Tritnite luminous hands. Tritnite is a luminous material with an extended glow exclusively developed by Invicta in Switzerland and added to their timepiece hands and markers. When exposed to regular daylight, it will hold its glow for about 20 hours. Quartz Movement The movement of a watch refers to the mechanics that power the ticking of the timepiece, and this watch is powered by a Swiss-made quartz movement. It utilizes the vibrations of a tiny quartz crystal to maintain timing, which provides a fast, steady oscillation (at more than 30,000 times per second) to provide excellent accuracy. SIMILAR PRODUCTS: Invicta Men's Slim Collection Square Stainless Steel Mesh Watch #5143:http://bestdeallocator.info/index.php?id=B001AQCRA4 DLO HipCase Leather Folio Case for iPod touch 1G and 2G (Black):http://bestdeallocator.info/index.php?id=B000WOIFO2 ACCESSORIES: Invicta Men's Slim Collection Square Stainless Steel Mesh Watch #5143:http://bestdeallocator.info/index.php?id=B001AQCRA4 Invicta Men's Slim Collection Stainless Steel Mesh Watch #5333:http://bestdeallocator.info/index.php?id=B001TH8U6O Invicta Men's Slim Collection Square Stainless Steel Mesh Watch #5146:http://bestdeallocator.info/index.php?id=B001AQAZLW Invicta Men's Slim Collection Square Stainless Steel Mesh Watch #5144:http://bestdeallocator.info/index.php?id=B001AQCRAE === Subject: Nikon MB40 Multi-Power Battery Pack for Nikon F6 Price:$445.00 Image: http://bestdeallocator.info/image.php?id=B000928LLO Best deal: http://bestdeallocator.info/index.php?id=B000928LLO 2/8/200615-19-32... SIMILAR PRODUCTS: Nikon MH-21 Quick Charger for Nikon EN-EL4 and EN-EL4a Rechargeable Li-Ion Batteries:http://bestdeallocator.info/index.php?id=B00017LTFC ACCESSORIES: Nikon EN-EL4 Rechargeable Li-Ion Battery for MB-D10 Battery Pack and Nikon D2 and D3 Digital SLR Cameras:http://bestdeallocator.info/index.php?id=B00016PB8E Power 2000 XP-333 NiMH Rapid AA and AAA Battery Charger:http://bestdeallocator.info/index.php?id=B0002LDGLC Lenmar DLN-EL4 Lithium-Ion Battery Equivalent to EN-EL4 for Nikon D2H:http://bestdeallocator.info/index.php?id=B0006B5242 Nikon MS 40 - Battery holder:http://bestdeallocator.info/index.php?id=B000I2PXSO AA Battery 1.5v Alkaline pack of 4:http://bestdeallocator.info/index.php?id=B0009MCSSQ === Subject: Re: collection of all self-independent sets is a sigma algebra posting-account=Jz4DtgkAAAAZkdWvJAd__jMF7l1N5_1V CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) On 13 May, 08:27, Robert Israel self-independent sets is a sigma-algebra. Proof: Let C be a collection of sets. Let S in C be a self-independent > member of C. So, mu(S n S)=mu(S)mu(S) and thus mu(S)=mu(S^2). > So either S equals the universal set X or the emptyset. Not quite. There may be e.g. nonempty sets of measure 0. And so also their complements, non-universal sets of probability measure 1. But a countable union of sets of measure 0 is a set of measure 0, and similarly a countable union of sets which includes a set of probabilty measure 1 is a set of probability measure 1. === Subject: Re: Drat -- Grid Game > In the first phase of the game players take turns, each placing ANY > integer from 1 to n^2 into any empty square of the grid on a turn. > (The same integer may be used more than once in the grid.) > Players thereafter take turns moving their markers from square to > orthogonally adjacent square. if it is possible, player B must move so that the > numbers of his/her consecutively-moved-on squares have either the same > greatest common divisor or same absolute difference as the last two > consecutively-moved-on squares of Player A -- where player A is the > player who moves just before player B (and where who exactly are > players A and B changes each move -- player B always being the player > currently moving). If there are no such adjacent squares that have the same GCD or > absolute difference, then player B may move to ANY orthogonally > adjacent square. If a player is forced to move onto a square already occupied by > another player's marker, then the SECOND player to occupy the square > is removed from competition, and his/her marker is removed from the > grid. (A player landing on an already occupied square is a 'drat', as > in Drat!.) > Play continues until there is one remaining player, who is then the > winner. It seems fairly easy to force a draw: I expect there are often cases where you by moving back and forth between two squares can force the other player to move back and forth between two other squares. But it is probably possible to place the numbers in the first phase so this can't happen. It sounds intersting, though I have a feeling it will be a bit too complicated for most players. A game with similar feel might be to have on each square one of N symbols that can be in one of N colours and one of N sizes. With N=4, you get 64 different combinations, so you can fill an 8x8 board with one of each. A move must preserve either shape, colour or size. Additionally, if the previous move preserved, say, colour, you must also preseve colour (and similar for shape and size). If the previous move preserved more than one property, you must preserve as many as possible of those. This includes that, if you can't preserve any of the previous moves preserved property, you can make any move, as long as it preserves _some_ property. An 8x8 board may be too big for this, so you could have N=2 and distribute two of each of the 8 combinations on a 4x4 board. You could even build a board out of tiles that have these markings. You could even use a Set deck: Take only the cards that have one symbols (that's 27 cards: three shapes, three colours and three shadings), shuffle them and lay 25 of them in a 5x5 grid. Alternatively, place them in a staggered grid like this: +---+ | | +-+-+-+-+ | | | +-+-+-+-+-+-+ | | | | +-+-+-+-+-+-+-+-+ | | | | | +-+-+-+-+-+-+-+-+-+-+ | | | | | | +-+-+-+-+-+-+-+-+-+-+ | | | | | +-+-+-+-+-+-+-+-+ | | | | +-+-+-+-+-+-+ | | | +-+-+-+-+ | | +---+ so each card has (up to) six neighbours. Torben === Subject: locally connected Let S = A / B. If A,B are locally connected, then A / B is locally connected. If A,B are locally connected and closed, then A / B is locally connected. The topologist sin curve shows closed is necessary. I doubt there's a counter example to either proposition. In fact, I think they're both true. However I'm fluster to come up with proofs. For the first I start by assuming U is open and x in U / A/B. Since A,B are locally connected, there's some open V,W with x in connected V / A subset U / A x in connected W / B subset U / B What to do next? Even using various components, I've gotten nowhere. === Subject: Re: locally connected posting-account=hhC2JwoAAAAQt9ZcdRPKAFNCTwZjbe1M 1.0.3705; .NET CLR 2.0.50727; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Let S = A / B. If A,B are locally connected, then A / B is locally connected. Let A be the closed upper half plane minus the rational points on the Let B be the closed lower half plane munus the rational points on the === Subject: Re: random actually I don't know what function matlab has so It's not simple traslate it from Matemathica. I need 1/5 of the lines: if the lines are 500 I need to extract 100 lines choosen ramdomly; of course if the lines are 21, I need about 4 lines.. === Subject: Re: random <30669760.97495.1242210733619.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=_jKG3QoAAADEMtE8HeaiS6MTKR3I7f2- MathPlayer 2.10b; SV1),gzip(gfe),gzip(gfe) > actually I don't know what function matlab has so It's not simple traslate it from Matemathica. > I need 1/5 of the lines: if the lines are 500 I need to extract 100 lines choosen ramdomly; of course if the lines are 21, I need about 4 lines.. Maybe this kind of code could inspire you : myRandomChoice[list_, n_]:= Module[{lg, chosen}, lg = 0; While[lg != n, chosen = Table[RandomInteger[{1,Length[list]}],{n+1}]//Union; lg=Length[chosen] ]; Return[chosen] ]; myRandomChoice[Range[500], 100] {2, 6, 11, 16, 20, 25, 31, 42, 46, 47, 50, 54, 58, 59, 75, 77, 78, 82, 84, 88, 89, 92, 122, 130, 132, 136, 139, 143, 146, 148, 153, 154, 155, 157, 159, 161, 164, 167, 168, 177, 181, 183, 209, 210, 211, 222, 231, 232, 236, 238, 247, 251, 258, 260, 263, 273, 282, 286, 291, 298, 301, 304, 308, 319, 325, 326, 341, 346, 347, 350, 351, 356, 358, 366, 371, 373, 385, 386, 387, 405, 408, 411, 415, 423, 424, 425, 435, 440, 443, 444, 446, 455, 462, 466, 482, 487, 489, 494, 499, 500} hth V.Astanoff === Subject: Re: What are hypercomplex numbers for? > It was just a shock to me to discover that so many topics such as > infinitesimals, nonstandard analysis, and Cartan's notation, among > other things, which are standard stuff in the toolbox of leading edge > mathematicians and physicists, have yet to receive much undergraduate > exposure with the exception of a few top schools Is nonstandard analysis used by many > leading edge mathematicians and physicists? leading edge is journalese and pretty-well devoid of meaning. > Also, I would have thought differential forms f(x,y)dx + g(x,y)dy, etc, > were fairly widely taught in undergraduate courses, > if that counts as infinitesimals. Nonstandard analysis is introduced in some 3rd year logic courses in some UK universities. -- ... when we came back, late, from the hyacinth garden, Your arms full, and your hair wet, I could not Speak, and my eyes failed... === Subject: Re: What are hypercomplex numbers for? > I understand there is an analogous number system to the complex > numbers as the hyperreal system is to the real numbers? If so, what > pure and/or applied mathematics is better off for hypercomplex numbers > and why? And are they considered the ultimate in describing physical > reality or are they just useful abstractions only? But I think the term hypercomplex numbers is just an old-fashioned term > for what now are called finite-dimensional algebras. > I think the term was used in van der Waerden's Algebra, IIRC. A ring A which is also a finite dimensional vector space over a field P and satisfies the condition (alpha u)v = u(alpha v) = alpha(uv) for alpha in P is called an associative algebra or a hypercomplex system over P. -- ... when we came back, late, from the hyacinth garden, Your arms full, and your hair wet, I could not Speak, and my eyes failed... === Subject: Re: What are hypercomplex numbers for? posting-account=Ud2SRgoAAABTrXAv25Os3AveXKvxrPcl AppleWebKit/525.27.1 (KHTML, like Gecko) Version/3.2.1 Safari/525.27.1,gzip(gfe),gzip(gfe) On Apr 24, 8:08am, Mariano Su.87rez-Alvarez > It was just a shock to me to discover that so many topics such as > infinitesimals, nonstandard analysis, and Cartan's notation, among > other things, which are standard stuff in the toolbox of leading edge > mathematicians and physicists, have yet to receive much undergraduate > exposure with the exception of a few top schools, especially since > what is traditionally taught in place of the leading edge (and some of > that's still 50+ years old) at the undergrad level is mostly passe, > except exclusively in academia, and very little of any of the > traditional math over 70 years old, aside from linear and higher > algebras, are being used to push mathematics or physics discoveries > forward. I think undergraduate students should at least be made aware > of the various paths to very specific places (electrical engineering, > civil, etc.) traditional mathematics is taking them, especially if > their goals are higher. I realize this is a miniscule protest in the > scheme of things, but maybe not given the person(s) who might be able > to prove or discard or discover something better than the various > unprovable things since QED in physics might not ever even think to > try without exposure to the most effective tools when their brains are > still relatively unfilled with soon to be obsolete mathematical ideas > and methods. Your minuscule protest is mostly absurd. There is essentially > nothing in the standard mathematical curriculum which can be > considered obsolete. While there *are*, in fact, subjects > that have become more or less irrelevant with time, your > 50+ year old and over 70 years old time frames are > utterly confused. It is quite sure that unless one masters > the so very passe ideas of a couple of centuries ago, there is > pretty much no chance of even understanding what QED is, let > alone more bizarre theories; at the same time, non-standard > analysis is more or less completely disjoint from, say, QED. -- m I know I tend to write too many run-on's, but I'm not confused about the time frames: Nonstandard analysis was a product of the 1960s and Cartan's notation used in exterior calculus for physics wasn't recognized as superior until the 1940s, and even now, few undergraduates have even heard of it. So, perhaps, my run-on's led you to take my timeframes out of the original context. And I didn't say one didn't need to know precalc or traditional calculus, and I excepted abstract and linear algebra from the timeline anyway. If you to things that are currently saved for graduate school or later: Mainly I meant in place of wasting undergraduates time with classical analysis. My conjecture is simply that since using nonstandard analysis (i.e., NSA accepts the infinitesimal and subsequent hyperreal number system) to prove the theorems of calculus takes a factor of 20 less in same size and spaced pages than using standard analysis, there must be better ways to do physics, even possibly extending QED since the infinitesimal is a quantum of sorts, due to NSA's inherent efficiencies over classical analysis. Some examples of NSA in action since the 60s: 1. Applications of non-standard analysis to relativistic quantum mechanics...Abstract. Some representations of the Dirac delta function are considered including a new representations. A new theory of Fourier transforms is developed which is better suited to use in physics than the standard theory. The work is of general interest as brief outline of the construction of field theory from quantum mechanics as facilitated by non-standard analysis and a theorem which enables the calculation of a cross section from plane wave states. http://www.iop.org/EJ/abstract/0305-4470/14/10/010 2. Nonstandard analysis in classical physics and quantum formal scattering - http://www.springerlink.com/content/tw01377l05746040/ 3. Nonstandard Feynman path integral for the harmonic oscillator...Using Nonstandard Analysis, we will provide a rigorous computation for the harmonic oscillator Feynman path integral. The computation will be done without having prior knowledge of the classical path. We will see that properties of classical physics falls out naturally from a purely quantum mechanical point of view. We will assume that the reader is familiar with Nonstandard Analysis. (c)1999 American Institute of Physics. This link along with others of the same variety: 4. Nonstandard Methods in Stochastic Analysis and Mathematical Physics...The Bulletin of the American Mathematical Society acclaimed this text as a welcome addition to the literature of nonstandard analysis, a field related to number theory, algebra, and topology. The first half presents a self-contained introduction to the subject, and the second part explores applications to stochastic analysis and mathematical physics. http://store.doverpublications.com/0486468992.html 5. Also see: A Course in Mathematics for Students of Physics: Volumes 1 & 2 - Bamberg and Sternberg, based on the course they taught at Harvard. Applied Exterior Calculus - Edelen There's more. Do you own searches if you'd care to explore further. I guess I'm merely saying that since the usual time to do great things in science and math is when one is young, why wade through classical analysis when NSA is conceptually superior and more in sync with the nature of the quantum concept? Also, whether you like your quantum mechanics Bohr or Bohm style, there's yet another area of mere historical prejudice in neglecting Bohm's causal version and its deterministic bent over the standard version and its indeterminancy (see James T. Cushing's Quantum Mechanics for the long well- researched discussion). === Subject: Re: What are hypercomplex numbers for? posting-account=Ud2SRgoAAABTrXAv25Os3AveXKvxrPcl AppleWebKit/525.27.1 (KHTML, like Gecko) Version/3.2.1 Safari/525.27.1,gzip(gfe),gzip(gfe) > It was just a shock to me to discover that so many topics such as > infinitesimals, nonstandard analysis, and Cartan's notation, among > other things, which are standard stuff in the toolbox of leading edge > mathematicians and physicists, have yet to receive much undergraduate > exposure with the exception of a few top schools Is nonstandard analysis used by many > leading edge mathematicians and physicists? Also, I would have thought differential forms f(x,y)dx + g(x,y)dy, etc, > were fairly widely taught in undergraduate courses, > if that counts as infinitesimals. What is Cartan's notation? > If it is just the use of spinors in relativity, > I would have thought it was fairly standard. -- > Timothy Murphy > e-mail: gayleard /at/ eircom.net > tel: +353-86-2336090, +353-1-2842366 > s-mail: School of Mathematics, Trinity College Dublin In answer, here's sampling of quotes with links, info on Cartan, his notation, two key books on Cartan's exterior calculus used in leading edge physics: 1. Nonstandard analysis and physics http://members.tripod.com/PhilipApps/physics.html 2. Nonstandard analysis in classical physics and quantum formal scattering - http://www.springerlink.com/content/tw01377l05746040/ 3. Nonstandard Feynman path integral for the harmonic oscillator...Using Nonstandard Analysis, we will provide a rigorous computation for the harmonic oscillator Feynman path integral. The computation will be done without having prior knowledge of the classical path. We will see that properties of classical physics falls out naturally from a purely quantum mechanical point of view. We will assume that the reader is familiar with Nonstandard Analysis. (c)1999 American Institute of Physics. http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMAPAQ0 00040000011005511000001&idtype=cvips&gifs=yes 4. Nonstandard Methods in Stochastic Analysis and Mathematical Physics...The Bulletin of the American Mathematical Society acclaimed this text as a welcome addition to the literature of nonstandard analysis, a field related to number theory, algebra, and topology. The first half presents a self-contained introduction to the subject, and the second part explores applications to stochastic analysis and mathematical physics. http://store.doverpublications.com/0486468992.html 5. Application of nonstandard analysis to quantum mechanics...Quantum mechanics is formulated using a nonstandard Hilbert space. The concept of an eigen vector of a linear operator, which applies to standard as well as nonstandard Hilbert spaces, is replaced by the more general concept of an ultra eigen vector, which applies to nonstandard Hilbert spaces alone. Ultra eigen vectors corresponding to all spectral points of internal self-adjoint operators are proved to exist. This result enables us to set up a formalism which is equally valid for the discrete, as well as the continuous spectrum. Finally, Dirac's formalism is reproduced, in a rigorous form within the nonstandard Hilbert space structure. Journal of Mathematical Physics is copyrighted by The American Institute of Physics. http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMAPAQ0 00016000002000177000001&idtype=cvips&gifs=yes 6. There's more. Do you own searches if you'd care to explore further. My conjecture is simply that since using nonstandard analysis (i.e., NSA accepts the infinitesimal and subsequent hyperreal number system) to prove the theorems of calculus takes a factor of 20 less in same size and spaced pages than using standard analysis, there must be better ways to do physics in there somewhere (such as samples above) due to the efficiencies. What is Cartan's notation? > If it is just the use of spinors in relativity, > I would have thought it was fairly standard. 7. The notation devised by Cartan, which employs invariant differential forms as elements in an algebra due to Grassman, combined with an operation called exterior differentiation. Cartan's recognition as a first rate mathematician came to him only in his old age; before 1930 Poincar.8e and Weyl were probably the only prominent mathematicians who correctly assessed his uncommon powers and depth. This was due partly to his extreme modesty and partly to the fact that in France the main trend of mathematical research after 1900 was in the field of function theory, but chiefly to his extraordinary originality. It was only after 1930 that a younger generation started to explore the rich treasure of ideas and results that lay buried in his papers. Since then his influence has been steadily increasing, and with the exception of Poincar.8e and Hilbert, probably no one else has done so much to give the mathematics of our day its present shape and viewpoints. http://www.gap-system.org/~history/Biographies/Cartan.html Also see: A Course in Mathematics for Students of Physics: Volumes 1 & 2 - Bamberg and Sternberg, based on the course they taught at Harvard. Applied Exterior Calculus - Edelen === Subject: Re: What are hypercomplex numbers for? > It was just a shock to me to discover that so many topics such as > infinitesimals, nonstandard analysis, and Cartan's notation, among > other things, which are standard stuff in the toolbox of leading edge > mathematicians and physicists, have yet to receive much undergraduate > exposure with the exception of a few top schools > Is nonstandard analysis used by many > leading edge mathematicians and physicists? > Also, I would have thought differential forms f(x,y)dx + g(x,y)dy, etc, > were fairly widely taught in undergraduate courses, > if that counts as infinitesimals. > What is Cartan's notation? > If it is just the use of spinors in relativity, > I would have thought it was fairly standard. > In answer, here's sampling of quotes with links, info on Cartan, his > notation, two key books on Cartan's exterior calculus used in leading > edge physics: I'm baffled by your mixture of nonstandard analysis (which was the subject matter of 6 or your 7 references) and E. Cartan. Did Cartan have some connection with nonstandard analysis? That seems improbable to me. Also, is Cartan's notation just the standard notation for differential forms, with the wedge product? If so, I would have thought it was widely taught in undergraduate courses, as I said. Incidentally, was the notation due to Cartan? I thought it went back much earlier. > Also see: > A Course in Mathematics for Students of Physics: Volumes 1 & 2 - > Bamberg and Sternberg, based on the course they taught at Harvard. > Applied Exterior Calculus - Edelen As to Cartan not being recognized, I would have thought his work (following that of Killing) on the classification of simple Lie algebras was seen almost at once as immensely significant. -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College Dublin === Subject: weighted least squares on multidimensional data I am trying to work out the paramaters A and B of a linear function: z=mx+b, where I already know z and x. The aim is to work out the parameters of a function g(x) which maps x to z, so: g(x)=mx+b=z. The parameters are to be worked out via weighted least squares regression. I have a total of N data samples, and each sample of x is a 36- dimensional vector. z is only one-dimensional. Their are N individual weights, w, which are one dimensional also. In order to work out the parameters m and b using the weights, would I need to modify the first equations for m and b at http://en.wikipedia.org/wiki/Linear_least_squares#Motivational_example by using weighted sums; i.e. sigma(w.x) instead of sigma(x) ? I beleive the parameters m and b would also work out to be 36- dimensional vectors. To get g(x) to output a one dimensional value, would i use the dot product of m.x as oppose to the standard matrix multiplication, which would give a 36x36 sized matrix? Adam === Subject: Re: weighted least squares on multidimensional data >I am trying to work out the paramaters A and B of a linear function: z=mx+b, where I already know z and x. The aim is to work out the parameters of >a function g(x) which maps x to z, so: g(x)=mx+b=z. The parameters are to be worked out via weighted least squares >regression. >I have a total of N data samples, and each sample of x is a 36- >dimensional vector. z is only one-dimensional. >Their are N individual weights, w, which are one dimensional also. In order to work out the parameters m and b using the weights, would I >need to modify the first equations for m and b at >http://en.wikipedia.org/wiki/Linear_least_squares#Motivational_example >by using weighted sums; i.e. >sigma(w.x) instead of sigma(x) ? I beleive the parameters m and b would also work out to be 36- >dimensional vectors. To get g(x) to output a one dimensional value, >would i use the dot product of m.x as oppose to the standard matrix >multiplication, which would give a 36x36 sized matrix? Take a look at and . They give examples of how to do multidimensional regression. Neither apply directly to your problem, but the methods there might be useful. Rob Johnson take out the trash before replying === Subject: Re: weighted least squares on multidimensional data posting-account=QATl1QoAAADZYVAWORqO_aubcdzudO6s Trident/4.0; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 1.1.4322; InfoPath.2; .NET CLR 3.5.30729; .NET CLR 3.0.30618; OfficeLiveConnector.1.3; OfficeLivePatch.0.0),gzip(gfe),gzip(gfe) On May 13, 11:57am, Adam Chapman I am trying to work out the paramaters A and B of a linear function: z=mx+b, where I already know z and x. The aim is to work out the parameters of > a function g(x) which maps x to z, so: g(x)=mx+b=z. The parameters are to be worked out via weighted least squares > regression. > I have a total of N data samples, and each sample of x is a 36- > dimensional vector. z is only one-dimensional. > Their are N individual weights, w, which are one dimensional also. In order to work out the parameters m and b using the weights, would I > need to modify the first equations for m and b athttp://en.wikipedia.org/wiki/Linear least squares#Motivational example > by using weighted sums; i.e. > sigma(w.x) instead of sigma(x) ? I beleive the parameters m and b would also work out to be 36- > dimensional vectors. To get g(x) to output a one dimensional value, > would i use the dot product of m.x as oppose to the standard matrix > multiplication, which would give a 36x36 sized matrix? > Adam When I talk about the weights, I mean in the calculation of m and b at http://en.wikipedia.org/wiki/Linear least squares#Motivational example , should both sigma(x) and sigma(y) be sigma(x i * w i) and sigma(y i * w i) respectively, or do you only use weight in x, so: sigma(x)*sigma(y) is actually sigma(w*x)*sigma(y) ??? === Subject: Re: Mathematical theater of the absurd A Dedekind cut is an infinite set. [NonBreakingSpace]Infinite sets have only a > potential existence. Hence, the length of the diagonal of a unit square exists only potentially. Did I say that right? Usually when we say something, we have a purpose for saying it. [NonBreakingSpace]What > would be the purpose for saying that? To clarify that the length (which is a number) of the diagonal does > not have an actual existence. [NonBreakingSpace]Is that what you mean? If, in your own mind, you equate the length with an infinite set, then > I guess you might say that the length has a potential existence. I > guess. But if you recognize that we have an algorithm to compute the > length to any desired accuracy, and you equate the length with that > algorithm, then you would probably want to say that the length has an > actual existence, although, again, I don't know why you would want to > say that. I'm just trying to figure out what are the consequences of your assertion that infinite sets have only a potential existence. E.g., the real numbers may be construed as Dedekind cuts of rational numbers, which cuts are, as you say, infinite, and therefore, according to your edict, have only a potential existence. This would suggest that you consider real numbers as having only a potential existence. But considering what you say above, and your previous posts, I'm guessing that you accept *some* real numbers as having actual existence -- e.g., sqrt(2), pi, the Louisville constant -- because the algorithm for their construction can be finitely given. So, I'm guessing that the breakdown for you is: Computable number = actually existent number Non-computable number = potentially existent number whereas my general impression was that people who subscribe to the potential/actual number distinction usually have it that rational number = actually existent number irrational number = potentially existent number or maybe algebraic number = actual non-algebraic number = potential. I must say that I don't really grasp what it means to describe a set or a number as having a potential existence, although I might understand something like There potentially exists an even number greater than 2 that is not the sum of two primes as an epistemological assertion. -- hz === Subject: Re: Mathematical theater of the absurd herbzet says... >I'm just trying to figure out what are the consequences of your >assertion that infinite sets have only a potential existence. David Petry's posts are pure expressions of emotion (in particular, hate). They have no propositional content. -- Daryl McCullough Ithaca, NY === Subject: Re: Mathematical theater of the absurd > herbzet says... I'm just trying to figure out what are the consequences of your >assertion that infinite sets have only a potential existence. David Petry's posts are pure expressions of emotion (in particular, > hate). They have no propositional content. As long as he confines himself to the maths, I'm not averse to discussing with him. I think he's trying to be mathematical. This potential/actual debate seems to have been infused with a lot of religious-type hysteria since Kronecker, at least. Like civilization is at stake. Not sure why that is. -- hz === Subject: Re: minimization problem posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) On May 12, 12:45pm, math.mud....@gmail.com hi..all.. i would like to get some help for this problem.... it is quite > difficult for me.. minimize with respect to A for the function: log det [ AXA'+AHA' AXB' ; BXA' BXB'] I think you are asking about a family of > block symmetric matrices: M = A(X+H)A' AXB' > BXA' BXB' though symmetry (and positive definiteness?) > will hinge on the properties of X and H. Note that M can be expressed as a product: / A 0 / X+H X / A' 0 > ( ) ( ) ( ) > 0 B / X X / 0 B'/ so that it would seem possible to rewrite > det M as (det AB)^2 times the determinant > of the middle factor above, and providing > that is determinant is positive, assuming > no constraint on A and B nonsingular, the > log det M can be made arbitrarily negative. Is that what you are trying to do? > hi .. chip actually i was expecting someone would find the determinant of the > matrix and differentiate with respect to A which i dont know how to > differentiate. > actually my precise problem is > Matrix M M= E X + A E S A' + A E P A' E X H' +A E S H' > H E X + H E S A' H E X H' + H E S H' + E Q where E X, E S, E P and E Q are covariance matrices.. i would like to minimize M with respect to A. this problem makes me > really headache.. But I'm unclear what it means to minimize > M with respect to A. The entries of A > give us (presumably) multiple parameters > to vary, and we need to clarify what the > objective function to be minimized is. Before you had log det(M), which is a > scalar function that (absent some > constraints on the submatrices) might > well take on arbitrarily negative > values. So please confirm that minimizing > log det(M) is what you mean, or give > a more accurate account... > Sorry for my mistake, Chip yea.. my objective function is log det (M). > and the values of Matrices A and H are samples taken from a particular > distribution. > Okay, you control A in some sense. You should probably tell what values A is allowed to have. Is H also something that you can vary, in the same fashion as A, in order to minimize log det(M)? Notice that log is a monotone function on the positive real numbers, so as long as det(M) stays positive, we may as well talk about minimizing det(M). The log would give a sum of logarithms of the eigenvalues of M, if M is pos. definite, where det(M) is the product of those eigenvalues. I'll assume all the submatrices are nxn. If H is nonsingular we can rewrite your M: M = / I 0 / E X + A E R A' E X + A E S / I 0 0 H / E X + E S A' E U / 0 H'/ where E X, E S, E P and E Q are covariance matrices... E R = E S + E P E T = H E X H' + H E S H' + E Q E U = H^-1 E T H'^-1 = E X + E S + H^-1 E Q H'^-1 Suppose it makes sense to consider H fixed, then minimize det(M) wrt A by: det(M) = det(H)^2 * det(F) where F is the middle matrix factor of M: F = / E X E X + / A E R A' A E S E X E U / E S A' 0 / Setting A = 0 would give: det(F) = det(E X) det(E U - E X) = det(E X) det(E S + H^-1 E Q H'^-1) det(M) = det(E X) det(H E S H' + E Q) === Subject: Re: Tom Potter is barking up the wrong tree. > And of course, if one just wants to mental masturbate > with a minimum of time, cost and energy, > and cost to the taxpayer's > they can just contemplate their naval. > Tom you used the word mental (an adjective) where the word mentally is gramatically required. Do not repeat this sort of heinous mistake. Second, you begin the last sentence with one as pronoun and subject (singular) and end with they (plural) as the pronoun and subject. Presumably the first subject did a lot more than contemplate his or her navel (correct spelling - naval being an adjective relating to a navy or having a navy). Third, the plural of taxpayer is taxpayers. Taxpayer's means belonging to one taxpayer. So many mistakes in one sentence? Well done! Would you consider having some of your puerile postings translated into English? === Subject: Re: Could high technology develop in an aquatic species? posting-account=TwCTWQgAAAC7hf6GV7aTGIk6mVGkiZ5c FDM),gzip(gfe),gzip(gfe) No problem in aquatic Why? Just like Fishes evolved to Frogs which have both gills and lungs. Then evolved to insects that can live on land. Incase theyt do not evolve lungs still they can research out. Just like we need Oxygen Atmosphere to live while they need Water to live. One can say Oxygen is very reactive so many events cannot take place on land that can in Water. They will develop different technologies unimaginable by humans with different chemistry. Chemestry, Physics, Maths, Biology has been develop-ed by Humanas. They have developed ffuu, dduu, kkll, mmuu. And Humans have never heard these branches of Science. They will say look how stupid are Humans They ewven dont have basic Knowledge of ffuu, dduu, kkll, mmuu. branches of Science. Bye Sanny Play Chess: http://www.GetClub.com/Chess.html A Descision making Game Enjoy & Chat: http://www.GetClub.com Talk with Computer Earn $1600/ month: http://www.getclub.com/salesjob.html Get $200-$400 per sale. Business Planning Software: http://www.softtanks.com/ 3 Versions of Software 1. Small Version: for Owner of Small Shops/ Companies 2. Medium Version: For: Managers working in Small/ Medium sized Companies or CEO/ Business Owners. 3. Large Version: For: Executives in Large Companies/ Business Owners === Subject: Re: Could high technology develop in an aquatic species? <8crOl.41049$5N7.21218@newsfe09.iad> Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) On May 12, 8:33pm, N:dlzc D:aol T:com (dlzc) My overall conclusion is this. All the major > branches of science would suffer if they had > to develop in an aquatic environment. > Chemistry in particular would be crippled. > All the world would be your reaction vessel. Containers can follow. And we can remove gases from our > containers. And they could go to the surface, having a near > vacuum (comparatively) nearby. Also, keeping notes would be tough, Wax. as the ink would smear and the paper > disintegrate. Hides were used for writing paper, at near the same time as cave > walls. The sodium factory would have issues. There you go. I imagine building a drybox would be a > challenge, too. Not above the surface... David A. Smith Star Trek was right - any civilisation that develops warp drive will surely be humanoid looking... Hardy === Subject: No Problem in aquatic species? posting-account=TwCTWQgAAAC7hf6GV7aTGIk6mVGkiZ5c FDM),gzip(gfe),gzip(gfe) No problem in aquatic Why? Just like Fishes evolved to Frogs which have both gills and lungs. Then evolved to insects that can live on land. Incase theyt do not evolve lungs still they can research out. Just like we need Oxygen Atmosphere to live while they need Water to live. One can say Oxygen is very reactive so many events cannot take place on land that can in Water. They will develop different technologies unimaginable by humans with different chemistry. Chemestry, Physics, Maths, Biology has been develop-ed by Humanas. They have developed ffuu, dduu, kkll, mmuu. And Humans have never heard these branches of Science. They will say look how stupid are Humans They ewven dont have basic Knowledge of ffuu, dduu, kkll, mmuu. branches of Science. Bye Sanny Play Chess: http://www.GetClub.com/Chess.html A Descision making Game Enjoy & Chat: http://www.GetClub.com Talk with Computer Earn $1600/ month: http://www.getclub.com/salesjob.html Get $200-$400 per sale. Business Planning Software: http://www.softtanks.com/ 3 Versions of Software 1. Small Version: for Owner of Small Shops/ Companies 2. Medium Version: For: Managers working in Small/ Medium sized Companies or CEO/ Business Owners. 3. Large Version: For: Executives in Large Companies/ Business Owners === Subject: I am Isaac Newton. I have been proven evidences that I am Isaac I am Isaac Newton, the true Messiah. Jesus was the alien. The following is about the great plague and the great star. terminate the Third World War, there is a great plague coming, more than two thirds of humanity will be died. I write will be died, not will die. Then will commence a persecution of the Churches the like of which was never seen. Meanwhile, such a plague will arise that more than two thirds of the world will be removed. One will be unable to ascertain the true owners of fields and houses, and weeds growing in the streets of cities will rise higher than the knees. For the clergy there will be but utter desolation. The warlords will usurp what is returned from chain of the port which wakes its name from the marine ox will be opened. 4:48 The fertile, spacious Ausonian plain Will produce so many gadflies and locusts, The solar brightness will become clouded, All devoured, great plague to come from them. 3:82 Fr.8ejus, Antibes, towns around Nice, They will be thoroughly devastated by sea and by land: The locusts by land and by sea the wind propitious, Captured, dead, bound, pillaged without law of war. 5:85 Through the Suevi and neighboring places, They will be at war over the clouds: Swarm of marine locusts and gnats, The faults of Geneva will be laid quite bare. 2.53 The great plague of the maritime city Will not cease until there be avenged the death Of the just blood, condemned for a price without crime, Of the great lady outraged by pretense. ... Then will commence a persecution of the Churches the like of which was never seen. referred to Falun Gong Cult. Through the Suevi and neighboring places, Suevi, Soviet, referred to Russia. Neighboring places referred to China. The fertile, spacious Ausonian plain Will produce so many gadflies and locusts referred to the great plague will be broken out from the Manchurian Plain. The great plague of the maritime city referred to the Sea of Japan, the Bohai Sea, the Yellow Sea, the East China Sea, and the South China Sea. The Creator wants to punish the evil nations though the great plague. According to Esdras II and Apocryphon of John, I will be protected the German People, the Latins and the Jews as the prophet. According to the following prophecies, I can tell you that the great star must be fallen down in China. 6:35 Near the Bear and close to the white wool, Aries, Taurus, Cancer, Leo, Virgo, Mars, Jupiter, the Sun will burn a great plain, Woods and cities letters hidden in the candle. 2:43 During the appearance of the bearded star. The three great princes will be made enemies: Struck from the sky, peace earth quaking, Po, Tiber overflowing, serpent placed upon the shore. 6:79 Near the Ticino the inhabitants of the Loire, Garonne and Saone, the Seine, the Tain and Gironde: They will erect a promontory beyond the mountains, Conflict given, Po enlarged, submerged in the wave. 10:65 O vast Rome, thy ruin approaches, Not of thy walls, of thy blood and substance: The one harsh in letters will make a very horrible notch, Pointed steel driven into all up to the hilt. 9:44 Leave, leave Geneva every last one of you, Saturn will be converted from gold to iron, Raypoz will exterminate all who oppose him, Before the coming the sky will show signs. 6:97 At forty-five degrees the sky will burn, Fire to approach the great new city: In an instant a great scattered flame will leap up, When one will want to demand proof of the Normans. 2:41 The great star will burn for seven days, The cloud will cause two suns to appear: The big mastiff will howl all night When the great pontiff will change country. 3:34 When the eclipse of the Sun will then be, The monster will be seen in full day: Quite otherwise will one interpret it, High price unguarded: none will have foreseen it. I will stop posting to UK yahoo answer when this UK yahoo account was banned. When you are traving to Israel, please tell the Jews that their Messiah is in Jerusalem now, he will choose the suitable time when he wants to appear. Because I am a saint, and the only son of the Creator, I must be the virgin for ever. I were the virgin in 1727 before I died. Even If I were reborn, I must be the virgin for ever. Nostradamus was not the virgin. I could get the messages same as Nostradamus. For a short time, when I had remembered my name, I had been seen that I lived in my house with my old body in England. I see, not remember. If the Creator likes, I could see the little future. 1:1 Sitting alone at night in secret study; it is placed on the brass tripod. A slight flame comes out of the emptiness and makes successful that which should not be believed in vain. 1:2 The wand in the hand is placed in the middle of the tripod's legs. With water he sprinkles both the hem of his garment and his foot. A voice, fear: he trembles in his robes. Divine splendor; the God sits nearby. The letters of Nostradamus: http://www.crystalinks.com/nostyepistle.html So much so that persons of future times may be seen in present ones, because God Almighty has wished to reveal them by means of images, together with various secrets of the future vouchsafed to orthodox astrology, as was the case in the past, so that a measure of power and divination passed through them, the flame of the spirit inspiring them to pronounce upon inspiration both human and divine. It is much like seeing in a burning mirror, with clouded vision, the great events, sad, prodigious and calamitous events that in due time will fall upon the principal worshipers. First, upon the temples of God; secondly, upon those who, sustained by the earth, approach such a decadence. Also a thousand other calamitous events which will be known to happen in due time. Because it is about the great plague, I will still post a lot when everyone see I will stop posting to UK yahoo answer when this UK yahoo account was banned. on yahoo answer. Read more: Isaac Newton Since 25th December, 2008 === Subject: Re: -1 x -1 ? posting-account=LChCFQoAAACR0FoxHzVn6GGERsr9zp8c Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) > If we can use the expression -1=e^{ipi}, we can show > -1*-1=e^{ipi}*e^{ipi}=1 readily: if we rotate once 1 (the vector 01) by pi rad around the origin anticlockwise on the complex plane, we obtain > -1. Further likewise if we rotate -1 by pi rad once more, we can obtain -1*-1=1. At least I understand -1*-1=1 in this way. In that way we can obtain i (imaginary unit), if we rotate 1 by pi/2 around the origin anticlockwise. If we cannnot allow to use the expression -1=e^[ipi}, I don't understand -1*-1=1 vividly. kudos to your question....it's a nice one.., I enjoyed it a lot. ciao nimo === Subject: Re: -1 x -1 ? posting-account=LChCFQoAAACR0FoxHzVn6GGERsr9zp8c Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) On May 12, 7:17pm, Bill Taylor ** The reason for this, though, we need not discuss! well said...Bill., keep it up....awesome best wishes ciao nimo === Subject: Re: Cantor's argument is erroneous The theory T = {Axy[x=y]} may be regarded either as a theory in {} > or a theory in {+,0}. > If by in you mean the fact that {} is a subset of {+,0} then sure. > But as I mentioned before, T is only vacuously written in L(+,0)! > And that has no bearing on the valid applications of rules of inference > on T's theorems! Right. Clear enough and obviously wrong. But how about if Chris > Menzel pipes in here to state clearly whether he agrees with you (as > you say) or not. I'd pay attention to what he might have to say. But do you have nothing of *your own* to refute my reasoning here? Remember one of a few supporting reasons I cited a few times is about our not being able to conclude more information from less. Are *you* be able to refute that, or not? -- To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician. (Shoenfield, Mathematical Logic) === Subject: Re: Cantor's argument is erroneous <%vONl.26977$0S.20110@newsfe22.iad> <87my9ixb5j.fsf@phiwumbda.org> posting-account=yKimjgoAAACk5WwPVD4l9HmbpoR6Hmy4 Presto/2.1.1,gzip(gfe),gzip(gfe) On the other hand, what you say in this debate is quite different. Your > notion of a formal system (theory) hence of proof is a mis-characterization > of Rogers' notation for a formal system T: T = . What you > said Rogers had said: of two things [duh!]: (a) a class of non-logical constants, and (b) > a class of non-logical axioms which contain no non-logical constants > other than those in the class mentioned in (a). Thus, two theories > T and T' are identical if and only if they are identical in their > non-logical constants and their non-logical axioms. (He uses the term non-logical constants to include both predicate > and function symbols, if I skimmed him right. I've never used the > text.) > First observation is Rogers didn't say one way or another on whether > or not the class of non-logical constants is equal to or different > from the union of all of those in the axioms. So he actually didn't > refute me here; you just mis-used his words! If he had such a restriction in mind, why would he not state it? If (a) is entirely determined by (b), why is (a) required at all? If (a) is entirely determined by (b), why does he say *** Thus, two theories T and T' are identical if and only if they *** are identical in their non-logical constants and their non- logical axioms. ? Under your restriction, if the non-logical axioms are identical then the non-logical constants are identical. So why would he say the above? You're not making much sense... > you could say T is vacuously written in *any* extension of L({Axy[x=y]}) > that you care to stipulate (and there are *infinite* of such > stipulated extensions), rules of inference can be applied only > to *the* minimal language of T Says who? === Subject: Re: Cantor's argument is erroneous <%vONl.26977$0S.20110@newsfe22.iad> <871vqu33uz.fsf@phiwumbda.org> posting-account=yKimjgoAAACk5WwPVD4l9HmbpoR6Hmy4 Presto/2.1.1,gzip(gfe),gzip(gfe) > Well then there's no disagreement between you and me on that point, which > is: > Axy[x+y=0], using the FOL syntactical proof machinery which includes > rules of inference. > Note the subtle change here. All of a sudden, Nam has dropped all > mention of the language for the formal system. > You're either too ignorant or borderline not telling the truth. Far > from dropping it, I've mentioned numerous times L(T = {Axy[x=y]}) > as *the language* in which *the formal system T* is written. > I have to go now but I'll be back. Meantime, why don't you stop something > you yourself would accuse a crank do: uttering nonsense repeatedly! What does a formal system consist of? Of its theorems. (Read: what sense of consist are you saying here?) Is there only one formal system for a given set of axioms? Of course. How many formal systems do *you* say there are, given > a set of axioms? two? three? What you you think the relationship > between a formal system and its set of axioms is? Infinitely many. Assuming we're fixing the inference rules/logical axioms, a formal system is a formal language AND a set of non-logical axioms. Two distinct formal systems can have the same axiom set, if their languages differ. > As far as I can see, most people (and every textbook cited) says that > we specify a language and a set of axioms, and combined they make a > theory/formal system. Nam says we specify a set of axioms, which > implies a (unique, minimal, signature) language, and that the > specified set of axioms along with the implied language makes a theory/ > formal system. Is this close to what you're saying? Of course not. You forgot the context I'm in for the phrase we specify > a set of axioms: the context is FOL syntactical proof. Did you read > what I said above: > Axy[x+y=0], using the FOL syntactical proof machinery which includes > rules of inference. Frankly, I have no idea what context you're in, or why you are insisting that the signature of the language of a formal system must be minimal w.r.t its non-logical axioms. Maybe you could explain what the devil you are talking about rather than calling everyone stupid? Maybe when you mean syntactical proof you're specifying that the language signature must be minimal? That's the only idea I've got that makes any sense of what you are saying. If that's it, I have no idea what you think the relevance is to anything, or why it has taken so long for you to fail to explain. > If so... I have > literally no idea where you're getting this idea from, or what it all > means. Why this restriction? What is wrong about considering more > expanded languages? You have no idea what you're arguing about to begin with, because you couldn't > read a simple paragraph describing what your opponent is arguing about! I think the communication problem is on your end, pal. === Subject: Re: Cantor's argument is erroneous <87r5z7thw5.fsf@phiwumbda.org> <87iqkjt4t3.fsf@phiwumbda.org> <87d4art0u0.fsf@phiwumbda.org> posting-account=yKimjgoAAACk5WwPVD4l9HmbpoR6Hmy4 Presto/2.1.1,gzip(gfe),gzip(gfe) > It's my right as a poster that I stay confined within the context > of syntactical/symbolical proof in axiom-set systems of FOL=, when > talking about valid application of rules of inference. > I've already agreed - from the beginning - that if '+' and '0' are not > in the language, then Axy x+y=0 is not derivable in such a system. > OK, so if you agreed with me ,in the context I've specified, Axy[x+y=0] > is not a theorem of T = {Axy[x=y]} then there's no argument between us > then. The context you've specifiied being that the language has no non- > logical symbols not used in the axioms? Nobody has disagreed with > that. What about the context where '+' and '0' are in the language? Do you > agree that it is a theorem in that context? I know the precise definition of theorem in my context, which is just > the definition of theorem of FOL. Can you cite the *precise definition* > of theorem in that context? Until you cite that definition , there's > no point to even think about anything, let alone agreeing. The same. The context is FOL. I am considering a formal system. The formal system I am considering has for its formal language the usual FOL symbols, plus 0, which is a 0-place predicate, and +, which is a 2- place predicate (written in infix notation for convenience). The formal system I am considering has for its axiom set the usual FOL logical axioms and the non-logical axiom AxAy x=y. In this formal system, x+y = 0 is a theorem. Yes or no? If no, why not? If yes, what the devil point are you trying to argue? === Subject: Re: Cantor's argument is erroneous <1od9gel9dx135.ef01fet6m530.dlg@40tude.net> posting-account=yKimjgoAAACk5WwPVD4l9HmbpoR6Hmy4 Presto/2.1.1,gzip(gfe),gzip(gfe) ... the following makes perfect sense to me: AxAy x=y ==> (any well formed expression) = (any other well formed > expression) Well, let's be picky (or more specific). I'd propose: AxAy(x = y) ==> = . well formed expression might be (mis)interpreted as well-formed formula > (wff), no? > And right, the rule universal elimination (UE) in, say, a calculus of > natural deduction (for FOPL) allows for (exactly) that move (though in > two steps). : > AxAy(x = y) > Ay(a + b = y) (by UE) > a + b = 0 (by UE) > : > After specifying a language that has the right symbols with the right > arities, what doesn't make sense about that? I'm not being mean, I > literally do not understand what intuition is supposed to be being > violated here. Well, the intuition of a crank, of course. I suppose so. === Subject: Curvature of spacetime posting-account=LChCFQoAAACR0FoxHzVn6GGERsr9zp8c Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) Hi.., straight to the point..., just 2 questions (1) Do spacetime has singular points & discontinuities ? if yes,then from Physics point of view what are they called as..? (2) what is the Curvature of spacetime? Ciao ___nimo __________ I have the impression that Einstein understands relativity theory very well. ChaimWeitzmann, first president of Israel === Subject: Re: Curvature of spacetime > Hi.., straight to the point..., > just 2 questions (1) Do spacetime has singular points & discontinuities ? No. > if yes,then from Physics point of view what are they called as..? It was No. (2) what is the Curvature of spacetime? Convex one way and concave the other. http://www.androcles01.pwp.blueyonder.co.uk/Shapiro/Crapiro.htm === Subject: Re: Curvature of spacetime Space-Time: Continuums & Curvature http://library.thinkquest.org/2890/spactim.htm General Relativity - A Novice's Ideas on the Curvature of Spacetime. http://www.math.gatech.edu/~berglund/GR.html Plate Tectonics Plate Tectonics Geology : Plate Tectonics http://www.ucmp.berkeley.edu/geology/tectonics.html Plate Tectonics: The Mechanism http://www.ucmp.berkeley.edu/geology/tecmech.html Plate Tectonics: The Rocky History of an Idea http://www.ucmp.berkeley.edu/geology/techist.html -- Ahmed Ouahi, Architect Nimo kirjoitti > Hi.., straight to the point..., > just 2 questions (1) Do spacetime has singular points & discontinuities ? > if yes,then from Physics point of view what are they called as..? > (2) what is the Curvature of spacetime? > Ciao > ___nimo __________ I have the impression that Einstein understands > relativity theory very well. > ChaimWeitzmann, first president of Israel > === Subject: Re: Mental Event & Mental Object: Is there a Difference? posting-account=hA77jwoAAABvWF820QwAlfYdLdF7G8UH Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > On May 11, 4:57pm, Patricia Aldoraz A tree is a thing, a tree growing or just withering away is an > event. Makes no difference what objects or events we are talking > about. Hint; Ewe clueless useless name calling arrogant bitch, the nature of > an entity is included in its identity. He says looking in a mirror. One question about mental events is what is it that has them. People > have them, of course. Your mission, should ewe ever choose to awaken from your Kantian > nightmare, is to identify what it is that your mental events are > about, BIG hint; if they are not about or if they are not triggered by > matter and matter's nature, then they do not matter a gnat's toss. Ewe > got that? Hint; An example of a matterless mental event is god and all that > other mystical religious mumbojumbo and an another example is that > nauseating nasal whining chant of ewe leftist retards, the greater > good Clear the way? Yes, Yes, off and come back when ewe have grown up and learnt at least > a little on HOW to think as against the arbitrary meaningless Kantian > garbage you regurgitate on here. MG === Subject: Re: Mental Event & Mental Object: Is there a Difference? <4A079CE1.192CA586@gmail.com> posting-account=hA77jwoAAABvWF820QwAlfYdLdF7G8UH Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) If only............. How very Kantian! Try begining with ... How very Kantian. A good point herb, but mickey is too dense to see it. -- > hz === Subject: Re: Mental Event & Mental Object: Is there a Difference? If only............. How very Kantian! Try begining with ... How very Kantian. A good point herb, but mickey is too dense to see it. Braying ass. Insufferable. -- hz === Subject: Re: Mental Event & Mental Object: Is there a Difference? posting-account=hA77jwoAAABvWF820QwAlfYdLdF7G8UH Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Yes, a 'mental state' is an object. > It is not an object. Mr. Jones, anything one observes of a state is a > tentative declaration of something within its moment. It is not that > moment, nor what it was (to you or any observer) at that moment. > Jones - your object is DEAD. > A mental state is an object, and because it is an object it is an > example of bad grammar. So your position is the same as mine. > A mental state is only an object to a machine or to another person. > Your own mental state is a subject. Your statement is an abuse of > language. > What's a 'mental state'? What's 'my own mental state'? > Why give this mental term a false credence by treating it as a physical > object? I am not saying it is an object. Why not READ what I said? You can > only make an object of the mental state of another. Your mental state > is a subject and subject to that to which is referred. I know you say 'it' or the mental state is a subject, but you are > phrasing it like it's an object. Perhaps you can demonstrate and show us all how you can refer to a thing without it becoming objectified? === Subject: Dimensions of our space ? posting-account=LChCFQoAAACR0FoxHzVn6GGERsr9zp8c Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) Hi.., well, I asked one of my friends to prove that our space has more than 2 dimensions. he had given me this example.., => if our space has only 2 dimensions, then it is not possible to tie the knot ( of a shoe lace or any other ) I understood what he means.., can you too help me in this regard.., by giving some more examples or any other explanations you would like. Ciao ___nimo === Subject: Re: Dimensions of our space ? > Hi.., well, I asked one of my friends to prove that > our space has more than 2 dimensions. he had given me this example.., => if our space has only 2 dimensions, then it is not > possible to tie the knot ( of a shoe lace or any other ) I understood what he means.., can you too help me in this regard.., by giving > some more examples or any other explanations > you would like. Ciao > ___nimo You cannot experimentally prove that space has more than two dimensions. You can only verify it for a particular case. This is not a failure of science, but a cornerstone. You cannot prove a theory, only disprove it. All empirical evidence (so far) indicates that we live in 3 spatial dimensions and one temporal. For an interesting and humorous discussion on perception of our physical world, and how one might perceive the world from the point of view of two dimensions, and beyond, read Flatland: A Romance in Many Dimensions, by Edwin Abbott Abbott. http://en.wikipedia.org/wiki/Flatland === Subject: Re: Dimensions of our space ? > Hi.., well, I asked one of my friends to prove that > our space has more than 2 dimensions. he had given me this example.., => if our space has only 2 dimensions, then it is not > possible to tie the knot ( of a shoe lace or any other ) I understood what he means.., can you too help me in this regard.., by giving > some more examples or any other explanations > you would like. > Have you heard of the well known Victorian Book 'Flatland' by Edwin A Abbott? text available here http://xahlee.org/flatland/flat1.html Ian Stewart has written a book 'Flatterland' taking the ideas further . http://en.wikipedia.org/wiki/Flatterland === Subject: Re: Dimensions of our space ? > Hi.., > well, I asked one of my friends to prove that > our space has more than 2 dimensions. > he had given me this example.., > => if our space has only 2 dimensions, then it is not > possible to tie the knot ( of a shoe lace or any other ) > I understood what he means.., > can you too help me in this regard.., by giving > some more examples or any other explanations > you would like. Have you heard of the well known Victorian Book 'Flatland' by Edwin A > Abbott? > text available here > http://xahlee.org/flatland/flat1.html Ian Stewart has written a book 'Flatterland' taking the ideas further . > http://en.wikipedia.org/wiki/Flatterland > Don't flatter yourself, Mighty dOG, you are still a moron. === Subject: Re: Dimensions of our space ? posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; InfoPath.2; .NET CLR 3.5.21022; Tablet PC 2.0; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Hi.., well, I asked one of my friends to prove that > our space has more than 2 dimensions. he had given me this example.., => if our space has only 2 dimensions, then it is not > possible to tie the knot ( of a shoe lace or any other ) I understood what he means.., can you too help me in this regard.., by giving > some more examples or any other explanations > you would like. Ciao > nimo forward(or backward) and also jump up (or down)... Tonio === Subject: Re: NY Times math problem Archie: You can win and achieve the adulatrion you so desperately seek. You simply take off at full speed running in a right handed helix until your speed begins to unwind your DNA and you run through your asshole and turn inside out. THEN, girls will talk to you ... maybe. === Subject: Re: NY Times math problem <12nb0554acbsf5dghh2ns000e4ce7dsd9n@4ax.com> <4a0acc6b.663883468@newsgroups.bellsouth.net> posting-account=O9zR9AkAAACmp918j6u5m5plppeILcze Filter 1.2.0.72; GTB6; .NET CLR 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) Archie: You can win and achieve the adulatrion you so desperately seek. You > simply take off at full speed running in a right handed helix until > your speed begins to unwind your DNA and you run through your asshole > and turn inside out. THEN, girls will talk to you ... maybe. adulatrion? What's that? === Subject: Re: NY Times math problem > Archie: > You can win and achieve the adulatrion you so desperately seek. =A0You > simply take off at full speed running in a right handed helix until > your speed begins to unwind your DNA and you run through your asshole > and turn inside out. =A0THEN, girls will talk to you ... maybe. adulatrion? What's that? It's the confusion caused by the extra r from another post about Archie's mental prowess where I spelled hemmorhoid (referencing a potential cause for his frequent headaches) incorrectly. My sincere apologies to the Mensa crew here. I trust no one has become so upset that their solution to the world's problems is put on hold. === Subject: Re: NY Times math problem We have a jackrabbit in the neighborhood. I see him every morning > running along the ridge as I'm pouring my coffee ;-) Sounds like you need to switch to decaffeinated! -- You can't have a sense of humor, if you have no sense! === Subject: Re: NY Times math problem > We have a jackrabbit in the neighborhood. I see him every morning > running along the ridge as I'm pouring my coffee ;-) > Sounds like you need to switch to decaffeinated! ..a hookah-smoking caterpillar has given you the call === Subject: Re: Diagonal wanderings (incongruent by construction) <66gl05tbaeuf0v5o8nt1atu26eg6kkcrcj@4ax.com> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > But in the present context > implying that people are insisting the theorem's true just > because they say so is either dishonest or stupid. Just a point of clarification: that or is inclusive, yes? Marshall === Subject: Re: Diagonal wanderings (incongruent by construction) <4A0A7795.4010B8FF@gmail.com> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) Just curious: do any of the sensible regulars think pre-formal > means anything? I am considering putting pre-formal on my list > of words that correlate highly with crankery. (Like potential and actual.) I like pre-formal. Pre-formal notions motivate formalization. Agree with you on potential and actual. OK. You're definitely on my sensible list. So, is an example of pre-formality like when an ancient Egyptian notices that 5 * 3 = 3 * 5, and so forth, and after a while inductively decides x*y = y*x without a formal proof? Perhaps then (if he's Greek) he's motivated to look for a formalism that supports the same. I can imagine the same thing in, say, bridge building. We have early conceptions of the arch, which seems to correlate with bridges not falling down. Eventually we have structural engineering. At that point, I see no reason why a structural engineer would want to re-examine pre-formal notions of bridges. Likewise, every use of pre-formal I see on usenet is a nutty questioning of axiomatic reasoning, or something similar. Not questioning of axioms, which would be one thing, but the axiomatic method itself. I am reminded of the crank version of RAA: A proof within (say) ZFC: ------------------- Assume sentence A proceed deductively ... Conclude ~A Therefore: ZFC is broken. QED. Marshall === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) You are the fool. Is that possible that the notion of a pre-formal > (founding the formal!) does not belong to your (and most people's, as > I see) background at all? What the hell do they teach in the schools > around there?? Just curious: do any of the sensible regulars think pre-formal > means anything? I am considering putting pre-formal on my list > of words that correlate highly with crankery. (Like potential and actual.) You have to put a lot of serious authors on your crankery list. But I > think, this should not be a problem for you. After all you seemingly > have decided to see only one half of the truth. Sorry, I was asking for replies from *sensible* people. But I haven't noticed anyone sensible on usenet use the terms potential and actual. And I haven't seen them in any math or logic books I've read (although I'm not claiming to be well read in the field.) I don't know what they would mean in a mathematical context. Mostly they seem to be favorite terms for infinity cranks such as yourself. Marshall === Subject: Re: Diagonal wanderings (incongruent by construction) ... > 0.1 > 0.11 > 0.111 > ... > is not distinct from every line. > > Pray tell to which line it is equal. > > If all lines exist, then N is in a line. > > By which rule? (And we were talking about the diagonal...) > > By the rule that the diagonal cannot exist without lines. The diagonal > is made from the last digits of the lines. If the diagonal were longer > than every line, then it could not exist. > > Wrong. > > No. The diagonal is made from the last digits of the lines. And? > The diagonal has more digits > than every line. > > Right. > > Hence there must be a line that has more digits than > every line. Or actual infinity is wrong a concept. > > On what is the hence based? > > The diagonal is made from the last digits of the lines. Yes, and? > But as WM has left this thread (as is customary when it becomes clear that > he cannot win the argument except by circularity), > > I left the thread because it will get confusing after exploding into > pieces beyond the 1001th contribution. Well, in that case you could also use a better news interface than the -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Diagonal wanderings (incongruent by construction) <6sj505tf0bif530lrt8bp2a9ds9a31k2qq@4ax.com> Do you now agree that we can prove uncountable sets exist without the > diagonal argument? ?Doesn't this invalidate your claim? > If you still mean, as I have to suppose, that you can prove it > axiomatically, the answer of course remains: no and no. As I have > already (and I'd think, clearly) stated, my position is that invoking > axioms (or properties like the LUB) to justify Cantor's arguments (any > of them) is simply incorrect. IOW, I am not denying such proofs exist, > neither I am dismissing tout court the formal approaches: what I am > saying is that they are irrelevant to a discussion of Cantor's > arguments and, more in general, to any discussion of pre-formal > assumptions/arguments/conclusions. How do you feel about invoking axioms to prove that 2+2 = 4? What does any of this have to do with your claim that the existence of uncountable sets ultimately depends on your stance on the diagonal argument? It most certainly does not, as I have demonstrated. -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === Subject: Re: Diagonal wanderings (incongruent by construction) set axiom is irrelevant to a discussion on the diagonal argument (just > as Levy's proof is irrelevant). That is not the current context. The current context *is* a discussion of the diagonal argument. One > just needs to abstract from the usual chaos around. You responded to a statement that the > reals are uncountable by claiming That ultimately depends on your stance > on the diagonal argument. I was pointing out that the existence of > uncountable sets does *not* depend on the diagonal argument. In fact, we > can also show the reals are uncountable without using a diagonal > argument. Right, from the power set axiom. So I could give a counter-power set > axiom and show that the list is complete while the diagonal does not > exist. Since axioms cannot fail, I have just proven all standard math > invalid. (Impeccable logic...) Quite right. The converse approach, to refuse to say what is acceptable as a proof, to demand a proof, and to indicate that any purported proof *must* involve invalid reasoning, is also impeccable -- and equally unconvincing. > -LV -- Alan Smaill === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > You are the fool. Is that possible that the notion of a pre-formal > (founding the formal!) does not belong to your (and most people's, as > I see) background at all? What the hell do they teach in the schools > around there?? Just curious: do any of the sensible regulars think pre-formal > means anything? I am considering putting pre-formal on my list > of words that correlate highly with crankery. (Like potential and actual.) You have to put a lot of serious authors on your crankery list. Either that or use a correlation coefficient of less than 1. In this case the use of the words potential and actual would be an indication of crankery but not a proof. - William Hughes === Subject: Re: Diagonal wanderings (incongruent by construction) posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > No, I agreed that any line can be reached by induction. > You can reach any line. However, reaching a line does > not mean finishing unless the line is the last line. > There is no last line so you can't finish. On the same line of reasoning, you cannot finish the anti-diagonal > either. You will never have a time of the last step, but if > you do the usual trick of squeezing an infinite number > of steps into a finite time, you can have a time after all > steps are done. The difference is that you can create an > anti-diagonal without doing a last step. You cannot create > a line containing all other lines without doing a last step. For me that's really a new and interesting aspect in the discussion: > do you really think, you can solve the potential/actual problem with > this trick? Let's take the program of Marshall to generate _all_ naturals: 10 LET I=0 > 20 PRINT I > 30 LET I = I + 1 > 35 Double Processor Speed > 40 GOTO 20 Now we should have all naturals in finite time. Please tell me, what > is the value of I if the program is ready (infinite ressources > assumed)? The program does not finish. Questions about the state of the program after it is finished are meaningless. Questions about the limit of the state of the program are not meaningless (but the answer may be that no limit exists). In the present case the answer is the limit of the value of I is aleph_0. (note if the question is What is the limit of the parity of I? the answer is no limit exists. ) - William Hughes === Subject: Re: P=NP Proof Published at CERN <29921474.87652.1242020365442.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) VMCM1905 a .8ecrit : Who is JSH?--Martin Musatov Oh please don't tell me Harris has moved from being a crank who thinks he > has a simple proof of FLT to a crank who thinks he can prove P vs NP? I don't see how that follows, but yes, he thinks he's solved >the Travelling Salesman Problem. It follows, but in a convoluted manner. > As to the OP's question, there are many sites that discuss JSH as a crank. When did JSH give up on FLT, and how was he finally convinced? For the record I have bo idea who JSH or what FLT is. ~~~~MMM~~~~|NNN === Subject: Re: P=NP Proof Published at CERN Victor Eijkhout a .8ecrit : How does that have to do with whether or not my proof is correct? In theory it shouldn't. In practice, your use of defective products > makes it impossible to read, and therefore assess, your proof. Victor. > -- > Victor Eijkhout -- eijkhout at tacc utexas edu What products are defective? 6P=NP. --MartinMusatov === Subject: Re: P=NP Proof Published at CERN <87k54ojw0r.fsf@phiwumbda.org> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) Martin Musatov a .8ecrit : [...] If you want help you have to make a readable copy of the > proof available. Er, guys? This is the same guy who announced his proof of P=NP on Apr. 26, That announcement happened to be a repost of a 1999 April Fool's > joke by a different poster (or a different alias, but I doubt it). He was having us on then. Surely, it is obvious he's having us on > again. -- > So why talk [about my factoring method] out on Usenet? Because it's a > highly public place so I'm unlikely to disappear[...] You people are > my protection. [...] You may be what's keeping me free and walking out > in the open air. -- James S. Harris, theory guy on the edge. >[ > Clearly now you have seen the light: P==NP is more than a repository. === Subject: johnreed take 26 Part-4, Modified May 14, 2009 posting-account=o56yPwoAAAB4uNxiO1WbAK5H7nOzlVbM Trident/4.0; GTB6; Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; yie8),gzip(gfe),gzip(gfe) johnreed take 26 Part-4, Modified May 14, 2009 Begin quote Mass is defined by the resistance that a body opposes to its acceleration (inert mass). It is also measured by the weight of the body (heavy mass). That these two radically different definitions lead to the same value for the mass of a body is, in itself, an astonishing fact. End quote Albert Einstein Rhetorical Questions: Why did Einstein, and why do most contemporary physicists, consider this equivalence astonishing? Why do they consider inert mass and gravitational mass, as radically different definitions? Why has the equivalence been invoked as an unexplained principle? Where all that is required is an explanation in words, for why gravitational mass, is equivalent to inert mass. Answer to Rhetorical Questions: The reason is: To rationally explain the equivalence requires a precise use of words, which requires in turn, more energy intensive thought, than we have been willing to invest. We have been content to rely on the convenient least action consistent mathematics. Its quantitative effectiveness in a least action [1] universe, provides us a pragmatic capability, far beyond an immediate necessity for the rational comprehension of what it means. notion of these forces, without considering their physical causes and seats. He continued with, ... the reader is not to imagine that ... I define the ... causes or the physical reasons ... thereof ... or that I attribute forces, in a true and physical sense, to centers (which are only mathematical points) when at any time I happen to speak of centers as attracting or as embued with attractive powers. (Principia) Indeed, to my knowledge, other than my work [1a], no one has since, even addressed the question of why the mathematics describes the universe so well, being content instead, to use the effective least action consistent mathematics, and leave the question unanswered. Consequently, humanity at large has come to believe that the mathematics is a crystal ball that mysteriously reveals the secrets of the (least action) universe. As an apparent justified consequence, the theoretical physicist mathematician professes a great disdain for words. The academic humanist, who thinks in words, has assumed a defacto, lowered intellectual status, or academic caste, and accepts the theoretical physicist's word interpretations for the mathematical models, with an intimidated silence. The academic humanist is tolerated and forgiven, and the theoretical physicist mathematician is considered a specially gifted, authoritative genius. A comparative position many theoretical physicist mathematicians have come to accept as deserved. The application of the least action consistent mathematics, to our least action universe, allows the theoretical physicist mathematician's imprecise use of words, to provide a quantitative illusion for a precise conceptual understanding, that is however, only attributed to the theoretical physicist mathematician. His/her disdain for words can be appreciated when one is introduced to the bizarre ideas generated by the nonetheless quantitatively effective, least action mathematical models. To see this, we need not review in this post, the more esoteric and incomprehensible notions spawned from the least action dependent, general relativity, and the least action dependent, quantum mechanics. We can understand why the so called gravitational mass is equivalent to inert mass by requiring a higher standard of precision for our mathematical word definitions. For example: I will show that the reason the increased or decreased inertia, exactly matches the so called gravitational force is: Inertia and force are each quantitative, sensory properties of the universe that we measure and feel. The quantity inert mass is generally (loosely) regarded as a measure of the amount of matter. This amount of matter is quantified in mass units that represent resistance, also a property of matter that we feel. We can leave it at that, just as though what we feel and call gravitational force is fundamental, because inert mass (resistance) is conserved locally (with respect to surface planet objects) in the least action universe. Consequently we presumptively assign the conservation of surface planet, inert object mass (resistance), to the celestial least action universe, where its (inert mass's) anonymous operation, as ranged mathematical points within Kepler's least action, law of areas, allows us to define the ranged mathematical points in the celestial least action universe in units that are proportional to what we, as planet surface inertial object's, feel and measure as force (as living planet surface inertial objects) quantified in planet surface units of accelerated inert mass, is the cause of the least action order that we observe as ranged mathematical points, in the celestial least action universe [3],[5]. We can carry this further as Albert Einstein did, again, just as though what we feel and measure (as living planet surface, inertial objects), is fundamental, by declaring that inert mass and so called gravitational mass are the same, as a matter of convenient, mathematically explained principle. Where gravity (the imprecise and erroneous, overly generalized definition of what we feel and measure, as living, planet surface inertial objects), is a consequence of the ranged mathematical points, now representing a curvature of spacetime, in a least action universe, which spacetime, least action ranged mathematical point curvature, is nonetheless, a consequence of inert mass (what we feel and measure quantitatively, as living planet surface inertial objects), in units that represent resistance (what we measure and feel). Either approach assumes that our feel of force, as living planet surface inertial objects, quantified in units of accelerated inert mass (resistance), is the cause of the least action order that we observe as ranged mathematical points in the celestial least action universe. Or, we can avail ourselves of the increased knowledge (that was not available to Isaac Newton), that we have gained in the last 350 years, and precisely define the physical cause of what we feel. Recall that Inertia and force and resistance, are each properties of the universe that we feel. To answer the question then requires that we define precisely what it is that we feel. oOo I say: If mass is the quantitative measure of the conserved cumulative resistance of a planet surface, inertial object's atoms (that we measure and feel), and if we are living, planet surface inertial objects; Then what we measure and feel as gravitational force is the accelerated, conserved cumulative resistance of a planet surface inertial object's atoms. This includes the atoms that make up our bodies and the atoms in the bowling ball that we lift. This defines inert mass in hard objective (quantitative) terms of what we feel, and compare and measure on the balance scale, and call weight. The comparative, cumulative resistance of a planet surface inertial object's atoms. Here, I have defined inert mass (what we measure and feel) precisely, prior to generalizing it (inert mass) to all bodies in the universe, on the basis of its (inert mass's) anonymous operation, in terms of ranged mathematical points, along least action, celestial trajectories. This definition explains why Newton's third law works with respect to the conserved interaction between planet surface inertial objects, in units of resistance called inert mass. The comparative measure of the conserved cumulative resistance of a planet surface inertial object's atoms. Where the cumulative resistance of ANY inertial object's atoms (like some planets, some moons, and stars, when viewed in terms of GR) may, or may not be, more or less than the sum of a theoretical composition of discrete atoms. In fact the state of matter at the cores of stars and some planets and moons, may not be the same state of matter we find at the planet surface [4], [5]. However, in the case of objects composed of pure elements, where each and every atom is directly proportional to a single, common, inert mass, the total mass magnitude of the object, provides us a straight forward means to calculate the number of atoms within the object [6]. This is also true for the molecules of objects of pure compounds. Either of these cases show that the inert mass of the surface planet object pretty near represents the cumulative resistance of that object's atoms. This, a fundamental aspect of the science of chemistry. Inert mass represents the magnitude of the intrinsic cumulative atomic resistance that we measure and feel (as living, planet surface inertial objects), but it does not tell us the physical cause of what we measure and feel [7]. Rather, we have continued to assume, that the least action order that we observe in the universe, is caused by what we measure and feel. This, even as it was initially disclaimed by Isaac Newton on the basis of his knowledge at the time, and even as we have since, increased our knowledge, which enables us to define the precise physical cause today. The cumulative resistance of atoms assimilates the quantities mass and energy and many other properties of planet surface matter, including many that may be heretofore undefined. The cumulative resistance of planet surface atoms is felt, and measured, in units of inert mass. We have further generalized inert mass, to describe the internal least action atomic electromagnetic properties of frequency and wavelength, in terms of energy, through the conversion factor called Planck's Constant (subjectively regarded as a constant of proportionality) [7a]. Where we are attracted to the Earth and where we feel and measure the conserved cumulative resistance of our planet surface atoms in units of inert mass, and the conserved cumulative resistance of a planet surface inertial object's atoms that we lift, in units of inert mass, does not show that an equal and opposite attractive force exists between our conserved inert mass (the conserved cumulative resistance of planet surface atoms) and the Earth's conserved inert mass (the theoretical, conserved cumulative resistance of atoms). Since this resistance does not occur (air resistance occurs, of course) when we travel in the direction our atoms are being pulled, during free fall (Einstein and peers called this natural motion and attributed it to a curved, mathematically functional, least action dependent, 4-D spacetime, ranged mathematical point geometry), we must wait to impact the Earth to experience that resistance from acceleration. Where the final velocity is zero. The idea that an increase or decrease in planet surface object inertial mass (resistance), is precisely met by an increase or decrease, in the so called gravitational force, where no resistance exists during free fall (natural? or least action motion, measured as a ranged series of mathematical points), is literal contrived nonsense. On the other hand we feel the resistance continually at the Earth's surface where vertical velocity is zero. This resistance is immediately increased as we accelerate against the direction of the pull on our atoms. This explains that the force we apply to lift the planet surface object, precisely matches the cumulative resistance of the planet surface object's atoms, that we lift. Inertia is the quantified, cumulative resistance of a planet surface inertial object's atoms. Therefore, we require no principle of equivalence. With respect to Newton's third law, equal and opposite is defined in units that represent the magnitude of the accelerated resistance of a planet surface inertial object's atoms, that we measure and feel, and compare against another planet surface object's atoms on the balance scale, and call weight. Where the conserved cumulative resistance of our planet surface atoms, or the planet surface atoms of the object that we lift, can in no objective (as opposed to subjective) way, be set equivalent to the theoretical conserved cumulative resistance of the atoms composing the Earth. We ask too much from acceleration, and ranged least action mathematical points, to say the least. Additionally, we are a part of the Earth's surface, and we do not know what state of matter exists at the core of the Earth, where pressure and temperature interact in ways we cannot presently describe [4],[5]. oOo Afterword: Consider a pulling force that acts on atoms individually, or in cases parts of atoms. Such a force does not act on resistance. More generally, if it did act on resistance, the more resistance we applied as physical effort, the more resistance we would have to overcome. The resistance we detect comes from the cumulative resistance of the object's atoms at the Earth's surface, where the pulling force acts on the object's atoms individually, and which we must overcome collectively to lift the object. The pulling force acts on the object's atoms individually where the force we apply acts on the cumulative resistance of the object's atoms [8]. Here we feel the accelerated (therefore increased) cumulative resistance of a planet surface inertial object's atoms, and here the measure is taken as the cumulative resistance of the planet surface inertial object's atoms at an instantaneous or final velocity. Note that where the Earth attractor acts on atoms individually, and not collectively on bodies (as we do, and as impacting inertial objects do), then all atoms falling at the same rate, cogently explains the measured equivalence. The subjective principle of equivalence is mathematically functional, with respect to our effort as surface planet inertial objects, and anonymously, with respect to the mathematical points ranged along least action celestial trajectories, but conceptually incorrect. And where the Earth attractor acts on planet surface atoms, it should be investigatively advantageous to examine the possibility for a form of super electromagnetism that acts on all planet surface atoms, not just those planet surface atoms with optimal structural characteristics and properties [4]. This super form of electromagnetism will replace our super so called, gravitationally caused blackhole, and provide a rational theoretical framework, for pulsars and quasars and whatever else is yet to be discovered out there. The so called universal force of gravity, singularities, and many other mathematical arguments, will no longer be viable. We will no longer seek blackholes and therefore, we will no longer find them. oOo It took me many years to figure this little bit out. Too many. It took many more years to articulate it. This is still ongoing. It will not meet with your expectations from Physics 101 scripture. If you consider your expectations from mathematical scripture as proved science, and if you think my little bit is inconsistent with that scripture, or otherwise incorrect from a rational perspective, please be my guest and try to blow it out of the water. As I have indicated before, I will appreciate your success if it occurs. However, if you cannot do this, without a sole reference and reliance, on science that is based solely on Physics 101 scripture, do your progeny a favor. Be one of the first to recognize the veracity of my little bit, or remain silent. Endnotes [1] The principle of least action can take many mathematical and many physical forms. For the academic humanist the easiest to understand mathematical form is probably the static representation of a Euclidean circle, where the circumference is the shortest line length to enclose the greatest area. For the first year physics student, Kepler's laws are easy to grasp in this regard. For the more advanced student, the Lagrangian and the Hamiltonian are other classical forms, and Feynman's sums of histories, yet another, albeit quantum mechanical form. Newton's first law and his laws of motion in general derive from a least action principle. Even the roll of a pair of dice can be shown to derive from a least action principle. For my purpose conceptually, least action is more clearly regarded as efficient action, where action includes, but takes on a broader meaning (frame specific) than [energy/time]. Stability in the field requires efficient action to the extent that stability is retained through perpetuity. [1a] See johnreed take 26 - The Principle of Equivalence - Part 1 and Part 2 and 3. Search Google.groups on johnreed take to access most of the rest of it. To acquire an attempt to account for the history of the logical development of these ideas see johnreed take 1A, 13 OCT 2005; johnreed take 1B, 12 NOV 2005; johnreed take 1C, 22 DEC 2005; johnreed take 1D, 23 FEB 2006. [2],[3],[6] See Is Mass an Emergent Quantity in an Electromagnetic Universe? ...Research Results on Centripetal Force, Part 2. [4] See johnreed take 23 24 and 25. [5] In the case of stable, least action orbital motion, the inertial resistance of the planet and the planet's momentum, are secondary to the time controlled least action orbit (where the orbital acceleration and free fall acceleration meets that time function), as are these properties with respect to our artificial satellites, once they are placed in orbit. Our so called universal orbit controlling gravitational force (while controlling us) conveniently (anonymously) operates within the time constraints of Kepler's law of areas. Where we have mathematically converted angular velocity [v^2/r], to angular momentum [mv^2/r], by multiplying both sides of a least action equation by unity in the convenient anonymous form of [m/m] [6]. In short we have defined the least action universe after our own planet surface inertial object image. It is functional for us, as planet surface inertial objects, but subjective and conceptually limiting. [6] Here [F=mg] can be written as [F=nNmg], where [n] represents the number of moles, [N] represents Avagadro's number, and [mg] represents the atomic weight of a single atom of the element. [7] A shovel full of dirt will weigh a specific magnitude. We cannot determine the composition of the dirt from its mass alone. We can however, determine its mass if we have its precise composition. [7a] See http://www.wbabin.net/science/michaud.pdf [8] This is a very simple concept actually. 100 atoms falling separately or 100 atoms falling as a cohesive object. They all arrive at the same time if they are dropped at the same time from the same height. The only question here* is whether the Earth attractor acts on the atom or on its mass. That question has been eliminated in this post. *As I pointed out earlier in johnreed take 1C there is an aspect here that I still have not completely worked out. Given that the Earth attractor acts on atoms, a question arises that is indicated by the Cavendish torsion bar experiment, the least action orbital behavior of a hollow sphere, the sense of nauseum we feel in a descending and ascending elevator, and possibly the action on objects within a returning to Earth, space vehicle. Where a falling astronaut will fall at the same rate as the vehicle, provided she is outside the vehicle. If she is inside the vehicle, unattached to any part of the vehicle, it would appear that she may be pressed against the inner part of the vehicle at its furthest point from the Earth during that descent, provided no internal atmosphere existed inside the ship and provided no external atmsophere slowed the descent of the ship. Indeed, even with an atmosphere a question still remains in my mind. Where her motive power for the fall would be increased by the far inner surface of the returning vehicle. These considerations lead me to suspect that in the case of surface planet inertial objects (also asteroids, comets, meteors, and some planets and moons), the actual attraction to the Earth results from an action on the surface atoms, of the planet surface inertial object, which carries the entire, planet surface object to the Earth. This could offer insight into the nature of the super-electromagnetic action of Celestial attractors, which planet Earth qualifies as.. johnreed === Subject: solutions manual for Intermediata Accounting 13e Kieso posting-account=ldzOOQoAAAALmEmlIRVd1wCu2DOhImIB CLR 2.0.50727),gzip(gfe),gzip(gfe) solutions manual and Test Bank Solutions Manuals and Test Bank in Electronic (PDF)Format! Just contact with , solutionsservice (at) hotmail.com (my email address,solutionsservice@hotmail.com ), these are parts of our solutions, if the solution you want isnÁøt on the list, please email to http://getsolutions.spaces.live.com is my blog. solution manuall for fundamentals of thermodynamics, 7th edition,sonntag,borgnak, test bank for managerial accounting 12th Edition authors Garrison Noreen Instructor's Manual Of Fundamental Methods Of Mathematical Economics Chiang & Wainwright 2005 Mc Graw Hill Fundamentals of Advanced Accounting 3rd Edition by Joe B. Hoyle, test bank for Operations Management 10e William J. Stevenson solutions manual for Financial and Mangerial Accounting 2e by Horngren test bank for Financial and Mangerial Accounting 2e by Horngren test bank for Finiancial Accounting 7e by Horngren solutions manual for Finiancial Accounting 7e by Horngren solutions manual for Intermediata Accounting 13e Kieso test bank for Intermediata Accounting 13e Kieso TB solutions manual for Fundamentals of financial management 12e Brigham SM test bank for Fundamentals of financial management 12e Brigham TB Solution manuall for Fundamentals of Engineering Thermodynamics 6th Edition by Michael Moran and Howard Shapiro A Computer System Architecture 3rd Edition by Morris Mano Solution Manual Complete Assignment of All Chapters A FIRST COURSE IN DIFFERENTIAL EQUATIONS WITH MODELING APPLICATION 7TH EDITION BY DENNIS G. ZILL 2500 Solved Problems in Fluid Mechanics and Hydraulics (Schaum's Solved Problems) by Jack B. Evett, Cheng Liu A Course in Game Theory Osborne and Rubinstein A First Course In Probability Solution Manual,Ross 6th A First Course in Abstract Algebra 7th by Fraleigh A First Course in Differential Equations with Modeling Applications (7th ed.) and Zill & ; Cullen s Diferential Equations with Boundary- Value Problems (5th ed.) zill A First Course in Probability: SOLUTIONS MANUAL (7th Edition) by sheldon ross A First Course in String Theory chapter 1 to 16 A First Course in the Finite Element Method, 4th Edition Daryl L. Logan A Friendly Introduction to Number Theory 3rd by Silverman A Guide to Physics Problems, Part 1 - Mechanics, Relativity, and Electrodynamics A Guide to Physics Problems, Part 2 - Thermodynamics, Statistical Physics, and Quantum Mechanics A Practical Introduction to Data Structures and Algorithm analysis 2nd edition by Clifford A. Shaffer A Quantum Approach to Condensed Matter Physics Solutions by philip l. Taylor Absolute Java, 3rd Ed by W. Savitch instructor manual and test bank Accounting Concepts and Applications (9th Ed.) by W. Steve Accounting 7e by horngren solution manual Accounting 7e by horngren TB accounting 7e by horngren TB (test generator File) Accounting 8th edition by horngren test bank and solution manual Accounting Information Systems - james hall 6ed sm Accounting Information Systems - james hall 6ed tb Accounting Information Systems 10E Romney solution manual Accounting Information Systems 10E Romney test bank Accounting Information Systems 11E Romney solution manual Accounting Information Systems 11E Romney test bank Accounting Information Systems 7E Edition Ulric J. Gelinas, Richard B. Dull instructor manual Accounting Information Systems 7E Edition Ulric J. Gelinas, Richard B. Dull solution manual Accounting Information Systems 7E Edition Ulric J. Gelinas, Richard B. Dull test bank Accounting Information Systems, 9E George H. Bodnar William S. Hopwood solution manual Accounting Information Systems, 9E George H. Bodnar William S. Hopwood test bank Accounting Principles 8E by Kieso SM chapter 1 to 10 Accounting Principles 8E by Kieso SM chapter 11 to 26 Accounting Principles, Edition 8E, Weygandt, Kieso, Kimmel (Test Bank) volume 1 Accounting Principles, Edition 8E, Weygandt, Kieso, Kimmel (Test Bank) volume 2 Accounting Text and Cases 12e by Anthony IM Accounting what number means 8e by Marshall Adaptive Control 2E. by Karl J. Astrom solution manual Adaptive Filter Theory, 4th edition S. Haykin Advance corporate finance 1e by Ogden Instructor manual and test bank Advanced accounting 10E by Flyd Beams (SM+IM+TB) Advanced Accounting 10th edition by Fischer (SolutionsManual) Advanced Accounting 10th edition by Fischer (test bank) Advanced Accounting 9e by Beams solution manual Advanced Accounting 9E Hoyle,Schaefer,Doupnik Solution Manual Advanced Accounting 9E Hoyle,Schaefer,Doupnik Test Bank Advanced Accounting 9th edition by Fischer (SolutionsManual) Advanced Accounting 9th edition by Fischer (test bank) ADVANCED CORPORATE FINANCE Policies and Strategies by Joseph P. Ogden,Frank C. Jen,Philip F. O£Àonnor ( solution manual) ADVANCED CORPORATE FINANCE Policies and Strategies by Joseph P. Ogden,Frank C. Jen,Philip F. O£Àonnor (test bank) Advanced Digital Design with the Verilog HDL Michael D. Ciletti selected solutions Advanced Dynamics by Donald T. Greenwood Advanced Engineering Mathematics 3rd Edition by Dennis G Zill and Michael R Cullen Advanced Engineering Mathematics by Erwin Kreyszig 8ed solutions manual Advanced Engineering Mathematics Dennis G Zill 2nd Solution Advanced Engineering Mathematics, 6th Edition Peter V. O'Neil - University of Alabama, Birmingham Advanced Engineering Mathematics, 9th Edition By Erwin Kreyszig Advanced Financial Accounting, 6th edition, by Baker, Lembke, and King solution manual Advanced Macroeconomics 1996 romer Advanced Macroeconomics, Solutions Manual 1996 Romer Advanced Modern Engineering Mathematics, 3rd Edt by Glyn James solution manual Aerodynamics for Engineers 5th ed. solution manual John J. Bertin Russell M. Cummings Algebra by Thomas W. Hungerford Published by Springer Algebra, Pure and Applied by Aigli Papantonopoulou An Introduction to Abstract Algebra with Notes to the Future Teacher by Nicodemi, Sutherland, and Towsley An Introduction to Economic Dynamics An Introduction to Economic Dynamics by Ronald Shone An Introduction to Mass and Heat Transfer Principles of Analysis and Design Middleman An Introduction to Mathematical Statistics and Its Application (4th Edition) by Richard J. Larsen An Introduction to Modern Astrophysics (2nd Ed., Bradley W. Carroll & Dale A. Ostlie) An Introduction to Numerical Analysis by Endre Suli An Introduction to Numerical Analysis by Endre S£Ài, David F. Mayers An Introduction to Ordinary Differential Equations James C. Robinson Publisher: Cambridge University Press An Introduction To The Finite Element Method, 3rd Edition by J. N. Reddy An introduction to the mathematics of financial derivatives Neftci solution manual Analysis and Design of Analog Integrated Circuits (4th Edition) Gray, Hurst, Lewis and Meyer analysis design of analog IC design Analytical Mechanics: Solutions Manual 7ed Grant R. Fowles, George L. Cassiday Anderson J.D. Fundamentals of aerodynamics, 2nd edition - problems and solutions Andrew Tanenbaum Structured Computer Organization Solutions Manual antenna balanis solution manual antenna balanis solution manual 2nd edition ANTENNAS FOR ALL APPLICATIONS, THIRD EDITION Antennas for all Applications 3rd Ed. by Kraus & Marhefka Anton calculus book+ solution manual + test bank 8th edition Anton, Bivens, Davis Calculus Early Transcendentals Combined, 8th Edition instructor solution manual Anton, Bivens, Davis Calculus Early Transcendentals Combined, 8th Edition instructor solution manual Anton, Bivens, Davis Calculus Multivariable, 8th Edition Applied Fluid Mechanics 6th Ed. by Robert L. Mott Applied Mechanics for Engineering Technology 8e Keith M Walker Applied Numerical Analysis 7Ed - Curtis F. Gerald, Patrick O. Wheatley - Solutions manual Applied Partial Differential Equations David Logan Applied Quantum Mechanics by A. F. J. Levi Applied Statistics and Probability for Engineers 3rd.Ed edition student manual Applied Statistics and Probability for Engineers by Douglas C. Montgomery 3rd edition complete Applied Statistics and Probability for Engineers, 4th Edition Montgomery, Runger Applied Strength of Materials (4th Edition) SOLUTION MANUAL by Robert L. Mott Artificial Intelligence A Modern Approach 2e Stuart Russell Peter Norvig Audit and Assurance service An Integrated Approach 11e TB auditing and assuance services by messier test bank 6th edition Auditing and Assurance Services An Intergrated Approach and ACL Software, 12e by Alvin Arens Randal J. Elder, ark Beasley solution manual Auditing and Assurance Services An Intergrated Approach and ACL Software, 12e by Alvin Arens Randal J. Elder, ark Beasley test bank Auditing Cases, 3E Mark S. Beasley solution manual Auditing Cases: An Interactive Learning Approach, 4/E Mark S BeasleyFrank A. BucklessSteven M GloverDouglas F Prawitt Bank Management & Financial Services, 7/e By Peter S. Rose, Sylvia C. Hudgins (IM+SM) Bank management 7e by peter s. rose TB Bank management 7e by Rose ( instructor manual ) Basic Electrical Engineering By Nagrath, D P Kothari, Nagrath D P Kothari I J Nagrath, I J Nagrath Published by Tata 2002 Basic Engineering Circuit Analysis, 8th Edition by J. David Irwin, R. Mark Nelms Basic Engineering Circuit Analysis, 9th Edition Irwin, Nelms Basic Technical Mathematics with Calculus 8e Allyn J. Washington Biological Science and CW+ Grade Tracker Access Card, 2/E test bank .bok file Biology Concepts and Connections 6e Neil A. Campbell Jane B. Reece Martha R. Taylor Eric J. Simon Jean L. Dickey test bank Biology with MasteringBiolog, 8E Neil A. Campbell Jane B. Reece Biomaterials - The Intersection of Biology and Materials Science (Temenoff & Mikos) Bioprocess Engineering Principles - Solutions Manual (Original) by pauline m. Doran Book keeping and Accounting 3e Joel J. Learnef sm + book Borgnakke, Sonntag Fundamentals of Thermodynamics, 7th Edition Brief History of Western Civilization 5e vol 1 im and tb Brief History of Western Civilization, A: The Unfinished Legacy, Volume 2 im and tb BUSINESS STATISTICS A Decision Making Approach 7e SM Business Communication Essentials 3rd edition bovee and thill test bank Business Data Networks and Telecommunications, 7/E Raymond R. Panko test bank Business Law by Cheeseman 6E (IM) Business Law by Cheeseman 6E (TB) Business Law Today: Comprehensive 8th edition Roger LeRoy Miller, Gaylord A. Jentz instructor manual Business Law Today: Comprehensive 8th edition Roger LeRoy Miller, Gaylord A. Jentz test bank Business Statistics (A Decision Making Approach), Groebner, Shannon, Fry, Smith, 7 sm and tb Business Statistics 4e by Leonard J. Kazmier book + sm Business Statistics Decision Making and Student CD Package test items and solution manual, 7E test bank and sm Butterworth Heinemann - Coulson And Richardson - Chemical Engineering Vol I (Solutions Manual V Edition) c ++ how to program deitel 6th edition solution manual and test bank C++ How to Program 3rd edition by deitel Calculus A Complete Course 6th by R.A. Adams calculus by gilbert strang calculus by leithold solution manual Calculus Early transcendentals 5Th Ed - Complete Instructor 's Solutions Manual by James Stewart 0534393217 CALCULUS early transcendentals 7th edition Anton Bivens Davis Calculus Early Transcendentals Single Variable, 8th Edition Howard Anton, Irl Bivens, Stephen Davis Calculus of Variations Solution Manual Russak Calculus Single Variable 4ed chapter 1 to 11 Hughes-Hallett, Gleason, McCallum, et al. Calculus Third Editon By Strauss, Bradley and Smith not complete Calculus With Analytic Geometry (6th) By Bruce E. Edwards, Ron Larson, Robert Hostetler Calculus With Analytic Geometry (7th) By Bruce E. Edwards, Ron Larson, Robert Hostetler student manual Calculus, Early Transcendentals, 7E by C. Henry Edwards ,David E. Penney Callister Fundamentals of Materials Science and Engineering An Integrated Approach, 2nd Edition Capital Budgeting and Long-Term Financing Decisions Neil Seitz, Mitch Ellison 4th Edition instructor manual Carey, Study guide and solution manual for organic chemistry Chapra Applied Numerical Methods With Matlab For Engieers Solutions Manual 1st edition nearly same with 2nd edition Chemical and Engineering Thermodynamics- 3rd Edition- Solutions Manual Chemical Engineering Design, Fourth Edition: Chemical Engineering Volume 6 (Coulson & Richardson's Chemical Engineering) Chemical Engineering Solutions manual for Volumes( 2 and 3) 3 edition Backurst J. R., Harker J.H. & Richardson J. F chemical Engineering: Solutions for Volumes 2 and 3 by coulson 2002-12-11 Chemical Reaction Engineering, 3rd Edition Levenspiel Chemical, Biochemical, and Engineering Thermodynamics, 4th Edition Sandler Chemistry: The Central Science (Hardcover, 2005) Author: Bruce E. Bursten, H. Eugene Lemay Jr., Lemay test bank 10th Classical Dynamics A Contemporary Approach by Jorge V. Jose, Eugene J. Saletan T. Thornton, Jerry B. Marion Classical Electrodynamics - 2nd Ed. John David Jackson byKasper van Wijk Classical Electrodynamics 3rd edition by Jackson Classical Mechanics (2nd Edition) by Herbert Goldstein Classical Mechanics by R. Douglas Gregory Classical Mechanics, 2ed Partial Solutions Manual by Safko Close, Frederick, Newell Modeling and Analysis of Dynamic Systems, 3rd Edition Cmos analog circuit design 2nd edition homework solutions by allen holberg CMOS Analog Circuit Design, 2ed Solutions by Phillip E. Allen, Douglas R. Holberg CMOS VLSI Design 3rd edition David Harris H E Weste College Accounting (Chapters 1-25), 10E by Jeffrey Slater sm and tb College Mathematics for Business, Economics, Life Sciences & Social Sciences (11th Edition) (Hardcover) by Raymond A. Barnett (Author), Michael R. Ziegler (Author), Karl E. Byleen (Author) Collins Mechanical Design of Machine Elements and Machines A Failure Prevention Perspective Communication Networks (2nd Edition) leon Communication Networks Fundamental concepts & key Architectures By Leon Garcia Widjaja not complete 3 4 5 6 7 8 10 Communication Networks Fundamental concepts & key Architectures Alberto Leon-Garcia Communication Systems (4th edt) by Simon Haykin Communication Systems 4Ed - A Bruce Carlson Solutions Manual communication systems engineering by proakis Communication Systems Engineering Proakis J (2002) Solutions Manual 2nd edition Compensation Management in a Knowledge-Based World, 10E Richard I Henderson instructor manual Complex Variables with Applications (Pie) by A.David Wunsch Computational Techniques for Fluid Dynamics: A Solutions Manual By Karkenahalli Srinivas, Clive A. J. Fletcher Computer Architecture A Quantitative Approach, 4th Edition, 2006 by John L. Hennessy, David A. Patterson Computer Architecture: Pipelined and Parallel Processor Design (Solutions Manual) by Michael J. Flynn selected solutions computer networking a top down approach 3rd edition solution manual by James F.Kurose, Keith W. Ross Computer Networking: A Top-Down Approach, 4/E solution manual and lab solutions Computer Networking: A Top-Down Approach, 5/E solution manual computer networks Andrew S. Tanenbaum 4th edition Computer Networks Systems Approach 3ed by davie peterson solutions manual Computer Networks: A Systems Approach 2nd edition Peterson and Davie£À Computer Organization and Architecture: Designing for Performance, 7/E William Stallings Computer Organization and Design, Revised Printing, 3rd Edition Solutions Manual By David A. Patterson, John L. Hennessy, Computer Organization and Design: The Hardware/Software Interface, 3rd Edition by David A. Patterson, John L. Hennessy Concepts In Federal Taxation 2007 (14thEd) - Murphy Solutions Manual Concepts of Genetics, 9e by Klug, Cummings, Spencer & Palladino test generator Concepts of Programming Languages, 8/E Robert W. Sebesta, University of Colorado, Colorado Springs Construction Surveying and layout 2nd edition by wesley g. Crawford Construction Surveying and layout 3rd edition by wesley g. Crawford Consumer Behavior, 8/E Michael R. Solomon test bank contemporary engineering economy by chan s. park 4th edition Contemporary Financial Management R. Charles Moyer, James R. McGuigan, William J. Kretlow 10th edition instructor manual Contemporary Financial Management R. Charles Moyer, James R. McGuigan, William J. Kretlow 10th edition test bank Contemporary Financial Management, 11th Edition R. Charles Moyer James R. McGuigan William J. Kretlow instructor manual Contemporary Financial Management, 11th Edition R. Charles Moyer James R. McGuigan William J. Kretlow test bank Control Systems Engineering by Nise 4ed£À? Corporate Computer and Network Security Raymond Panko Corporate Finance By Stephen A. Ross 6 edition Corporate Finance By Stephen A. Ross 8th edition corporate finance 1e by berk sm corporate finance 1e by berk tb Corporate Finance, 8e Stephen A. Ross Randolph W. Westerfield Jeffrey Jaffe instructor manual and solution manual Corporate Finance-7th Edition by Stephen A. Ross , Randolph W. Westerfield , Jeffrey Jaffe cost Accounting 12e by Horngren Test Bank cost accounting 12e by Charles T. Horngren Srikant Datar George Foster Madhav Rajan Chris Ittner solution manual cost accounting 13e by Charles T. Horngren Srikant Datar George Foster Madhav Rajan Chris Ittner solution manual cost accounting 13e by Charles T. Horngren Srikant Datar George Foster Madhav Rajan Chris Ittner test bank Cost Accounting: Foundations and Evolutions 7E By Kinney solution manual Cost Accounting; Foundations and Evolutions, Edition 7, Kinney, Raiborn Cost Management A Strategic Emphasis, 4e Blocher Cost ManagementMeasuring Monitoring and Motivating Performance by Eldenburg Wolcott SM TB Cryptography & network security 4e william stallings Daniel Shanks - Solved And Unsolved Problems In Number Theory (2Nd Ed), 1978 Data and Computer Communications William Stallings 8th edition William Stallings Data and Computer Communications, 7th Edition by William Stallings Data Communications and Networking fourth edition by Behrouz A.Forouzan odd numbered solutions Data structure and Problem Solving using Java 3rd Mark Allen Weiss, Data Structures and Algorithm Analysis in C++, 3/E Data Structures with Java by John R. Hubbard Anita Huray University of Richmond Database Concepts, 3E david kroenke, david auer tb and im DATABASE MANAGEMENT SYSTEMS 3rd Edition by Ramakrishnan, Gehrke, Derstad, Seliko, Zhu- Solution Manual only odd solutions Database Processing Fundamentals, Design, and Implementation, 10E David Kroenke test bank Database System Concepts, Fifth Edition by Avi Silberschatz ,Henry F. Korth solutions to exercises Database Systems: An Application Oriented Approach, Compete Version, 2/ E Michael Kifer Arthur Bernstein Philip M. Lewis Databases systems: An Application-Oriented Approach 2nd edition Michael Kifer, Arthur Bernstein, Philip M. Lewis test bank + solution manual David Kroenke's Database Processing: Fundamentals, Design and Implementation (10th Edition) test bank Derivatives Markets 2nd edition by Yufeng Guo Solution Manual Derivatives Markets 2nd by Rober L. McDonald solution manual Derivatives Markets 2nd by Rober L. McDonald test bank Design and Analysis of Experiments Solutions Manual 6th edition Design and Analysis of Experiments, 6th Edition Montgomery complete all chapters Design of Analog CMOS Integrated Circuits McGraw Hill Solutions Manual Design of Fluid Thermal Systems, 2nd Edition William S. Janna design of machinary by norton 3rd edition Design with Operational Amplifiers and Analog Integrated Circuits, 3rd edt. by Franco Device Electronics for Integrated Circuits 3Edition Muller Kamins Device Electronics for Integrated Circuits Solutions Manual 3ed DIGITAL DESIGN FOURTH EDITION by M. MORRIS MANO DIGITAL SIGNAL PROCESSING: Signals, Systems, and Filters Andreas Antoniou Differential Equations & Linear Algebra, 2nd ed., Farlow Differential Equations & Linear Algebra, edition 2, by Edwards Penny differential equations 5th edition by zill classic fifth edition Differential Equations And Boundary Value Problems C. Henry Edwards - David E. Penney 2nd edition Differential Equations and Boundary Value Problems Computing and Modeling, 4E C. Henry Edwards David E. Penney Differential Equations and Linear Algebra, 3e by Stephen W. Goode and Scott A. Annin instructor manual Digital & Analog Communication Systems - Leon Couch (7th ed) (ISBN 0131424920) Digital Communications, 4th edition, 2000-08 book+solution by John Proakis Digital Communications: Fundamentals And Applications (2nd Edition)- Bernard Sklar Digital Design (3rd Edition) by M. Morris Mano Digital Electronics with VHDL (Quartus II Version) By William Kleitz Digital Fundamentals (10th Edition) floyd Digital Integrated Circuits by Rabaey 2nd edition solution manuel chapter 3,5,6,10 Digital image processing - Gonzalez 2Ed- Solutions Manual (209p) Digital Signal Processing - A Modern Introduction, 1st Edition Cengage learning Ashok Ambardar Digital Signal Processing - Proakis & Manolakis - Solutions Manual 3ed Digital Signal Processing (2nd Ed.) (Mitra) Solution Manual? Digital Signal Processing A Computer-Based Approach 1st ed Solutions Manual mitra Digital Signal Processing by Thomas J. Cavicchi - solution manuel Digital Signal Processing Principles, Algorithms and Applications (International Edition) by John Proakis ,Dimitris Manolakis Digital Signal Processing Using Matlab- Solution Manual Vinay K Ingle Proakis 2nd edition Discrete and Combinatorial Mathematics 5e (Solutions Manual Only) by Ralph P. Grimaldi Discrete Mathematics (5th Edition) By Dossey, Otto, Spence, Vanden Eynden Discrete Mathematics (5th Edition) by John A. Dossey , Albert D. Otto, Lawrence E. Spence ,Charles Vanden Eynden Discrete Mathematics, 5e John A. Dossey Albert D. Otto Lawrence E. Spence Charles Vanden Eynden Discrete-Event System Simulation 3rd edition by Jerry Banks John S. Carson II Barry L. Nelson David M. Nicol Discrete-Time Signal Processing 2nd Edition, 1999-02 by oppenheim Distributed Systems, Concepts and Design (Exercise Solutions) - G. Coulouris, J. Dollimore and T. Kindberg Doets & Eijck - The Haskell Road To Logic, Math and Programming - Solutions to Exercises Dorf, Svoboda Introduction to Electric Circuits, 7th Edition DSP First: A Multimedia Approach-Mclellan, Schafer & Yoder Solution Manual Dynamics of Mechanical Systems Solutions Manual (Horwood Engineering Science Series) by C. T. F. Ross Econometric Analysis Solutions Manual to the 6th Edition By William H. Greene Econometric Analysis, 5th edition william h. Greene Econometrics - [Instructor Solution Manual] The Econometrics of Financial Markets john y. campbell, andrew w. Lo Economics for Managers by Paul Farnham, 2008 custom edition sm + tb Economists Solution Manual (Blume, 1994) Effective Writing 8e May & May instructor manual Electric Circuits, Nilsson Riedel , 7th edition Electric Circuits,Nilsson Riedel , 8th edition Electric Machinery and Power System Fundamentals Electric Machinery and Power System Fundamentals Stephen J. Chapman first edition Electric Machinery by A. E. Charles Kingsley, Jr.Fitzgerald 6th edition electric machinery fundamentals 4th edition stephen j chapman Electric Machines By D. P. Kothari, I. J. Nagrath Electrical Engineering Principles and Applications 4th Allan R. Hambley Electrical Engineering: Principles and Applications 3ed Allan R. Hambley Electrical Machines Drives and Power Systems 6th edition by Theodore Wildi Electrical Machines, Drives and Power Systems 6th edition ISBN 0131969188 Electrical Power and controls Skvarenina 2nd Electromagnetics for Engineers by Fawwaz T. Ulaby Electronic Circuit Analysis and Design 2nd edt. by Donald A. Neamen - solution manuel Electronic Devices (Electron Flow Version), 8e floyd Electronic Devices and Circuit Theory 8th Ed Instructors Resource Manual with Text Solutions, Lab Solutions, and Test Item File Electronic Devices and Circuit Theory, 10e Boylestad & Nashelsky Electronic Devices and Circuit Theory, 9e Boylestad & Nashelsky Electronic Physics Strabman Electronics Fundamentals Circuits Devices and Applications by Thomas Floyd 7th edition Electronics, 2nd ed. by Allan R. Hambley Elementary Algebra with Applications, 3rd Edition Author: Terry H. Wesner Harry L. Nustad Elementary Differential Equations and Boundary Value Problems , 8th Edition Elementary Differential Equations And Boundary Value Problems, 7Th Ed - Boyce And Diprima Student Solutions Manual, Charles W Haines Ode Architect Companion Elementary Differential Equations with Boundary Value Problems, 6E Henry Edwards avid Penney Elementary Differential Equations With Boundary Value Problems, 4E,Edwards, Penney Elementary Linear Algebra with Applications 9 edition by Howard Anton, Chris Rorres Instructor Solutions Manual and Instructor Testbank Elementary Linear Algebra with Applications, 9/E Bernard Kolman David Hill Elementary Mechanics & Thermodynamics [2000 by Professor Jhon W. Norbury Elementary Number Theory (5th Edition) by Kenneth H. Rosen Elementary Principles of Chemical Processes Solutions Manual 3 ed By Richard M.Felder Elementary Principles of Chemical Processes Solutions Manual 3 ed By Richard M.Felder chapters 2 to 14 Elementary Statistics by Mario F. Triola, 10th Elementary Statistics With Multimedia Study Guide, 10/E solution manual Elementary Statistics With Multimedia Study Guide, 10/E test bank Elements of Chemical Reaction Engineering By H Fogler, 3rd ed Elements of Electromagnetics by Sadiku 2nd Ed. Elements of Electromagnetics, 3rd Ed., Matthew N.O. Sadiku Elements of Electromagnetics, 4th Ed., Matthew N.O. Sadiku homework and midterm problems Elements Of Information Theory - Solution Manual by thomas m. cover and joy a. Thomas E-Marketing, 4e Judy Strauss, Adel El-Ansary, Raymond Frost sm E-Marketing, 4e Judy Strauss, Adel El-Ansary, Raymond Frost tb E-Marketing, 5e Judy Strauss, Adel El-Ansary, Raymond Frost sm E-Marketing, 5e Judy Strauss, Adel El-Ansary, Raymond Frost tb Embedded Microcomputer Systems: Real Time Interfacing, 2nd Edition Jonathan W. Valvano Energy Management 5 ed 2005-12 Klaus-Dieter E. Pawlik Engineering - Materials Science, Milton Ohring Solutions Manual Engineering and Chemical Thermodynamics [Solution Manual] by Milo D. Koretsky Engineering and Chemical Thermodynamics Milo D. Koretsky Engineering biomechanics statics by Beatriz Guevarez, Joshua R?os, Nayka Rivera, Sharon V?zquez and Melvia Villegas Engineering Circuit Analysis 6Ed - Hayt Solutions Manual.pdf Engineering Circuit Analysis 7Ed William Hart Hayt Jack E. Kemmerly Solutions Manual contains chapters 2,3,4,5,7,9,10,11 Engineering Economy - Leland Blank & Anthony Tarquin 6th Edition selected solutions ( student solution) Engineering Economy 14e Sullivan solution manual Engineering Electromagnetics -Hayt (2001) Engineering Electromagnetics -Hayt (2001).rar Engineering Electromagnetics Nathan Ida 2nd edition Engineering Electromagnetics, 7th Ed., Hayt, Buck Engineering Fluid Mechanics, 7th Edition - Student Solutions Manual by Clayton T. Crowe, Donald F. Elger, John A. Roberson Engineering Fluid Mechanics, 9th Edition Crowe, Elger, Roberson, Williams Engineering Mathematics, 4th edt. by John Bird - solution manual Engineering Mechanics - Statics (11th ) by R.C.HIBBELER Engineering mechanics statics 12th edition by hibbeler Engineering Mechanics Dynamics (11th Edition) by Russell C. Hibbeler Engineering Mechanics Dynamics 3rd edition solution manual Hibbeler R.C. updated fixed 09-2006 Engineering Mechanics Dynamics, 4E Anthony M. Bedford and Wallace Fowler Engineering Mechanics Dynamics, 5E Anthony M. Bedford and Wallace Fowler Engineering Mechanics Dynamics, SI 6th Edition Meriam, Kraige Engineering Mechanics Statics 11th Edition By R.C.Hibbeler Engineering Mechanics Statics, 4E Anthony M. Bedford and Wallace Fowler Engineering Mechanics Statics, 5E Anthony M. Bedford and Wallace Fowler Engineering Mechanics, Dynamics 5E - Solutions manual By J. L. Meriam, L. G. Kraige, chapter 1-8 Engineering Mechanics, statics 5th edition Solutions manual By J. L. Meriam, L. G. Kraige Engineering Mechanics, statics 6th edition Solutions manual By J. L. Meriam, L. G. Kraige Engineering Mechanics: Dynamics 2 Ed. by Riley and Sturges contains chapters 13,14,15,16,17 chapters Engineering Mechanics: Statics: Solutions Manual (10th edition) by R.C. Hibbeler Engineering Problem Solving with C, 3E Delores M. Etter Engineering Problem Solving with Matlab 2nd edition by etter sm-tb- quiz Engineering Statistics, 4th Edition Montgomery, Runger, Hubele Engineering Vibration, 3e Daniel J. Inman Entrepreneurial Finance Chris Leach, Ronald W. Melicher 2nd edition instructor manual Entrepreneurial Finance Chris Leach, Ronald W. Melicher 2nd edition test bank Entrepreneurial Finance, 3rd Edition Chris Leach Ronald W. Melicher test bank Entrepreneurial Finance, 3rd Edition Chris Leach Ronald W. Melicher instructor manual Equilibrium and Non-Equilibrium Statistical Thermodynamics By Michel Le Bellac Essentials of Accounting for Governmental and Not for Profit Organizations 9e by Paul A. Copley Essentials of Entrepreneurship and Small Business Management 5e tb and im Essentials of Fluid Mechanics: Fundamentals and Applications, 1st Ed., by Cengel & Cimbala ISBN-13 9780073301129 Essentials of Investments 7th edition Zvi Bodie Alex Kane, Alan marcus test bank Essentials of Investments Zvi Bodie, Alex Kane, Alan J. Marcus Essentials of Managerial Finance 14e Brigham TB Essentials of Managerial Finance 13th Edition Scott Besley, Eugene F. Brigham instructor manual Essentials of Managerial Finance 13th Edition Scott Besley, Eugene F. Brigham test bank Essentials of Managerial Finance, 14th EditionScott Besley Eugene F. Brigham instructor manual Essentials of Managerial Finance, 14th EditionScott Besley Eugene F. Brigham spreadsheet problem solutions Essentials of Managerial Finance, 14th EditionScott Besley Eugene F. Brigham test bank Essentials of Organizational Behavior 9e Stephen P. Robbins Tim Judge Essentials of Statistics, by Triola, 3rd edition sm Essentials of Statistics, by Triola, 3rd edition tb Essentials of Strategic Management 4E DAVID HUNGER THOMAS L. WHEELEN im with tb Ethical Theory and Business, 8/E Tom L. Beauchamp Norman Bowie Denis Arnold Federal Taxation 2008 Corporations Partnerships Estates and Trusts 21E Anderson Pope Kramer test bank Feedback Control of Dynamic Systems 4th edition Franklin - Solutions Manual Feedback Control of Dynamic Systems, 5/E franklin Festo didactic Process Control System Field and Wave Electromagnetics, 2nd edition, Cheng Finance Management Test Bank brigham 11 test bank Financial & managerial Accounting 13E By william Haka bettner Financial accounting an introduction to concepts, methods, and uses by clyde Stickney and Roman Weil solutions manual 11th edition financial Accounting 4e by John Wild Financial Accounting 6e by kieso solution manual Financial Accounting 6e by kieso test bank Financial Accounting 6e Harrison Horngren Financial Accounting 6th edition by Harrison Solution Manual financial accounting 6th edition harrison test bank financial accounting 7th edition harrison solution manual financial accounting 7th edition harrison test bank Financial Accounting; Tools for Business Decision Making, 4th Edition, Kimmel.Weygandt.Kieso instructor manual Financial Accounting; Tools for Business Decision Making, 4th Edition, Kimmel.Weygandt.Kieso solution manual Financial Accounting; Tools for Business Decision Making, 4th Edition, Kimmel.Weygandt.Kieso test bank Financial Analysis with Microsoft?Excel 4th Edition Timothy R. Mayes, Todd M. Shank instructor manual Financial Analysis with Microsoft?Excel 4th Edition Timothy R. Mayes, Todd M. Shank spreadsheet problems Financial Analysis with Microsoft?Excel?2007 Timothy R. Mayes, Todd M. Shank Instructor Spreadsheet Files Financial and managerial accounting 14e by williams haka bettner SM Financial management 2e by jae K shim Financial management Principles and application 10e by By Arthur J. Keown, John D. Martin, John W. Petty, David F. Scott solution manual Financial Management Theory & Practice,Eugene Brigham,12th edition [ Test Bank ] financial management theory and practice 10e by Brigham solution manual Financial Management Theory And Practice 11e by Brigham solution manual Financial management theory and practice 12e by Brigham sm financial management theory and practice 12e by Brigham TB financial Management Theory and Practice, 11e By Eugene F. Brigham test bank Financial Markets And Institution (7thEd) Madura TestBank Financial Reporting Analysis 10th edition TB by gibson Finite Element Method: Volume 1 The Basis 5th edition by O. C. Zienkiewicz, R. L. Taylor Finite Mathematics ?nstructor's Resource Guide and Solutions Manual, 8E Margaret L. Lial Raymond N. Greenwell Nathan P. Ritchey Fluid Mechanics Solutions Manual by yunus a. cengel Fluid Mechanics Fundamentals and Applications by Cengel & Cimbala Fluid Mechanics With Engineering Applications -- Solutions Manual by E. John Finnemore, Joseph B Franzini Fluid Mechanics with Student Resources, 5th edition 2002-12 by Frank M. White Fluid Power with Applications, 7E esposito Foundations of Finance The Logic and Practice of Financial Management Art J Keown John D Martin 6th edition im + tb Foundations of Financial Management, 12e By Stanley B. Block Geoffrey A. Hirt Foundations of Financial Markets and Institutions 4e Fabozzi, Modigliani & Jones instructor manual and test gen Fourier and Laplace Transform - Antwoorden Fractal Geometry Mathematical Foundations and Applications Solutions Fracture mechanics fundamentals and applications 2nd edition northam anderson solution manual framework for Marketing Management, A - Philip Kotler (4th ed) (ISBN 0136026605) framework for Marketing Management, A - Philip Kotler (4th ed) (ISBN test bank Friendly Introduction to Analysis - Witold A.J. Kosmala (2nd ed) fundamental accounting principles 17th edition larson solution manual Fundamental Accounting Principles, 18/e John J. Wild Barbara Chiappetta Kermit D. Larson solution manual and test bank fundamentals accounting principles by larson 18e SM Fundamentals of Actuarial Mathematics?by D. Promislow Fundamentals of Advanced Accounting 1e Fisher taylor chang solution manual Fundamentals of Advanced Accounting 1e Fisher taylor chang test bank fundamentals of advanced accounting 3e by hoyle Fundamentals of Advanced Accounting, 2e Joe B. Hoyle Thomas F. Schaefer Timothy S. Doupnik ( Solution Manual ) Fundamentals of Applied Electromagnetics 5th edition by Fawwaz T. Ulaby solution manual Fundamentals of Biochemistry Life at the Molecular Level, 3rd Edition Fundamentals of Chemical Reaction Engineering - Solutions Manual By Mark E. E. Davis, Robert J. J. Davis Fundamentals of Communication Systems by John G. Proakis ,Masoud Salehi computer solutions Fundamentals of Contemporary Financial Management, 2nd Edition R. Charles Moyer James R. McGuigan Ramesh P. Rao instructor manual Fundamentals of Contemporary Financial Management, 2nd Edition R. Charles Moyer James R. McGuigan Ramesh P. Rao test bank fundamentals of corporate finance 8e by Ross Ross, Westerfield,jordan Fundamentals of Corporate Finance, 4th Edition (Brealey, Myers, Marcus) by Bruce Swenson Fundamentals of Database Systems, 5E Ramez Elmasri,Shamkant B. Navathe Fundamentals of Differential Equations and Boundary Value Problems (4th Ed., Kent B. Nagle, Late, Edward B. Saff & Arthur David Snider) Fundamentals of Digital Logic with Verilog Design by s. Brown z vranesic Fundamentals of Digital Logic with VHDL Design-1st edition by S. Brown, Z. Vranesic Fundamentals of Electric Circuits 2nd by Alexander Sadiku Fundamentals of Electric Circuits 3rd edition by Alexander Sadiku Fundamentals of Electromagnetics for Electrical and Computer Engineering Nannapaneni Narayana Rao Fundamentals of Electromagnetics with Engineering Applications by Stuart M. Wentworth Fundamentals of Electronic Circuit Design by David and Donal Comer Fundamentals of Engineering Electromagnetics--Cheng Fundamentals of engineering thermodynamics moran shapiro Fundamentals of engineering thermodynamics moran shapiro 6th edition Fundamentals of Engineering Thermodynamics: Si Version / 5th Edition by Michael J. Moran Howard N. Shapiro Fundamentals of Financial Management 12th edition instructor manual Fundamentals of Financial Management 12th edition test bank Fundamentals of Financial Management 10th Edition by Eugene Brigham Solution manual Fundamentals of Financial Management 10th Edition by Eugene Brigham test bank Fundamentals of Financial Management 11e by Eugene F. Brigham, Joel F. Houston spreadsheet problems Fundamentals of Financial Management 11e by Eugene F. Brigham, Joel F. Houston test bank and cyberproblems Fundamentals of Financial Management 11e by Brigham Instructor manual Fundamentals of financial management 12e by james c. van horne Fundamentals of Financial Management Eugene F. Brigham, Joel F. Houston 12th edition instructor manual Fundamentals of Financial Management Eugene F. Brigham, Joel F. Houston 12th edition spreadsheets problems and web appendix solutions Fundamentals of Financial Management Eugene F. Brigham, Joel F. Houston 12th edition test bank Fundamentals of Financial Management, 12th Edition (Instructors guide ONLY) by James C. Van Horne, John M Wachowic Fundamentals of Financial Management, Concise Edition 5th Eugene F. Brigham, Joel F. Houston test bank Fundamentals of Financial Management, Concise Edition 5th Eugene F. Brigham, Joel F. Houston spreadsheets problems Fundamentals of Financial Management, Concise Edition 5th Eugene F. Brigham, Joel F. Houston instructor manual Fundamentals of Financial Management, Concise Edition Eugene F. Brigham, Joel F. Houston 6th edition instructor manual Fundamentals of Financial Management, Concise Edition Eugene F. Brigham, Joel F. Houston 6th edition spreadsheet problem solutions Fundamentals of Financial Management, Concise Edition Eugene F. Brigham, Joel F. Houston 6th edition test bank Fundamentals Of Fluid Mechanics 3Rd And 4Th Edition Solutions Manual Fundamentals of Fluid Mechanics, 6th Edition Munson, Young, Okiishi, Huebsch Fundamentals of Heat and Mass Transfer, 5th Edition by Frank P. Incropera Fundamentals of Heat and Mass Transfer, 6th Edition Incropera, DeWitt, Bergman, Lavine Fundamentals of Investing, 10th Edition by Gitman and Joehnk Fundamentals of Investments 3e Gordon J. Alexander William F. Sharpe Fundamentals of Logic Design 5th edition by charles roth Fundamentals of Machine Component Design, 3rd ed - Student Solutions Manual by Juvinall, Marshek Fundamentals of manufacturing 2nd edition by philip d. rufe solutions manual Fundamentals of Momentum, Heat and Mass Transfer, 5th Edition Welty, Wicks, Rorrer, Wilson Fundamentals of Multinational Finance, 2nd edition by Michael H. Moffett test bank and solution manual Fundamentals of Multinational Finance, 3E by Michael H. Moffett (Instructor's Manual) Fundamentals of Multinational Finance, 3E by Michael H. Moffett test bank Fundamentals of Organic Chemistry, 5E, Study Guide and Solutions Manual By T. W. Graham Solomons Fundamentals of Organizational Communication 7e Pamela S. Shockley- Zalabak Fundamentals of physics Halliday Resnick 8th student solution not complete Fundamentals of Physics, 7th Edition - Instructor's SOLUTIONS MANUAL halliday and resnick Fundamentals of Physics, Edition 8, Halliday, Resnick, Walker (Solution Manual) Fundamentals of Probability With stochastic processes 3/e (Solutions Manual ) By Saeed Ghahramani Fundamentals of Quantum Mechanics 0521829526 sols Fundamentals of Quantum Mechanics For Solid State Electronics and Optics by C. L. Tang fundamentals of selling 10e by futrell TB Fundamentals of Semiconductor Devices - Anderson Fundamentals of Signals and systems using web and matlab third edition by Edward W. Kamen, Bonnie S Heck Fundamentals of Solid State Electronics by chih tang sah Fundamentals of Thermal-Fluid Sciences Solution Manual 2ed By Yunus A. Cengel, Robert H. Turner Fundamentals of Thermodynamics [Sonntag-Borgnakke-Van Wylen Solutions Manual volume 1 and volume 2 Fundamentals of Thermodynamics SOLUTION MANUAL 6ed By Richard E. Sonntag, Claus Borgnakke, Gordon J. Van Wylen, Fundamentals of Wireless Communication by Tse and Viswanath Fundementals of differential equations 7E & Fundementals of differential equations and boundary value problems 5E Nagle , Saff , Snider even problems Fundementals of engineering economics 2E Chan S. Park Further Mathematics for Economic Analysis General Chemistry, 8th Edition - Solution Manual by Ralph H. Petrucci William S. Harwood; Geoffrey Herring General Chemistry: Principles and Modern Application & Basic Media Pack, 9/E Ralph H Petrucci test bank Geology for Engineers & Environmental Scientists by Alan Kehew 3rd edition Gilat MATLAB An Introduction with Applications, 3rd Edition Government and Nonprofit Accounting Theory & Practice - 8th edition by Freeman, Shoulders, Allison, Patton, Smith sm Government and Nonprofit Accounting Theory & Practice - 8th edition by Freeman, Shoulders, Allison, Patton, Smith tb Government and Nonprofit Accounting Theory & Practice - 9th edition by Freeman, Shoulders, Allison, Patton, Smith instructor manual Government and Nonprofit Accounting Theory & Practice - 9th edition by Freeman, Shoulders, Allison, Patton, Smith tb Government and Nonprofit Accounting Theory & Practice - 9th edition by Freeman, Shoulders, Allison, Patton, Smith sm Government and Not-for-Profit Accounting: Concepts & Practices (4th edition) by Granof and Wardlow sm Government and Not-for-Profit Accounting: Concepts & Practices (4th edition) by Granof and Wardlow tb Gravity An Introduction to Einstein's General Relativity by hartle Gravity An Introduction to Einstein's General Relativity James B. Hartle Haberman Applied Partial Differential Equations 4e Instructor's Manual Harcourt mathematics 12 Advanced Functions and Introductory Calculus - Solutions Manual by Brian / Nelson Harcourt Mathematics 12 Geometry and Discrete Mathematics Solutions Manual By McGraw-Hill Heat and Mass Transfer 3e SM Yunus A. Cengel Heat Transfer: A Practical Approach. Solution Manual ONLY by cengel 2nd edition Heat Transfer-Fundamentals of Heat and Mass Transfer-Incropera & Dewitt Solution Manual Heating, Ventilating and Air Conditioning Analysis and Design, 6th Edition McQuiston, Parker, Spitler High-Speed Digital System Design A Handbook of Interconnect Theory and Design Practicesbby Stephen H. Hall Human Biology by Colleen Belk Virginia Borden-Maier test bank Human Culture Highlights of Cultural Anthropology By Melvin R Ember, Carol R. Ember Human resources management 10e Gary dessler (IM+TB) Hydraulics in Civil and Environmental Engineering By Andrew Chadwick Hydrology and Floodplain Analysis (4th Ed., Philip Bedient, Wayne Huber & Baxter Vieux) Hydrology and Floodplain Analysis, 4e Philip B. Bedient Wayne C. Huber Baxter E. Vieux IBM WebSphere RFID Handbook A Solution Guide by IBM Redbooks In Experiments with Economic Principles, Instructor's Manual By Bergstrom Industrial Safety and Health Management 5e C. Ray Asfahl all resources Instructor Manual to A Course in Modern Mathematical Physics By S. M. SZE 2nd edition Instructor Manual to An Introduction to Thermodynamics and Statistical Mechanics 2ed By Keith Stowe Instructor Manual to Introduction to Solid State Physics Eighth Edition By Charles Kittel Instructor Manual to Introductory Quantum Optics By Christopher Gerry and Peter Knight Instructor Manual to Quantum Physics 3rd ediiton by Gasiorowicz, S. - Instructor Manual to Quantum Physics Third Edition by Stephen Gasiorowicz Instructor Manual to SEMICONDUCTOR DEVICES Physics and Technology Second Edition By S.M.Sze Instructor Manual to Special Relativity by Patricia M. Schwarz and John H. Schwarz Instructor Solutions Manual for Building Java Programs Stuart Reges,Martin Stepp Instructor Solutions Manual to accompany Boyce Elementary Differential Equations and Boundary Value Problem 8ed by Charles W. Haines, William E. Boyce INSTRUCTOR£À SOLUTIONS MANUAL Basic Technical Mathematics with Calculus SI Version John R. Martin Eighth Canadian Edition Instructor's Manual for Solving ODEs with MATLAB By L. F. Shampine, I. Gladwell, S. Thompson Instructor's Manual for Sipser's Introduction to the Theory of Computation by Ching Law Instructor's Manual for Solving ODEs with MATLAB By L. F. Shampine, I. Gladwell, S. Thompson Instructor's Manual Im Experiments with Economic Principles By Bergstrom Instructor's Manual Of Fundamental Methods Of Mathematical Economics Chiang & Wainwright 2005 Mc Graw Hill Instructor's Manual(Information Technology Project Management 3Rd Edition) by Kathy Schwalbe Instructor's Manual: Im Experiments with Economic Principles By Bergstrom INSTRUCTOR'S SOLUTIONS MANUAL (to accompany Elementary Statistics Ninth Edition) by milton loyer Instructor's solution manual ISBN 0534382150 A Transition to Advanced Mathematics solution manual by douglas smith 5th edition Instructor's Solutions for: Design of Analog CMOS Integrated Circuits by razavi Instructors Solutions Manual for Differential Equations with Boundary Value Problems, 2/E by john polking Instructor's Solutions Manual for Serway and Jewett's Physics for Scientists and Engineers Sixth Edition Volume One by Ralph V. McGrew Instructor's Solutions Manual for Serway and Jewett's Physics for Scientists and Engineers Sixth Edition Volume two by Ralph V. McGrew Intermediate Accounting 10e by Nikolai sm Intermediate Accounting 11e by Kieso Intermediate Accounting 12e by Kieso Intermediate accounting 12th Updated by Kieso Solution manual Intermediate accounting 12th Updated by Kieso test bank Intermediate Accounting 2e by Baruch Englard Intermediate Accounting 3e by J. David Spiceland Intermediate Accounting 4e revised by J. David Spiceland solution manual Intermediate accounting by Spiceland 4e Solution manual Intermediate Accounting, Update, 12th Edition international student solution manual Intermediate Algebra, 8th Edition By Margaret L. Lial, John Hornsby, Terry McGinnis International Accounting 1e by Doupnik solution manual International Accounting 6e Frederick D. Choi Gary K. Meek International Business The Challenges of Globalization, 4E John J. Wild,Kenneth L. Wild tb and im International Economics, 7e Husted Melvin test bank and solution manual International Financial Management 9th Edition jeff madura instructor manual International Financial Management 9th Edition jeff madura test bank International Financial Management Geert Bekaert Robert J. Hodrick test bank and sol manual International Management Managing Across Borders and Cultures 6e test bank and instructor manual Introduction Fluid Mechanics, 6Th Edition Solution by fox CHAPTER 1-8 Introduction To Analysis (3rd) Wade Solution Manual Introduction to Chemical Engineering Thermodynamics 7th edition (solution manual) By J.M. Smith, Hendrick C Van Ness Introduction to Corporate Finance William L. Megginson, Scott B. Smart 1st edition instructor manual Introduction to Corporate Finance William L. Megginson, Scott B. Smart 1st edition test bank Introduction to Corporate Finance, 2nd Edition William L. Megginson Scott B. Smart test bank Introduction to Corporate Finance, 2nd Edition William L. Megginson Scott B. Smart solution manual Introduction To Electric Circuits 6th Ed [Solutions Manual] By R. C. Dorf and J. A. Svoboda Introduction to Electrodynamics (Third Edition) by David J. Griffiths Introduction to Engineering Experimentation 2e wheeler and ganji Introduction to Engineering Thermodynamics, Edition 2, Sonntag, Borgnakke (Solution Manual) Introduction to Environmental Engineering and Science, 3E Gilbert M. Masters Wendell P. Ela, Introduction to Financial Accounting, 9E Charles T. Horngren Gary L. Sundem John A. Elliott Donna Philbrick test bank + solution manual Introduction to Fluid Mechanics (Fox, 5th ed) Solutions Manual Introduction to Fluid Mechanics, Edition 7, Fox, Pritchard, McDonald (Solutions Manual) Introduction to Fourier Optics Third Edition Problem Solutions by Joseph W. Goodman Introduction to Government and Non-for-Profit Accounting 6th edition by martin ives sm Introduction to Government and Non-for-Profit Accounting 6th edition test bank by martin ives Introduction To Graph Theory, 2nd ed. Douglas B. West Introduction to Heat Transfer - 013391061X Solution's Manual By peyman pourmoghaddam , Vedat S. Arpaci not complete Introduction to Heat Transfer 4th Edition SOLUTION MANUAL By Frank P. Incropera, David P. DeWitt Introduction to Heat Transfer, 5th Edition Incropera, DeWitt, Bergman, Lavine Introduction to Java Programming, Comprehensive Version, 7/E Y. Daniel Liang Introduction To Linear Algebra 3Ed - Gilbert Strang Solutions Manual Introduction To Management Accounting horngren 14e TB Introduction to management sciences 9e by Taylor solution manaul Introduction to Managerial Accounting 2nd ed Brewer test bank Introduction to Mathematical Statistics 6th Robert V. Hogg, allen t. Craig Introduction to Operations and Supply Chain Management 2e Cecil Bozarth Introduction to Operations and Supply Chain Management 2e Cecil Bozarth Robert B. Handfield test bank and sol manual Introduction to Ordinary Differential Equations by James C. Robinson Introduction to Probability by Dimitri P. Bertsekas - solution isbn: 1-886529-40-X Introduction to Quantum Mechanics (Second Edition) - Solutions Manual By David J. Griffiths Introduction to Statistical Quality Control, 6th Edition Montgomery Introduction to VLSI Circuits and Systems (2001 draft) - John P Uyemura - Solutions Manual Introduction To Wireless Systems - P M Shankar - Solutions Manual Introductory Circuit Analysis, 11E Robert L. Boylestad Introductory Econometrics A Modern Approach 2ed Jeffrey Wooldridge Introductory Econometrics A Modern Approach, 3Ed (with Economic Applications Online, Econometrics Data Sets with Solutions Manual Web Site Printed Access Card) by Jeffrey Wooldridge Introductory Linear Algebra An Applied First Course, 8e by Kolman Hill Introductory Mathematical Analysis for Business, Economics and the Life and Social Sciences, 12/E Ernest F. Haeussler Richard S. Paul Richard J. Wood test bank and sol manual Introductory Quantum Optics by Knight and Gerry Investment analysis and portfolio management 8e by Reily Brown Investment Analysis and Portfolio Management- Solutions Manual, 7th Edition by Frank K. et al. Reily Investment analysisnvestment analysis and management 9e by charles p. jones & management, 9e By jones some chapters missings sm IR: The New World of International Relations, 7e Michael G. Roskin, Lycoming College Nicholas O. Berry instructor manual and test bank instructor s manual with powerpoints to accompany pic microcontroller and embedded systems by muhammed ali mazidi rolin d mckinlay instructor s resource manual to accompany electronic devices,6th edition and electronic devices electron flow version 4th edition thomas floyd intermediate Accounting 12 E Kieso (TB) intermediate accounting 5e spiceland test bank and solution manual intermediate Accounting 12e by Keiso sm intermediate accounting 4th edition spiceland test bank introduction to algorithms 2nd edition instructors manual McGraw-Hill by thomas h. Cormen introduction to Environmental Engineering and Science (2nd Edition) (Hardcover) by Gilbert M. Masters introduction to linear algebra (5th, johnson) introduction to management accounting 14e Charles T. Horngren, Gary L. Sundem, William O. Stratton, Jeff Schatzberg, Dave Burgstahler solution manual introduction to mechatronics and measurement systems 2nd edition David G. Alciatore and Michael B. Histand introduction to Operations Research - 7th by Frederick Hillier, Gerald Lieberman introduction to probability Charles M. Grinstead and J. Laurie Snell odd solutions investment Analysis & Portfolio Management, 7th edition by Reilly and Brown Java, an Introduction to problem solving and programming , fifth Ed. by W. Savith and F. Carrano john E. Freund's Mathematical Statistics with Applications, 7th edition, by Miller and Miller Journey into Mathematics: An Introduction to Proofs (with solution manual) by Joseph J. Rotman Kc's Problems and Solutions for Microelectronic Circuits by k.c smith 4th edition Kinematics, Dynamics, and Design of Machinery by K. J. Waldron (Author), G. L. Kinzel (Author) Kinetics of Catalytic Reactions--Solutions Manual by: M. Albert Vannice Labor Relations, 12E Arthur A Sloane instructor manual laboratory manual to accompany Introductory Circuit Analysis 11e boylestad Lagrangian and Hamiltonian Mechanics Solutions to the Exercises by M. G. Calkin laser fundamentals 2nd edition by william t. Silfvast lectures on corporate finance 2e by Peter Bossaerts Lectures on Corporate Finance, Second Edition by Peter Bossaerts and Bernt Arne ??degaard legal environmental A critical thinking approach by KUBASKE, BRENNAN and BROWNE 5th edition test bank ?m legal environmental: A critical thinking approach by KUBASKE, BRENNAN and BROWNE 5th edition test bank Linear algebra and it s applications 3rd edition by david c. Lay linear algebra Juan de Burgos Solutions Manual spanish Linear algebra with applications 3e Otto Bretcher - Solutions Manual Linear Algebra with Applications 6 edition by Leon Linear Algebra with Applications, 7E by Steven J. Leon Linear Algebra, 4E Stephen H. Friedberg Arnold J. Insel Lawrence E. Spence Linear circuit analysis by R. A. DeCarlo and P. Lin - solution manuel 2nd edition linear systems and signals by lathi solutions manual blue covered Living Religions, 7E Mary Pat Fisher test bank Logic And Computer Design Fundamentals (4thEd) - Mano - SolutionsManual Lu, Likos Unsaturated Soil Mechanics Machine Design, An Integrated Approach, 3rd edition, by Robert L. Norton Macroeconomics (8thEd) - Froyen - Solutions Manual Macroeconomics 6e Andrew B. Abel, Ben S. Bernanke, Dean Croushore Macroeconomics, 4E Olivier Blanchard instructor manual Macroeconomics, 4E Olivier Blanchard test bank Macroeconomics, 5E Olivier Blanchard instructor manual and test bank Management 5th Edition Chuck Williams instructor manual Management 9E Stephen P. Robbins Mary Coulter test bank and instructor manual management accounting 5e Anthony A. Atkinson Robert S. Kaplan Ella Mae Matsumura S. Mark Young instructor manual test bank and solution manual Management accounting 5E atkinson solution manual Management accounting 5E atkinson test bank Management Control Systems Performance Measurement, Evaluation and Incentives 2e Merchant & Van der Stede Management Information Systems; Managing the Digital Firm, Edition 10, Laudon test bank and solution manual Management Robbins Coulter 9th edition (test bank ) Management10E Stephen P. Robbins Mary Coulter Management10E Stephen P. Robbins Mary Coulter test bank Managerial Accounting 12e By Garrison Noreen ( Solution Manual ) Managerial Accounting 12e By Garrison Noreen (Test Bank) Managerial Accounting Bamber, L. S., K. W. Braun, and W. T. Harrison, Jr. 2008 Managerial Accounting international edition Garrison11 ( Solution Manual) Managerial Accounting, 11th Edition by Ray H Garrison, Eric Noreen, Peter C. Brewer Managerial Accounting: An Introduction to Concepts, Methods and Uses 10th Edition Michael W. Maher, Clyde P. Stickney, Roman L. Weil instructor manual Managerial Accounting: An Introduction to Concepts, Methods and Uses 10th Edition Michael W. Maher, Clyde P. Stickney, Roman L. Weil solution manual Managerial Accounting: An Introduction to Concepts, Methods and Uses 10th Edition Michael W. Maher, Clyde P. Stickney, Roman L. Weil test bank Managerial Economics Applications, Strategies, and Tactics 11th Edition James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris instructor manual Managerial Economics Applications, Strategies, and Tactics 11th Edition James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris test bank Managerial Finance, Gitman,Lawrence 12e [SM] Managing Human Resources by Luis R. Gomez-Mejia; David B. Balkin; Robert L. Cardy. 5th Edition im and tb Manufacturing Engineering and Technology 5E by Kalpakjian and Schimid Marketing Kotler Armstrong 11th edition (Test bank ) Marketing Management (13th Edition) by Philip Kotler, Kevin Keller im Marketing Management (13th Edition) by Philip Kotler, Kevin Keller tb Materials and Processes in Manufacturing, 9 edition, [Solution manual] by E. Paul DeGarmo, Solutions Manual by: Barney E. Klamecki Materials Science and Engineering An Introduction 6E By William D. Callister materials science and engineering,callister 7th edition Mathematical Methods for Physics and Engineering A Comprehensive Guide 3rd edition by riley Mathematical Methods in the Physical Sciences, 3rd Edition Mary L. Boas Mathematical Models in Biology An Introduction by Elizabeth S. Allman john a. Rhodes Mathematical Models in Biology Sols by Elizabeth S. Allman, John A. Rhodes Mathematics for Economists Solution Manual - Simon and Blume (ver 2) Mathematics with Applications (9th Edition) Margaret L. Lial McgrawHill - William H. Hayt, John A. Buck - Engineering Electromagnetics, 6ed Solutions Manual Mechanical Behavior of Materials, 3E Norman E Dowling Mechanical Measurements 6th Edition by Thomas G. Beckwith, Roy D. Marangoni, John H. Lienhard V Mechanical Vibrations - Singiresu Rao - Solutions Manual 3rd edition chapters missing 6 9 12 ( all links same ) 35 mb Mechanics of Fluids by Merle C. Potter Mechanics of Fluids Solutions Manual 8ed By John Ward-Smith Mechanics of Materials [solutions manual] hibbeler 7th edition Mechanics of Materials An Integrated Learning System Philpot Mechanics of Materials Sixth Edition by R.C.Hibbeler Mechanics Of Materials Solution Manual (3Rd Ed , By Beer) nearly same 4th edition just numbered different Mechanics of Materials, 2nd Edition Craig Mechanics of Materials, 5th Edition by James M. Gere Mechanics of Materials, 6th Edition by James M. Gere (missing small portion, section 8.5) Mechanics of Materials, 7th Edition James M. Gere - Stanford University (Professor Emeritus) mechanics of solids by carl t. t. ross, d.sc. Microeconomic Analysis Solution Manual - Varian 3rd edition Microeconomic Analysis Third Edition by Hal R. Varian Microeconomic Theory Solutions Manual for Mas-Colell Microeconomics 7e Robert Pindyck Daniel Rubinfeld Microeconomics Theory and Applications with Calculus by perloff insrructor manual Microeconomics Theory and Applications with Calculus by perloff test bank Microelectronic Circuit Design 2nd and 3rd edition by R. Jaeger selected solutions answers not solution microelectronic circuits 5th ed. By Adel S. Sedra, Kenneth C. Smith microelectronics 3rd edition by jeager Microelectronics Circuit Analysis Desing 3rd Edition by Donald A. Neamen Microelectronics I & II 1st edition by by Dr. Wen-Ching Chang Microelectronics-Digital and Analog Circuits and Systems by Millman Microprocessors and Interfacing,Revised Second Edition Douglas V Hall Microwave And Rf Design Of Wireless Systems - Solution Manual (D M Pozar) Microwave Engineering 3e - David M Pozar Microwave Transistor Amplifiers Analysis and Design, 2nd Edition (Solutions Manual) by Guillermo Gonzalez Miller and Freund's Probability and Statistics for Engineers 7th edition by Richard A. Johnson Modern Advanced Accounting , 10th Edition, by Larsen, (publisher McGraw Hill) Modern Advanced Accounting 10th edition Test Bank LARSEN Modern Auditing Assurance Services 8e Boynton sm and tb modern control engineering by katsuhiko ogata 3rd edition ISBN: 0132273071 modern control engineering by katsuhiko ogata 4th edition Modern Control System 11th edition by richard c dorf, robert H bishop Modern Control System 9th by richard c dorf, robert H bishop Modern Database management 9e by Jeffrey A. Hoffer im Modern Database management 9e by Jeffrey A. Hoffer TB Modern Database Management, Ninth Edition Jeffrey A. Hoffer, University of Dayton Mary Prescott Heikki Topi Modern Digital and Analog Communications Systems - B P Lathi Solutions Manual 3rd edition Modern Digital Electronics,3E R P JAIN modern electronic communication 8th edition by gary miller and beasley Modern Electronic Communication, Beasly and Miller, 9th edition instructor manual Modern Electronic Communication, Beasly and Miller, 9th edition test bank Modern elementary statistics 12 ed freund e john test bank Modern Elementary Statistics 12e John E. Freund Benjamin M. Perles solution manual Modern Elementary Statistics 12e John E. Freund Benjamin M. Perles test bank modern Operating Systems 3rd edition - Andrew S. Tanenbaum Modern Organic Synthesis: An Introduction By Michael H. Nantz, Hasan Palandoken, George S. Zweifel Modern Physics, 2/E Randy Harris Modern Quantum Mechanics - J. J. Sakurai - Solution Modern Systems Analysis and Design, 5E Jeffrey A. Hoffer Joey F George Joseph S Valacich test bank Money, Banking, & Financial Markets 2nd edition Roger LeRoy Miller David D. VanHouse tb Multinational Business Finance 11E by David K. Eiteman SM Multinational Business Finance 11E by David K. Eiteman tb Multivariable Calculus: Student Solutions Manual 4th edition by James Stewart (Author) Nanoengineering of Structural Functional and Smart Materials Nanoengineering of Structural, Functional and Smart Materials Network flows theory, algorithms, and applications by Ravindra K. Ahuja (Author), Thomas L. Magnanti (Author), James B. Orlin (Author) odd solutions Network Security Essentials: Applications and Standards, 3/e stallings Nonlinear programming 2nd edition solution manual by dimitri bertsekas Numerical Methods For Engineers Solutions Manual by chapra 4th edition Numerical Methods for Engineers,5E,Steven C Chapra Numerical Solution of Partial Differential Equations: An Introduction 2ed By K. W. Morton, D. F. Mayers Operating System Concepts (6Th Ed)-Instructor'S Manual (A Silberschatz) Operating System Concepts, 7th Edtion, Instructor's Manual by A Silberschatz Operating systems Internals and Design principles 4th by william stallings Operating Systems Internals and Design principles( 5th Ed,William Stallings) Operating systems internals and design principles William STALLINGS 5.edition test bank Operating Systems: Internals and Design Principles, 6/e stallings solution manual and test bank Operations Management 9e heizer test bank Operations Management 5th edition Slack Nigel Instructors Manual Operations Management 8e by Jay Heizer Barry Render sm Operations Management 8e by Jay Heizer Barry Render tb Operations Management 9th by Jay Heizer Barry Render sm Operations Management 9th by Jay Heizer Barry Render tb Operations Management for MBAs, 3rd Edition Meredith, Shafer tb+im Operations Management Reid, Sanders An Integrated Approach 2nd edition Operations Management Reid, Sanders An Integrated Approach 2nd edition test bank Operations Management, 10e William J. Stevenson instructor manual with solutions to supplementary problems Operations Management, 10e William J. Stevenson test bank Operations Management, 9e William J. Stevenson instructor manual with solutions to supplementary problems Operations Management, William J Stevenson, 9e [TB] operations managenment processes and value chains 8e by Lee J. Krajewski solution manual operations managenment processes and value chains 8e by Lee J. Krajewski test bank Operations Research An Introduction, 8E Hamdy A. Taha Optimal Control Theory An Introduction By Donald E. Kirk Options, Futures and Other Derivatives, 4th Edition (Solutions Manual) by John Hull, John C. Hull Options, Futures and Other Derivatives, Fifth Edition (Solutions Manual) by John Hull, John C. Hull Options, Futures, and Other Derivatives 7E by JOHN C HULL TB Organic Chemistry by Robert C. Atkins, Francis A Carey, Robert Atkins, Francis Carey Organic Chemistry 2nd ed [Student SOLUTIONS MANUAL and Study Gde] - J. Hornback, B. Murugaverl (Thomson, 2006) WW Organic Chemistry and CW+ GradeTracker Access Card Package, 6/E Leroy G. Wade, JR., Whitman College test bank Organic Chemistry, 5/E Paula Y. Bruice test bank Organizational Behavior Stephen P Robbins 12th edition (test bank) Organizational Behavior, by Stephen P. and Timothy A. 13th Edition Robbins Judge im and tb Organizational Behavior: Managing People and Organizations, 9th Edition Ricky W. Griffin Gregory Moorhead instructor manual Organizational Behavior: Managing People and Organizations, 9th Edition Ricky W. Griffin Gregory Moorhead test bank organizational behaviour 12e by Robbins ( instructor manual ) organizational behaviour 12e by Robbins ( Test bank ) Organizational Theory, Design and Change, 5/E Gareth R. Jones Organizational Theory, Design and Change, 5E Gareth R. Jones Instructor's Manual and test bank Organizational Theory, Design and Change, 5E Gareth R. Jones im and tb Papathomas T.V. - Solutions Manual to Accompany Millman Integrated Electronics Analog and Digital Circuits and Systems (McGraw-Hill) Partial Differential Equations and Boundary Value Problems with Fourier Series (2nd Edition) (Student Solutions Manual) by asmar Pattern Recognition and Machine Learning (Solution Manual) - Bishop Personal Finance Turning Money into Wealth 5e Arthur J. Keown Personal Finance Turning Money into Wealth and Student Workbook, 4E keown instructor manual and test bank Physical Chemistry (Instructor's Solutions Manual) Peter Atkins & Julio de Paula 7ed Physics by Resnick Halliday Krane, 5th Ed. Vol 2 physics concepts and connections combined edition solutions manual by igor nowikow brian heimbecker christopher t. Howes Physics for Scientist & Engineers with Modern Physics - A strategic Approach by Randall D. Knight chapter 1-42 Physics for Scientist & Engineers with Modern Physics - A strategic Approach chapter by Randall D. Knight 1-35 Physics for Scientists and Engineers 5th edition by Serway And Jewett Physics for Scientists and Engineers Extended Version, 5th edition, Physics for Scientists and Engineers with Modern Physics (3rd Edition) Physics for Scientists and Engineers with Modern Physics (3rd Edition) by giancoli Physics For Scientists Engineers With Modern Physics (4thEd) - Giancoli - Solutions Manual Physics of the Solar Corona: An Introduction with Problems and Solutions by Markus Aschwanden Physics Principles with Applications Instructor's Solutions Manual (Giancoli, Volume 1and 2) 6th edition power analysis and design by glover , sarma 3rd ediiton Power Electronics, Converters, Applications, and Design By Ned Mohan, Tore M. Undeland, William P. Robbins Power System Analysis and Design Glover and Sarma 4e Thomson Learning Glover J. Duncan, Sarma Mulkutla .S. Power System Analysis Hadi Saadat 2nd Edition Power System Analysis Solution Manual John Grainger, William D. Stevenson Practical Financial Management 5th Edition William R. Lasher instructor manual Practical Financial Management 5th Edition William R. Lasher test bank Practical Financial Management William R. Lasher 4th edition instructor manual Practical Financial Management William R. Lasher 4th edition test bank Prentice Hall - Solutions Manual; Communication Systems Engineering Prentice Hall's Federal Taxation 2008 Individual , 21th Edition By Thomas R. Pope, Kenneth instructor guide Prentice Hall's Federal Taxation 2009 Comprehensive, 22nd Edition By Thomas R. Pope, Kenneth E. Anderson, John L. Kramer solution manual Prentice Hall's Federal Taxation 2009 Comprehensive, 22nd Edition By Thomas R. Pope, Kenneth E. Anderson, John L. Kramer test bank Prentice Hall's Federal Taxation 2009 Individual , 22th Edition By Thomas R. Pope, Kenneth instructor guide Prentice Hall's Federal Taxation 2009: Corporations, 22/E Kenneth E. Anderson Thomas R. Pope John L. Kramer instructor guide Prentice Hall's Federal Taxation 2009: Corporations, 22/E Kenneth E. Anderson Thomas R. Pope John L. Kramer tb +sm Prentice Hall's Federal Taxation 2009: Individuals, 22/E Kenneth E. Anderson Thomas R. Pope John L. Kramer tb +sm Principles and Applications of Electrical Engineering giorigo rizzoni 4th edition Principles and Applications of Electrical Engineering giorigo rizzoni 5th edition Principles Geotechnical Engineering Braja Das 6th ed solution manual Principles of Auditing 15e by Whittington TB Principles of Communication: Systems, Modulation and Noise (5th Ed) by R. E. Ziemer, W. H. Tranter, solution manual Principles of Communications, 6th Edition Ziemer, Tranter Principles of corporate finance 7e Principles of corporate finance 7e by brealy mayers Principles Of Corporate Finance 8E By Brealey Myers Allen Principles of corporate finance 9e by brealy mayers allen (SM+TB) Principles of Electric Circuits Conventional Current Version 8e by floyd Principles of Electric Circuits Conventional Current Version by Floyd 8th edition principles of electric circuits- electron flow version Floyd 8th edition Principles of Electronic Materials and Devices, Solutions Manual ONLY Safa O. Kasap 2nd edition principles of managerial finance 10e by gitman Lawrence principles of managerial finance 11e by gitman Lawrence Principles of managerial finance 11e by gitman Lawrence solution manual Principles of Managerial Finance 11e by Gitman Lawrence test bank Principles of managerial finance 12e by gitman Lawrence test bank Principles of Managerial Finance Brief plus 5e sm Principles of Managerial Finance Brief plus 5e tb Principles of Managerial Finance Brief plus My Finance Lab Student Access Kit, 5E Lawrence J. Gitman principles of marketing 11e by Kotler ( instructor manual ) principles of marketing 11e by Kotler ( test bank ) principles of marketing 12e by Kotler TB Principles of Microeconomics, 9/e Case, Fair & Oster instructor manual Principles of Microeconomics, 9/e Case, Fair & Oster test bank Principles of Microeconomics, 9e Case, Fair & Oster instructor manual ad test bank Principles of Neurocomputing for Science and Engineering, 1st Edition Probability & Statistics for Engineers & Scientists, 8th Edition: Instructors Solution Manual ONLY by Sharon Myers , Keying Ye, Walpole Probability and Statistical Inference 7th edition Hogg Tanis solution Probability and Statistics for Engineering and the Sciences [Solutions Manual] 6th edition by Jay L. Devore Probability and Statistics for Engineers and Scientists Manual HAYLER Solutions Manual Probability, Random Variables, and Stochastic Processes [Only Solutions Manual] by Athanasios Papoulis ,S.Unnikrishna Pillai 4th edition Probability, Statistics, and Random Processes For Electrical Engineering - Alberto Leon-Garcia (3rd ed) (ISBN 0131471228) Probability, Statistics, and Random Processes for Engineers, 1st Edition Richard H. Williams - University of New Mexico Problem Solving and Programming Concepts, 8/E Maureen Sprankle Jim Hubbard Problem Solving with C++ The Object of Programming, 5E Walter Savitch tb and im Problems and Solutions on Electromagnetism by zhao shu ping - you jun han Process Dynamics and Control, 2nd Edition Seborg, Edgar, Mellichamp PROCESS SYSTEMS ANALYSIS AND CONTROL - DONALD R. COUGHANOWR Solution Manual Process Systems Analysis and Control by Donald R Coughanowr Programming the World Wide Web, 4E Robert W. Sebesta Quantitative Analysis for management TENTH EDITION by Barry Render, Ralph M. Stair, Jr, Michael E.Hanna solution manual Quantitative Analysis for management TENTH EDITION by Barry Render, Ralph M. Stair, Jr, Michael E.Hanna test bank Quantum Field Theory (draft version) & Instructor's Manual by Mark Srednicki book + solution manual Radiation Protection in the Health Sciences: With Problem Solutions Manual by: Marilyn E. Noz, Gerald Q. Maguire Sargen Recursive Methods in Economic Dynamics By Claudio Irigoyen ,Esteban Reinforced Concrete: Mechanics and Design by James MacGregor and James Wight, either 5th edition Research in Education 10th edition by Best, J., & Kahn, J instructor manual and test bank Retail Management A Strategic Approach, 10E Barry Berman, Joel R. Evans tb and sm RF circuit Design Theory and Application by Ludwig bretchko solution manuel Roy D. Yates and David J. Goodman, Probability and Stochastic Processes - A Friendly Introduction for Electrical and Computer Engineers, 2nd edition, Satellite Communications By Timothy Pratt, Charles W. Bostian Schaums Mathematical Handbook of Formulas and Tables Schaum's Outline of Discrete Mathematics, 2nd edition, 1997 by Seymor Lipschutz, Marc Lipson Schaum's Outline of Discrete Mathematics, 3rd Ed. (Schaum's Outlines) by Seymour Lipschutz, Marc Lipson Schaum's Outline of Probability, 2nd Edition by Seymour Lipschutz Schaum's Outline of Theory and Problems of Linear Algebra, 2nd Edition (Schaum's Outlines) by Seymour Lipschutz Schaum's Outline of Theory and Problems of Managerial Accounting. [1998.ISBN0070580413] book + sm Selected Answers-Basic Engineering Circuit Analysis-7th Ed. by J. David Irwin Semiconductor Device Fundamentals 1st edition by Robert F Semiconductor Physics and Devices: Basic Principles 3rd edition by neamen separation Process Principles, 2nd Ed by Seader, Henley Shigley's Mechanical Engineering Design 7th Ed - Solution Manual By by Richard Budynas, J.Charles Mischke Shigley's Mechanical Engineering Design 8th Ed - Solution Manual By by Richard Budynas, J. Keith Nisbett Signal , System and transforms fourth edition BY charles L. philips , John M Parr, Eve A. Riskin Signal , System and transforms fourth edition BY charles L. philips , John M Parr, Eve A. Riskin international edition Signal Processing and Linear Systems by lathi Signal Processing First-Mclellan, Schafer & Yoder Solution Manual chapter 3-12 Signals and Systems - Analysis using Transform methods and MATLAB M. Signals and Systems 2nd edition simon Haykin Solutions Manual Signals and Systems, Prentice-Hall Oppenheim, Willsky, Young 2nd edition Silicon VLSI Technology: Fundamentals, Practice, and Modeling James D. Plummer Michael D. Deal Peter B. Griffin sociology John J. Macionis 12th edition test bank and instructor manual Soil Mechanics Concepts and Applications By William Powrie Soil Mechanics Solutions Manual (2nd Edition) By William Powrie soils and foundations 7th by cheng liu and jack b evett Solid State Electronic Device by Ben Streetman Solid State Electronic Devices, 6E Ben Streetman Sanjay Banerjee Solid State Physics - Solutions Manual by Ashcroft & Mermin some chapters missings SOLUTIONS MANUAL AND WORKBOOK to accompany Quantitative Methods for Management Solution Automatic Control Systems 8Ed by Kuo and Golnaraghi Solution to Skill - Assessment Exercises to Accompany Control Systems Engineering 3rd edt. by Norman S. Nise Solution To Two-Dimensional Incompressible Navier-Stokes Equations- Maciej Matyka Solutions & Supplements Introductory Chemical Engineering Thermodynamics by J.R. Elliott & C. T. Lira selected solutions Solutions Irodov's Prob. Gen. Physics Volume 1 by Abhay K. Singh Solutions Irodov's Prob. Gen. Physics Volume 2 by Abhay K. Singh South-Western Federal Taxation 2009 (Individual), Edition 32, Hoffman, Smith, Wills (Test Bank) South-Western Federal Taxation 2009 Corporations - William Hoffman (Solutions Manual) (32nd ed) (ISBN 0324660219) South-Western Federal Taxation 2009 Corporations - William Hoffman (test bank) (32nd ed) (ISBN 0324660219) South-Western Federal Taxation 2009: Comprehensive - William Hoffman (Solutions Manual) (32nd ed) (ISBN 0324660529) South-Western Federal Taxation 2009: Comprehensive - William Hoffman (Test Bank) (32nd ed) (ISBN 0324660529 South-Western Federal Taxation Individual Income Taxes, 32nd by William Hoffman, Jr. James Smith Eugene WillisEdition sm 2009 STATISTICS FOR THE SCIENCES by Martin Buntinas & Gerald M. Funk, Statics & Mechanics of Materials SI 2nd edition Russell Hibbeler Jul Statics and Mechanics of Materials Anthony M. Bedford Kenneth M. Liechti Wallace Fowler Statistical Digital Signal Processing and Modeling - SOLUTIONS MANUAL By Monson H. Hayes Statistical Inference - Casella & Berger 2nd edition statistical Quality Design and Control, 2/E Richard E. DeVor, Tsong- how Chang, John W. Sutherland Statistics 4e by Murray R Spiegel, Larry J. Stephens book + sm Statistics for Business and Economics and Student CD (6th Edition) (Hardcover) by Paul Newbold (Author), William L. Carlson (Author), Betty Thorne (Author Statistics for Engineering and the Sciences, 5e by William Mendenhall and Terry Sincich Statistics for Engineers and Scientists by William Navidi Steel Design, 4th Edition solution Manual William T. Segui Steel Structures Design and Behavior, 5th Edition by Charles G. Salmon Strategic Brand Management 3e keller Strategic management 11e by Fred R. David ( instructor manual ) Strategic management 11e by Fred R. David ( test bank ) Strategic Management and Business Policy, 11e Thomas L. Wheelen David L. Hunger instructor manual with test items Strategic Management and Competitive Advantage: Concepts and Cases, 2/ E instructor manual with test items Jay Barney William S Hesterly Strength of Materials 4th Ed. by Ferdinand L. Singer & Andrew Pytel Strength of Materials 4th Ed. by Singer & Pytel Structural Analysis by Hibbeler 5th edition Structural Analysis by Hibbeler 7th edition Structured computer organization 5E Andrew S. Tanenbaum Student Solution Manual for Mathematical Methods for Physics and Engineering Third Edition by K. F. Riley, M. P. Hobson not complete Student Solutions Manual and Study Guide to accompany Fundamentals of Fluid Mechanics, 5th Edition by Bruce R. Munson Donald F. Young, Theodore H. Okiishi Study Guide with Solutions Manual for McMurry's Organic Chemistry, 7th Ed. by John E. McMurry Fawcett Lisa M. Ellram Jeffrey A. Ogden Survey of Accounting, Fourth Edition Carl S. Warren System Dynamics 3rd Ed By Katsuhiko Ogata System Dynamics and Response, 1st Edition S. Graham Kelly Techniques of Problem Solving by luiz fernandez TestGen for Physics for Scientists and Engineers - A Strategic Approach 2E knight The Analysis and Design of Linear Circuits Laplace Early, 4th Edition Thomas, Rosa The Art of Electronics. Thomas C. Hayes, Paul Horowitz The Economic Way of Thinking 11e Peter J. Boettke Peter J. Boettke David L. Prychitko test bank and solution manual The Economics of Financial Markets By Roy E. Bailey The Economics of Financial Markets by Roy E. Bailey The Economics of Money,Banking,and Financial Market 8th edition by Frederic S. Mishkin instructor manual and test bank The Legal Environment of Business, 5th edition Kubasek The Science and Engineering of Materials by Donald R. Askeland Frank Haddleton 4th edition The Spacetime Frontier Science and Society in the 21st Century by Stewart Swain Theory & Design for Mechanical Measurements 4th edition by Richard S. Figliola & Donald E. Beasley Thermodynamics an engineering approach sixth edition ( SI units ) : solutions manual by Yunus A. Cengel, Michael A. Boles Thermodynamics Of TurboMachinery 5th edition Thermodynamics: An Engineering Approach by: Yunus A. Cengel 5th edition thomas calculus 10th edition instructor solution manual volume 1and 2 Thomas Calculus 11th edition Thomas Calculus updated 10E by finney, weir giordano Thomas, Rosa The Analysis and Design of Linear Circuits Laplace Early, 4th Edition Transport Phenomena - 2nd edition by Bird, Stewart and Lightfoot Solution Manual Ulaby Applied Electromagnetics Undergraduate Econometrics Solutions Manual - Hill, Judge and Griffiths Understanding Corporate Annual Reports, 6e William R. Pasewark, Texas Tech University Understanding Financial Statements 8e Lyn M. Fraser Aileen Ormiston test bank and sol manual Understanding Financial Statements 8e Lyn M. Fraser sm and tb Unit Operations of Chemical Engineering, 6th Edition, Solutions Manual by Warren McCabe, Julian Smith, Peter Harriott Unit Operations of Chemical Engineering, 7th Edition, Solutions Manual by: Warren McCabe, Julian Smith, Peter Harriott University Calculus, Part Two Multivariable, Chap 9-14 Joel Hass Maurice D. Weir sm University Calculus, Part Two Multivariable, Chap 9-14 Joel Hass Maurice D. Weir test generator University physics 11th edition solution manual by Young and Freedman University Physics with Modern Physics (12th Edition) by Hugh D. Young, Roger A. Freedman Unsaturated Soil Mechanics by Ning Lu and William J. Likos Using Econometrics A Practical Guide, 5th edition by Studenmund Vector Calculus (3rd Ed., Susan J. Colley) R. Johnston Jr., E. R. Eisenberg, & G. H. Staab Vector Mechanics for Engineers STATICS, 7th Edition by F. P. Beer, E. R. Johnston Jr., E. R. Eisenberg, & G. H. Staab Vector Mechanics for Engineers; Dynamics 8th edition Beer Johnston Vector Mechanics for Engineers; Statics 8th edition Beer Johnston solution manual Wankat & Oreovicz - Teaching Engineering water and wastewater technology mark j. hammer mark J. Hammer 6th edition Water Supply and Pollution Control, 8E Warren Viessman, Jr. Mark J. Hammer Elizabeth M. Perez Paul A. Chadik WHO-DUN-IT Fifth Edition Shari L. DeMarco wiley applied corporate finance 2nd ed Wiley Chemical And Engineering Thermodynamics 3Ed Solutions Manual Wireless Communication and Networks second edition William Stallings solutions manual Wireless Communications: Principles and Practice, 2nd edition theodore rappaport solutions manual === Subject: Solutions manuals for $36 each. posting-account=WMGg9woAAAB3bYTClCVTyzuu-K8I_Byc Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) These solutions manuals/ test bank are for $36 each. To buy email me at esol...@gmail.com Introductory Circuit Analysis 11e Boylestad Solution manual Mathematical Thinking problem solving Angelo & West Instructors manual ISBN -0130144126 Elementary Differential Equations Boundary Value(2ndEd) - Kohler [CapitalEth] Solutions Manual Introduction to Econometrics 2E stock watson solution manual Human Anatomy & Physiology 7E TEST BANK ISBN 0805373810 Principles Of Managerial Finance Brief (5thEd) - Gitman [CapitalEth] Solutions Manual Elementary Linear Algebra(6thEd) - Larson, Falvo [CapitalEth] Solutions Manual Cornerstone Of Managerial Accounting(2ndEd) - Mowen [CapitalEth] Solutions Manual Electronic Devices and Circuit Theory 9e Instructors resource manual ISBN 0132214466 test bank Strategic Management Concepts and Cases 12E Fred David Prince Medical Imaging Signals and Systems Instructors manual Strategic Management Test Bank Hitt 8th Auditing Cases Interactive Learning Approach(4thEd) - Beasley [CapitalEth] Solutions Manual Test Bank Macroeconomics Principles and Policy Baumol 10th Prebles' Artforms TestBank Data Structures Algorithm Analysis In CPP(3rdEd) - Weiss [CapitalEth] Solutions Manual Management Information Systems Managing the Digital Firm 10th Edition by Laudon Understanding and Managing Organizational Behavior 5E Test bank Discrete Mathematics with Applications by Susanna S. Epp 3rd Edition - Test Bank, Exam questions and Review questions with answers Linear Algebra: A Modern Introduction by David Poole 2nd Edition Instructors Guide Auditing and Assurance Services An Intergrated Approach and ACL Software12E -ISBN 0136128300 Solution Manual & Test bank Organic Chemistry 6E Wade Test Bank Strategic Management and Competitive Advantage Concepts and Cases 2E barney hesterly ISBN 0136036112 Pearson TESTGEN file Biology with Mastering Biology 8E Campbell Reece ISBN -0321494334 Test Bank Fundamentals of Differential Equations 7thE Nagle Snider Instructors resource manual ISBN -0321388445 Brock Biology of Microorganisms 12E ISBN -0132324997 Test Bank Corporate Partnership Estate Gift Tax 2009 - Pratt [CapitalEth] Solutions Manual Introduction to the Design and Analysis of Algorithms 2E Levitin ISBN 0321428102 Prebles' Artforms 9E patrick Frank TESTGEN file ISBN 0136044166 International Economics Theory and Policy 8E Krugman Obstfeld Management Information Systems 11E Laudon 0136078907 test bank Friendly Introduction To Analysis (2ndEd) - Kosmala [CapitalEth] Solutions Manual Fundamentals of Communication Systems Java Foundations- Introduction to Program Design and Data Structures- ES zip Object Oriented Programming in C++ 4E suplement robert Lafore Operations Management 9E Jay Heizer ISBN 0132342979 test bank Operations Management 9E Jay Heizer ISBN 0131585576 Statistics 11E James T. McClave Solution manual Introduction to Econometrics 2E Stock Watson Solution manual Physics Principles with Applications with Mastering Physics Giancoli 6E ISM Solutions Manual Business Statistics First Course Levine Strategic Management Concepts and Cases 12E Fred David ISBN 0138132178 test bank Business Statistics First Course Levine test bank Physics with Mastering Physics 3E James walker E-Marketing 5E Strauss Frost ISBN 0136154417 Test Bank University Chemistry with Student Access Kit siska 1e test bank Bond Markets, Analysis and Strategies 6E Instructors manual Fabozzi Instructor's Edition LAN Switching and Wireless CCNA Exploration Labs and Study Guide Allan Johnson EBOOK ElementaryDifferentialEquationsBoundaryValue(2ndEd) - Kohler [CapitalEth] SolutionsManual Busines statistics Decision making 7E David F Groebner Solution manual & test bank Analysis- With an Introduction to Proof, 4-E-Instructors SM Deitel &Deitel How to Program C++ 6th E Code solutions +solution manual John Hull Options, Futures and Other Derivatives 7E test bank Cutnell John, Physics 7th E, Instructors manual (all solutions even +odd) Prentice Hall's Federal Taxation 22/e 2009 Corporations test bank and solution manual Cost accounting by Hongren 13/e test bank and Solution manual Instructors Solution Manual Gas Dynamics John & Keith Aerodynamics for Engineers, 5E Solution manual Bertin Russ Cummings Java Foundations- Introduction to Program Design and Data Structures- project solutions Instructor's Manual Contemporary Engineering economics 4e Park Advanced Accounting 10e Beams Oppenheim - Signals And Systems 2Ed- Solution Manual DATA COMMUNICATION BY FOROUZAN Solution manual (Artech 05) Signal Detection And Estimation - Solution Manual (McGraw-Hill) (Instructors Manual) Electric Machinery Fundamentals 4th Edition (Stephen J Chapman) A Course in Game Theory Solution Manual - Martin J. Osborne A First Course in Probability - Sheldon Ross - 7th Ed - Solution Manual Alpha C. Chiang - Fundamental Methods of Econometrics Solution Manual An Introduction To The Finite Element Method - Solution Manual (J N Reddy) Calculus - Jerrold Marsden & Alan Weinstein - Student Solution Manual Calculus of Variations + Solution Manual-01--Russak Chapra - Applied Numerical Methods with matlab Solutions Cormen - Introduction To Algorithms 2nd Edition Solutions (Instructors.Manual) Cheng - Field And Wave Electromagnetics 2Ed Solution Manual Cover - Elements of Information Theory - solution manual Discrete Time Signal Processing 2Ed - Oppenheim Solution Manual E Springer - Probability And Statistics For Engineering And The Sciences - Solution Manual - Jay L Devore ELECTRIC CIRCUITS + SOLUTION MANUAL - NILSSON - 7TH ED (EN) Electric Machinary Fundamentals Instructors Manual - Maquinas Electrical Machines, Drives and Power Systems 6th ed [INSTRUCTORS MANUAL] - T. Wildi (PTC, 2006) === Subject: Re: completing the square problem >Hello I'm trying to solve this problem: Given that x^2 + 4x + c = (x + a)^2 + b where a, b and c are constants: i) find the value of a >ii) find b in terms of c Well, it depends on what you mean. If you mean simple equality, then: x^2 + 4x + c = x^2 +2ax + a^2 +b So (4-2a)x = a^2 +b-_c Then when x = (a^2 +b - c)/(4-2a), you have equality for every a,b, and c (except a=2 If you mean identically equal for all x, then 2a = 4 and a^2 + b = c. Then a=2 and b = c -a^2 = c-4 According to the book where this problem comes from, the answers are a >= 2 and b = c - 4. I realise that by completing the square of the quadratic expression on >the left of the equal sign I get: (x + 2)^2 - 4 + c = (x + a)^2 + b from this I can see that a = 2 if b = -4 + c. But in my mind, I don't see why a can't be less than 2, if b was >greater than c - 4 (to maintain the equality). In the same manner, I don't see why a can't be greater than 2, if b >was less than c - 4 (to maintain the equality). Am I wrong? TIA === Subject: Re: completing the square problem > Hello I'm trying to solve this problem: Given that x^2 + 4x + c = (x + a)^2 + b where a, b and c are constants: i) find the value of a > ii) find b in terms of c According to the book where this problem comes from, > the answers are a > = 2 and b = c - 4. I realise that by completing the square of the > quadratic expression on > the left of the equal sign I get: (x + 2)^2 - 4 + c = (x + a)^2 + b from this I can see that a = 2 if b = -4 + c. But in my mind, I don't see why a can't be less than > 2, if b was > greater than c - 4 (to maintain the equality). In the same manner, I don't see why a can't be > greater than 2, if b > was less than c - 4 (to maintain the equality). Am I wrong? TIA You can't compensate a choice of a less/greater than 2 by a choice of b greater/less than c - 4. The reason is that the choice of a does not only affect the constant term of the polynomial on the left-hand side, but also the coefficient of x. Note that two polynomials are identical if and only if they agree in all their coefficients. Best wishes Torsten. === Subject: Re: completing the square problem <25436388.105873.1242368931574.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=BLsY7gkAAACx8YsqGvzDRj-HT50qzG30 1.1.4322; InfoPath.1),gzip(gfe),gzip(gfe) two polynomials are identical if and only if they agree in all their coefficients. I didn't know this. Can you elaborate on this? Can you prove it? === Subject: Re: completing the square problem >two polynomials are identical if and only if they agree in all their >coefficients. I didn't know this. Can you elaborate on this? Can you prove it? Do you know a little calculus? One way to verify it is by taking derivatives. Say you are given two polynomials: p(x) = sum[k=0..n] (a_k x^k) q(x) = sum[k=0..m] (b_k x^k) I'm going to assume that what you mean by identical is p(x) = q(x) for all x. Put x = 0. p(0) = a_0 and q(0) = b_0 so a_0 = b_0 Now take derivatives: p'(x) = sum[k=1..n] (k a_k x^(k-1)) q'(x) = sum[k=1..m] (k b_k x^(k-1)) Put x = 0: p'(0) = a_1 and q'(0) = b_1 so the second coefficients are equal. Continue this process. Each time you differentiate and evaluate at x = 0 you get the next coefficients are equal. --Lynn http://math.asu.edu/~kurtz === Subject: Re: completing the square problem >I'm going to assume that what you mean by identical is p(x) = q(x) for >all x. Put x = 0. p(0) = a_0 and q(0) = b_0 so a_0 = b_0 Now take derivatives: p'(x) = sum[k=1..n] (k a_k x^(k-1)) >q'(x) = sum[k=1..m] (k b_k x^(k-1)) Put x = 0: p'(0) = a_1 and q'(0) = b_1 >so the second coefficients are equal. I should have mentioned here that p(x) = q(x) for all x implies that p'(x) = q'(x) for all x. --Lynn http://math.asu.edu/~kurtz === Subject: Re: completing the square problem two polynomials are identical if and only if they agree in all their > coefficients. I didn't know this. Can you elaborate on this? Can you prove it? How do you _define_ two polynomials are identical? -- ... when we came back, late, from the hyacinth garden, Your arms full, and your hair wet, I could not Speak, and my eyes failed... === Subject: Re: completing the square problem >Hello I'm trying to solve this problem: Given that x^2 + 4x + c = (x + a)^2 + b where a, b and c are constants: i) find the value of a >ii) find b in terms of c According to the book where this problem comes from, the answers are a >= 2 and b = c - 4. I realise that by completing the square of the quadratic expression on >the left of the equal sign I get: (x + 2)^2 - 4 + c = (x + a)^2 + b from this I can see that a = 2 if b = -4 + c. But in my mind, I don't see why a can't be less than 2, if b was >greater than c - 4 (to maintain the equality). In the same manner, I don't see why a can't be greater than 2, if b >was less than c - 4 (to maintain the equality). Hint: Expand the right hand side of the given equation, then equate the coefficients of like powers of x. Am I wrong? TIA === Subject: Re: completing the square problem > Hello I'm trying to solve this problem: Given that x^2 + 4x + c = (x + a)^2 + b where a, b and c are constants: i) find the value of a > ii) find b in terms of c According to the book where this problem comes from, the answers are a > = 2 and b = c - 4. I realise that by completing the square of the quadratic expression on > the left of the equal sign I get: (x + 2)^2 - 4 + c = (x + a)^2 + b from this I can see that a = 2 if b = -4 + c. But in my mind, I don't see why a can't be less than 2, if b was > greater than c - 4 (to maintain the equality). In the same manner, I don't see why a can't be greater than 2, if b > was less than c - 4 (to maintain the equality). Am I wrong? TIA > how about taking a=1.5 or sqrt of 2 and see what you get. how are you going to produce 4 in 4x? === Subject: P vs. NP posting-account=Xkwj_wkAAAAlnSv9LxZGz8fhtKwlFYMB Gecko/2009040821 Firefox/3.0.9,gzip(gfe),gzip(gfe) What is the difference between P=0 N=1 and P=1 N=1? === Subject: N and P > What is the difference between P=0 N=1 and P=1 N=1? > Something is weird. Is this what you mean? What is the difference between P=0, N=1 and P=1, N=1? Clearly the difference is the change of the value of P. -- Is this what you don't mean? What is the difference between P = 0N = 1 and P = 1N =1? The difference is glaring. 0 = 1 is a contradiction and the other is equivalent to P = N = 1. === Subject: Re: JSH: The Simple Lie posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) The fact that with integers x^2 - Dy^2 = 1, can always be solved by > integer solutions to j^2 - Dk^2 = -1, using x = 2j^2 + 1, is just a > mathematical reality. The belief that modern mathematicians give a damn about their field > and value knowledge over basic human politics is a simple lie. So I can show 9^2 - 5*4^2 = 1, is given by x = 2*2^2 + 1, as 2^2 - > 5*1^2 = -1, but you can ignore that because you wish. And modern mathematicians who pathetically don't give a damn about > anything but their grants so they can pay their mortgages I guess, can > get away with a simple lie. Check online sources on Pell's Equation: http://en.wikipedia.org/wiki/Pell's equation http://mathworld.wolfram.com/PellEquation.html See if you can see something so simple mentioned. I selected that > result as it reduces to saying that given an integer solution to the > equation often called the negative Pell's Equation you ALWAYS have a > solution to Pell's Equation in integers. Always. A mathematical absolute. Now that result probably was known to Fermat and Euler but > incompetence entered the math field in the late 1800's and it kind of > slipped through the cracks. But if modern mathematicians were what they claim to be, so what? > They'd just kind of laugh it off, note the obvious result and give me > credit...oh. Now you see where it's politics. Can you imagine physicists stopping with a result in an area with two > thousand years of research interest to prevent one man from getting > credit? Why do you admire mathematicians so much? When they lie to you in > return? Maybe some of you need to value yourselves and your knowledge better > as that is nothing compared to the bigger lies that cover Galois > Theory and shift what you think you know about Group Theory. Your loss is just about history. When humanity adjusts and corrects, > and much of your research is tossed on the heap, students later may > read about you in a paragraph and imagine there is no way they could > have been like you. But you are you. History is waiting to happen. The destruction of > your research is today. James Harris > The fact that with integers x^2 - Dy^2 = 1, can always be solved by > integer solutions to j^2 - Dk^2 = -1, using x = 2j^2 + 1, is just a > mathematical reality. Nope. Your forgot a constraint you had before. > j^2 - Dk^2 = -1 has no integer solutions for D = 7 or 19 Enrico It goes the other way: if you have a solution to j^2 - Dk^2 = -1, then necessarily you have a solution to x^2 - Dy^2 = 1 from x = 2j^2 + 1. Read what I said more carefully. James Harris === Subject: JSH: Learning from the negative Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) For me the chilling proof that math society itself willfully lies can be seen with some really trivial algebra, Pell's Equation and the negative Pell's Equation which is why I keep mentioning it, as I can beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a contemporary mathematical textbook. What I like about this result is how clearly it shows the political nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring this result! I like beating up on them. James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=WlifZwoAAADn4Qc008FhhuRE4Syn8J58 3.011; GTB6; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Given ANY set of non-zero integer solutions to the negative Pell's > equation > j^2 - Dk^2 = -1 > you will ALWAYS have a solution to Pell's Equation > x^2 - Dy^2 = 1 > from x = 2j^2 + 1. Hey Dumbo! Given ANY set of non-zero integer solutions to the equation j^2 - Dk^2 = A you will ALWAYS have a solution to the equation x^2 - Dy^2 = A^2 from x = 2j^2 - A Unfortunately, James is too stupid to realize that if I take A=-1, I recover his result. Even more unfortunately, James is too stupid to realize that the more general result is---and always has been---obvious to anyone who knows even a little algebraic number theory (i.e. not James) and therefore is as unlikely to be recorded as any other blindingly obvious identity. Modern algebra books tend not to include the identity 3x^2 + 7x^2 = 10x^2 Is that because modern mathematicians are unaware of this fact? === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=WlifZwoAAADn4Qc008FhhuRE4Syn8J58 3.011; GTB6; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. > Hey Dumbo! Given ANY set of non-zero integer solutions to the equation j^2 - Dk^2 = A you will ALWAYS have a solution to the equation x^2 - Dy^2 = A^2 from x = 2j^2 - A >Unfortunately, James is too stupid to realize that >if I take A=-1, I recover his result. Even more unfortunately, James is too stupid to >realize that the more general result is---and >always has been---obvious to anyone who knows >even a little algebraic number theory (i.e. >not James) and therefore is as unlikely to be >recorded as any other blindingly obvious >identity. Modern algebra books tend not to include the >identity 3x^2 + 7x^2 = 10x^2 Is that because modern mathematicians are unaware >of this fact? === Subject: Re: JSH: Learning from the negative Pell's Equation >For me the chilling proof that math society itself willfully lies can >be seen with some really trivial algebra, Pell's Equation and the >negative Pell's Equation which is why I keep mentioning it, as I can >beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's >equation However, there is a problem... j^2 - Dk^2 = -1 This equation does not have an integer solution for all non-square D. What is the solution for D = 7 for example? you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Provided you also have a solution to the first equation, which is not always the case. That is a mathematical absolute. Your method does not give an answer for all values of D. That is also a mathematical absolute. >Now go try to find it in a >contemporary mathematical textbook. You have been given references to Brahmagupta and to a 20th century textbook. What I like about this result is how clearly it shows the political >nature of the modern field of number theory. I think not. It shows that an inferior method that is incapable of solving the Pell equation for D = 7 has been discarded in favour of a superior method, continued fractions, that can solve the Pell equation for all non-square values of D. Nothing to do with politics, merely replacing a less good solution with a better solution. No need to look for sinister hidden motives. rossum Number theorists, quite simply, lie. I dare them to keep ignoring >this result! I like beating up on them. >James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation <72ut05tjjbs0b7uf61dn369tbhl1mrlphd@4ax.com> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) For me the chilling proof that math society itself willfully lies can >be seen with some really trivial algebra, Pell's Equation and the >negative Pell's Equation which is why I keep mentioning it, as I can >beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's >equation > However, there is a problem... > j^2 - Dk^2 = -1 > This equation does not have an integer solution for all non-square D. > What is the solution for D = 7 for example? >you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. > Provided you also have a solution to the first equation, which is not > always the case. >That is a mathematical absolute. > Your method does not give an answer for all values of D. That is also > a mathematical absolute. Now go try to find it in a >contemporary mathematical textbook. > You have been given references to Brahmagupta and to a 20th century > textbook. >What I like about this result is how clearly it shows the political >nature of the modern field of number theory. > I think not. It shows that an inferior method that is incapable of > solving the Pell equation for D = 7 has been discarded in favour of a > superior method, continued fractions, that can solve the Pell equation > for all non-square values of D. Nothing to do with politics, merely > replacing a less good solution with a better solution. No need to > look for sinister hidden motives. rossum > peg iin a round hole, the process hath required a little elbow grease.--Martin Musatov [P=NP, Just for the taste of it! Diet Coke!] === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) For me the chilling proof that math society itself willfully lies can >be seen with some really trivial algebra, Pell's Equation and the >negative Pell's Equation which is why I keep mentioning it, as I can >beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's >equation > However, there is a problem... > j^2 - Dk^2 = -1 > This equation does not have an integer solution for all non-square D. > What is the solution for D = 7 for example? >you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. > Provided you also have a solution to the first equation, which is not > always the case. >That is a mathematical absolute. > Your method does not give an answer for all values of D. That is also > a mathematical absolute. Now go try to find it in a >contemporary mathematical textbook. > You have been given references to Brahmagupta and to a 20th century > textbook. >What I like about this result is how clearly it shows the political >nature of the modern field of number theory. > I think not. It shows that an inferior method that is incapable of > solving the Pell equation for D = 7 has been discarded in favour of a > superior method, continued fractions, that can solve the Pell equation > for all non-square values of D. Nothing to do with politics, merely > replacing a less good solution with a better solution. No need to > look for sinister hidden motives. rossum peg iin a round hole, the process hath required a little elbow === Subject: Re: JSH: Learning from the negative Pell's Equation <72ut05tjjbs0b7uf61dn369tbhl1mrlphd@4ax.com> posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) For me the chilling proof that math society itself willfully lies can >be seen with some really trivial algebra, Pell's Equation and the >negative Pell's Equation which is why I keep mentioning it, as I can >beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's >equation However, there is a problem... j^2 - Dk^2 = -1 This equation does not have an integer solution for all non-square D. > What is the solution for D = 7 for example? you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Provided you also have a solution to the first equation, which is not > always the case. I didn't say it was. The negative Pell's Equation j^2 - Dk^2 = -1 does not always have all non-zero integer solutions. But whenever it does it solves Pell's Equation as well from x = 2j^2 + 1. That is a mathematical fact. It is a mathematical absolute. Now cite that result if it exists in the literature. James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation > Provided you also have a solution to the first equation, which is not > always the case. I didn't say it was. Agreed. I was pointing out why the method you give here is inferior to the continued fraction method, which does work for cases like D = 7. Given that a superior method exists it is not surprising that the inferior method gets less mention in the textbooks. The negative Pell's Equation j^2 - Dk^2 = -1 does not always have all >non-zero integer solutions. But whenever it does it solves Pell's Equation as well from x = 2j^2 + >1. That is a mathematical fact. It is a mathematical absolute. Agreed. Now cite that result if it exists in the literature. You are obviously reading this thread. IIRC there are three references to the literature in this thread from 630 CE to 2009 CE. References to this method are less common than references to the continued fraction method but they do exist. rossum >James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=sKfmEQkAAAC8kI3Pv6_U_nt9sVsxZ_ou 1.1.4322),gzip(gfe),gzip(gfe) > Provided you also have a solution to the first equation, which is not > always the case. I didn't say it was. Agreed. I was pointing out why the method you give here is inferior > to the continued fraction method, which does work for cases like D = > 7. Given that a superior method exists it is not surprising that the > inferior method gets less mention in the textbooks. The negative Pell's Equation j^2 - Dk^2 = -1 does not always have all >non-zero integer solutions. But whenever it does it solves Pell's Equation as well from x = 2j^2 + >1. That is a mathematical fact. It is a mathematical absolute. Agreed. Now cite that result if it exists in the literature. You are obviously reading this thread. IIRC there are three > references to the literature in this thread from 630 CE to 2009 CE. > References to this method are less common than references to the > continued fraction method but they do exist. Shortcuts to a solution are not a method of solution. Even the Indian mathematicians had the cyclic method that would take them to x^2 --Dy^2 =1 whether or not they encountered -1, 2,-2, 4 or -4 on the way. If you consider the cubic Pell x^3+ky^3 +kkz^3 -3kxyz =1 Knowing that a solution to x^3+ky^3 +kkz^3 -3kxyz =3 implies a solution to x^3+ky^3 +kkz^3 -3kxyz =1 exists is not a great help as solving the first equation is just as difficult as solving the first - there is no simple ternary continued fraction algorithm available. Anyone who has tried to solve the cubic Pell using composition of forms knows that JSH is talking nonsense. === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=sKfmEQkAAAC8kI3Pv6_U_nt9sVsxZ_ou 1.1.4322),gzip(gfe),gzip(gfe) > For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. Number theorists have known this result for centuries. If you look on page 27 of Carmichael's book Diophantine Analysis you will find Exercise 1 Show how all integral solutions of the equation x^2-Dy^2 =-1 may be obtained from one of them. D being as usual a non-sqaure positive integer. SUGGESTION - Observe that the relations a^2-Db^2 =-1 c^2 -Dg^2 =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1 This is utterly trivial and you make your pitiful lack of understanding clear to all. You are simply too stupid to find the references. === Subject: Re: JSH: Learning from the negative Pell's Equation > For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. Number theorists have known this result for centuries. If you look on page 27 of Carmichael's book Diophantine Analysis you will find Exercise 1 Show how all integral solutions of the equation x^2-Dy^2 =-1 may be obtained from one of them. D being as usual a non-sqaure positive integer. SUGGESTION - Observe that the relations a^2-Db^2 =-1 c^2 -Dg^2 =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1 This is utterly trivial and you make your pitiful lack of understanding clear to all. You are simply too stupid to find the references. > Google has scanned this book in, you can download the entire Book for FREE, JSH, and Study it!! page 27 => page 33 How HAPPY!!! you must be JSH to get your hands on this reference published just a short 93 years ago, to update and correct your failed research, and now get you further ahead from where your are now, (a century ago). Google Book Search Carmichael Diophantine Analysis OR, http://tiny.cc/LMcn7 Number theorists, quite simply, lie. I dare them to keep ignoring this result! I like beating up on them, said the weak-brain sheepish failed troll before getting his lying mouth stuffed and choking with large black fat CROW, on the internet, dip. > === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) > For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. Number theorists have known this result for centuries. If you look on page 27 of Carmichael's book > Diophantine Analysis you will find Exercise 1 Show how all integral solutions of the equation > x^2-Dy^2 =-1 may be obtained from one of them. > D being as usual a non-sqaure positive integer. > SUGGESTION - Observe that the relations a^2-Db^2 =-1 > c^2 -Dg^2 =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1 Work an example. I'll work one with what I show: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1. > This is utterly trivial and you make your pitiful lack > of understanding clear to all. > You are simply too stupid to find the references. I've repeatedly said it is trivial. Now then, work an example, with what you gave. I dare you. James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=sKfmEQkAAAC8kI3Pv6_U_nt9sVsxZ_ou 1.1.4322),gzip(gfe),gzip(gfe) For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. Number theorists have known this result for centuries. If you look on page 27 of Carmichael's book > Diophantine Analysis you will find Exercise 1 Show how all integral solutions of the equation > x^2-Dy^2 =-1 may be obtained from one of them. > D being as usual a non-sqaure positive integer. > SUGGESTION - Observe that the relations a^2-Db^2 =-1 > c^2 -Dg^2 =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1 Work an example. I'll work one with what I show: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1. This is utterly trivial and you make your pitiful lack > of understanding clear to all. > You are simply too stupid to find the references. I've repeatedly said it is trivial. Now then, work an example, with what you gave. I dare you. Carmichael and anyone who knows anything about the composition of quadratic forms knows thar a^2-Db^2 =-1 and c^2 -Dg^2 =-1 imply (ac +Dbg) -D(ag+bc)^2 =1 In your delusinal world you obviuosly have some inkling that somebody is lying to somebody about something. But you don;t realize that you ae lying to yourself about your claim that the identity for the Negative Pell doesn't appear in modern maths books. It has appeared in maths books throughout the centuries. Only a cretin would find this difficult to understand. If you know what a book title an author and a page number are, you could check this for yourself. === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) > For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. Number theorists have known this result for centuries. If you look on page 27 of Carmichael's book > Diophantine Analysis you will find Exercise 1 Show how all integral solutions of the equation > x^2-Dy^2 =-1 may be obtained from one of them. > D being as usual a non-sqaure positive integer. > SUGGESTION - Observe that the relations a^2-Db^2 =-1 > c^2 -Dg^2 =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1 Work an example. I'll work one with what I show: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1. This is utterly trivial and you make your pitiful lack > of understanding clear to all. > You are simply too stupid to find the references. I've repeatedly said it is trivial. Now then, work an example, with what you gave. I dare you. Carmichael and anyone who knows anything about the > composition of quadratic forms knows thar a^2-Db^2 =-1 > and c^2 -Dg^2 =-1 imply (ac +Dbg) -D(ag+bc)^2 =1 Work an example. I'll work one with what I show: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1. > In your delusinal world you obviuosly have some inkling that I don't know when math became insults to some of you, but the simple request to work an example should be honored. James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=sKfmEQkAAAC8kI3Pv6_U_nt9sVsxZ_ou 1.1.4322),gzip(gfe),gzip(gfe) For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. Number theorists have known this result for centuries. If you look on page 27 of Carmichael's book > Diophantine Analysis you will find Exercise 1 Show how all integral solutions of the equation > x^2-Dy^2 =-1 may be obtained from one of them. > D being as usual a non-sqaure positive integer. > SUGGESTION - Observe that the relations a^2-Db^2 =-1 > c^2 -Dg^2 =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1 Work an example. I'll work one with what I show: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1. This is utterly trivial and you make your pitiful lack > of understanding clear to all. > You are simply too stupid to find the references. I've repeatedly said it is trivial. Now then, work an example, with what you gave. I dare you. Carmichael and anyone who knows anything about the > composition of quadratic forms knows thar a^2-Db^2 =-1 > and c^2 -Dg^2 =-1 imply (ac +Dbg) -D(ag+bc)^2 =1 Work an example. I'll work one with what I show: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1. In your delusinal world you obviuosly have some inkling that I don't know when math became insults to some of you, but the simple > request to work an example should be honored. A statement of the truth is not an insult. The truth may be unpleasant for you to hear, but that is another matter. There is no need to work and example because the identity in question given by Carmichael and known by many others is generally valid. For some reasson you seem unable to grasp the idea of a trivial result. Trivial means unimportant, simple, obvious, not going beyond first principles. This is why it was the subject of an EXCERCISE in Carmichael. Doing a EXCERCISE in maths is like doing SCALES when practising a musical instrument. Only a fool would mistake someone doing SCALES for someone playing a composition, original or not. In all candour, I cannot think of an explanation simpler than the one above - even a child could understand it. Perhaps you do in fact suffer from the mathematical equivalent of tone deafness === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) > For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. Number theorists have known this result for centuries. If you look on page 27 of Carmichael's book > Diophantine Analysis you will find Exercise 1 Show how all integral solutions of the equation > x^2-Dy^2 =-1 may be obtained from one of them. > D being as usual a non-sqaure positive integer. > SUGGESTION - Observe that the relations a^2-Db^2 =-1 > c^2 -Dg^2 =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1 Work an example. I'll work one with what I show: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1. This is utterly trivial and you make your pitiful lack > of understanding clear to all. > You are simply too stupid to find the references. I've repeatedly said it is trivial. Now then, work an example, with what you gave. I dare you. Carmichael and anyone who knows anything about the > composition of quadratic forms knows thar a^2-Db^2 =-1 > and c^2 -Dg^2 =-1 imply (ac +Dbg) -D(ag+bc)^2 =1 Work an example. I'll work one with what I show: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1. In your delusinal world you obviuosly have some inkling that I don't know when math became insults to some of you, but the simple > request to work an example should be honored. A statement of the truth is not an insult. > The truth may be unpleasant for you to hear, but > that is another matter. > There is no need to work and example because the > identity in question given by Carmichael and known by > many others is generally valid. I didn't say it wasn't. It is, however, not what I'm explaining. And The result I'm noting is that given a solution to the negative Pell's Equation: j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. You made a false claim saying that was given by equations you cited, and I impeached your claim by giving an example: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 and asking you to work an example, which is simpler than a back-and- forth where I just note that you are wrong, and you keep claiming you are not. However, you continually refuse to work an example, which indicates you KNOW you are wrong, bu are intent on simply repeating false statements, as if that matters. To me that's behavior that shows a disdain for mathematics as well as readers. It's like, you don't give a damn about anything important except repeatedly babbling in reply, but why? What do you believe you're accomplishing? James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=aLpfCwoAAACh4BOs3HOlQBCoxUpEgyxc Gecko/2009032609 Firefox/3.0.8,gzip(gfe),gzip(gfe) [...] There is no need to work and example because the > identity in question given by Carmichael and known by > many others is generally valid. I didn't say it wasn't. It is, however, not what I'm explaining. And The result I'm noting is that given a solution to the negative Pell's > Equation: j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. You made a false claim saying that was given by equations you cited, > and I impeached your claim by giving an example: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 and asking you to work an example, which is simpler than a back-and- > forth where I just note that you are wrong, and you keep claiming you > are not. OK, I'll work an example. The quote from Carmichael's book given by Juandiego says [o]bserve that the relations a^2-Db^2 =-1, c^2 -Dg^2 =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1. So let's apply this to the same example you give above, namely 2^2 - 5*1^2 = -1. Take a = c = 2, b = g = 1. Then the quote tells us to take x = a*c + D*b*g = 2*2 + 5*1*1 = 9, and y = a*g + b*c = 4, which gives 9^2 - 5*4^2 = 1 i.e. it gives the same result as your example above. More generally, suppose that j^2 - D*k^2 = -1. Then taking a = c = j, b = g = k and substituting this into the result given by Carmichael tells us that we always have a solution to Pell's equation x^2 - D*y^2 = 1 with x = a*c + D*b*g = j^2 + D*k^2 = j^2 + (j^2 + 1) = 2*j^2 + 1. In other words, your result is a special case of that given in Carmichael as an exercise. Which is what people have been saying. === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) > [...] There is no need to work and example because the > identity in question given by Carmichael and known by > many others is generally valid. I didn't say it wasn't. It is, however, not what I'm explaining. And The result I'm noting is that given a solution to the negative Pell's > Equation: j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. You made a false claim saying that was given by equations you cited, > and I impeached your claim by giving an example: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 and asking you to work an example, which is simpler than a back-and- > forth where I just note that you are wrong, and you keep claiming you > are not. OK, I'll work an example. The quote from Carmichael's book given by > Juandiego says [o]bserve that the relations a^2-Db^2 =-1, c^2 -Dg^2 > =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1. So let's apply this > to the same example you give above, namely 2^2 - 5*1^2 = -1. Take a = c = 2, b = g = 1. Then the quote tells us to take x = a*c + > D*b*g = 2*2 + 5*1*1 = 9, and y = a*g + b*c = 4, which gives 9^2 - 5*4^2 = 1 i.e. it gives the same result as your example above. More generally, Ok. Was that so hard? > suppose that j^2 - D*k^2 = -1. Then taking a = c = j, b = g = k and substituting this into the result > given by Carmichael tells us that we always have a solution to Pell's > equation x^2 - D*y^2 = 1 with x = a*c + D*b*g = j^2 + D*k^2 = j^2 + (j^2 + 1) = 2*j^2 + 1. Yeah, I know it's true. > In other words, your result is a special case of that given in > Carmichael as an exercise. Which is what people have been saying. And the result is more simply stated as, given a solution to the negative Pell's Equation j^2 - Dk^2 = -1 you always have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. A trivial result as I noted. It is one of several minor results about Pell's Equation not clearly stated in modern mathematical literature. James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=aLpfCwoAAACh4BOs3HOlQBCoxUpEgyxc Gecko/2009032609 Firefox/3.0.8,gzip(gfe),gzip(gfe) [...] OK, I'll work an example. The quote from Carmichael's book given by > Juandiego says [o]bserve that the relations a^2-Db^2 =-1, c^2 -Dg^2 > =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1. So let's apply this > to the same example you give above, namely 2^2 - 5*1^2 = -1. Take a = c = 2, b = g = 1. Then the quote tells us to take x = a*c + > D*b*g = 2*2 + 5*1*1 = 9, and y = a*g + b*c = 4, which gives 9^2 - 5*4^2 = 1 i.e. it gives the same result as your example above. More generally, Ok. Was that so hard? Too hard for you to figure out, apparently, since you needed somebody else to work an example for you. > [...] In other words, your result is a special case of that given in > Carmichael as an exercise. Which is what people have been saying. And the result is more simply stated as, given a solution to the > negative Pell's Equation j^2 - Dk^2 = -1 you always have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. A trivial result as I noted. It is one of several minor results about > Pell's Equation not clearly stated in modern mathematical literature. It IS clearly stated in modern mathematical literature, for example the quote from Carmichael given in this thread. The fact that you're unable to recognise that the above is a special case of the more general result without somebody walking you through it doesn't mean it isn't clear to the rest of us. === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) > [...] OK, I'll work an example. The quote from Carmichael's book given by > Juandiego says [o]bserve that the relations a^2-Db^2 =-1, c^2 -Dg^2 > =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1. So let's apply this > to the same example you give above, namely 2^2 - 5*1^2 = -1. Take a = c = 2, b = g = 1. Then the quote tells us to take x = a*c + > D*b*g = 2*2 + 5*1*1 = 9, and y = a*g + b*c = 4, which gives 9^2 - 5*4^2 = 1 i.e. it gives the same result as your example above. More generally, Ok. Was that so hard? Too hard for you to figure out, apparently, since you needed somebody > else to work an example for you. Nope. [...] In other words, your result is a special case of that given in > Carmichael as an exercise. Which is what people have been saying. And the result is more simply stated as, given a solution to the > negative Pell's Equation j^2 - Dk^2 = -1 you always have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. A trivial result as I noted. It is one of several minor results about > Pell's Equation not clearly stated in modern mathematical literature. It IS clearly stated in modern mathematical literature, for example > the quote from Carmichael given in this thread. The fact that you're > unable to recognise that the above is a special case of the more > general result without somebody walking you through it doesn't mean it > isn't clear to the rest of us. Then why not just say that in the first place? I suggest that you are lying and that it was NOT clear to you before, and in fact, you didn't know it before I stated it. You simply worked back from what was known after the fact. Readers who are curious should do a web search on negative Pell's Equation. This trivial result easily *derivable* does not appear to have been generally known to *modern* mathematicians. But it was, as I've repeatedly said, probably well-known to Fermat and Euler. Denials in this area are the curious lies I'm studying now with these posts!!! In mathematics, any number of results can be derived from what is known. That is how all math advances. But someone has to know where to look!!! James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation > [...] OK, I'll work an example. The quote from Carmichael's book given by > Juandiego says [o]bserve that the relations a^2-Db^2 =-1, c^2 -Dg^2 > =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1. So let's apply this > to the same example you give above, namely 2^2 - 5*1^2 = -1. Take a = c = 2, b = g = 1. Then the quote tells us to take x = a*c + > D*b*g = 2*2 + 5*1*1 = 9, and y = a*g + b*c = 4, which gives 9^2 - 5*4^2 = 1 i.e. it gives the same result as your example above. More generally, Ok. Was that so hard? Too hard for you to figure out, apparently, since you needed somebody > else to work an example for you. Nope. [...] In other words, your result is a special case of that given in > Carmichael as an exercise. Which is what people have been saying. And the result is more simply stated as, given a solution to the > negative Pell's Equation j^2 - Dk^2 = -1 you always have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. A trivial result as I noted. It is one of several minor results about > Pell's Equation not clearly stated in modern mathematical literature. It IS clearly stated in modern mathematical literature, for example > the quote from Carmichael given in this thread. The fact that you're > unable to recognise that the above is a special case of the more > general result without somebody walking you through it doesn't mean it > isn't clear to the rest of us. Then why not just say that in the first place? I suggest that you are lying and that it was NOT clear to you before, and in fact, you didn't know it before I stated it. You simply worked back from what was known after the fact. Readers who are curious should do a web search on negative Pell's Equation. This trivial result easily *derivable* does not appear to have been generally known to *modern* mathematicians. But it was, as I've repeatedly said, probably well-known to Fermat and Euler. Denials in this area are the curious lies I'm studying now with these posts!!! In mathematics, any number of results can be derived from what is known. That is how all math advances. But someone has to know where to look!!! *** LIES *** Google for it Thief, and Lier. James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=aLpfCwoAAACh4BOs3HOlQBCoxUpEgyxc Gecko/2009032609 Firefox/3.0.8,gzip(gfe),gzip(gfe) [...] Ok. Was that so hard? Too hard for you to figure out, apparently, since you needed somebody > else to work an example for you. Nope. So why did you keep asking for an example? Had you understood the result you would have known that it would turn out to give exactly the same result as your own example - it's hard to see why you imagine that would be a point in your favour. > It IS clearly stated in modern mathematical literature, for example > the quote from Carmichael given in this thread. The fact that you're > unable to recognise that the above is a special case of the more > general result without somebody walking you through it doesn't mean it > isn't clear to the rest of us. Then why not just say that in the first place? Why not say what? Do you mean, why doesn't Carmichael say that there is a solution to Pell's equation with x = 2*j^2 + 1 whenever j^2 - D*k^2 = -1? If so then the reason he or she doesn't say it is because it's standard practice when stating a result in a book to not also state every special case of that result. This is because most results have an infinite number of special cases and books are finite in length. > I suggest that you are lying and that it was NOT clear to you before, > and in fact, you didn't know it before I stated it. Oh, indeed I didn't. That's because I have never read anything about Pell's equation outside the threads you start about it, since there are many more interesting things I'd rather read about with the limited amount of time I have to spend learning real maths. But had I, for example, read Carmichael's book then I would know about the result, and also about every other special case that I went to the trouble of working through. > [...] In mathematics, any number of results can be derived from what is > known. That is how all math advances. Perhaps. But some results, unlike the one you keep going on about, are /difficult/ to derive from what is already known. If you ever go to the trouble of reading an actual text book or paper you'll notice that the authors mostly spend their time working towards the latter sort. === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=sKfmEQkAAAC8kI3Pv6_U_nt9sVsxZ_ou 1.1.4322),gzip(gfe),gzip(gfe) [...] There is no need to work and example because the > identity in question given by Carmichael and known by > many others is generally valid. I didn't say it wasn't. It is, however, not what I'm explaining. And The result I'm noting is that given a solution to the negative Pell's > Equation: j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. You made a false claim saying that was given by equations you cited, > and I impeached your claim by giving an example: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 and asking you to work an example, which is simpler than a back-and- > forth where I just note that you are wrong, and you keep claiming you > are not. OK, I'll work an example. The quote from Carmichael's book given by > Juandiego says [o]bserve that the relations a^2-Db^2 =-1, c^2 -Dg^2 > =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1. So let's apply this > to the same example you give above, namely 2^2 - 5*1^2 = -1. Take a = c = 2, b = g = 1. Then the quote tells us to take x = a*c + > D*b*g = 2*2 + 5*1*1 = 9, and y = a*g + b*c = 4, which gives 9^2 - 5*4^2 = 1 i.e. it gives the same result as your example above. More generally, Ok. Was that so hard? suppose that j^2 - D*k^2 = -1. Then taking a = c = j, b = g = k and substituting this into the result > given by Carmichael tells us that we always have a solution to Pell's > equation x^2 - D*y^2 = 1 with x = a*c + D*b*g = j^2 + D*k^2 = j^2 + (j^2 + 1) = 2*j^2 + 1. Yeah, I know it's true. In other words, your result is a special case of that given in > Carmichael as an exercise. Which is what people have been saying. And the result is more simply stated as, given a solution to the > negative Pell's Equation j^2 - Dk^2 = -1 you always have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. A trivial result as I noted. It is one of several minor results about > Pell's Equation not clearly stated in modern mathematical literature. But it is stated clearly in the modern mathematical literature. The only problem is that you do not understand the statement. === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=sKfmEQkAAAC8kI3Pv6_U_nt9sVsxZ_ou 1.1.4322),gzip(gfe),gzip(gfe) For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. Number theorists have known this result for centuries. If you look on page 27 of Carmichael's book > Diophantine Analysis you will find Exercise 1 Show how all integral solutions of the equation > x^2-Dy^2 =-1 may be obtained from one of them. > D being as usual a non-sqaure positive integer. > SUGGESTION - Observe that the relations a^2-Db^2 =-1 > c^2 -Dg^2 =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1 Work an example. I'll work one with what I show: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1. This is utterly trivial and you make your pitiful lack > of understanding clear to all. > You are simply too stupid to find the references. I've repeatedly said it is trivial. Now then, work an example, with what you gave. I dare you. Carmichael and anyone who knows anything about the > composition of quadratic forms knows thar a^2-Db^2 =-1 > and c^2 -Dg^2 =-1 imply (ac +Dbg) -D(ag+bc)^2 =1 Work an example. I'll work one with what I show: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1. In your delusinal world you obviuosly have some inkling that I don't know when math became insults to some of you, but the simple > request to work an example should be honored. A statement of the truth is not an insult. > The truth may be unpleasant for you to hear, but > that is another matter. > There is no need to work and example because the > identity in question given by Carmichael and known by > many others is generally valid. I didn't say it wasn't. >It is, however, not what I'm explaining. If it is a result that is already known and trivial why are you mentioning it ? What do you think you are explaining that is not covered in the Carmichael excercise ? Carmichael covers everything you claim and more. Are you really too stupid to see that ? === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. Number theorists have known this result for centuries. The real ones have. I said so myself, noting that it was probably known to Fermat and Euler. The result is trivial. > If you look on page 27 of Carmichael's book > Diophantine Analysis you will find Exercise 1 Show how all integral solutions of the equation > x^2-Dy^2 =-1 may be obtained from one of them. > D being as usual a non-sqaure positive integer. > SUGGESTION - Observe that the relations a^2-Db^2 =-1 > c^2 -Dg^2 =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1 This is utterly trivial and you make your pitiful lack > of understanding clear to all. > You are simply too stupid to find the references. You crawl on your belly. The result is, given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Mathematics is a hard discipline. Too many of you have gotten away with lying about your belief in it, your belief in proof. Lying is easy. Doing correct mathematics, now that is what is hard. I think many of you lack real mathematical ability. But you are very good liars as are many number theorists around the world. They lie for grants and math prizes. And screw the world like it was some stupid whore who doesn't know better. James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. Number theorists have known this result for centuries. The real ones have. I said so myself, noting that it was probably known to Fermat and Euler. The result is trivial. > If you look on page 27 of Carmichael's book > Diophantine Analysis you will find Exercise 1 Show how all integral solutions of the equation > x^2-Dy^2 =-1 may be obtained from one of them. > D being as usual a non-sqaure positive integer. > SUGGESTION - Observe that the relations a^2-Db^2 =-1 > c^2 -Dg^2 =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1 > This is utterly trivial and you make your pitiful lack > of understanding clear to all. > You are simply too stupid to find the references. >You crawl on your belly. with lying about your belief in it, your belief in proof. You lie all the time, you're projecting again. >Lying is easy. Doing correct mathematics, now that is what is hard. You always lie, it is easy for you. You have no idea how to do Mathematics, any of it. >I think many of you lack real mathematical ability. But you are very >good liars as are many number theorists around the world. This is you in spades; http://en.wikipedia.org/wiki/Psychological_projection >They lie for grants and math prizes. And screw the world like it was >some stupid whore who doesn't know better. You always make mistakes in your post, cause you like to get *screwed* like stupid bitch whore. >James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; InfoPath.1),gzip(gfe),gzip(gfe) > Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You lying little bitch! You have been spewing your nonsense for 14 years now. The sum total of everything you have done is pure useless garbage - you have nothing. You are a cheat, a charlatan, a quack. You delusional narcissist! Get your head out of your ass! ~A === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=wVv_VwoAAAAVTfUuyxLzug5SzYWCgHj1 Gecko/20081217 Firefox/2.0.0.20,gzip(gfe),gzip(gfe) > For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution uniformly for all possible values of N, rather he was only able to show that if x^2 - Ny^2 = k has an integral solution for k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. Brahmagupta did not know the continued fraction solution, but it is absolutely clear from the above that he only knew what you claim as a great discovery. By modern standards with modern notation, Brahmagupta's result is a triviality. So is yours, and clearly it is well known. Marcus. === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution > uniformly for all possible values of N, rather he was only able > to show that if x^2 - Ny^2 = k has an integral solution for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. That is not the same as, given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the first solution to Pell's Equation. > Brahmagupta did not know the continued fraction solution, but > it is absolutely clear from the above that he only knew what you > claim as a great discovery. By modern standards with modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known to Fermat and Euler. > notation, Brahmagupta's result is a triviality. So is yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search on the subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number theorists, lie. James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=wVv_VwoAAAAVTfUuyxLzug5SzYWCgHj1 Gecko/20081217 Firefox/2.0.0.20,gzip(gfe),gzip(gfe) > For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution > uniformly for all possible values of N, rather he was only able > to show that if x^2 - Ny^2 = k has an integral solution for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. That is not the same as, given j^2 - Dk^2 = -1 you will ALWAYS have a > solution to Pell's Equation > x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the first > solution to Pell's Equation. Brahmagupta did not know the continued fraction solution, but > it is absolutely clear from the above that he only knew what you > claim as a great discovery. By modern standards with modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known to > Fermat and Euler. notation, Brahmagupta's result is a triviality. So is yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search on the > subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number theorists, lie. James Harris Cite a reference where someone has lied about this. Marcus. === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution > uniformly for all possible values of N, rather he was only able > to show that if x^2 - Ny^2 = k has an integral solution for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. That is not the same as, given j^2 - Dk^2 = -1 you will ALWAYS have a > solution to Pell's Equation > x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the first > solution to Pell's Equation. Brahmagupta did not know the continued fraction solution, but > it is absolutely clear from the above that he only knew what you > claim as a great discovery. By modern standards with modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known to > Fermat and Euler. notation, Brahmagupta's result is a triviality. So is yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search on the > subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number theorists, lie. James Harris Cite a reference where someone has lied about this. Marcus. Readers can simply look at your previous reply in this thread and contrast it with what I've said, where I'll repeat the math yet again. Given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. The result is fairly trivial but the point here is that with something not seen in the mainstream literature rather than behave like real researchers who value knowledge, you and posters like you, lie. Ergo, you do not value knowledge! Your intentions in posting must then be about something else. In my opinion you post simply to coerce the crowd in a direction of your choosing, so your postings are political!!! So what you do in posting has nothing to do with mathematics. It is all about a darker side in human nature, and a disdain of knowledge. James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=wVv_VwoAAAAVTfUuyxLzug5SzYWCgHj1 Gecko/20081217 Firefox/2.0.0.20,gzip(gfe),gzip(gfe) > For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution > uniformly for all possible values of N, rather he was only able > to show that if x^2 - Ny^2 = k has an integral solution for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. That is not the same as, given j^2 - Dk^2 = -1 you will ALWAYS have a > solution to Pell's Equation > x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the first > solution to Pell's Equation. Brahmagupta did not know the continued fraction solution, but > it is absolutely clear from the above that he only knew what you > claim as a great discovery. By modern standards with modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known to > Fermat and Euler. notation, Brahmagupta's result is a triviality. So is yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search on the > subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number theorists, lie. James Harris Cite a reference where someone has lied about this. Marcus. Readers can simply look at your previous reply in this thread and > contrast it with what I've said, where I'll repeat the math yet again. Given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's > Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. > Notice that if you had bothered to read the Wikipedia which generalize yours. Nothing is being covered up here. > The result is fairly trivial but the point here is that with something > not seen in the mainstream literature rather than behave like real > researchers who value knowledge, you and posters like you, lie. > I did not lie at all. Your result is well-known and well- explained in the literature from 1500 years ago. > Ergo, you do not value knowledge! Your intentions in posting must > then be about something else. > Ergo, you do not read any references, even those that are most easily accessible. > In my opinion you post simply to coerce the crowd in a direction of > your choosing, so your postings are political!!! > Oh sure. Telling the truth about this is a political act. > So what you do in posting has nothing to do with mathematics. > Are we now talking about your post, where the objectives are to obtain recognition for your great genius and to show that mathematicians lie? Is that the part that nothing to do with mathematics? > It is all about a darker side in human nature, and a disdain of > knowledge. > Again I'm getting confused. I would think that your a disdain for knowledge. Looks like we are talking about you and your darker side, not that of mathematicians. Marcus. > James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution > uniformly for all possible values of N, rather he was only able > to show that if x^2 - Ny^2 = k has an integral solution for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. That is not the same as, given j^2 - Dk^2 = -1 you will ALWAYS have a > solution to Pell's Equation > x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the first > solution to Pell's Equation. Brahmagupta did not know the continued fraction solution, but > it is absolutely clear from the above that he only knew what you > claim as a great discovery. By modern standards with modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known to > Fermat and Euler. notation, Brahmagupta's result is a triviality. So is yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search on the > subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number theorists, lie. James Harris Cite a reference where someone has lied about this. Marcus. Readers can simply look at your previous reply in this thread and > contrast it with what I've said, where I'll repeat the math yet again. Given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's > Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Notice that if you had bothered to read the Wikipedia > which generalize yours. Nothing is being covered up > here. Then work an example relying on what you quoted. I'll work an example based on what I gave: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 Note that here D=5. j=2, and since x= 2j^2 + 1, you have x = 9. Now YOU work an example. I hate how some of you babble on in these long-winded replies that are just straight lies. Do math, not sophistry. James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=wVv_VwoAAAAVTfUuyxLzug5SzYWCgHj1 Gecko/20081217 Firefox/2.0.0.20,gzip(gfe),gzip(gfe) > For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution > uniformly for all possible values of N, rather he was only able > to show that if x^2 - Ny^2 = k has an integral solution for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. That is not the same as, given j^2 - Dk^2 = -1 you will ALWAYS have a > solution to Pell's Equation > x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the first > solution to Pell's Equation. Brahmagupta did not know the continued fraction solution, but > it is absolutely clear from the above that he only knew what you > claim as a great discovery. By modern standards with modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known to > Fermat and Euler. notation, Brahmagupta's result is a triviality. So is yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search on the > subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number theorists, lie. James Harris Cite a reference where someone has lied about this. Marcus. Readers can simply look at your previous reply in this thread and > contrast it with what I've said, where I'll repeat the math yet again. Given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's > Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Notice that if you had bothered to read the Wikipedia > which generalize yours. Nothing is being covered up > here. Then work an example relying on what you quoted. I'll work an example based on what I gave: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 Note that here D=5. j=2, and since x= 2j^2 + 1, you have x = 9. Now YOU work an example. I hate how some of you babble on in these long-winded replies that are > just straight lies. > Do math, not sophistry. James Harris immediately implies that if (x, y) is a solution to u^2 - Dv^2 = =/-1, then (x^2 + Dy^2, 2xy) is a solution to u^2 - Dv^2 = +1. Hence, starting with your example: (2, 1) is a solution to u^2 - 5v^2 = -1, therefore (2^2 + 5*1^2, 2*2*1) = (9, 4) is a solution to u^2 - 5v^2 = 1. Period. Trivial application of Brahmagupta's well-known result. Marcus. === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution > uniformly for all possible values of N, rather he was only able > to show that if x^2 - Ny^2 = k has an integral solution for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. That is not the same as, given j^2 - Dk^2 = -1 you will ALWAYS have a > solution to Pell's Equation > x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the first > solution to Pell's Equation. Brahmagupta did not know the continued fraction solution, but > it is absolutely clear from the above that he only knew what you > claim as a great discovery. By modern standards with modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known to > Fermat and Euler. notation, Brahmagupta's result is a triviality. So is yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search on the > subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number theorists, lie. James Harris Cite a reference where someone has lied about this. Marcus. Readers can simply look at your previous reply in this thread and > contrast it with what I've said, where I'll repeat the math yet again. Given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's > Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Notice that if you had bothered to read the Wikipedia > which generalize yours. Nothing is being covered up > here. Then work an example relying on what you quoted. I'll work an example based on what I gave: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 Note that here D=5. j=2, and since x= 2j^2 + 1, you have x = 9. Now YOU work an example. I hate how some of you babble on in these long-winded replies that are > just straight lies. > Do math, not sophistry. James Harris immediately implies that if (x, y) is a solution to u^2 - Dv^2 = =/-1, then (x^2 + Dy^2, 2xy) is a solution to u^2 - Dv^2 = +1. Hence, starting with your example: (2, 1) is a solution > to u^2 - 5v^2 = -1, therefore (2^2 + 5*1^2, 2*2*1) = (9, 4) is a solution to u^2 - 5v^2 = 1. Period. Trivial application of Brahmagupta's well-known > result. Good. Progress. Then you agree that given a solution to the negative Pell's Equation j^2 - Dk^2 = -1, you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1, from x = 2j^2 + 1? Simple question. Only requires a simple answer. James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=wVv_VwoAAAAVTfUuyxLzug5SzYWCgHj1 Gecko/20081217 Firefox/2.0.0.20,gzip(gfe),gzip(gfe) > For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution > uniformly for all possible values of N, rather he was only able > to show that if x^2 - Ny^2 = k has an integral solution for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. That is not the same as, given j^2 - Dk^2 = -1 you will ALWAYS have a > solution to Pell's Equation > x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the first > solution to Pell's Equation. Brahmagupta did not know the continued fraction solution, but > it is absolutely clear from the above that he only knew what you > claim as a great discovery. By modern standards with modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known to > Fermat and Euler. notation, Brahmagupta's result is a triviality. So is yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search on the > subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number theorists, lie. James Harris Cite a reference where someone has lied about this. Marcus. Readers can simply look at your previous reply in this thread and > contrast it with what I've said, where I'll repeat the math yet again. Given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's > Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Notice that if you had bothered to read the Wikipedia > which generalize yours. Nothing is being covered up > here. Then work an example relying on what you quoted. I'll work an example based on what I gave: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 Note that here D=5. j=2, and since x= 2j^2 + 1, you have x = 9. Now YOU work an example. I hate how some of you babble on in these long-winded replies that are > just straight lies. > Do math, not sophistry. James Harris immediately implies that if (x, y) is a solution to u^2 - Dv^2 = =/-1, then (x^2 + Dy^2, 2xy) is a solution to u^2 - Dv^2 = +1. Hence, starting with your example: (2, 1) is a solution > to u^2 - 5v^2 = -1, therefore (2^2 + 5*1^2, 2*2*1) = (9, 4) is a solution to u^2 - 5v^2 = 1. Period. Trivial application of Brahmagupta's well-known > result. Good. Progress. Then you agree that given a solution to the negative Pell's Equation > j^2 - Dk^2 = -1, you will ALWAYS have a solution to Pell's Equation > x^2 - Dy^2 = 1, from x = 2j^2 + 1? Simple question. Only requires a simple answer. James Harris Of course. No one denies this. It isn't new or interesting. It's old, trivial and boring. What is your point??? Marcus. === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution > uniformly for all possible values of N, rather he was only able > to show that if x^2 - Ny^2 = k has an integral solution for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. That is not the same as, given j^2 - Dk^2 = -1 you will ALWAYS have a > solution to Pell's Equation > x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the first > solution to Pell's Equation. Brahmagupta did not know the continued fraction solution, but > it is absolutely clear from the above that he only knew what you > claim as a great discovery. By modern standards with modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known to > Fermat and Euler. notation, Brahmagupta's result is a triviality. So is yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search on the > subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number theorists, lie. James Harris Cite a reference where someone has lied about this. Marcus. Readers can simply look at your previous reply in this thread and > contrast it with what I've said, where I'll repeat the math yet again. Given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's > Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Notice that if you had bothered to read the Wikipedia > which generalize yours. Nothing is being covered up > here. Then work an example relying on what you quoted. I'll work an example based on what I gave: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 Note that here D=5. j=2, and since x= 2j^2 + 1, you have x = 9. Now YOU work an example. I hate how some of you babble on in these long-winded replies that are > just straight lies. > Do math, not sophistry. James Harris immediately implies that if (x, y) is a solution to u^2 - Dv^2 = =/-1, then (x^2 + Dy^2, 2xy) is a solution to u^2 - Dv^2 = +1. Hence, starting with your example: (2, 1) is a solution > to u^2 - 5v^2 = -1, therefore (2^2 + 5*1^2, 2*2*1) = (9, 4) is a solution to u^2 - 5v^2 = 1. Period. Trivial application of Brahmagupta's well-known > result. Good. Progress. Then you agree that given a solution to the negative Pell's Equation > j^2 - Dk^2 = -1, you will ALWAYS have a solution to Pell's Equation > x^2 - Dy^2 = 1, from x = 2j^2 + 1? Simple question. Only requires a simple answer. James Harris Of course. No one denies this. It isn't new or > interesting. It's old, trivial and boring. I agree it is old as in previously well-known. I agree it is trivially proven. But is it really boring? > What is your point??? Marcus. You say it's well-known. Then EVERY reader who did NOT know that given a solution to the negative Pell's Equation j^2 - Dk^2 = -1, you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1, from x = 2j^2 + 1 is simply badly educated about number theory? Is that correct? Is everyone who did not know this result simply not as smart as you and others like you who claim you knew it already? And what about skeptics who do a search on negative Pell's Equation, and see my research dominating everything, who try as hard as they can, cannot find the result you call boring outside of me saying it? Are they simply not smart enough to understand? Are you just one of the smartest people on planet earth and all of them are idiots in comparison to you? James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation For me the chilling proof that math society itself > willfully lies can > be seen with some really trivial algebra, Pell's > Equation and the > negative Pell's Equation which is why I keep mentioning > it, as I can > beat up on math society worldwide with this result > indefinitely. Given ANY set of non-zero integer solutions to the > negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it > in a > contemporary mathematical textbook. What I like about this result is how clearly it shows > the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep > ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here > it is Unfortunately, Brahmagupta was not able to apply his > solution > uniformly for all possible values of N, rather he was only > able > to show that if x^2 - Ny^2 = k has an integral solution > for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a > solution. That is not the same as, given j^2 - Dk^2 = -1 you will > ALWAYS have a > solution to Pell's Equation > x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the > first > solution to Pell's Equation. Brahmagupta did not know the continued fraction solution, > but > it is absolutely clear from the above that he only knew > what you > claim as a great discovery. By modern standards with > modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known > to > Fermat and Euler. notation, Brahmagupta's result is a triviality. So is > yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search > on the > subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number > theorists, lie. James Harris Cite a reference where someone has lied about this. Marcus. Readers can simply look at your previous reply in this thread > and > contrast it with what I've said, where I'll repeat the math yet > again. Given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's > Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Notice that if you had bothered to read the Wikipedia > which generalize yours. Nothing is being covered up > here. Then work an example relying on what you quoted. I'll work an example based on what I gave: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 Note that here D=5. j=2, and since x= 2j^2 + 1, you have x = 9. Now YOU work an example. I hate how some of you babble on in these long-winded replies that > are > just straight lies. > Do math, not sophistry. James Harris immediately implies that if (x, y) is a solution to u^2 - Dv^2 = =/-1, then (x^2 + Dy^2, 2xy) is a solution to u^2 - Dv^2 = +1. Hence, starting with your example: (2, 1) is a solution > to u^2 - 5v^2 = -1, therefore (2^2 + 5*1^2, 2*2*1) = (9, 4) is a solution to > .> u^2 - 5v^2 = 1. > .> Period. Trivial application of Brahmagupta's well-known > result. > Good. Progress. > Then you agree that given a solution to the negative Pell's Equation > j^2 - Dk^2 = -1, you will ALWAYS have a solution to Pell's Equation > x^2 - Dy^2 = 1, from x = 2j^2 + 1? > Simple question. Only requires a simple answer. > James Harris > Of course. No one denies this. It isn't new or > interesting. It's old, trivial and boring. >I agree it is old as in previously well-known. I agree it is >trivially proven. >But is it really boring? > What is your point??? Marcus. Trivial, discovered 1500 years ago, in print 1915, and again in 2009, still Trivial On this one, YOU LOSE JSH !! >James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=wVv_VwoAAAAVTfUuyxLzug5SzYWCgHj1 Gecko/20081217 Firefox/2.0.0.20,gzip(gfe),gzip(gfe) > For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution > uniformly for all possible values of N, rather he was only able > to show that if x^2 - Ny^2 = k has an integral solution for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. That is not the same as, given j^2 - Dk^2 = -1 you will ALWAYS have a > solution to Pell's Equation > x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the first > solution to Pell's Equation. Brahmagupta did not know the continued fraction solution, but > it is absolutely clear from the above that he only knew what you > claim as a great discovery. By modern standards with modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known to > Fermat and Euler. notation, Brahmagupta's result is a triviality. So is yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search on the > subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number theorists, lie. James Harris Cite a reference where someone has lied about this. Marcus. Readers can simply look at your previous reply in this thread and > contrast it with what I've said, where I'll repeat the math yet again. Given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's > Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Notice that if you had bothered to read the Wikipedia > which generalize yours. Nothing is being covered up > here. Then work an example relying on what you quoted. I'll work an example based on what I gave: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 Note that here D=5. j=2, and since x= 2j^2 + 1, you have x = 9. Now YOU work an example. I hate how some of you babble on in these long-winded replies that are > just straight lies. > Do math, not sophistry. James Harris immediately implies that if (x, y) is a solution to u^2 - Dv^2 = =/-1, then (x^2 + Dy^2, 2xy) is a solution to u^2 - Dv^2 = +1. Hence, starting with your example: (2, 1) is a solution > to u^2 - 5v^2 = -1, therefore (2^2 + 5*1^2, 2*2*1) = (9, 4) is a solution to u^2 - 5v^2 = 1. Period. Trivial application of Brahmagupta's well-known > result. Good. Progress. Then you agree that given a solution to the negative Pell's Equation > j^2 - Dk^2 = -1, you will ALWAYS have a solution to Pell's Equation > x^2 - Dy^2 = 1, from x = 2j^2 + 1? Simple question. Only requires a simple answer. James Harris Of course. No one denies this. It isn't new or > interesting. It's old, trivial and boring. I agree it is old as in previously well-known. I agree it is > trivially proven. But is it really boring? > In math, trivial is boring. So yes. > What is your point??? Marcus. You say it's well-known. Then EVERY reader who did NOT know that given a solution to the > negative Pell's Equation j^2 - Dk^2 = -1, you will ALWAYS have a > solution to Pell's Equation x^2 - Dy^2 = 1, from x = 2j^2 + 1 is > simply badly educated about number theory? Is that correct? > Yes. If they did not know it already they should be able to see it immediately. It's several dozen orders of magnitude more trivial than, say, Dirichlet's theorem on primes in arithmetic sequences. > Is everyone who did not know this result simply not as smart as you > and others like you who claim you knew it already? > Smart? That's not the issue at all. People may be very smart but not aware of a given fact. > And what about skeptics who do a search on negative Pell's Equation, > and see my research dominating everything, who try as hard as they > can, cannot find the result you call boring outside of me saying it? > Searches like this prove nothing. You are a well-known crank. You attract attention and notoriety by being stupid. You get lots of hits because people want to see in what way you are currently making a fool of yourself. I wouldn't brag about it. > Are they simply not smart enough to understand? > Almost anyone can understand it. Not everyone is aware of it. That's not because it isn't out there in textbooks, websites, etc.. It's because there is too much information out there for everyone to know every- thing. > Are you just one of the smartest people on planet earth and all of > them are idiots in comparison to you? > Not at all. I never said any such thing. That's you putting words in my mouth. Or is it what you think about yourself? You asked me to do the math. You didn't want more propaganda and politics. So I showed you how your method follows trivially from well-known results of Brahmagupta (~500 AD). Now here you are back again trying to turn it into propaganda and politics and personal comments. So I assume you know you have lost the math argument and are back to your old whining about society, etc. Right? Marcus. > James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=sKfmEQkAAAC8kI3Pv6_U_nt9sVsxZ_ou 1.1.4322),gzip(gfe),gzip(gfe) > For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution > uniformly for all possible values of N, rather he was only able > to show that if x^2 - Ny^2 = k has an integral solution for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. That is not the same as, given j^2 - Dk^2 = -1 you will ALWAYS have a > solution to Pell's Equation > x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the first > solution to Pell's Equation. Brahmagupta did not know the continued fraction solution, but > it is absolutely clear from the above that he only knew what you > claim as a great discovery. By modern standards with modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known to > Fermat and Euler. notation, Brahmagupta's result is a triviality. So is yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search on the > subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number theorists, lie. James Harris Cite a reference where someone has lied about this. Marcus. Readers can simply look at your previous reply in this thread and > contrast it with what I've said, where I'll repeat the math yet again. Given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's > Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Notice that if you had bothered to read the Wikipedia > which generalize yours. Nothing is being covered up > here. Then work an example relying on what you quoted. I'll work an example based on what I gave: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 Note that here D=5. j=2, and since x= 2j^2 + 1, you have x = 9. Now YOU work an example. I hate how some of you babble on in these long-winded replies that are > just straight lies. > Do math, not sophistry. James Harris immediately implies that if (x, y) is a solution to u^2 - Dv^2 = =/-1, then (x^2 + Dy^2, 2xy) is a solution to u^2 - Dv^2 = +1. Hence, starting with your example: (2, 1) is a solution > to u^2 - 5v^2 = -1, therefore (2^2 + 5*1^2, 2*2*1) = (9, 4) is a solution to u^2 - 5v^2 = 1. Period. Trivial application of Brahmagupta's well-known > result. Good. Progress. Then you agree that given a solution to the negative Pell's Equation > j^2 - Dk^2 = -1, you will ALWAYS have a solution to Pell's Equation > x^2 - Dy^2 = 1, from x = 2j^2 + 1? Simple question. Only requires a simple answer. But this was not disputed. You were claiming that there were no previous references or clear references to this in the modern mathematical literature. Why else would I give the Carmicahel reference ? Clearly Carmicahel disproves your claim and you have shown youself to be a fool because,without an example, you could not understand that Carmicheal includes and goes beyond your statement above. === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution > uniformly for all possible values of N, rather he was only able > to show that if x^2 - Ny^2 = k has an integral solution for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. That is not the same as, given j^2 - Dk^2 = -1 you will ALWAYS have a > solution to Pell's Equation > x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the first > solution to Pell's Equation. Brahmagupta did not know the continued fraction solution, but > it is absolutely clear from the above that he only knew what you > claim as a great discovery. By modern standards with modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known to > Fermat and Euler. notation, Brahmagupta's result is a triviality. So is yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search on the > subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number theorists, lie. James Harris Cite a reference where someone has lied about this. Marcus. Readers can simply look at your previous reply in this thread and > contrast it with what I've said, where I'll repeat the math yet again. Given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's > Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Notice that if you had bothered to read the Wikipedia > which generalize yours. Nothing is being covered up > here. Then work an example relying on what you quoted. I'll work an example based on what I gave: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 Note that here D=5. j=2, and since x= 2j^2 + 1, you have x = 9. Now YOU work an example. I hate how some of you babble on in these long-winded replies that are > just straight lies. > Do math, not sophistry. James Harris immediately implies that if (x, y) is a solution to u^2 - Dv^2 = =/-1, then (x^2 + Dy^2, 2xy) is a solution to u^2 - Dv^2 = +1. Hence, starting with your example: (2, 1) is a solution > to u^2 - 5v^2 = -1, therefore (2^2 + 5*1^2, 2*2*1) = (9, 4) is a solution to u^2 - 5v^2 = 1. Period. Trivial application of Brahmagupta's well-known > result. Good. Progress. Then you agree that given a solution to the negative Pell's Equation > j^2 - Dk^2 = -1, you will ALWAYS have a solution to Pell's Equation > x^2 - Dy^2 = 1, from x = 2j^2 + 1? Simple question. Only requires a simple answer. But this was not disputed. Then why didn't you just agree before? > You were claiming that there were no previous references > or clear references to this in the modern mathematical literature. There aren't. > Why else would I give the Carmicahel reference ? I don't know. > Clearly Carmicahel disproves your claim and you have shown Then cite in modern literature the result that given a solution to the negative Pell's Equation j^2 - Dk^2 = -1, you always have a solution to Pell's Equation x^2 - Dy^2 = 1, from x = 2j^2 + 1. > youself to be a fool because,without an example, you could not Your insults betray your lack of confidence here. > understand that Carmicheal includes and goes beyond your statement > above. Then why argue? Why not just agree that the result follows? I dare you, just agree. Given a solution to the negative Pell's Equation j^2 - Dk^2 = -1, you always have a solution to Pell's Equation x^2 - Dy^2 = 1, from x = 2j^2 + 1. Agree? James Harris === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=sKfmEQkAAAC8kI3Pv6_U_nt9sVsxZ_ou 1.1.4322),gzip(gfe),gzip(gfe) > For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution > uniformly for all possible values of N, rather he was only able > to show that if x^2 - Ny^2 = k has an integral solution for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. That is not the same as, given j^2 - Dk^2 = -1 you will ALWAYS have a > solution to Pell's Equation > x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the first > solution to Pell's Equation. Brahmagupta did not know the continued fraction solution, but > it is absolutely clear from the above that he only knew what you > claim as a great discovery. By modern standards with modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known to > Fermat and Euler. notation, Brahmagupta's result is a triviality. So is yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search on the > subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number theorists, lie. James Harris Cite a reference where someone has lied about this. Marcus. Readers can simply look at your previous reply in this thread and > contrast it with what I've said, where I'll repeat the math yet again. Given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's > Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Notice that if you had bothered to read the Wikipedia > which generalize yours. Nothing is being covered up > here. Then work an example relying on what you quoted. I'll work an example based on what I gave: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 Note that here D=5. j=2, and since x= 2j^2 + 1, you have x = 9. Now YOU work an example. I hate how some of you babble on in these long-winded replies that are > just straight lies. > Do math, not sophistry. James Harris immediately implies that if (x, y) is a solution to u^2 - Dv^2 = =/-1, then (x^2 + Dy^2, 2xy) is a solution to u^2 - Dv^2 = +1. Hence, starting with your example: (2, 1) is a solution > to u^2 - 5v^2 = -1, therefore (2^2 + 5*1^2, 2*2*1) = (9, 4) is a solution to u^2 - 5v^2 = 1. Period. Trivial application of Brahmagupta's well-known > result. Good. Progress. Then you agree that given a solution to the negative Pell's Equation > j^2 - Dk^2 = -1, you will ALWAYS have a solution to Pell's Equation > x^2 - Dy^2 = 1, from x = 2j^2 + 1? Simple question. Only requires a simple answer. But this was not disputed. Then why didn't you just agree before? You were claiming that there were no previous references > or clear references to this in the modern mathematical literature. There aren't. You obviously have not looked. Solving the Pell Equation Williams and Jacobson Published 2009 See pages 3 to 32 for history Page 32 states Brahmagupta's identities discovered AD 628. Brahmagupta even had two more alternates than you bously claim to have discovered Is 2009 modern enough for you ? > Why else would I give the Carmicahel reference ? I don't know. You know nothing and understand nothing. Even the glaringly obvious escapes you. === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) For me the chilling proof that math society itself willfully lies can > be seen with some really trivial algebra, Pell's Equation and the > negative Pell's Equation which is why I keep mentioning it, as I can > beat up on math society worldwide with this result indefinitely. Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a > contemporary mathematical textbook. What I like about this result is how clearly it shows the political > nature of the modern field of number theory. Number theorists, quite simply, lie. I dare them to keep ignoring > this result! I like beating up on them. James Harris You have been shown this before. You keep denying it. Here it is Unfortunately, Brahmagupta was not able to apply his solution > uniformly for all possible values of N, rather he was only able > to show that if x^2 - Ny^2 = k has an integral solution for > k = +/- 1, +/-2, +/- 4 then x^2 - Ny^2 = 1 has a solution. That is not the same as, given j^2 - Dk^2 = -1 you will ALWAYS have a > solution to Pell's Equation > x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Further note that if j is the first solution then x is the first > solution to Pell's Equation. Brahmagupta did not know the continued fraction solution, but > it is absolutely clear from the above that he only knew what you > claim as a great discovery. By modern standards with modern Nope, it's not a great discovery. It's completely trivial. Easily proven. Probably well-known to > Fermat and Euler. notation, Brahmagupta's result is a triviality. So is yours, > and clearly it is well known. Marcus. Lies. What is remarkable to me is that readers can easily search on the > subject. I don't claim this result is some great discovery. It's not. I simply claim it's an easy way to watch modern number theorists, lie. James Harris Cite a reference where someone has lied about this. Marcus. Readers can simply look at your previous reply in this thread and > contrast it with what I've said, where I'll repeat the math yet again. Given j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's > Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. Notice I GIVE the solution for x. Notice that if you had bothered to read the Wikipedia > which generalize yours. Nothing is being covered up > here. Then work an example relying on what you quoted. I'll work an example based on what I gave: 2^2 - 5*1^2 = -1, so x = 2*2^2 + 1 = 9, and 9^2 - 5*4^2 = 1 Note that here D=5. j=2, and since x= 2j^2 + 1, you have x = 9. Now YOU work an example. I hate how some of you babble on in these long-winded replies that are > just straight lies. > Do math, not sophistry. James Harris immediately implies that if (x, y) is a solution to u^2 - Dv^2 = =/-1, then (x^2 + Dy^2, 2xy) is a solution to u^2 - Dv^2 = +1. Hence, starting with your example: (2, 1) is a solution > to u^2 - 5v^2 = -1, therefore (2^2 + 5*1^2, 2*2*1) = (9, 4) is a solution to u^2 - 5v^2 = 1. Period. Trivial application of Brahmagupta's well-known > result. Good. Progress. Then you agree that given a solution to the negative Pell's Equation > j^2 - Dk^2 = -1, you will ALWAYS have a solution to Pell's Equation > x^2 - Dy^2 = 1, from x = 2j^2 + 1? Simple question. Only requires a simple answer. But this was not disputed. Then why didn't you just agree before? You were claiming that there were no previous references > or clear references to this in the modern mathematical literature. There aren't. Why else would I give the Carmicahel reference ? I don't know. Clearly Carmicahel disproves your claim and you have shown Then cite in modern literature the result that given a solution to the > negative Pell's Equation j^2 - Dk^2 = -1, you always have a solution > to Pell's Equation x^2 - Dy^2 = 1, from x = 2j^2 + 1. youself to be a fool because,without an example, you could not Your insults betray your lack of confidence here. understand that Carmicheal includes and goes beyond your statement > above. Then why argue? Why not just agree that the result follows? I dare you, just agree. Given a solution to the negative Pell's Equation j^2 - Dk^2 = -1, you > always have a solution to Pell's Equation x^2 - Dy^2 = 1, from x = > 2j^2 + 1. Agree? > James Harris Mar 2009 ... which reveal the 'why' of Pell's Equation, as for composite D there are ... not know there was such an easy route to solving the negative Pell's Equation, ... Ever stranger story .81E Why P=NP .81E Mystery with Pell's Equation ...http://mymath.blogspot.com/ 2009/03/negative-pells-equation.html: My Math: Mystery with Pell's Equation parametric solution More surprisingly though is my result on the negative Pell's Equation and two other alternates, which shows that in general the alternates should be solved ...http://mymath.blogspot.com/2009/04/mystery-with- pells-equation-parametric.html: Hallgren's Quantum Algorithm for Pell's Equation Quantum ...NP.81Àco-NP. Regulator. Bigger picture. * Hallgren's can be expanded to solve other ... algorithms for Pell's Equation and the prinicipal ...http:// www.cs.washington.edu/homes/nchernia/new_pell.pdf: Pell's equationStudy the solutions of the following equations: x2 - 2y2 = k with k = -2, -8, 4, 1022. ... For each prime number p, examine = ( fV BIn fact, Theorem 2 also holds if p , q satisfy the negative Pell's equation and more generally, if p, q satisfy. (1.5) q. 2. - Np ...http://journals.cambridge.org/production/action/cjoGetFulltext %3Ffulltextid%3D4912580: PELL'S EQUATION AND TWO GENERATOR FREE MOBIUS GROUPS: Pell's equation is. =q*-Np. (4.1) where N is a given positive integer (not .... a finite number of possible values of / (positive or negative) satisfying ...http://blms.oxfordjournals.org/cgi/reprint/ 25/6/527.pdf: Polynomial-Time Quantum Algorithms for Pell's Equation and the ...The first is Pell's equation. Given a positive nonsquare integer d, Pell's equation .... In addition, while finding multiples of the regulator is in NP, ...... a negative component. A 1/8 fraction have y-component at most 1/ ...http://portal.acm.org/citation.cfm%3Fdoid %3D1206035.1206039: Polynomial-Time Quantum Algorithms for Pell's Equation and the ...problem, which is at least as hard as solving Pell's equation, .... is in NP, finding the regulator itself is only known to be in NP under the GRH [BW89a]. ...http://www.cse.psu.edu/~hallgren/ pell.pdf: Chapter 8 : Continued Fraction and Pell's EquationAlgorithm for continued fraction expansion and solution of Pell equation ... Qn = (n- Pn2) / Qn-1. pn, p1 = a1 p2 = a1 * a2+1 ...... pn = an * pn-1+pn-2 ...http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math08/ pell02.htm: The Limits of Quantum Computers (or: What We Can't Do With ...Ideas extend to computing discrete logarithms, solving Pell's equation, breaking elliptic curve cryptography... But these problems aren't believed to be NP- ...http://www.scottaaronson.com/talks/sipbtalk.ppt: But we have proved they are:--Musatov === Subject: Re: JSH: Learning from the negative Pell's Equation Then you agree that given a solution to the negative Pell's Equation > j^2 - Dk^2 = -1, you will ALWAYS have a solution to Pell's Equation > x^2 - Dy^2 = 1, from x = 2j^2 + 1? > Simple question. Only requires a simple answer. > But this was not disputed. >Then why didn't you just agree before? He gave you a reference that totally solved this problem was solved 100 YEARS ago. > You were claiming that there were no previous references > or clear references to this in the modern mathematical literature. >There aren't. WRONG, there are many. You claim Others Work, thief. > Why else would I give the Carmicahel reference ? >I don't know. Perhaps, you JSH, could DOWNLOAD the ENTIRE 100 year old BOOK and READ IT, dummy. > Clearly Carmicahel disproves your claim and you have shown >Then cite in modern literature the result that given a solution to the >negative Pell's Equation j^2 - Dk^2 = -1, you always have a solution >to Pell's Equation x^2 - Dy^2 = 1, from x = 2j^2 + 1. He already did that, you JSH, are boxed in as an outright lier and thief. > youself to be a fool because,without an example, you could not >Your insults betray your lack of confidence here. They underscore what a technical Thief and Lier JSH is. > understand that Carmicheal includes and goes beyond your statement > above. >Then why argue? Why not just agree that the result follows? Perhaps you would READ THE ING BOOK ? >I dare you, just agree. You, JSH are Wrong again, an error in every post. >Given a solution to the negative Pell's Equation j^2 - Dk^2 = -1, you >always have a solution to Pell's Equation x^2 - Dy^2 = 1, from x = >2j^2 + 1. >Agree? Trivial, done 100 years ago, Carmicheal, also done by Brahmagupta, 1500 years ago ? >James Harris JSH => THIEF and LIER. (Read a book, moron) === Subject: Re: JSH: Learning from the negative Pell's Equation Û8. On The Equation x2-Dy2=z2 We shall now develop a general theory by means of which the solutions of the equation may be found. Naturally D is assumed to be an integer. Without loss of generality it may be taken positive; for if it were negative the equation might be written in the form s2 - (- D)y2 = x2, where - D is positive. If D is the square of an integer, say, D = d2, the equation may be written x2 = z2+(dy)2, so that the theory becomes essentially that of the Pythagorean equation x2+y2 = z2. Accordingly, we shall suppose that D is not a square. By suitably specializing Eq. (2) of the preceding section we readily obtain the following two-parameter solution of (1) : x = m2+Dn2, y = 2mn, z = m2-Dn2. But there is no ready means for determining whether this is the general solution. Consequently we shall approach from another direction the problem of finding the solution of (1). We shall first show that Eq. (1) possesses a non-trivial solution for which z - i; that is, we shall prove the existence of a solution of the equation x2-Dy2 = i. (2) different from the trivial solutions x = ±i, y=o. For this purpose we shall first show that integers u, v exist such that the absolute value * of the (positive or negative) real quantity u-vvD is less than i/v and also less than any pre- assigned positive constant e. (By VD we mean the positive square root of D.) Let / be an integer such that te> i. Now give to v successively the integral values from o to t and in each case choose for u the least integral value greater than * By the absolute value of A is meant A itself when A is positive and - A when A is negative. We denote it by 'A'. In each case the quantity u-v^/D lies between o and i and in no two cases are its values equal. If we divide the interval from o to i into t subintervals, each of length i/t, then two_pf the above values of u-vVD, say u'-v'VD and u - vVD, must lie in the same interval. Then the expression (u'-u)-(v'-v)VD is different from zero, is of the form u-vVD and has an absolute value less than i/t and hence less than e. That this absolute value is less than that of i/(v'-v) follows from the fact that the difference of v' and v is not greater than t. This completes the proof of the above statement concerning the existence of u, v with the assigned properties. there is an infinite number of such sets. For, let u, v be one such set. Let ei be a positive constant less than u-i)VL>|. Then integers ui, vi can be determined such that ui-viVD is in absolute value less than i/vi, and also less than 'i. It is then less than e. Thus we have a second set ui, vi satisfying the original conditions. Then, letting e2 be a positive constant less than ui - viV7)l, we may proceed as before to find a third set u2, v2 with the required properties. It is obvious that this process may be continued indefinitely and that we are thus led to an infinite number of sets of integers u, v such that u-v/D is in absolute value less than ? and also less than the absolute value of i/v. Now let u and v be a pair of integers determined as above. Then we have 2vV7)l< v Hence r i v so that I + 2VD. + |*'V5|}, Since u2-Dv2l is less than i + 2VZ) for every one of the infinite number of sets u, v in consideration, and since its value is always integral, it follows . that an integer / exists such that u2-Dv2 = l for an infinite number of sets of values u, -u. It is then obvious that there is an infinite number of these pairs Mi, ér, u2, V2; Ms, vz; ., such that ut - Uj and vt - vj are both divisible by / for every i and j. Let u', v'; u, v be two pairs belonging to this last infinite subset and chosen so the equations u'2-Dv'2 = l, u2-Dv2=l, we have (by Formula (2) of Û 7) : (u'u-Dv'v)2-D(u'v-uv')2=l2. Here we take u' =m, u = p, v' = n, v = -q, D= -b, in applying the formula referred to. Setting u'u -Dv'v u'v-uv' f . *- - -, y=- - - , (3) we have 1. (4) It remains to show that the values of x and y in (3) are integers. On account of (4) it is obviously sufficient to show that y is an integer. That y is an integer follows at once from the equations .u'=u+1Jil, v = v'+vl, by multiplication member by member. We show further that y^o. If we suppose that y = o, we have u'v-uv' = o, u'u-Dv'v = ±l. These equations are satisfied only if u = ±u', v = ±z/, relations which are contrary to the hypothesis concerning u', u, v', v. We have thus established the fact that Eq. (2) has at least one integral solution which is not trivial. Since we may associate with any solution x, y of (2) the other solutions - x, y; -x, -y; x, -y; it is clear that there is at least one solution of (2) in which x and y are positive. Let xi, yi, and X2, y2 be any solutions of (2), whether the same or different. Then we have i = (xi2-Dyi2)(x22-Dy22) = (xix2+Dyiy2)2-D(xiy2+x2yi)2, so that Xix2+Dyiy2 and Xiy2+x2yi afford a solution of (2). Hence from the solution x, y, whose existence has already been proved, we have a second solution x2-{-Dy2, 2xy. It is easy to show that this process may be continued and that it will lead to an infinite number of solutions of (2). But this problem is a special case of one to be treated presently; and hence will not be further pursued now. In order to come upon the more general problem let us seek solutions of Eq. (1) in which z shall have the positive value [Crap, as usual - what else can he write?] Learning? That would a be first with you. === Subject: Re: JSH: Learning from the negative Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) On May 15, 5:19pm, Jens Stueckelberger [Crap, as usual - what else can he write?] > Learning? That would a be first with you. LOL. See how your society crawls on its belly? No longer able to answer math with math, you simply rely on insults. Insults? What insults? Over the many years in Usenet you have only written crap. That's not an insult: It's a fact. Also, you have been unable to learn from your many mistakes - another fact. > Your hatred of the world is now complete. I would pity you if you were not so stupid. === Subject: e: JSH: Learning from the negative Pell's Equation posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) On May 15, 5:19pm, Jens Stueckelberger [Crap, as usual - what else can he write?] > Learning? That would a be first with you. LOL. See how your society crawls on its belly? No longer able to answer math with math, you simply rely on insults. Insults? What insults? Over the many years in Usenet you have > only written crap. That's not an insult: It's a fact. Also, you have been unable to learn from your many mistakes - > another fact. Your hatred of the world is now complete. I would pity you if you were not so stupid. It is not natural to carry on with such angry and intensity. You sound like children slinging mud in the sandbox. I am waiting for your mothers to show up and separate the two of you while one gets a snack and the other a diaper change. (No comments on which is which *please*) --Martin Musatov === Subject: JSH: What's remarkable posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) Years ago I discovered that posters would routinely lie to me in reply and they'd lie about even the most basic mathematics. But what was remarkable to me was, other readers would side with them, even on trivial errors. And the denial about the publication, withdrawal and destruction of It's just basic human psychology though. It's a group effect. A human being can be convinced of ANYTHING. And groups of human beings are more easily convinced than a single person!!! So your knowledge of mathematics can be completely flawed, but if you have a group of people telling you otherwise, you believe. But not if you're a true mathematician. THAT is the test. It always was. And most of you failed, long ago. All that has remained is the clean-up. James Harris === Subject: Re: What's remarkable > Years ago I discovered that posters would routinely lie to me in reply > and they'd lie about even the most basic mathematics. But what was remarkable to me was, other readers would side with them, > even on trivial errors. And the denial about the publication, withdrawal and destruction of It's just basic human psychology though. It's a group effect. A human being can be convinced of ANYTHING. And groups of human > beings are more easily convinced than a single person!!! So your knowledge of mathematics can be completely flawed, but if you > have a group of people telling you otherwise, you believe. But not if you're a true mathematician. THAT is the test. It always was. And most of you failed, long ago. All that has remained is the clean-up. *yawn* I heard that people with schizophrenia sometime improve as they get older. How old are you Harris? === Subject: Re: JSH: What's remarkable > Years ago I discovered that posters would routinely lie to me in reply > and they'd lie about even the most basic mathematics. But what was remarkable to me was, other readers would side with them, > even on trivial errors. And the denial about the publication, withdrawal and destruction of It's just basic human psychology though. It's a group effect. A human being can be convinced of ANYTHING. And groups of human > beings are more easily convinced than a single person!!! So your knowledge of mathematics can be completely flawed, but if you > have a group of people telling you otherwise, you believe. But not if you're a true mathematician. THAT is the test. It always was. And most of you failed, long ago. All that has remained is the clean-up. And you are just the man for the job. Get to it! === Subject: Re: JSH: What's remarkable > Years ago I discovered that posters would routinely lie to me in reply > and they'd lie about even the most basic mathematics. > Yet you cannot display a single example. > But what was remarkable to me was, other readers would side with them, > even on trivial errors. Such as? And the denial about the publication, withdrawal and destruction of > (a) stylistically deficient, even to the standards one applies to undergraduate homework papers, let alone a presumed refereed journal. (b) mathematically deficient, in that it failed to state and prove a single result. (c) mathematically flawed, in that its main result was demonstrably incorrect, using ordinary arithmetic. On item (c), we all know you attempted to backpedal and claim that your argument showed that the ring of algebraic integers constituted some sort of error in mathematics. You persist in thinking that a definition [a mere convention: a naming for a certain thing] has the capacity to introduce error. To be sure, if we name something, and assume that it exists when it does not, or assume that it has a specific property that it fails to have, that additional assumption can introduce error. However, this is simply not the case for the ring of algebraic integers: *every* claimed property of the ring of algebraic integers that is used is *first* proven to hold. Your assertion of units that are not units in the ring of algebraic integers is prima facie evidence of the depths of misunderstanding that you bring to your alleged work. It goes without saying that a black hole can be considered shallow in comparison. > It's just basic human psychology though. > Apparently something you grasp at as an explanation for why no one accepts any argument of yours without extensive testing. > It's a group effect. A human being can be convinced of ANYTHING. And groups of human > beings are more easily convinced than a single person!!! So your knowledge of mathematics can be completely flawed, but if you > have a group of people telling you otherwise, you believe. > I see. A number of people agree to what constitutes a proof, but since Google has you place at the top of the heap for the phrase definition of mathematical proof, you are the world authority. That and five bucks will get you a cappuccino. > But not if you're a true mathematician. No true Scotsman .... Right. THAT is the test. It always was. THAT? Having a completely flawed knowledge of mathematics? THAT'S THE TEST for being a true mathematician?? And most of you failed, long ago. > Let's see. We failed to believe your assertion that the integers are irrational. We failed to believe that the rationals with denominator equal to a power of 2 constitute the entire set of real numbers. We failed to buy your claim that your prime-counting algorithm was substantially different from Lagrange's method. We failed to buy your claim that the passage Show how all integral solutions of the equation x^2-Dy^2 =-1 may be obtained from one of them. D being as usual a non- sqaure positive integer. SUGGESTION - Observe that the relations a^2-Db^2 =-1 c^2 -Dg^2 =-1 imply the relation (ac +Dbg) -D(ag+bc)^2 =1 as an exercise in Carmichaels's 1915 text on Diophantine analysis doesn't subsume your more recent claim of originality of the same exact result. After all, what true Scotsman would lay claim to an *exercise* in a text that's close to 100 years old? > All that has remained is the clean-up. > Well, just as long as you wipe, things'll be just fine. Don't forget to flush & wash yer hands. James Harris All of this, and still you can't come up with single example of a lie. One wonders whether all this attention is worth the bother of humiliating yourself. I'd say it's about time to reach out to your contacts in the Powers That Be, and schedule them Congressional hearings. Dale === Subject: Re: JSH: What's remarkable <_mrPl.10563$im1.6745@nlpi061.nbdc.sbc.com> posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) On May 15, 9:34pm, W. Dale Hall > Years ago I discovered that posters would routinely lie to me in reply > and they'd lie about even the most basic mathematics. Yet you cannot display a single example. But what was remarkable to me was, other readers would side with them, > even on trivial errors. Such as? Given ANY set of non-zero integer solutions to the negative Pell's equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a contemporary mathematical textbook. James Harris === Subject: Re: JSH: What's remarkable > Years ago I discovered that posters would routinely lie to me in reply > and they'd lie about even the most basic mathematics. > Yet you cannot display a single example. > But what was remarkable to me was, other readers would side with them, > even on trivial errors. > Such as? Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a contemporary mathematical textbook. http://mathforum.org/kb/plaintext.jspa?messageID=6709152 I gave up on the math world. What happened? Did you miss us? :-) Jose Carlos Santos === Subject: Re: JSH: What's remarkable sha1:kANfrBNeCpNdFtyE1ZjW27TVaeA= > We failed to believe that the rationals with denominator equal to a > power of 2 constitute the entire set of real numbers. Ooh, that's a good one! I don't remember that one. Do you have a link? -- You got more out of it than I put into it last night. Who were you thinking of when we were loving last night? -- Texas Tornadoes === Subject: Re: JSH: What's remarkable > We failed to believe that the rationals with denominator equal to a > power of 2 constitute the entire set of real numbers. Ooh, that's a good one! I don't remember that one. Do you have a > link? > I had to piece it out of a few sources: First, grabbed from Math Forum, JSH claims that if you can express a convergent series in an infinite ring, then the ring must support the summation to yield an element of the ring. http://mathforum.org/kb/plaintext.jspa?messageID=4508447 I couldn't locate this in the Google archives of sci.math. A person wonders why. === Subject: Re: JSH: Issue of counterexamples ... The existence of the object ring does not in and of itself contradict Galois Theory or ideal theory, but once you have it, you can then easily find results, using non-polynomial factorizations that do contradict with standard usage of Galois Theory--doesn't refute Galois Theory itself, but just conclusions derived from parts of it--and does refute ideal theory. The problem with ideal theory isn't terribly complicated as there mathematicians tried to just define out convergent infinite series, when you can't. Just like mathematicians tried to define away two solutions for the sqrt() function, and you can't. I think part of the problem is that some mathematicians got the idea that if you have something you find inconvenient, like sqrt(2) having two solutions, a positive and a negative one, you can just say, by definition, it only has one. But the mathematics doesn't change because you try to define something away, and that's what sinks ideal theory. Convergent infinite series cannot be blocked from an infinite ring, so ideal theory fails. ... Then, the main event: putting it all together. Local: Thurs, Mar 9 2006 8:32 pm === Subject: JSH: Putting it all together ... You see, if you append 1/2 to the ring of integers, you get the field of reals. If you append 1/2 and i to the ring of integers, you get the field of complex numbers. I attempted to convince JSH that there is a *minimal* ring containing Z and 1/2, but he couldn't be wrestled off the cardinality notion of minimal. Ditto for alternate metrics (and with them, alternate forms of convergence). Dale === Subject: Re: JSH: What's remarkable sha1:eUKPw9z2OA6mPWnWXqIyfKdadAE= > Then, the main event: putting it all together. Local: Thurs, Mar 9 2006 8:32 pm === > Subject: JSH: Putting it all together ... You see, if you append 1/2 to the ring of integers, you > get the field of reals. If you append 1/2 and i to the > ring of integers, you get the field of complex numbers. > I attempted to convince JSH that there is a *minimal* ring containing Z > and 1/2, but he couldn't be wrestled off the cardinality notion of > minimal. Ditto for alternate metrics (and with them, alternate forms > of convergence). -- Jesse F. Hughes The Cantorians are conducting a campaign of psychological warfare against humanity. -- David Petry, on why set theory is evil. === Subject: Re: JSH: What's remarkable posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; InfoPath.1),gzip(gfe),gzip(gfe) > James Harris What's remarkable is that you have had your head deeply implanted in your ass. It is so deep, you have not seen sunlight for 14 years. So, you lie, cheat and try to convince people with the garbage you call math. You are a liar, a cheat and a charlatan. You are a super troll, a crank and a kook! Delusional narcissist! Eat you little mathurbating bitch! ~A === Subject: Re: Solution manual to Strength of Materials; 4 edition by: Ferdinand L. Singer, Andrew Pytel posting-account=PsLPIQoAAAAIWMQ0G85FCjIXDub5FY6K SV1),gzip(gfe),gzip(gfe) > Can someone upload the complete solution manual for Solution manual > to Strength of Materials; 4 edition by: Ferdinand L. Singer, Andrew > Pytel. === Subject: Re: JSH: What's remarkable posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) Years ago I discovered that posters would routinely lie to me in reply > and they'd lie about even the most basic mathematics. > Yet you cannot display a single example. But what was remarkable to me was, other readers would side with them, > even on trivial errors. > Such as? Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a contemporary mathematical textbook. > http://mathforum.org/kb/plaintext.jspa?messageID=6709152 I gave up on the math world. What happened? Did you miss us? :-) The poster I replied to was suggesting some circuitous route to publication where I used a pseudonym to try and get published in a print journal, and later came forward (assuming I got published I guess) with a paper in my real name, which to me is about needing to convince people I see as corrupted. Corrupted people can't be convinced. I'm sure mathematicians who come across my research know it's correct anyway. So, in answer, I gave up on the math world a long time ago. And am not willing to jump through hoops in a vain attempt at convincing people who are willfully wrong! However I can beat up on your with simple results!!! Maybe some of your cow-like followers who waste their lives and mental energy on junk math people like you spew into their minds will give up their blind servitude. It doesn't take much effort to toss out a post, so why not try? Those not cows can notice once again an avoidance of the simple math result. James Harris === Subject: Re: JSH: What's remarkable Years ago I discovered that posters would routinely lie to me in reply > and they'd lie about even the most basic mathematics. > Yet you cannot display a single example. > But what was remarkable to me was, other readers would side with them, > even on trivial errors. > Such as? > Given ANY set of non-zero integer solutions to the negative Pell's > equation > j^2 - Dk^2 = -1 > you will ALWAYS have a solution to Pell's Equation > x^2 - Dy^2 = 1 > from x = 2j^2 + 1. > That is a mathematical absolute. > Now go try to find it in a contemporary mathematical textbook. > http://mathforum.org/kb/plaintext.jspa?messageID=6709152 > I gave up on the math world. What happened? Did you miss us? :-) The poster I replied to was suggesting some circuitous route to > publication where I used a pseudonym to try and get published in a > print journal, and later came forward (assuming I got published I > guess) with a paper in my real name, which to me is about needing to > convince people I see as corrupted. Corrupted people can't be convinced. I'm sure mathematicians who come > across my research know it's correct anyway. Even that research that you acknowledge that it is wrong? > So, in answer, I gave up on the math world a long time ago. And am > not willing to jump through hoops in a vain attempt at convincing > people who are willfully wrong! However I can beat up on your with simple results!!! Maybe some of your cow-like followers who waste their lives and mental > energy on junk math people like you spew into their minds will give up > their blind servitude. It doesn't take much effort to toss out a post, so why not try? Those not cows can notice once again an avoidance of the simple math > result. When will the non-cows start to rate each of your posts with five stars? They can do that anonymously, but, even so, they don't do it. Do you have a reason for that? BTW, how many copies of your book have you sold so far, besides the one bought by you? Jose Carlos Santos === Subject: Re: JSH: What's remarkable <778cclF1fjbdqU1@mid.individual.net> posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) Years ago I discovered that posters would routinely lie to me in reply > and they'd lie about even the most basic mathematics. > Yet you cannot display a single example. > But what was remarkable to me was, other readers would side with them, > even on trivial errors. > Such as? > Given ANY set of non-zero integer solutions to the negative Pell's > equation > j^2 - Dk^2 = -1 > you will ALWAYS have a solution to Pell's Equation > x^2 - Dy^2 = 1 > from x = 2j^2 + 1. > That is a mathematical absolute. > Now go try to find it in a contemporary mathematical textbook. >http://mathforum.org/kb/plaintext.jspa?messageID=6709152 > I gave up on the math world. What happened? Did you miss us? :-) The poster I replied to was suggesting some circuitous route to > publication where I used a pseudonym to try and get published in a > print journal, and later came forward (assuming I got published I > guess) with a paper in my real name, which to me is about needing to > convince people I see as corrupted. Corrupted people can't be convinced. I'm sure mathematicians who come > across my research know it's correct anyway. Even that research that you acknowledge that it is wrong? Of course not. > So, in answer, I gave up on the math world a long time ago. And am > not willing to jump through hoops in a vain attempt at convincing > people who are willfully wrong! However I can beat up on your with simple results!!! Maybe some of your cow-like followers who waste their lives and mental > energy on junk math people like you spew into their minds will give up > their blind servitude. It doesn't take much effort to toss out a post, so why not try? Those not cows can notice once again an avoidance of the simple math > result. When will the non-cows start to rate each of your posts with five stars? Group behavior. So your group has obsessive people who rate very single post I make with a single star, so what? Their obsession is telling. Oh, for those who don't know, there is at least one poster who (last time I checked, I did quit checking) immediately rates every single post I make, with a single star. He's quite an obsessive fellow. Doesn't usually take him long to get to a post!!! > They can do that anonymously, but, even so, they don't do it. Do you > have a reason for that? I see it as basic human psychology. > BTW, how many copies of your book have you sold so far, besides the one > bought by you? 0 last time I checked. Jose Carlos Santos So the arrogant mathematician cites group behavior as his best evidence? I don't get high ratings in Google's star system. I have yet to sell a single copy of my book on my math research. I'm not winning in the court of public opinion. But I make my point that modern mathematicians rely on it. Your world is corrupted. You NEED people to believe in you. Here's some simple math: Given integer solutions to j^2 - Dk^2 = -1, you have solutions to Pell's Equation x^2 - Dy^2 = 1, from x = 2j^2 + 1. How many math texts can you give voting for that result? How many books can you list that give it? Want to vote down mathematical knowledge too? Human beings make mistakes. That's why mathematicians supposedly use a higher standard. You, clearly, care more about voting. I say, that's because you NEED group effects. You're a politician. James Harris === Subject: Re: JSH: What's remarkable posting-account=aLpfCwoAAACh4BOs3HOlQBCoxUpEgyxc Gecko/2009032609 Firefox/3.0.8,gzip(gfe),gzip(gfe) [...] Group behavior. So your group has obsessive people who rate very single post I make > with a single star, so what? Their obsession is telling. Oh, for those who don't know, there is at least one poster who (last > time I checked, I did quit checking) immediately rates every single > post I make, with a single star. He's quite an obsessive fellow. > Doesn't usually take him long to get to a post!!! They can do that anonymously, but, even so, they don't do it. Do you > have a reason for that? I see it as basic human psychology. BTW, how many copies of your book have you sold so far, besides the one > bought by you? 0 last time I checked. > Jose Carlos Santos So the arrogant mathematician cites group behavior as his best > evidence? I don't get high ratings in Google's star system. I have yet to sell > a single copy of my book on my math research. I'm not winning in the court of public opinion. But I make my point > that modern mathematicians rely on it. Your world is corrupted. You NEED people to believe in you. Here's some simple math: Given integer solutions to j^2 - Dk^2 = -1, you have solutions to > Pell's Equation x^2 - Dy^2 = 1, from x = 2j^2 + 1. How many math texts can you give voting for that result? How many books can you list that give it? Want to vote down mathematical knowledge too? Human beings make mistakes. That's why mathematicians supposedly use > a higher standard. You, clearly, care more about voting. I say, that's because you NEED group effects. It's most peculiar to hear you decry voting and group effects in this way - after all, there was a time when you /urged/ people to use Google's rating system to downrate posts they didn't like, and also speculated that the resulting group effect could work to your advantage: People coming in with Google Groups can rate posts, and I'd like more people to start rating to help me pick which people to reply to. [...] I think these people are more thin-skinned than many of you realize, and will fear getting down-rated with their postings--if enough people do it! Then when a poster has hundreds of people rating their posts as worthless, it's harder for that person to believe they are the voice of the newsgroup. It takes a will to do it, and an understanding of how the process can work, as it should become second nature for readers through Google Groups to rate posts after they read them. ALL of them, and not just ones in my threads. Rate everybody, as it's easy. And then step back to see what happens... (from http://mathforum.org/kb/message.jspa?messageID=5394820 ) Evidently you've changed your mind about that. I wonder why? === Subject: Re: JSH: What's remarkable posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) [...] Group behavior. So your group has obsessive people who rate very single post I make > with a single star, so what? Their obsession is telling. Oh, for those who don't know, there is at least one poster who (last > time I checked, I did quit checking) immediately rates every single > post I make, with a single star. He's quite an obsessive fellow. > Doesn't usually take him long to get to a post!!! They can do that anonymously, but, even so, they don't do it. Do you > have a reason for that? I see it as basic human psychology. BTW, how many copies of your book have you sold so far, besides the one > bought by you? 0 last time I checked. > Jose Carlos Santos So the arrogant mathematician cites group behavior as his best > evidence? I don't get high ratings in Google's star system. I have yet to sell > a single copy of my book on my math research. I'm not winning in the court of public opinion. But I make my point > that modern mathematicians rely on it. Your world is corrupted. You NEED people to believe in you. Here's some simple math: Given integer solutions to j^2 - Dk^2 = -1, you have solutions to > Pell's Equation x^2 - Dy^2 = 1, from x = 2j^2 + 1. How many math texts can you give voting for that result? How many books can you list that give it? Want to vote down mathematical knowledge too? Human beings make mistakes. That's why mathematicians supposedly use > a higher standard. You, clearly, care more about voting. I say, that's because you NEED group effects. It's most peculiar to hear you decry voting and group effects in this > way - after all, there was a time when you /urged/ people to use > Google's rating system to downrate posts they didn't like, and also > speculated that the resulting group effect could work to your > advantage: Yeah, I was wrong. > People coming in with Google Groups can rate posts, and I'd like > more people to start rating to help me pick which people to reply > to. [...] I think these people are more thin-skinned than many of you > realize, and will fear getting down-rated with their postings--if > enough people do it! Then when a poster has hundreds of people rating their posts as > worthless, it's harder for that person to believe they are the > voice of the newsgroup. It takes a will to do it, and an understanding of how the process > can work, as it should become second nature for readers through > Google Groups to rate posts after they read them. ALL of them, and not just ones in my threads. Rate everybody, as it's easy. And then step back to see what > happens... (fromhttp://mathforum.org/kb/message.jspa?messageID=5394820) Evidently you've changed your mind about that. I wonder why? It didn't work in my favor!!! Why else? JSH === Subject: Re: JSH: What's remarkable posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) [...] Group behavior. So your group has obsessive people who rate very single post I make > with a single star, so what? Their obsession is telling. Oh, for those who don't know, there is at least one poster who (last > time I checked, I did quit checking) immediately rates every single > post I make, with a single star. He's quite an obsessive fellow. > Doesn't usually take him long to get to a post!!! They can do that anonymously, but, even so, they don't do it. Do you > have a reason for that? I see it as basic human psychology. BTW, how many copies of your book have you sold so far, besides the one > bought by you? 0 last time I checked. > Jose Carlos Santos So the arrogant mathematician cites group behavior as his best > evidence? I don't get high ratings in Google's star system. I have yet to sell > a single copy of my book on my math research. I'm not winning in the court of public opinion. But I make my point > that modern mathematicians rely on it. Your world is corrupted. You NEED people to believe in you. Here's some simple math: Given integer solutions to j^2 - Dk^2 = -1, you have solutions to > Pell's Equation x^2 - Dy^2 = 1, from x = 2j^2 + 1. How many math texts can you give voting for that result? How many books can you list that give it? Want to vote down mathematical knowledge too? Human beings make mistakes. That's why mathematicians supposedly use > a higher standard. You, clearly, care more about voting. I say, that's because you NEED group effects. It's most peculiar to hear you decry voting and group effects in this > way - after all, there was a time when you /urged/ people to use > Google's rating system to downrate posts they didn't like, and also > speculated that the resulting group effect could work to your > advantage: Yeah, I was wrong. People coming in with Google Groups can rate posts, and I'd like > more people to start rating to help me pick which people to reply > to. [...] I think these people are more thin-skinned than many of you > realize, and will fear getting down-rated with their postings--if > enough people do it! Then when a poster has hundreds of people rating their posts as > worthless, it's harder for that person to believe they are the > voice of the newsgroup. It takes a will to do it, and an understanding of how the process > can work, as it should become second nature for readers through > Google Groups to rate posts after they read them. ALL of them, and not just ones in my threads. Rate everybody, as it's easy. And then step back to see what > happens... (fromhttp://mathforum.org/kb/message.jspa?messageID=5394820) Evidently you've changed your mind about that. I wonder why? It didn't work in my favor!!! Why else? > JSH/JSH PIN: 3037a7ab MSISDN: 818 430 4586 Device Type: BlackBerry 8330 Application Version: v4.5.0.77 Platform Version: 3.2.0.51 Service Books: WAPPushConfig,MMS,WPTCP,IPPP,BrowserConfig,BBIMConfig,YHO,LbsConfig,BBIMConf ig,BBIMConfig,IPPP,BrowserConfig,BBIM,BBIM,PROVISIONING,CMIME,CICAL,CMIME,CI C AL Free File Space: 34452974 bytes Radio Data Activation: No Signal Level: Unknown Radio Access: Unknown Network: Unknown IP Address: Unknown ICMP Ping Echo: No BlackBerry Registration: No Connected to BlackBerry: No BlackBerry PIN Email: No Server Name: GBISXNAC01S03 Email Address: m.mm@vzw.blackberry.net Connection to m.mm@vzw.blackberry.net : No Server Name: GBISXNAC01S07 Email Address: marty.musatov@gmail.com Connection to marty.musatov@gmail.com : No Sent from my Verizon Wireless BlackBerry And so random fodder for the cannon: Recent balance: é$6,244 as of Apr 2009 scheduled to March 2012. Address identification 399690786. Alpha+Omega=Primes. P6=PN. === Subject: Re: JSH: What's remarkable posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) Years ago I discovered that posters would routinely lie to me in reply > and they'd lie about even the most basic mathematics. > Yet you cannot display a single example. > But what was remarkable to me was, other readers would side with them, > even on trivial errors. > Such as? > Given ANY set of non-zero integer solutions to the negative Pell's > equation > j^2 - Dk^2 = -1 > you will ALWAYS have a solution to Pell's Equation > x^2 - Dy^2 = 1 > from x = 2j^2 + 1. > That is a mathematical absolute. > Now go try to find it in a contemporary mathematical textbook. >http://mathforum.org/kb/plaintext.jspa?messageID=6709152 > I gave up on the math world. What happened? Did you miss us? :-) The poster I replied to was suggesting some circuitous route to > publication where I used a pseudonym to try and get published in a > print journal, and later came forward (assuming I got published I > guess) with a paper in my real name, which to me is about needing to > convince people I see as corrupted. Corrupted people can't be convinced. I'm sure mathematicians who come > across my research know it's correct anyway. Even that research that you acknowledge that it is wrong? Of course not. So, in answer, I gave up on the math world a long time ago. And am > not willing to jump through hoops in a vain attempt at convincing > people who are willfully wrong! However I can beat up on your with simple results!!! Maybe some of your cow-like followers who waste their lives and mental > energy on junk math people like you spew into their minds will give up > their blind servitude. It doesn't take much effort to toss out a post, so why not try? Those not cows can notice once again an avoidance of the simple math > result. When will the non-cows start to rate each of your posts with five stars? Group behavior. So your group has obsessive people who rate very single post I make > with a single star, so what? Their obsession is telling. Oh, for those who don't know, there is at least one poster who (last > time I checked, I did quit checking) immediately rates every single > post I make, with a single star. He's quite an obsessive fellow. > Doesn't usually take him long to get to a post!!! They can do that anonymously, but, even so, they don't do it. Do you > have a reason for that? I see it as basic human psychology. BTW, how many copies of your book have you sold so far, besides the one > bought by you? 0 last time I checked. > Jose Carlos Santos So the arrogant mathematician cites group behavior as his best > evidence? I don't get high ratings in Google's star system. I have yet to sell > a single copy of my book on my math research. I'm not winning in the court of public opinion. But I make my point > that modern mathematicians rely on it. Your world is corrupted. You NEED people to believe in you. Here's some simple math: Given integer solutions to j^2 - Dk^2 = -1, you have solutions to > Pell's Equation x^2 - Dy^2 = 1, from x = 2j^2 + 1. How many math texts can you give voting for that result? How many books can you list that give it? Want to vote down mathematical knowledge too? Human beings make mistakes. That's why mathematicians supposedly use > a higher standard. You, clearly, care more about voting. I say, that's because you NEED group effects. You're a politician. > James Harris subjected to abuse JSH does not speak for me : 2nd, I will take this opportunity to mathematically correct a few unwritten assumptions: a)If the equation x2 -- dy2 = k is solvable in integers, then in particular x2 ... For each prime number p, examine the structure of Quantum Algorithm for Pell's Equation Quantum NPÁ.83co-NP. Regulator. Bigger picture. * Hallgren's can be expanded to solve other problems of fundamental algebraic number algorithms for Pell's Equation and the prinicipal: 2s 2 Z b/c p > 2 in denominator of s ) p ...http:// www.cs.washington.edu/homes/nchernia/new pell.pdf Chapter 8 : Continued Fraction and Pell's EquationQ2 = n-a12 ...... Qn = (n-Pn2) / Qn-1. pn, p1 = a1 p2 = a1 * a2+1 ... When Qn = 1, then pn-1, qn-1is the solution of Pell equation. http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math08/pell02.htm PELL'S EQUATION AND TWO GENERATOR FREE MOBIUS GROUPS and so if we choose // to be of the form 2 sin (pn/q), where p and q are coprime Pell's equation is. =q*-Np. (4.1) where N is a given positive integer http://blms.oxfordjournals.org/cgi/reprint/25/6/527.pdf Polynomial-Time Quantum Algorithms for Pell's Equation and he first is Pell's equation. Given a positive nonsquare integer d, .... In addition, while finding multiples of the regulator is in NP with .a6' Á K. The set of principal ideals will be denoted P, and it is a subgroup http://portal.acm.org/citation.cfm%3Fdoid%3D1206035.1206039 A = ( fV BIn fact, Theorem 2 also holds if p , q satisfy the negative Pell's equation and more generally, if p, q satisfy. (1.5) q. 2. -Np http://journals.cambridge.org/production/action/cjoGetFulltext%3Ffulltextid% 3D4912580 Fermat's Last Theorem: A Genetic Introduction to Algebraic Number then: 9 = nP ÁË p i2 = nQ ÁË q where the signs are both plus if Kk < 0 and both of Pell's equation when: A = 149. Zoo:B - Qwiki10 Feb 2009 ... BPPpath contains MA and P, and is contained in PP and BPP. Pell's equation and principal ideal problems [Hal02], and some other: http://qwiki.stanford.edu/wiki/Complexity Zoo:B Signed, Martin Musatov [P=NP, there is too much good in the world to absolutely insist on the impossibility of reason. --Martin Musatov] === Subject: Re: JSH: What's remarkable JSH, now you have NINE references to research. Read up, absorb the intellectualism and hard work of others, and show respect, dumbass. === Subject: Re: JSH: What's remarkable posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.19 (KHTML, like Gecko) Chrome/1.0.154.65 Safari/525.19,gzip(gfe),gzip(gfe) > Years ago I discovered that posters would routinely lie to me in reply > and they'd lie about even the most basic mathematics. > Yet you cannot display a single example. But what was remarkable to me was, other readers would side with them, > even on trivial errors. > Such as? Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a contemporary mathematical textbook. >http://mathforum.org/kb/plaintext.jspa?messageID=6709152 I gave up on the math world. What happened? Did you miss us? :-) The poster I replied to was suggesting some circuitous route to > publication where I used a pseudonym to try and get published in a > print journal, and later came forward (assuming I got published I > guess) with a paper in my real name, which to me is about needing to > convince people I see as corrupted. Corrupted people can't be convinced. I'm sure mathematicians who come > across my research know it's correct anyway. So, in answer, I gave up on the math world a long time ago. And am > not willing to jump through hoops in a vain attempt at convincing > people who are willfully wrong! However I can beat up on your with simple results!!! Should be you not your. But I think it interesting for readers who ever deluded themselves that with today's modern math society they could have their own math discoveries as reality is: not unless they let you. It's a political field and for those who wonder how you lie about math; it's how people lie about anything else!!! Given ANY set of non-zero integer solutions to the negative Pell's equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is mathematical fact. The result has no emotion. It doesn't care who delivered it. But math people DO so you get lies about a trivial result probably known to Fermat and Euler because posters know you can't go find this easy result in current mathematical literature, which means it should be noted, even if it is trivial, as mathematics is recorded. But then they need to say WHO noted it in modern times, which is me. So they make my point that it's not about whether or not I'm right. Their issue is with me. And they will lie about ANY result, no matter how easily you can verify they are lying. And what is done to me can be done to others. None of you can get a result through without the permission of the people who rule modern mathematics. None of you. James Harris === Subject: Re: JSH: What's remarkable posting-account=sKfmEQkAAAC8kI3Pv6_U_nt9sVsxZ_ou 1.1.4322),gzip(gfe),gzip(gfe) Years ago I discovered that posters would routinely lie to me in reply > and they'd lie about even the most basic mathematics. > Yet you cannot display a single example. But what was remarkable to me was, other readers would side with them, > even on trivial errors. > Such as? Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a contemporary mathematical textbook. >http://mathforum.org/kb/plaintext.jspa?messageID=6709152 I gave up on the math world. What happened? Did you miss us? :-) The poster I replied to was suggesting some circuitous route to > publication where I used a pseudonym to try and get published in a > print journal, and later came forward (assuming I got published I > guess) with a paper in my real name, which to me is about needing to > convince people I see as corrupted. Corrupted people can't be convinced. I'm sure mathematicians who come > across my research know it's correct anyway. So, in answer, I gave up on the math world a long time ago. And am > not willing to jump through hoops in a vain attempt at convincing > people who are willfully wrong! However I can beat up on your with simple results!!! Should be you not your. But I think it interesting for readers who ever deluded themselves > that with today's modern math society they could have their own math > discoveries as reality is: not unless they let you. It's a political field and for those who wonder how you lie about > math; it's how people lie about anything else!!! Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is mathematical fact. The result has no emotion. It doesn't > care who delivered it. But math people DO so you get lies about a trivial result probably > known to Fermat and Euler because posters know you can't go find this > easy result in current mathematical literature, which means it should > be noted, even if it is trivial, as mathematics is recorded. But then > they need to say WHO noted it in modern times, which is me. What about Solving the Pell Equation by Williams and Jacobson Page 32. Book published in 2009 Ed Barbeau;s book Pell's Equation , page 25 Excercise 3.4 Wllliams, Jacobson and Barbeau have therefore noted it in modern times. Are you saying they haven't, boted it in modern times ? === Subject: Re: JSH: What's remarkable posting-account=sKfmEQkAAAC8kI3Pv6_U_nt9sVsxZ_ou 1.1.4322),gzip(gfe),gzip(gfe) Years ago I discovered that posters would routinely lie to me in reply > and they'd lie about even the most basic mathematics. > Yet you cannot display a single example. But what was remarkable to me was, other readers would side with them, > even on trivial errors. > Such as? Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a contemporary mathematical textbook. >http://mathforum.org/kb/plaintext.jspa?messageID=6709152 I gave up on the math world. What happened? Did you miss us? :-) The poster I replied to was suggesting some circuitous route to > publication where I used a pseudonym to try and get published in a > print journal, and later came forward (assuming I got published I > guess) with a paper in my real name, which to me is about needing to > convince people I see as corrupted. Corrupted people can't be convinced. I'm sure mathematicians who come > across my research know it's correct anyway. So, in answer, I gave up on the math world a long time ago. And am > not willing to jump through hoops in a vain attempt at convincing > people who are willfully wrong! However I can beat up on your with simple results!!! Should be you not your. But I think it interesting for readers who ever deluded themselves > that with today's modern math society they could have their own math > discoveries as reality is: not unless they let you. It's a political field and for those who wonder how you lie about > math; it's how people lie about anything else!!! Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is mathematical fact. The result has no emotion. It doesn't > care who delivered it. But math people DO so you get lies about a trivial result probably > known to Fermat and Euler because posters know you can't go find this > easy result in current mathematical literature, which means it should > be noted, even if it is trivial, as mathematics is recorded. But then > they need to say WHO noted it in modern times, which is me. So they make my point that it's not about whether or not I'm right. Their issue is with me. And they will lie about ANY result, no matter > how easily you can verify they are lying. And what is done to me can be done to others. None of you can get a result through without the permission of the > people who rule modern mathematics. None of you. James Harris- Hide quoted text - - Show quoted text - === Subject: Re: JSH: What's remarkable > Years ago I discovered that posters would routinely lie to me in > reply > and they'd lie about even the most basic mathematics. > Yet you cannot display a single example. But what was remarkable to me was, other readers would side with > them, > even on trivial errors. > Such as? Given ANY set of non-zero integer solutions to the negative Pell's > equation j^2 - Dk^2 = -1 you will ALWAYS have a solution to Pell's Equation x^2 - Dy^2 = 1 from x = 2j^2 + 1. That is a mathematical absolute. Now go try to find it in a contemporary mathematical textbook. >http://mathforum.org/kb/plaintext.jspa?messageID=6709152 I gave up on the math world. What happened? Did you miss us? :-) > The poster I replied to was suggesting some circuitous route to > publication where I used a pseudonym to try and get published in a > print journal, and later came forward (assuming I got published I > guess) with a paper in my real name, which to me is about needing to > convince people I see as corrupted. > Corrupted people can't be convinced. I'm sure mathematicians who come > across my research know it's correct anyway. > So, in answer, I gave up on the math world a long time ago. And am > not willing to jump through hoops in a vain attempt at convincing > people who are willfully wrong! > However I can beat up on your with simple results!!! >Should be you not your. === Subject: Canon Pixma Pro9500 Canon CLI-8 8-Color Multipack Ink Tanks Price:$99.99 Image: http://bestdealfinder.us/image.php?id=B000IBPD0S Best deal: http://bestdealfinder.us/index.php?id=B000IBPD0S Canon PGI-9 Value Pack (1033B005) Price:$139.99 Image: http://bestdealfinder.us/image.php?id=B000P1BEFE Best deal: http://bestdealfinder.us/index.php?id=B000P1BEFE Canon Pixma Pro9000 Professional Large Format Inkjet Printer (9995A001) Price:$499.99 Image: http://bestdealfinder.us/image.php?id=B000J1HPK8 Best deal: http://bestdealfinder.us/index.php?id=B000J1HPK8 Epson R1900 Large Format Photo Printer Price:$915.00 Image: http://bestdealfinder.us/image.php?id=B0011G47PQ Best deal: http://bestdealfinder.us/index.php?id=B0011G47PQ Canon PGI-9 Photo Black Ink Tank (1034B002) Price:$15.99 Image: http://bestdealfinder.us/image.php?id=B000O9UUUM Best deal: http://bestdealfinder.us/index.php?id=B000O9UUUM Epson R2880 Large Format Photo Printer (C11CA16201) Price:$899.99 Image: http://bestdealfinder.us/image.php?id=B001A11KA2 Best deal: http://bestdealfinder.us/index.php?id=B001A11KA2 Canon Pixma PRO9000MkII Inkjet Photo Printer (3295B002) Price:$499.99 Image: http://bestdealfinder.us/image.php?id=B001R4BTIA Best deal: http://bestdealfinder.us/index.php?id=B001R4BTIA Epson Stylus Pro 3800 Printer Standard Model Photo Printer Price:$1,295.00 Image: http://bestdealfinder.us/image.php?id=B000ID3L50 Best deal: http://bestdealfinder.us/index.php?id=B000ID3L50 Canon Digital Art Paper Variety Pack (1822B001) Price:$50.60 Image: http://bestdealfinder.us/image.php?id=B000II3HW2 Best deal: http://bestdealfinder.us/index.php?id=B000II3HW2 HP Photosmart Pro B9180 Printer Price:$800.00 Image: http://bestdealfinder.us/image.php?id=B000GWMK8C Best deal: http://bestdealfinder.us/index.php?id=B000GWMK8C Canon CLI-8 8-Color Multipack Ink Tanks Price:$99.99 Image: http://bestdealfinder.us/image.php?id=B000IBPD0S Best deal: http://bestdealfinder.us/index.php?id=B000IBPD0S Canon PGI-9 Value Pack (1033B005) Price:$139.99 Image: http://bestdealfinder.us/image.php?id=B000P1BEFE Best deal: http://bestdealfinder.us/index.php?id=B000P1BEFE Canon Pixma Pro9000 Professional Large Format Inkjet Printer (9995A001) Price:$499.99 Image: http://bestdealfinder.us/image.php?id=B000J1HPK8 Best deal: http://bestdealfinder.us/index.php?id=B000J1HPK8 Epson R1900 Large Format Photo Printer Price:$915.00 Image: http://bestdealfinder.us/image.php?id=B0011G47PQ Best deal: http://bestdealfinder.us/index.php?id=B0011G47PQ Canon PGI-9 Photo Black Ink Tank (1034B002) Price:$15.99 Image: http://bestdealfinder.us/image.php?id=B000O9UUUM Best deal: http://bestdealfinder.us/index.php?id=B000O9UUUM Epson R2880 Large Format Photo Printer (C11CA16201) Price:$899.99 Image: http://bestdealfinder.us/image.php?id=B001A11KA2 Best deal: http://bestdealfinder.us/index.php?id=B001A11KA2 Canon Pixma PRO9000MkII Inkjet Photo Printer (3295B002) Price:$499.99 Image: http://bestdealfinder.us/image.php?id=B001R4BTIA Best deal: http://bestdealfinder.us/index.php?id=B001R4BTIA Epson Stylus Pro 3800 Printer Standard Model Photo Printer Price:$1,295.00 Image: http://bestdealfinder.us/image.php?id=B000ID3L50 Best deal: http://bestdealfinder.us/index.php?id=B000ID3L50 Canon Digital Art Paper Variety Pack (1822B001) Price:$50.60 Image: http://bestdealfinder.us/image.php?id=B000II3HW2 Best deal: http://bestdealfinder.us/index.php?id=B000II3HW2 HP Photosmart Pro B9180 Printer Price:$800.00 Image: http://bestdealfinder.us/image.php?id=B000GWMK8C Best deal: http://bestdealfinder.us/index.php?id=B000GWMK8C === Subject: Waterproof Case Canon Powershot A630 Canon WP-DC8 Waterproof Case for the Powershot A640 and A630 Price:$240.00 Image: http://bestdealfinder.us/image.php?id=B000KIR9F6 Best deal: http://bestdealfinder.us/index.php?id=B000KIR9F6 Underwater Housing Camera Case for Canon Powershot A720, A650, A630, A590, A580, A570, & A560 Digital Cameras (Rated Depth of 30') Price:$49.99 Image: http://bestdealfinder.us/image.php?id=B001B6469S Best deal: http://bestdealfinder.us/index.php?id=B001B6469S Canon Waterproof Case for the Canon PowerShot A630 and A640 -PLUS- 4 AA Rechargeable Batteries With Charger -PLUS- Canon WWDC1 Weight -PLUS- Accessory Kit Price:$199.95 Image: http://bestdealfinder.us/image.php?id=B000U527WE Best deal: http://bestdealfinder.us/index.php?id=B000U527WE === Subject: Re: JSH: EMIS has my old paper back up? > Submit your paper to a print journal under a pseudonym. > Leave out explicit statements that it proves there are > errors in the foundations of number theory. > Wait until your paper is published and irretrievably > distributed in printed copies of the journal. > Write another paper under your real name, citing your > earlier paper, showing that there are errors in the > foundations of number theory. > The evil, lying mathematicians won't have a chance. I gave up on the math world. I'm working now on financial problems-- Yours? -- Michael Press === Subject: Re: JSH: EMIS has my old paper back up? posting-account=mgs1FwoAAABD3j5T_RLZ06yrgt2dghDu Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > But instead we're muddling around here burning up our resources and > over populating--stupid species. to the solution by not reproducing. --