mm-4479 === Subject: I need a solution manual I need a solution manual for Dynamics by Meriam & Kraige for 6th edition THank you yjh8474(at)gmail.com === Subject: Re: Misnomer being taught in all sorts of math texts? What annoys me is really bad line breaks in Usenet posts. Unfortunately this is often related to the buggy nature of Outlook Express and > is not always visible to the user prior to posting. === Subject: Re: Misnomer being taught in all sorts of math texts? > What about using Google Groups, like I do? What about it? There is no need for you to impose your self-chosen masochism on those who have some choice, such as installing OE-QuoteFix or switching to a less ill-behaved news client. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Misnomer being taught in all sorts of math texts? >What annoys me is really bad line breaks in Usenet posts. > Unfortunately this is often related to the buggy nature of Outlook > Express and is not always visible to the user prior to posting. > What about using Google Groups, like I do? Google seems to aggravate the problem. For example, if I use Outlook Express and my lines exceed in length the character limit imposed by OE, then the text in Google groups will be screwed up. If you quote such a response, it will again be screwed up. I just checked some of my responses. They appear ok using my newserver, but they appear screwed up on Google. Just to test this, /this/ response, should show ok on independent newservers, but I bet that the line feeds will be screwed on Google. I don't know what Google does with line feeds. -- I.N. Galidakis === Subject: Re: Misnomer being taught in all sorts of math texts? <1194117810.647747@athprx04> <1194139267.651990@athprx04> What annoys me is really bad line breaks in Usenet posts. > Unfortunately this is often related to the buggy nature of Outlook > Express and is not always visible to the user prior to posting. What about using Google Groups, like I do? Google seems to aggravate the problem. For example, if I use Outlook Express and > my lines exceed in length the character limit imposed by OE, then the text in > Google groups will be screwed up. If you quote such a response, it will again be screwed up. I just checked some of my responses. They appear ok using my newserver, but they > appear screwed up on Google. Just to test this, /this/ response, should show ok > on independent newservers, but I bet that the line feeds will be screwed on > Google. I don't know what Google does with line feeds. > -- > I.N. Galidakis I know. Google does seem to be screwing it up :( And guess what, I only have Outlook Express as my newsreader for using my ISP's server... gah... === Subject: Re: Misnomer being taught in all sorts of math texts? <472c10ba$0$21148$7a628cd7@news.club-internet.fr> On Nov 2, 11:10 pm, Denis Feldmann mike3 a ?crit : Should one go back and edit all the old books > and then retroactively republish them? No way! > It's proven, so it's a theorem now, just make > that retroactive. This is not what he said. For 350 years, it was called FLT, not FLC... > Maybe it was a misnomer, but what I'm saying is that there's no way to change it as the past can't be changed, nor should it be changed now. > And what do you call the zero vector? (It's clearly not a vector, > since it has no direction and no magnitude!) > --- Christopher Heckman I call it a zero vector, of course, since it > *is* a vector, as a vector is defined as an > element of a vector space. By who? Otoh, what do you think of the Dirac function? How do you call it? > What definition are you using for a vector? As for Dirac's delta function, I'll call it Dirac's delta, or the Dirac delta. Is there something else using that name that I'm not aware of, though? === Subject: Re: Misnomer being taught in all sorts of math texts? mike3 a .8ecrit : > On Nov 2, 11:10 pm, Denis Feldmann mike3 a .8ecrit : > Should one go back and edit all the old books > and then retroactively republish them? No way! > It's proven, so it's a theorem now, just make > that retroactive. > This is not what he said. For 350 years, it was called FLT, not FLC... Maybe it was a misnomer, but what I'm saying is that > there's no way to change it as the past can't be > changed, nor should it be changed now. ' And what do you call the zero vector? (It's clearly not a vector, > since it has no direction and no magnitude!) > --- Christopher Heckman > I call it a zero vector, of course, since it > *is* a vector, as a vector is defined as an > element of a vector space. > By who? Otoh, what do you think of the Dirac function? How do you call it? What definition are _you_ using for a vector? Depends. Vectors of (classical) geometry are equivalence classes of couples of points, for instance. > As for Dirac's delta function, I'll > call it Dirac's delta, or the Dirac delta. > Is there something else using that name that I'm > not aware of, though? > === Subject: Re: Misnomer being taught in all sorts of math texts? <472c10ba$0$21148$7a628cd7@news.club-internet.fr> <472d6825$0$21145$7a628cd7@news.club-internet.fr> On Nov 3, 11:35 pm, Denis Feldmann mike3 a .8ecrit : couples of points, for instance. > Well, alright, but I fail to see how that would exclude the zero vector. === Subject: Re: Misnomer being taught in all sorts of math texts? <472c10ba$0$21148$7a628cd7@news.club-internet.frOn Nov 2, 11:10 pm, Denis Feldmann Should one go back and edit all the old books > and then retroactively republish them? No way! > It's proven, so it's a theorem now, just make > that retroactive. This is not what he said. For 350 years, it was called FLT, not FLC... Maybe it was a misnomer, but what I'm saying is that > there's no way to change it as the past can't be > changed, nor should it be changed now. Then why are you making such a big stink about multivalued functions being a misnomer? The same argument applies to IT. > And what do you call the zero vector? (It's clearly not a vector, > since it has no direction and no magnitude!) I call it a zero vector, of course, since it > *is* a vector, as a vector is defined as an > element of a vector space. By who? Otoh, what do you think of the Dirac function? How do you call it? What definition are you using for a vector? > As for Dirac's delta function, I'll > call it Dirac's delta, or the Dirac delta. Then just call a multivalued function a multivalued. Geez. What a whiner. --- Christopher Heckman > Is there something else using that name that I'm > not aware of, though? === Subject: Re: Misnomer being taught in all sorts of math texts? <472c10ba$0$21148$7a628cd7@news.club-internet.fr > On Nov 2, 11:10 pm, Denis Feldmann Should one go back and edit all the old books > and then retroactively republish them? No way! > It's proven, so it's a theorem now, just make > that retroactive. This is not what he said. For 350 years, it was called FLT, not FLC... Maybe it was a misnomer, but what I'm saying is that > there's no way to change it as the past can't be > changed, nor should it be changed now. Then why are you making such a big stink about multivalued functions > being a misnomer? The same argument applies to IT. > Because that cannot be changed, like I said, and it is now no longer a misnomer. It would be silly to whine about it. That is not the case with multivalued function -- it's still as misnomery as ever, I'm afraid! As for Dirac's delta function, I'll > call it Dirac's delta, or the Dirac delta. Then just call a multivalued function a multivalued. Geez. What a whiner. > The point of the Dirac delta thing was to show an alternative for that case, not to suggest a general scheme (Hey! Let's drop the offending term!), especially not when there's a term that would easily drop in in place of function and make a perfectly good name: relation. So one speaks of multivalued relations. One can even say the relation is multivalued, too. === Subject: Re: Misnomer being taught in all sorts of math texts? > This is not what he said. For 350 years, it was called FLT, not FLC... > Maybe it was a misnomer, but what I'm saying is that > there's no way to change it as the past can't be > changed, nor should it be changed now. > Then why are you making such a big stink about multivalued functions > being a misnomer? The same argument applies to IT. > Because that cannot be changed, like I said, and it is now > no longer a misnomer. It would be silly to whine about it. > That is not the case with multivalued function -- it's > still as misnomery as ever, I'm afraid! Fermat's last theorem is also a misnomer, since it suggests that the theorem is *Fermat's* theorem. But that can be true only if Fermat is one credited with proving the theorem and the overwhelming majority of mathematicians think this is very unlikely. So the fact that it is now a theorem does not make the term FLT any less problematic. Anyway, nice to see that a man passionate about language doesn't mind using terms like misnomery. -- These mathematicians are worse than communists, as how do you explain their behavior? I *am* the American Dream, fighting for what should be mine, having to get past weak-minded academics who are fighting to block my success. But I shall prevail!!! -- James S. Harris === Subject: Re: Misnomer being taught in all sorts of math texts? > Anyway, nice to see that a man passionate about language doesn't mind > using terms like misnomery. Language pedantry as a hobby is often enlivened by healthy linguistic incompetence and idiosyncratic pet peeves. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Misnomer being taught in all sorts of math texts? |Did you know that the term multivalued function is a misnomer? No, it's not. I do know that a multivalued function is not always a function. I remember a guy describing to me his thesis advisor's high standards for thesis writing, which included among other things requiring that any term of the form {adjective} {noun} be used only when the term given by the {noun} is already defined, and that it be a special case of it. I can understand why one might prefer it that way. But it's not a requirement for everybody. (Who would have the authority to require it?) So it's not a misnomer to apply the term to cases where the base term does not apply; one is merely using the compound term as it is defined. It's not so rare in mathematical terminology for an adjective to take away a property that is assumed as part of the definition of the base term, as with multivalued functions. A weak equivalence is not necessarily an equivalence. Noncommutative rings and fields are not necessarily rings or fields. A virtual group is not always a group. Some adjectives are just for emphasis. A formal power series is a power series, but there's no such thing as as an informal power series. Keith Ramsay === Subject: Re: Misnomer being taught in all sorts of math texts? > |Did you know that the term multivalued function is a misnomer? No, it's not. I do know that a multivalued function is not > always a function. > I do too. But that does not make it any less of a misnomer. See below. > I remember a guy describing to me his thesis advisor's high > standards for thesis writing, which included among other > things requiring that any term of the form {adjective} > {noun} be used only when the term given by the {noun} is > already defined, and that it be a special case of it. I can > understand why one might prefer it that way. But it's not > a requirement for everybody. (Who would have the authority > to require it?) So it's not a misnomer to apply the term > to cases where the base term does not apply; one is merely > using the compound term as it is defined. It's not so rare in mathematical terminology for an > adjective to take away a property that is assumed > as part of the definition of the base term, as with > multivalued functions. A weak equivalence is not > necessarily an equivalence. Noncommutative rings and > fields are not necessarily rings or fields. A virtual > group is not always a group. Some adjectives are just for emphasis. A formal power > series is a power series, but there's no such thing as as > an informal power series. Keith Ramsay Well alright, a multivalued function is not a type of function, but the terminology may be confusing. Also, I don't like misnomers. It is not whether or not it is a function that makes the term a misnomer, it's that the name seems to say something about what it refers to that is not so, namely that that it refers to a type of function, even though there is no such thing as a function that gives multiple values. Anything that does that is not a function! What's perhaps much worse though is crap like this function may be multivalued, which is indeed using multivalued like it qualifies a specific type of function, even though there is NO type of function which is multivalued -- the very definition of function forbids it! This is DEFINITELY a no-no no matter what. What is the rationale behind the choosing of the term multivalued function, anyway? === Subject: Re: Misnomer being taught in all sorts of math texts? It is not whether or not it is a function that makes > the term a misnomer, it's that the name seems to > say something about what it refers to that is not > so, namely that that it refers to a type of function, > even though there is no such thing as a function > that gives multiple values. Anything that does that > is not a function! You're being way to literal. Adjectives *usually* work like that, but they're not *limited* to working like that. > What's perhaps much worse though is crap like this > function may be multivalued, which is indeed using > multivalued like it qualifies a specific type of > function, even though there is NO type of function > which is multivalued -- the very definition of > function forbids it! This is DEFINITELY a no-no > no matter what. Are you having some trouble understanding what the sentence is supposed to mean? No? I don't think anyone else is either. > What is the rationale behind the choosing of the term > multivalued function, anyway? Where did you get the idea that language formation happens because of some rationale? Marshall === Subject: Re: Misnomer being taught in all sorts of math texts? It is not whether or not it is a function that makes > the term a misnomer, it's that the name seems to > say something about what it refers to that is not > so, namely that that it refers to a type of function, > even though there is no such thing as a function > that gives multiple values. Anything that does that > is not a function! You're being way to literal. Adjectives *usually* > work like that, but they're not *limited* to working > like that. > So then you think that it should be taken less literally, and that multivalued need not be a qualifier? Then what do you call ...the function becomes multivalued...? Is not that using it as a qualifier? > What's perhaps much worse though is crap like this > function may be multivalued, which is indeed using > multivalued like it qualifies a specific type of > function, even though there is NO type of function > which is multivalued -- the very definition of > function forbids it! This is DEFINITELY a no-no > no matter what. Are you having some trouble understanding what > the sentence is supposed to mean? No? I don't > think anyone else is either. > No, I have no trouble understanding it, but it is still an abuse of the language, as you don't split a compound term like that and treat it like a term plus a qualifier, which it cannot be (as multivaluedness is not a property any functions have.). > What is the rationale behind the choosing of the term > multivalued function, anyway? Where did you get the idea that language formation > happens because of some rationale? > Somebody coined the term. They had to have a rationale. But then again perhaps it was never recorded, in which case we'll never know. > Marshall === Subject: Re: Misnomer being taught in all sorts of math texts? > What is the rationale behind the choosing of the term > multivalued function, anyway? That it conveys what it should to almost everyone? Indeed, it only seems to give problems to those people who try their best to misunderstand it. It doesn't seem that puzzling to me. A multivalued function from A to B is really just a function A -> PB (P the powerset functor), right? Seems a pretty harmless abuse of nomenclature to call it a multivalued function A -> B. No more than three minutes instruction will dispel any confusion. -- Jesse F. Hughes Of course, my ability to admit my mistakes and correct them is a trait that many of you seem to never have properly appreciated. -- JSH, discussing his 1463rd proof of Fermat's Last Theorem. === Subject: solutions manual of david romer I want the solutions manual of (advanced macroeconomics, david romer) that has all the problems not just the odds or even problems, If you have this solution manual please email and let me know === Subject: identity matrix Q.? if i have a SVD decomposition vector of a matrix into UEV* does UU*=Identity matrix and does U*U= identity matrix === Subject: Re: identity matrix Q.? > if i have a SVD decomposition vector of a matrix into UEV* does > UU*=Identity matrix ' and does U*U= identity matrix It depends on your convention. If your E has the same shape as the matrix you're decomposing, then U and V are (square) unitary matrices, so U U* = U* U = I. However, sometimes a compact SVD is used, with only the columns of U and rows of V* corresponding to the nonzero singular values. Then U is only a partial isometry, and U U* will not be the identity matrix (it can't have full rank, if U has more rows than columns). U* U, however, should still be the identity matrix. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Solutions Manuals really need Kittel Introduction to Solid State solutions manual please email at sheelshah@gmail.com === Subject: Re: Problem related to a linear regression All I mean is that the numerical value of Xf that you have at this > stage is not supposed to be close to the actual value of Y. For > example, in Test_1.gif, you have a point at, say, Xf = 1.395, Y = 15, > to pick an example at random. And 1.395 is nowhere near 15. All you've > done so far is arrange the Xf to be closely correlated to Y, which > means that there is a good linear relationship between Xf and Y -- or > as good as you can get with your model. This relationship could be > anything of the form y = A + B*x, where A and B might be any numbers > at all (you'll find them in the next step). Only in the special case A > = 0, B = 1 will Xf be the final estimate of Y. Ok, I got this finally; yes, the Xf values do not have to correspond > directly to the Y values. So, which method should you use? Well, since you want to estimate Y > from Xf', it would seem that minimising squared y-distances (or > possibly minimising absolute y-differences) makes more sense than > minimising squared x-differences or ODR, even though it may look > wrong. Well, that's how I started by minimising - ignoring now u and v - > SUM(Y-F)^2> Given the F-parameters that you have, you want the generated > Xf' to be as near as possible to the corresponding real Y; > yes > you are not > interested in the extent to which you need to vary the parameters > supplied to F in order to get Xf' to exactly equal the real Y. If I understand you right here: yes; the Y values have a 'natural' > standard deviation, i.e. it would be wrong to reduce the standard > deviation obtained by an increased number of observations below their > 'natural' limit. I'm not sure that's exactly what I meant... though maybe we're not > disagreeing here. Anyway, I put a diagram athttp://img100.imageshack.us/img100/1439/regression2dn1.gif. You are > given Xf, and you estimate Y as Y = A + B*Xf. It seems to me that you > want to choose A and B so as to minimise the dy's (or actually dy^2 if > you're using least squares) -- so that the predicted Y is as close as > possible to the actual Y. If you do it the other way and choose A and > B to minimise the dx's (or dx^2) then you are minimising a quantity > that, as far as I can tell, isn't very relevant to what you're trying > to achieve. The dx's only came in question when I saw the graphs and had the > impression that there would be a better regression than the one I > already had. The regression line obtained by minimising the dx's also > had a noticeable lower standard deviation which I interpreted as > having a higher probability for the estimate to be close to the real > Y. Finally, it strikes me that this is a kind of roundabout way of doing > things (probably you have very good reasons!). That's why I mentioned at the very beginning that I probably make a > detour. > As mentioned above I started finding the parameters of a correlating > function by minimising SUM(Y-F)^2, then I tried to 'match' the results > to the size range of the observed values and gaining the impression > that the estimations could be further improved when looking at the > graph.> I don't really see why > you don't fix up your optimisation of F so that it gives you the final > estimate for Y (using whatever best fit criteria you choose), thus > obviating the need for the separate regression step. But how? The problem is still how the values of F providing the 'best' > correlation can be 'matched' to the observed values Y? It's hard to be specific because I don't know the details of what > you're doing there, but let me try to illustrate with an example. Say you have a parameter T (I think you mentioned you had several, but > let's stick with one). Let's say your non-linear model is Xf = F(T) = > D*T^2 + E*T + F, and in the first step you optimise D, E and F to get > the best correlation between Xf and the actual Y. This gives you the > best linear relationship between Xf and Y. Then in the subsequent > regression step you find out what that linear relationship actually > is, and you finally estimate Y = A*Xf + B. But this is just equal to > (A*D)*T^2 + (A*E)*T + (A*F + B), which is exactly the same functional > form as your original model. Therefore, I don't see why you don't just > optimise D, E and F in the first place so as to minimise sum (Y - > Xf)^2 (or whatever best-fit criteria you choose), and then you're > done. I don't see what you've gained by doing it in two stages. The F consists of a product with several factors of the type (T+To)^t > with T being a different observed physical parameter than Y and To and > t the fitted parameters; one of the factors is a SIN-function. (Note: > not knowing the physical relationships the 'best' function had to be > found by trial and error.) I think I tried to do with this function > exactly what you suggest in your example for F(T), this by adding u > and v to F, thus Xj=u*F+v. However, I noticed afterwards that by the > iterative procedure for minimising SUM(Y-Xj)^2 those two parameters (u > and v) remained completely untouched. That is also the reason for > having had u and v replaced *after* the iteration with A and B from > the regression. If you optimise Xf = u*F(T) + v so that the *correlation* between Xf and Y is maximised then it's quite plausible that u and v wouldn't be touched, since the (unsigned) correlation of u*F(T) + v with Y is independent of u and v. If, however, you optimise Xf = u*F(T) + v so that SUM(Y - Xf)^2 is minimised, and you later find that you can choose A and B such that SUM(Y - Xf')^2 is less than SUM(Y - Xf)^2, where Xf' = A + B*Xf, then I would say that the optimisation hasn't worked (unless the less than is very near an equality, in which case we might forgive it). In particular, if after optimising Xf = u*F(T) + v to minimise SUM(Y - Xf)^2 you end up with anything remotely like Test_1.gif then something has gone very wrong because the Xf's are nowhere even close to the corresponding Y's, and we know that a vastly better match can be achieved by a simple linear transformation. === Subject: Re: Problem related to a linear regression The situation corresponds to the last of the three cases you mention above. Following your comment I had again a look at it, realising that > However, I noticed afterwards that by the > iterative procedure for minimising SUM(Y-Xj)^2 those two parameters (u > and v) remained completely untouched. That is also the reason for > having had u and v replaced *after* the iteration with A and B from > the regression. Correct is: since I was so much focusing to find a useful (reasonably well) correlating fitting function, my attention was concentrated on the correlation. Realising that u and v didn't affect the correlation, the two parameters were left out of the fitting procedure. Somewhere en-route I must have realised that the iterative procedure found always values for u and v which corresponded very closely to a regression line with slope one and offset zero. Further I found that the values for u and v resulting in such a regression line (slope one and offset zero) could be obtained without the iterative procedure by setting u=1 and v=0; the values of this regression line corresponded to those values for u and v resulting in an regression line with 'exact' slope one and 'exact' offset zero. === Subject: Re: Problem related to a linear regression I realise now my mistake assuming that u and v should (always) be selected in such a way that the resulting regression line would have slope one and offset zero. Depending on the data that might not always be the case. This suggests that for finding the 'best' *correlation* u and v can indeed be neglected, but once the 'best' correlation has been found the values for u and v have to been found with an additional iterative run for finding the 'best' values for only those two parameters. === Subject: Re: Problem related to a linear regression What was thought to be a problem related to regression, turned finally out to be a consequence of a mistake done before doing the regression... especially for being so insistent on the details. It really needed that in order to reveal the real problem. Again, thank you! Marcel temporary in K'nopel/I'bul === Subject: Re: sense, existence, knowledge <2ju4i3dbtil6qqujqf3sba50cq7lsvqhn4@4ax.com> You, and I was talking about not forgetting the point in the original > post; It does not make sense to think of truth or falsity of a > mathematical statement independently of our knowledge concerning the > statement. this in reference to your non-contradictory- > identification; the knowledge about such a statement. You're still not making the slightest bit of sense. Math without something to count aint worth getting your tits in a > tangle over. The formula for knowledge = existence - sense - non-contradictory identification (using math can > help in the identifiation process but you have to remember that math > only exists as a man made tool of measurement and doesn't exist in > reality). > How do you justify your beliefs and theories about existence, sense, and non-contradictory-identification? The focus of my response was on the customs you use when formulating non-contradictory-identification, which you some how believe is a self-justified basic belief that needs no further evidence to satisfy the condition. Are you saying that the process of doing math doesn't happen in reality? > I am saying, math without it being applied to something sensory and > existing is NOTHING more than an exercise of the mind and can NOT > achieve anything in reality. HINT; Reason is man's ONLY means to knowledge, reason defined as.... > Sounds like man is a hostage then. > HINT; In math, no matter matter's not. MG === Subject: Re: sense, existence, knowledge <2ju4i3dbtil6qqujqf3sba50cq7lsvqhn4@4ax.comHow do you justify your beliefs and theories about existence, sense, > and non-contradictory-identification? Who to? A twit who claims he has doubt in any and everything anyone, including himself, FFS, thinks? Are you bieng serious? MG === Subject: Re: sense, existence, knowledge <2ju4i3dbtil6qqujqf3sba50cq7lsvqhn4@4ax.com > arbitrarily sense, existence, knowledge Whereas in reality there has to be something to sense AND identified. Therefore Existence, sense, non-contradictory identification, knowledge. It does not make sense to think of > truth or falsity of a mathematical statement > independently of our knowledge > concerning the statement. Likewise you idea that something is necessary depends upon memory, > introspection, testimony, or memories of introspection and testimony; > protocals of predicate logic and grammar. Plus the inference > transmission mechanism seems to be missing a premise or two since to > say that because logic works a certain way give warrent to the > deductive conclusion that concepts about existence are justified, seem > lacking. Michael Gordge- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text - WHO the are you talking to Mortal? > I was typing to you about something you said and how I was of the opinion that the idea lacked justification. > MG- Hide quoted text - - Show quoted text - === Subject: Re: Pythagorean Septuple Find integer solutions for a^2 + b^2 +c^2 +d^2 +e^2 +f^2 = g^2, Bill Jones 3^2 + 4^2 + 12^2 + 84^2 + 3612^2 + 6526884^2 = 6526885^2 suggests itself immediately or simply 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 = 3^2 How quickly does the solution starting with > 5^2 + 12^2 + ... come to mind? > What comes quicker is 3^3 + 4^2 + 25^2 + 3^3 + 4^2 + 25^2 = 10^2 If a^2 + b^2 = c^2 then a^2 + b^2 + c^2 + a^2 + b^2 + c^2 = 4c^2 For instantant gratification 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2 = 0^2 === Subject: [] Pythagorean Septuple <4233099.1194119861504.JavaMail.jakarta@nitrogen.mathforum.org1^2+3^2+5^2+6^ 2+ 7^2+7^2=13^2 3^2+3^2+4^2+5^2+6^2+7^2=12^2 > Youusetoomanyspaces.Usefewertomakeithardertoread. > of course 2^2+6^2+10^2+12^2+14^2+14^=26^2 6^2+6^2+8^2+10^2+12^2+14^2=24^2 18^2+18^2+24^2+30^2+36^2+42^2=72^2 Notbeinginthespaciousstyleofothers,isalsodismissedasunreadable. === Subject: Re: Probability Challenge <9478309.1194125927502.JavaMail.jakarta@nitrogen.mathfo rum.org>, > A test will be 6 out of 10 possible questions. I however only know 7 of the 10. If the actual test is indeed 6 questions but I only have to answer any 5 of the 6, what is the probability that I ace the test? binom(10,6) different tests. binom(7,5).binom(3,1) + binom(7,6) tests in which you can answer 5 questions correctly. binom(10,6) = 210 binom(7,5) = 21 binom(3,1) = 3 binom(7,6) = 7 Probability of acing the test = 70/210. Please learn how to line wrap you messages to 72 or fewer characters. -- Michael Press === Subject: Re: Probability Challenge >A test will be 6 out of 10 possible questions. I however only know 7 of >the 10. If the actual test is indeed 6 questions but I only have to >answer any 5 of the 6, what is the probability that I ace the test? The probability of picking zero questions you don't know is 7/10*6/9*5/8*4/7*3/6*2/5 = .03333 The probability of picking exactly one question you don't know is 1st pick 3/10*7/9*6/8*5/7*4/6*3/5 = .05 2nd pick 7/10*3/9*6/8*5/7*4/6*3/5 = .05 3rd pick 7/10*6/9*3/8*5/7*4/6*3/5 = .05 4th pick 7/10*6/9*5/8*3/7*4/6*3/5 = .05 5th pick 7/10*6/9*5/8*4/7*3/6*3/5 = .05 6th pick 7/10*6/9*5/8*4/7*3/6*3/5 = .05 These total to .33333 which covers all the ways you can ace the test. Phil H === Subject: Paid for your interests Hey i want to introduce you to a site where i published my book and i get paid when people read my posts. You also can get paid for your interests. Go to http://r.yuwie.com/drmu and have fun. === Subject: Re: Fermat's Last Theorem simple proof impossible? Apart from the other responses... ' DidFermathimself have a realprooflike he > claimed? to others that they did show something for which he claimed to have a >proof(this clearly shows his background, i.e. not mathematics, he was > just an amateur). Why does one need to be something other than an amateur to have > a background (ie. knowledge, experience) in mathematics? Knowing what other people have tried is important, so you don't end up > reinventing the wheel. Uh, wouldn't that be implied in what I said? I said why does one need > to be something other than an amateur to have a background (ie. > *knowledge, experience*) in mathematics. See: knowledge was > mentioned right there. But an amateur, by definition, does NOT have all the background knowledge. > And is that knowledge hard to get, financially, by the way, > and/or is it difficult to get rich enough to get it? --- Christopher Heckman === Subject: Re: Fermat's Last Theorem simple proof impossible? Uh, wouldn't that be implied in what I said? I said why does one need > to be something other than an amateur to have a background (ie. > *knowledge, experience*) in mathematics. See: knowledge was > mentioned right there. But an amateur, by definition, does NOT have all the background > knowledge. Isn't the definition of an amateur just that one doesn't get paid? === Subject: Re: Fermat's Last Theorem simple proof impossible? > Isn't the definition of an amateur just that one doesn't get paid? That's one meaning. Often 'amateur' is understood to carry connotations of non-professionalism in a more general sense, much like 'hobbyist' or 'enthusiast'. We may also recall that in the strict sense most mathematicians were amateurs at Fermat's time, and that he was not at all alone in issuing challenges, presenting results without proofs, and so on. Members of the Royal Society were not unknown to present their results in a cryptographic form, to establish priority without revealing important information to their rivals, for example. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: how to start solving it > Let A ? Cn[Times]n, B ? Cn[Times]m and C ? Cm[Times]m be given matrices. Define the > following block-upper triangular matrix: M = ?A B > 0 C? Let ? be a complex number. Prove that the following statements are > equivalent: > 1. ? is an eigenvalue of M. > 2. ? is an eigenvalue of A or C. any hints on how to start solving it ...... please.... thanx. First of all, state the problem in normal ASCII. --- Christopher Heckman === Subject: Re: Mandelbrot set midgets: why do they exist? mike3 a .8ecrit : > On Nov 2, 11:15 pm, Denis Feldmann mike3 a .8ecrit : >Hmm. Can't get the paper you referenced, though, as I don't have > access to a university library (I need to know how to get rich so I'll > be able to.). Anyway, I read the website, and it was interesting. > Is that also why Mandelbrot midgets may often appear in other > iterative maps? Yes ' I also noticed on this page for the logistic map, > another quadratic map: ' http://mathworld.wolfram.com/LogisticMap.html ' The period doubling bifurcations come faster and faster (8, 16, > 32, ...), then suddenly break off. Beyond a certain point known as the > accumulation point, periodicity gives way to chaos, as illustrated > below. In the middle of the complexity, a window suddenly appears with > a regular period like 3 or 7 as a result of mode locking. ' And a midget appears right at the spot in the complex > graph where that happens. So then does this phenomenon > of mode locking also have something to do with the > appearance of midgets in the Mandelbrot set? ' Yes. You may be interested by this site : http://www.ibiblio.org/e-notes/MSet/Contents.htm There are there a lot of interesting results, and a few proofs... > === Subject: Re: Mandelbrot set midgets: why do they exist? <472c1200$0$21150$7a628cd7@news.club-internet.fr> <472d66f1$0$21150$7a628cd7@news.club-internet.fr> On Nov 3, 11:30 pm, Denis Feldmann mike3 a ?crit : On Nov 2, 11:15 pm, Denis Feldmann mike3 a ?crit : Hmm. Can't get the paper you referenced, though, as I don't have > access to a university library (I need to know how to get rich so I'll > be able to.). Anyway, I read the website, and it was interesting. > Is that also why Mandelbrot midgets may often appear in other > iterative maps? Yes I also noticed on this page for the logistic map, > another quadratic map: http://mathworld.wolfram.com/LogisticMap.html The period doubling bifurcations come faster and faster (8, 16, > 32, ...), then suddenly break off. Beyond a certain point known as the > accumulation point, periodicity gives way to chaos, as illustrated > below. In the middle of the complexity, a window suddenly appears with > a regular period like 3 or 7 as a result of mode locking. And a midget appears right at the spot in the complex > graph where that happens. So then does this phenomenon > of mode locking also have something to do with the > appearance of midgets in the Mandelbrot set? Yes. You may be interested by this site :http://www.ibiblio.org/e-notes/MSet/Contents.htm There are there a lot of interesting results, and a few proofs... > Interesting stuff. It is kind of convenient that a piece of ornamentation -- the real axis spike, happens to lie along the real axis, as it makes it easy to study. Based on the behavior of the bifurcation diagram there, would it make any sense then to consider all ornamentation as the complex analogue of the chaotic region of the real bifurcation diagram, and the remaining parts of the set (cardioid and all bulbs/sub-bulbs/sub-sub-bulbs/etc.) as analogous to the stable region of the diagram? It seems that with complex numbers, the very shape of the chaotic region has become incredibly intricate! Also, has any research been done to determine the precise nature of ornamentation (not bulbs, but ornamentation)? Why does ornamentation seem to contain motifs that seem related to the periods of the bulbs it adorns? It is in the ornamentation, after all, that all the grand Complexity is found... === Subject: Re: Mandelbrot set midgets: why do they exist? mike3 a .8ecrit : > On Nov 3, 11:30 pm, Denis Feldmann mike3 a .8ecrit : >On Nov 2, 11:15 pm, Denis Feldmann mike3 a .8ecrit : > Hmm. Can't get the paper you referenced, though, as I don't have > access to a university library (I need to know how to get rich so I'll > be able to.). Anyway, I read the website, and it was interesting. > Is that also why Mandelbrot midgets may often appear in other > iterative maps? > Yes >I also noticed on this page for the logistic map, > another quadratic map: > http://mathworld.wolfram.com/LogisticMap.html > The period doubling bifurcations come faster and faster (8, 16, > 32, ...), then suddenly break off. Beyond a certain point known as the > accumulation point, periodicity gives way to chaos, as illustrated > below. In the middle of the complexity, a window suddenly appears with > a regular period like 3 or 7 as a result of mode locking. > And a midget appears right at the spot in the complex > graph where that happens. So then does this phenomenon > of mode locking also have something to do with the > appearance of midgets in the Mandelbrot set? > Yes. You may be interested by this site :http://www.ibiblio.org/e-notes/MSet/Contents.htm > There are there a lot of interesting results, and a few proofs... > Interesting stuff. It is kind of convenient that a > piece of ornamentation -- the real axis spike, happens to > lie along the real axis, as it makes it easy to study. > Based on the behavior of the bifurcation diagram there, > would it make any sense then to consider _all_ ornamentation > as the complex analogue of the chaotic region of the > real bifurcation diagram, Yes, and it can even be proved, quite easily, for things like, say, the branch ending at i. and the remaining parts of > the set (cardioid and all bulbs/sub-bulbs/sub-sub-bulbs/etc.) > as analogous to the stable region of the diagram? It > seems that with complex numbers, the very _shape_ of > the chaotic region has become incredibly intricate! ' Also, has any research been done to determine the precise > nature of ornamentation (not bulbs, but ornamentation)? Yes, a lot. > Why does ornamentation seem to contain motifs that > seem related to the periods of the bulbs it adorns? > It is in the ornamentation, after all, that all the grand Complexity > is found... > All this is well known (you should really get that book by Tan Lei). The m.9dain tool is the correspondance between M and the Julia sets. What is still mysterious is , say, the locally connected nature of M, or the presence (or not) of other components that the midget M's ... http://www.u-cergy.fr/rech/pages/tan/papers/similarity.ps (the ps format may be somewhat annoying, I admit) === Subject: Re: Mandelbrot set midgets: why do they exist? <472c1200$0$21150$7a628cd7@news.club-internet.fr> <472d66f1$0$21150$7a628cd7@news.club-internet.fr> <472da6dc$0$21148$7a628cd7@news.club-internet.fr> On Nov 4, 4:03 am, Denis Feldmann mike3 a ?crit : On Nov 3, 11:30 pm, Denis Feldmann mike3 a ?crit : On Nov 2, 11:15 pm, Denis Feldmann mike3 a ?crit : > Hmm. Can't get the paper you referenced, though, as I don't have > access to a university library (I need to know how to get rich so I'll > be able to.). Anyway, I read the website, and it was interesting. > Is that also why Mandelbrot midgets may often appear in other > iterative maps? > Yes I also noticed on this page for the logistic map, > another quadratic map: >http://mathworld.wolfram.com/LogisticMap.html > The period doubling bifurcations come faster and faster (8, 16, > 32, ...), then suddenly break off. Beyond a certain point known as the > accumulation point, periodicity gives way to chaos, as illustrated > below. In the middle of the complexity, a window suddenly appears with > a regular period like 3 or 7 as a result of mode locking. > And a midget appears right at the spot in the complex > graph where that happens. So then does this phenomenon > of mode locking also have something to do with the > appearance of midgets in the Mandelbrot set? > Yes. You may be interested by this site :http://www.ibiblio.org/e-notes/MSet/Contents.htm > There are there a lot of interesting results, and a few proofs... Interesting stuff. It is kind of convenient that a > piece of ornamentation -- the real axis spike, happens to > lie along the real axis, as it makes it easy to study. > Based on the behavior of the bifurcation diagram there, > would it make any sense then to consider all ornamentation > as the complex analogue of the chaotic region of the > real bifurcation diagram, Yes, and it can even be proved, quite easily, for things like, say, the > branch ending at i. > How does one do that, even though that branch does not lie along an easy path like how that real axis spike lies along the path it does? > and the remaining parts of the set (cardioid and all bulbs/sub-bulbs/sub-sub-bulbs/etc.) > as analogous to the stable region of the diagram? It > seems that with complex numbers, the very shape of > the chaotic region has become incredibly intricate! Also, has any research been done to determine the precise > nature of ornamentation (not bulbs, but ornamentation)? Yes, a lot. > What is the stuff (ornamentation), then? > Why does ornamentation seem to contain motifs that > seem related to the periods of the bulbs it adorns? > It is in the ornamentation, after all, that all the grand Complexity > is found... All this is well known (you should really get that book by Tan Lei). The > m?ain tool is the correspondance between M and the Julia sets. What is > still mysterious is , say, the locally connected nature of M, or the > presence (or not) of other components that the midget M's ... may be somewhat annoying, I admit) Hmm. So then is it like bits and pieces of the shapes of Julia sets all amalgamated together? Which then of course raises the question of why the Julia sets look the way they do. But I'd like to know why exactly the period of the bulb influences the ornamentation the way it does. Why that p-1 motif appears all throughout. What mechanism is responsible for creating this? What is meant by other components than midgets, anyway? You mean like if, somewhere, deep in the Mandelbrot set, there lurks a bizarre, frightening blob that looks like nothing like a midget? === Subject: Re: Mandelbrot set midgets: why do they exist? mike3 a .8ecrit : > On Nov 4, 4:03 am, Denis Feldmann mike3 a .8ecrit : > Yes. You may be interested by this site :http://www.ibiblio.org/e-notes/MSet/Contents.htm > There are there a lot of interesting results, and a few proofs... > Interesting stuff. It is kind of convenient that a > piece of ornamentation -- the real axis spike, happens to > lie along the real axis, as it makes it easy to study. > Based on the behavior of the bifurcation diagram there, > would it make any sense then to consider _all_ ornamentation > as the complex analogue of the chaotic region of the > real bifurcation diagram, > Yes, and it can even be proved, quite easily, for things like, say, the > branch ending at i. How does one do that, even though that branch does not lie > along an easy path like how that real axis spike lies along > the path it does? The main idea here is conformal transformations : the fact that (locally) there is a multiplication by a (complex) constant leaving the set invariant explains the spirals, for instance. ' and the remaining parts of >the set (cardioid and all bulbs/sub-bulbs/sub-sub-bulbs/etc.) > as analogous to the stable region of the diagram? It > seems that with complex numbers, the very _shape_ of > the chaotic region has become incredibly intricate! > Also, has any research been done to determine the precise > nature of ornamentation (not bulbs, but ornamentation)? > Yes, a lot. What is the stuff (ornamentation), then? Mostly, images of the real axis by (multivalued) functions... ' Why does ornamentation seem to contain motifs that > seem related to the periods of the bulbs it adorns? > It is in the ornamentation, after all, that all the grand Complexity > is found... > All this is well known (you should really get that book by Tan Lei). The > m.9dain tool is the correspondance between M and the Julia sets. What is > still mysterious is , say, the locally connected nature of M, or the > presence (or not) of other components that the midget M's ... > may be somewhat annoying, I admit) ' Hmm. So then is it like bits and pieces of the shapes of Julia sets > all > amalgamated together? Yes :-) You got it Which then of course raises the question of > why the Julia sets look the way they do. Much easier : those are really self similar, and the mapping is *exactly* z->z^2+c But I'd like to know why > exactly > the period of the bulb influences the ornamentation the way it does. > Why that p-1 motif appears all throughout. What mechanism is > responsible for creating this? This is harder (the idea is that roots of unity lurks there, but it is not so easy to see why : a lot to do with, for instance, the doubling cascade in the Feigenbaum sequences) ' What is meant by other components than midgets, anyway? You mean > like if, somewhere, deep in the Mandelbrot set, there lurks a > bizarre, frightening blob that looks like nothing like a midget? > Exactly (well, it could be a M_3 set, for instance (this is what results from the sequence z--> z^3+c ), but perhaps something really new. Otoh, almost evry specialist believes this is not teh case...) And for frightening, have a look at http://www.ansible.co.uk/writing/c-b-faq.html (the BLIT's Faq) === Subject: Re: Mandelbrot set midgets: why do they exist? <472c1200$0$21150$7a628cd7@news.club-internet.fr> <472d66f1$0$21150$7a628cd7@news.club-internet.fr> <472da6dc$0$21148$7a628cd7@news.club-internet.fr> <472e15d2$0$21144$7a628cd7@news.club-internet.fr> great dyscussion, but this was answered by monsieur M. at a general audience talk at Royce Hall, UCLA, 10ya: It blew up. thing is, he could have meant, Eet Bleu Windowscreened (tm), considering his long affiliation with IBM (and their OSes .-) he came back, a few months ago, and still held the same anomalous opinion! > This is harder (the idea is that roots of unity lurks there, but it is > not so easy to see why : a lot to do with, for instance, the doubling > cascade in the Feigenbaum sequences) > What is meant by other components than midgets, anyway? You mean > like if, somewhere, deep in the Mandelbrot set, there lurks a > bizarre, frightening blob that looks like nothing like a midget? Exactly (well, it could be a M_3 set, for instance (this is what results > from the sequence z--> z^3+c ), but perhaps something really new. Otoh, > almost evry specialist believes this is not teh case...) And for frightening, have a look athttp://www.ansible.co.uk/writing/c-b-faq.html(the BLIT's Faq) --n~nerfman~n! === Subject: Re: Maybe Exponential? check it out > the time to make a pizza is exponentially distributed with average > mean 6 minutes. If 250 people want pizza in one week, what is the probability that AT > LEAST half of them need to wait more than 6 minutes to get their > pizza? Looks like it might be binomial to me. what do you think? Would that be exponentially growing or decaying? I think your problem is incorrectly stated. Phil H === Subject: Re: Maybe Exponential? check it out > the time to make a pizza is exponentially distributed with average > mean 6 minutes. > If 250 people want pizza in one week, what is the probability that AT > LEAST half of them need to wait more than 6 minutes to get their > pizza? > Looks like it might be binomial to me. what do you think? Would that be exponentially growing or decaying? I think your problem is >incorrectly stated. Your objection is based on a lack of understanding. Look up exponential distribution. quasi === Subject: Re: Maybe Exponential? check it out the time to make a pizza is exponentially distributed with average > mean 6 minutes. If 250 people want pizza in one week, what is the probability that > AT > LEAST half of them need to wait more than 6 minutes to get their > pizza? Looks like it might be binomial to me. what do you think? >Would that be exponentially growing or decaying? I think your problem >is >incorrectly stated. Your objection is based on a lack of understanding. Look up exponential distribution. > Yeah probably, but enough to know that a rate parameter is missing. Phil H === Subject: Re: Maybe Exponential? check it out <15773529.1194132330990.JavaMail.jakarta@nitrogen.mathforum.org>, > You have only one pun for to prepare these pizzas ? If so, it must involve pi. === Subject: Re: Maybe Exponential? check it out ><15773529.1194132330990.JavaMail.jakarta@nitrogen.mathforum.org>, > You have only one pun for to prepare these pizzas ? If so, it must involve pi. That's a really cheesy pun. === Subject: Re: Third dimension... > What is to the third dimension as a point is to the first dimension > and as > a line is to the second dimension? As I noted in my original response, the answer should be plain to > see. David Well put. Now ... borrowing that... What is to the fourth dimension > as a point is to the first dimension, > as a line is to the second dimension and > as a plain is to the third dimension? Jay Bala. When you have a line, you need 2 points to make from it an abscissa. > When you have two lines (parallel), you need 2 lines to make from it > a square or a rectangle. > When you have a tube with a square profile, you need 2 squares to make > from it a cube. > When you have a tube with a cubical profile, you need 2 cubes to make > from it a 4-dimensional cube. Two free ends in the new dimension must > be closed, always. Plugs in (n + 1) dimensions have n-dimensions. > Write all vertices of 4-dimensional cube as (0,0,0,0) till (1,1,1,1). > You get 16 vectors giving position of vertices. 8 from them have on > the last place 0. They form 3 dimensional cube, the first side of the > higher dimensional cube. > kunzmilan http://en.wikipedia.org/wiki/Gimbal_lockhttp://www.hq.nasa.gov/alsj/gimbals. > htmlhttp://en.wikipedia.org/wiki/Image:Gyroscope_operation.gif > Is gimbal lock a hint that the definition of physical space as three > dimensional instead of four dimensional is just a case of too much > brevity by mathematicians? The reason I ask is that you can define physical space as four > dimensional like the Synergetics coordinate system, which is from the > tetrahedron, described at:http://bfi.org/node/574 > and a method to overcome gimbal lock uses four dimensional unit > quaternions. And the Pythagoreons might have had the right idea at: http://kmr.nada.kth.se/files/gok/firstproto/index.php?gallery=Fenomen... > _Begrepp/Pythagoras/Misc&image=Number_related_to_form.jpg Cliff Nelson Dry your tears, there's more fun for your ears, > Forward Into The Past 2 PM to 5 PM, Sundays, > California time,http://www.geocities.com/forwardintothepast/ > Don't be a square or a blockhead; > see:http://bfi.org/node/574http://library.wolfram.com/infocenter/search/?sea > rch_results=1;search... > son_id=607 ' I am not sure, what you want. Your publications on internet are not I think that the average person ought to know why they put a four-gimbal-platform on Gemini http://www.hq.nasa.gov/alsj/gimbals.html because it could mean that physical space is at minimum really four-dimensional, not three-dimensional. Cliff Nelson > more than 30 years old. I already published my first results in > scientific journals before this time. Thus you can not claim priority. > Tetrahedrons are only four dimensional planes, only ones from > different n possibilities of multidimensional planes. These planes > form comlexes. You limited yourself only on one posibility. > I was chemist, and I tried to solve some chemical problems. Physical > properties of molecules, as boiling points of alkanes can be explained > using my results, some, and even more important were known even before > I was born. Similarly, most of mathematics I use is older than I am. > The couting of products n^m as sums of products of two polynomial > coefficient was described in textbooks before I rediscovered it and > realized its importance. > kunzmilan === Subject: Re: Third dimension... <18101346.1193840391486.JavaMail.jakarta@nitrogen.mathforum.org> <20071031102807.843$Gp@newsreader.com> What is to the third dimension as a point is to the first dimension and as > a line is to the second dimension? As I noted in my original response, the answer should be plain to see. David Well put. Now ... borrowing that... What is to the fourth dimension > as a point is to the first dimension, > as a line is to the second dimension and > as a plain is to the third dimension? Jay Bala. When you have a line, you need 2 points to make from it an abscissa. > When you have two lines (parallel), you need 2 lines to make from it > a square or a rectangle. > When you have a tube with a square profile, you need 2 squares to make > from it a cube. > When you have a tube with a cubical profile, you need 2 cubes to make > from it a 4-dimensional cube. Two free ends in the new dimension must > be closed, always. Plugs in (n + 1) dimensions have n-dimensions. > Write all vertices of 4-dimensional cube as (0,0,0,0) till (1,1,1,1). > You get 16 vectors giving position of vertices. 8 from them have on > the last place 0. They form 3 dimensional cube, the first side of the > higher dimensional cube. > kunzmilan http://en.wikipedia.org/wiki/Gimbal_lockhttp://www.hq.nasa.gov/alsj/g... > Is gimbal lock a hint that the definition of physical space as three > dimensional instead of four dimensional is just a case of too much > brevity by mathematicians? The reason I ask is that you can define physical space as four > dimensional like the Synergetics coordinate system, which is from the > tetrahedron, described at:http://bfi.org/node/574 > and a method to overcome gimbal lock uses four dimensional unit > quaternions. And the Pythagoreons might have had the right idea at: http://kmr.nada.kth.se/files/gok/firstproto/index.php?gallery=Fenomen... > _Begrepp/Pythagoras/Misc&image=Number_related_to_form.jpg Cliff Nelson Dry your tears, there's more fun for your ears, > Forward Into The Past 2 PM to 5 PM, Sundays, > California time,http://www.geocities.com/forwardintothepast/ > Don't be a square or a blockhead; see:http://bfi.org/node/574http://library.wolfram.com/infocenter/search/?... . .. > son_id=607 I am not sure, what you want. Your publications on internet are not > more than 30 years old. I already published my first results in > scientific journals before this time. Thus you can not claim priority. > Tetrahedrons are only four dimensional planes, No, they aren't; they are three-dimensional solids. If you take the convex hull of the points (0,0,0), (0,0,1), (0,1,0), and (1,0,0), you get a tetrahedron. --- Christopher Heckman > only ones from > different n possibilities of multidimensional planes. These planes > form comlexes. You limited yourself only on one posibility. > I was chemist, and I tried to solve some chemical problems. Physical > properties of molecules, as boiling points of alkanes can be explained > using my results, some, and even more important were known even before > I was born. Similarly, most of mathematics I use is older than I am. > The couting of products n^m as sums of products of two polynomial > coefficient was described in textbooks before I rediscovered it and > realized its importance. > kunzmilan === Subject: is this feasible?? pls check out and comment on de design http://createthefuturecontest.com/pages/view/entriesdetail.html?entryID=798 === Subject: Re: lation between parameters for continuous osculation locus > Please indicate how, as much as possible analytically or upto an > unsolved equation, to find a relation between constants a and b that > can be perturbed (parametrically varied) each in 2-D implicit curves > f(x,y,a) = 0 and g(x,y,b) = 0. Equal slope at tangent point gives rise > to a third curve - dy/dx = fx/fy = gx/gy = 0, or fx/fy - gx/gy = 0, a > locus of continuous osculations between the two sets of curves. An > example is given of two conics for some values of a and b, without > exact tangency: ' http://tinypic.com/view.php?pic=67809xf&s=1 ' In other words, required is a concurrency condition of the three > curves, when the third curve passes through tangential contact point > of first two touching curves. Would it be expressed as a vanishing > Jacobian etc, and if so how? ' Narasimham What you want is f_x/f_y = g_x/g_y at a point where f = g = 0. If f and g are polynomials, you can try taking lexicographic-order Groebner bases for the ideal generated by f_x g_y - f_y g_x, f, and g. Using Maple 11, I did your example as follows: > f:= x^2+a*y^2 - x*y+5: g:= x^2+3*y^2-8*x+b: L:= [diff(f,x)*diff(g,y)-diff(f,y)*diff(g,x), f, g]: G:= Groebner[Basis](L, plex(x,y,a,b)); G[1]; # this is the relation between the parameters a and b 13780800+1274400*a+44880*b^2*a-2134080*b*a-43200*b+31680*b^2+16*b^4*a^4 -104*b^4*a^3+1392*b^3+1413120*a^4+241*b^4*a^2-40608*b^2*a^3-7167600*a^2 +338720*b*a^3-5520*b^3*a+12*b^4+2312*b^3*a^3+807840*b*a^2+36856*b^2*a^2 +2321920*a^3-672*b^3*a^4-728*b^3*a^2+14736*b^2*a^4-217600*b*a^4-102*b^4*a -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: lation between parameters for continuous osculation locus On Nov 4, 12:19 pm, Robert Israel > Please indicate how, as much as possible analytically or upto an > unsolved equation, to find a relation between constants a and b that > can be perturbed (parametrically varied) each in 2-D implicit curves > f(x,y,a) = 0 and g(x,y,b) = 0. Equal slope at tangent point gives rise > to a third curve - dy/dx = fx/fy = gx/gy = 0, or fx/fy - gx/gy = 0, a > locus of continuous osculations between the two sets of curves. An > example is given of two conics for some values of a and b, without > exact tangency: http://tinypic.com/view.php?pic=67809xf&s=1 In other words, required is a concurrency condition of the three > curves, when the third curve passes through tangential contact point > of first two touching curves. Would it be expressed as a vanishing > Jacobian etc, and if so how? Narasimham What you want is f_x/f_y = g_x/g_y at a point where f = g = 0. > If f and g are polynomials, you can try taking lexicographic-order > Groebner bases for the ideal generated by f_x g_y - f_y g_x, f, and g. > Using Maple 11, I did your example as follows: f:= x^2+a*y^2 - x*y+5: g:= x^2+3*y^2-8*x+b: > L:= [diff(f,x)*diff(g,y)-diff(f,y)*diff(g,x), f, g]: > G:= Groebner[Basis](L, plex(x,y,a,b)); > G[1]; > # this is the relation between the parameters a and b 13780800+1274400*a+44880*b^2*a-2134080*b*a-43200*b+31680*b^2+16*b^4*a^4 > -104*b^4*a^3+1392*b^3+1413120*a^4+241*b^4*a^2-40608*b^2*a^3-7167600*a^2 > +338720*b*a^3-5520*b^3*a+12*b^4+2312*b^3*a^3+807840*b*a^2+36856*b^2*a^2 > +2321920*a^3-672*b^3*a^4-728*b^3*a^2+14736*b^2*a^4-217600*b*a^4-102*b^4*a > -- > Robert Israel isr...@math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada twice or in several instances, and, also say if one of f or g is implicit transcendental, would Groebner still produce a relation between a and b? I chose second degree polynomials f,g for a simple example. Also, since Mathematica has the Groebner basis command, can this be written in much the same way there also? Narasimham === Subject: Re: lation between parameters for continuous osculation locus > On Nov 4, 12:19 pm, Robert Israel > Please indicate how, as much as possible analytically or upto an > unsolved equation, to find a relation between constants a and b that > can be perturbed (parametrically varied) each in 2-D implicit curves > f(x,y,a) = 0 and g(x,y,b) = 0. Equal slope at tangent point gives rise > to a third curve - dy/dx = fx/fy = gx/gy = 0, or fx/fy - gx/gy = 0, a > locus of continuous osculations between the two sets of curves. An > example is given of two conics for some values of a and b, without > exact tangency: http://tinypic.com/view.php?pic=67809xf&s=1 In other words, required is a concurrency condition of the three > curves, when the third curve passes through tangential contact point > of first two touching curves. Would it be expressed as a vanishing > Jacobian etc, and if so how? Narasimham What you want is f_x/f_y = g_x/g_y at a point where f = g = 0. > If f and g are polynomials, you can try taking lexicographic-order > Groebner bases for the ideal generated by f_x g_y - f_y g_x, f, and g. > Using Maple 11, I did your example as follows: f:= x^2+a*y^2 - x*y+5: g:= x^2+3*y^2-8*x+b: > L:= [diff(f,x)*diff(g,y)-diff(f,y)*diff(g,x), f, g]: > G:= Groebner[Basis](L, plex(x,y,a,b)); > G[1]; > # this is the relation between the parameters a and b 13780800+1274400*a+44880*b^2*a-2134080*b*a-43200*b+31680*b^2+16*b^4*a^4 > -104*b^4*a^3+1392*b^3+1413120*a^4+241*b^4*a^2-40608*b^2*a^3-7167600*a^2 > +338720*b*a^3-5520*b^3*a+12*b^4+2312*b^3*a^3+807840*b*a^2+36856*b^2*a^2 > +2321920*a^3-672*b^3*a^4-728*b^3*a^2+14736*b^2*a^4-217600*b*a^4-102*b^4*a > -- > Robert Israel isr...@math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada ' twice or in several instances, and, also say if one of f or g is > implicit > transcendental, would Groebner still produce a relation between a and > b? I chose second degree polynomials f,g for a simple example. Several instances is no problem. But f and g must be polynomials (or at least the equations must be reducible to polynomials). > Also, since Mathematica has the Groebner basis command, can this be > written in much the same way there also? Yes, I think so, but I don't have Mathematica. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: lation between parameters for continuous osculation locus On Nov 4, 12:19 pm, Robert Israel > Please indicate how, as much as possible analytically or upto an > unsolved equation, to find a relation between constants a and b that > can be perturbed (parametrically varied) each in 2-D implicit curves > f(x,y,a) = 0 and g(x,y,b) = 0. Equal slope at tangent point gives rise > to a third curve - dy/dx = fx/fy = gx/gy = 0, or fx/fy - gx/gy = 0, a > locus of continuous osculations between the two sets of curves. An > example is given of two conics for some values of a and b, without > exact tangency: http://tinypic.com/view.php?pic=67809xf&s=1 In other words, required is a concurrency condition of the three > curves, when the third curve passes through tangential contact point > of first two touching curves. Would it be expressed as a vanishing > Jacobian etc, and if so how? Narasimham What you want is f_x/f_y = g_x/g_y at a point where f = g = 0. > If f and g are polynomials, you can try taking lexicographic-order > Groebner bases for the ideal generated by f_x g_y - f_y g_x, f, and g. > Using Maple 11, I did your example as follows: f:= x^2+a*y^2 - x*y+5: g:= x^2+3*y^2-8*x+b: > L:= [diff(f,x)*diff(g,y)-diff(f,y)*diff(g,x), f, g]: > G:= Groebner[Basis](L, plex(x,y,a,b)); > G[1]; > # this is the relation between the parameters a and b 13780800+1274400*a+44880*b^2*a-2134080*b*a-43200*b+31680*b^2+16*b^4*a^4 > -104*b^4*a^3+1392*b^3+1413120*a^4+241*b^4*a^2-40608*b^2*a^3-7167600*a^2 > +338720*b*a^3-5520*b^3*a+12*b^4+2312*b^3*a^3+807840*b*a^2+36856*b^2*a^2 > +2321920*a^3-672*b^3*a^4-728*b^3*a^2+14736*b^2*a^4-217600*b*a^4-102*b^4*a > -- > Robert Israel isr...@math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada twice or several instances, and also say if one of f or g is implicit transcendental, would Groebner still produce a relation between a and b? I chose second degree polynomials f,g for a simple example. Also, since Mathematica has the Groebner basis command, can this be written in much the same way there also? Narasimham === Subject: Simplifying/reducing vector expression: (a + b) x (c + d) Can the expression (a + b) x (c + d), where a, b,c a,c d are vectors be written differently? I'm doing some algebra/analysis and I'd like to simplify this expression. === Subject: Re: Simplifying/reducing vector expression: (a + b) x (c + d) > Can the expression (a + b) x (c + d), where a, b,c a,c d are vectors > be written differently? I'm doing some algebra/analysis and I'd like > to simplify this expression. I believe I've found the answer. Using the relationships: u x ( v + w ) = ( u x v ) + ( u x w ), and u x v = -v x u (a + b) x ( c + d ) = [ (a + b) x c ] + [ (a + b) x d ] = [ -c x (a + b) ] + [ -d x (a + b) ] = (-c x a ) + (-c x b) + (-d x a) + (-d x b) = (a x c) + (b x c) + (a x d) + (b x d) === Subject: Re: Simplifying/reducing vector expression: (a + b) x (c + d) > Can the expression (a + b) x (c + d), where a, b,c a,c d are vectors > be written differently? I'm doing some algebra/analysis and I'd like > to simplify this expression. I believe I've found the answer. Using the relationships: u x ( v + w ) = ( u x v ) + ( u x w ), > and u x v = -v x u (v + w)u = vu + vw > (a + b) x ( c + d ) = [ (a + b) x c ] + [ (a + b) x d ] > = [ -c x (a + b) ] + [ -d x (a + b) ] > = (-c x a ) + (-c x b) + (-d x a) + (-d x b) > = (a x c) + (b x c) + (a x d) + (b x d) Thus directly (a + b) x ( c + d ) = [ (a + b) x c ] + [ (a + b) x d ] = (a x c) + (b x c) + (a x d) + (b x d) Vectors with + and x are a ring, almost like algebra except for ab = ba. === === Subject: Re: Complex analysis with constant..f > Hello sir~ ' f : C -> C is entire. ' Let f(z) = f(iz) = f(e^z) for all z in C. ' Show that f is constant. ' -------------------------------------------- > Maybe... for Liouville's theorem. > I need that f is bounded on C. > I can't induce this. > so, I need your advice. > Actually you don't need the f(iz), and you don't need f to be entire. Note that there are some w where w = e^w (namely the branches of -LambertW(-1)). I claim that any f that is analytic in a connected neighbourhood of such w and satisfies f(z) = f(exp(z)) for all z in that neighbourhood is constant there. We have |w| > 1 (you can check that Re(exp(z)-z) > 0 for |z| <= 1), and |exp'(w)| = |exp(w)| = |w| so w is an unstable fixed point of exp. Thus for every z_0 in a suitable disk around w, there is a sequence z_n with exp(z_{n+1}) = z_n and z_n -> w as n -> infty. But f(z_{n+1}) = f(exp(z_{n+1})) = f(z_n). Thus w is the limit of a sequence on which f is constant. It's well known that this implies f is constant. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === === Subject: Psychology and mathematics The problem of psychology is how to describe a multicurved structure such as the brain in terms of linear logic. The dishonest psychology misrepresents the mind, either judging it for difference from linearity or from difference from one or another normative construct. But the honest psychology does have a chance at it, and that is through what is known by mathematics: That linear functions can be made to approximate curves - when taken in infinitesily small approximations, in which the lines become vanishingly small until they become small enough to fit the curve. Psychology thus has a potential - by doing just that. At which point linear logic becomes advanced enough to be curved, and the scientific and the poetic merge into one. Which transforms the understanding of logic as much as it enhances it, as well as allowing it to actually do its job of explaining the mind. Ilya Shambat. === Subject: Re: Psychology and mathematics > The problem of psychology is how to describe a multicurved structure > such as the brain in terms of linear logic. The dishonest psychology misrepresents the mind, either judging it for > difference from linearity or from difference from one or another > normative construct. But the honest psychology does have a chance at > it, and that is through what is known by mathematics: That linear functions can be made to approximate curves - when taken > in infinitesily small approximations, in which the lines become > vanishingly small until they become small enough to fit the curve. Psychology thus has a potential - by doing just that. At which point linear logic becomes advanced enough to be curved, and > the scientific and the poetic merge into one. Which transforms the understanding of logic as much as it enhances it, > as well as allowing it to actually do its job of explaining the mind. Ilya Shambat. They used to talk about this on Lawrence Welk. === Subject: Re: Psychology and mathematics > The problem of psychology is how to describe a multicurved structure > such as the brain in terms of linear logic. ' The dishonest psychology misrepresents the mind, either judging it for > difference from linearity or from difference from one or another > normative construct. But the honest psychology does have a chance at > it, and that is through what is known by mathematics: ' That linear functions can be made to approximate curves - when taken > in infinitesily small approximations, in which the lines become > vanishingly small until they become small enough to fit the curve. ' Psychology thus has a potential - by doing just that. ' At which point linear logic becomes advanced enough to be curved, and > the scientific and the poetic merge into one. ' Which transforms the understanding of logic as much as it enhances it, > as well as allowing it to actually do its job of explaining the mind. ' Ilya Shambat. > Did your legs go numb while penning this blather from the loo? === Subject: Re: Psychology and mathematics i learned that on the street === Subject: Re: Psychology and mathematics > i learned that on the street Where else have you learnt anything? -- Teilhard Knight The Extraterrestrial I'm not screwed up .... It's all in my mind === Subject: Re: Psychology and mathematics <5p5fbiFphi3eU2@mid.individual.neti learned that on the street Where else have you learnt anything? Don't underestimate what can be learnt on the street. A wise man will learn more from a fool than a fool from a wise man > I'm not screwed up .... It's all in my mind And what are you doing about it? === Subject: x^2 + y^2 = n (mod m) Conjecture: If n is an integer for which the equation x^2 + y^2 = n has no integer solutions, then there exists an integer m > 1 such that the congruence x^2 + y^2 = n (mod m) has no integer solutions. quasi === Subject: Re: x^2 + y^2 = n (mod m) > Conjecture: If n is an integer for which the equation x^2 + y^2 = n has no integer > solutions, then there exists an integer m > 1 such that the congruence > x^2 + y^2 = n (mod m) has no integer solutions. quasi Sure, m = n^2 will do. The condition on n is equivalent to saying that there exists a prime p == 3 (mod 4) and an integer k such p^(2k+1) divides n but p^(2k+2) doesn't. An^2 + n will of course have the same property for any positive integer A. --- J K Haugland http://home.no.net/zamunda === Subject: Re: x^2 + y^2 = n (mod m) > Conjecture: If n is an integer for which the equation x^2 + y^2 = n has no integer > solutions, then there exists an integer m > 1 such that the congruence > x^2 + y^2 = n (mod m) has no integer solutions. quasi ' Sure, m = n^2 will do. ' The condition on n is equivalent to saying that there exists a prime p > == 3 (mod 4) and an integer k such p^(2k+1) divides n but p^(2k+2) > doesn't. An^2 + n will of course have the same property for any > positive integer A. Screaming for a (n,m)=1 condition, then? Phil -- -- Microsoft voice recognition live demonstration === Subject: Re: x^2 + y^2 = n (mod m) > Conjecture: > If n is an integer for which the equation x^2 + y^2 = n has no integer > solutions, then there exists an integer m > 1 such that the congruence > x^2 + y^2 = n (mod m) has no integer solutions. > quasi Sure, m = n^2 will do. The condition on n is equivalent to saying that there exists a prime p >== 3 (mod 4) and an integer k such p^(2k+1) divides n but p^(2k+2) >doesn't. An^2 + n will of course have the same property for any >positive integer A. Very nice. However, it seems that your argument only works for positive integers n. In the conjecture, n is not required to be positive. quasi === Subject: Re: x^2 + y^2 = n (mod m) Conjecture: > If n is an integer for which the equation x^2 + y^2 = n has no integer > solutions, then there exists an integer m > 1 such that the congruence > x^2 + y^2 = n (mod m) has no integer solutions. > quasi Sure, m = n^2 will do. The condition on n is equivalent to saying that there exists a prime p >== 3 (mod 4) and an integer k such p^(2k+1) divides n but p^(2k+2) >doesn't. An^2 + n will of course have the same property for any >positive integer A. Very nice. However, it seems that your argument only works for positive integers > n. In the conjecture, n is not required to be positive. quasi You are right. However, the method for positive n helps us solve the general case: Suppose n is negative. If -n is not the sum of two squares, take m = (- n)^2 as before. Otherwise, take m = -4n. Observe that A(-4n) + n is the product of a number that is the sum of two squares (-n) and one that isn't (4A-1). --- J K Haugland http://home.no.net/zamunda === Subject: Re: x^2 + y^2 = n (mod m) > Conjecture: >If n is an integer for which the equation x^2 + y^2 = n has no integer > solutions, then there exists an integer m > 1 such that the congruence > x^2 + y^2 = n (mod m) has no integer solutions. >quasi >Sure, m = n^2 will do. >The condition on n is equivalent to saying that there exists a prime p >== 3 (mod 4) and an integer k such p^(2k+1) divides n but p^(2k+2) >doesn't. An^2 + n will of course have the same property for any >positive integer A. > Very nice. > However, it seems that your argument only works for positive integers > n. In the conjecture, n is not required to be positive. > quasi You are right. However, the method for positive n helps us solve the >general case: Suppose n is negative. If -n is not the sum of two squares, take m = (- >n)^2 as before. Otherwise, take m = -4n. Observe that A(-4n) + n is >the product of a number that is the sum of two squares (-n) and one >that isn't (4A-1). Great solution. quasi === Subject: Re: x^2 + y^2 = n (mod m) Conjecture: > If n is an integer for which the equation x^2 + y^2 = n has no integer > solutions, then there exists an integer m > 1 such that the congruence > x^2 + y^2 = n (mod m) has no integer solutions. > quasi Sure, m = n^2 will do. The condition on n is equivalent to saying that there exists a prime p >== 3 (mod 4) and an integer k such p^(2k+1) divides n but p^(2k+2) >doesn't. An^2 + n will of course have the same property for any >positive integer A. Very nice. However, it seems that your argument only works for positive integers > n. In the conjecture, n is not required to be positive. quasi He he...I noticed n could be negative afterwards and thought it was just a slip. ;-) --- J K Haugland http://home.no.net/zamunda === Subject: Re: x^2 + y^2 = n (mod m) > Conjecture: >If n is an integer for which the equation x^2 + y^2 = n has no integer > solutions, then there exists an integer m > 1 such that the congruence > x^2 + y^2 = n (mod m) has no integer solutions. >quasi >Sure, m = n^2 will do. >The condition on n is equivalent to saying that there exists a prime p >== 3 (mod 4) and an integer k such p^(2k+1) divides n but p^(2k+2) >doesn't. An^2 + n will of course have the same property for any >positive integer A. > Very nice. > However, it seems that your argument only works for positive integers > n. In the conjecture, n is not required to be positive. > quasi He he...I noticed n could be negative afterwards and thought it was >just a slip. ;-) In any case, you quickly and dispatched the upper half, and very cleanly. quasi === Subject: Re: Encouraging quotes > I am teaching MCAS math in high school ( MCAS is the Massachusetts > standardized test that all students must pass to graduate, my students > are those whom the school thinks are in danger of failing). I have > taken to putting up quotes on the whiteboard...the first when the the > Red Sox were losing to cleveland and I put up Do not go gentle into > that good night Currently I have Be ye doers of the word and not > just hearers deceiving only yourselves. I need some more. They should > be short ( not to take up too much whiteboard space) with some > reference to the joys of learning/understanding math Around 1977, the co-inventors of RSA put out a challenge RSA-encrypted message, which AFAIK appeared in Scientific American. It was solved in 1994 by a large team, and an announcement went out including this passage: Using the decoding scheme 01=A, 02=B, ..., 26=Z, and 00 a space between words, the decoded message reads Cf.: Derek Atkins's announcement on a crypto mailing list: < http://cypherpunks.venona.com/date/1994/04/msg01407.html > David Bernier === Subject: Re: we can only gather finite collections of electrons [sci.lang removed from cross-posting as requested] What is mathematics? Search me. >the belief in the possibility of identity? Do you mean identity as a relationship between two things (er, > one thing)? The philosophical half of my brain is temporarily > (I hope!) out of commission, but I'm still interested to hear > what you and others think (so I can perhaps make sense of it > later). > -- its about belief in the idea of equality? - so A is identical with A - though in reality its not - symbolically it would be nice if it was. But post-modernists would not have that. On the topic i thought electrons were all (thought to be) identical and that John Barrow proposed that there could therefore just be one electron? Its a vague memory. A couple of other passing thoughts - it might be a mistake to think of electrons as discreet countable objects, (as they can be in two places at once counting might be difficult) and ignoring that - then there is no reason to suppose there is an infinite number of them? though looking at the ruler on my desk - it is potentially infinitely divisible- though i thought some mathematicians only count up to the totality of finite things... > Angus Rodgers > Contains mild peril === Subject: Re: we can only gather finite collections of electrons >No - the only group i subscribe to is alt.postmodern - if i reply to a >post >its because someone - in this case galathaea et al are cross posting to >alt.postmodern - and hey postmodernism is in their topic!... i suggest you >either spend sometime with a filter or tell these others. I know how you feel (because the only group in the list that > I subscribe to is sci.math), and I was embarrassed when Peter > replied in similar fashion to one of my posts (incidentally, > we sort-of know one another from rec.puzzles.crosswords), but > I tried not to take it personally; also, when replying to a > one chooses to crosspost the reply. But i've no way of knowing if say in one of the groups that there are not people interested in the topic, if someone is not they can simply filter, but the benefit of cross posting is that it widens the gene pool :-) > It's awkward, I know. Still, I expect no-one in alt.anarchism minds. :-) > -- > Angus Rodgers > Contains mild peril === Subject: Re: A First Course In Probability 7th Ed by Sheldon Ross if you have got the solutions then please share those solutions with me. kinshuk.saurabh@gmail.com === Subject: Re: The negative dimension > Give an example of a space which has -1 dimension. Are there any spaces which are more negatively dimensional? According to the physicists negative dimensions take away length from a mixed form of positive and negative dimensions. This does not seem to be what you are talking about on this thread but under this rendition you just keep changing the signature. I think it's a piece of cheese but they don't. Is a metric a free choice? Where is the outrage? As if there weren't already enough wrong in the world... The damn Jews are doing it to us going all the way back to Albert Einstein who encouraged the US to build the bomb. Oh yeah, isn't he the one who took this negative dimension idea so seriously too? It's not just the Jews- it's all of these Abrahamics- faction after faction of splinters of forks of a trunk of a dying tree. Half half mathematics. I second the second and meter the meter. Whose components are one part negative according to current theory- Oh no, I'm fading... -Tim === Subject: Re: The negative dimension <11095909.1194036548761.JavaMail.jakarta@nitrogen.mathforum.org> effectively falls back on known definitions. But its motivating virtue > is that it can take negative values, which measure usefully the degree > of emptiness of empty sets.[...] And negative fractals dimensions [...] > are latent (= hidden, but present). (Benoit B. Mandelbrot, 1990) Indeed. Have you a fractal of dimension -1/2 or -1 to present? Again, what definition of dimension are you referring to? I refer you to the handbook of Dimensional Dementia wherein is stated Since dimension resides in the mind of the beholder, dimension, like beauty, is a multi-dimensional ambiguity. > For fractal dimension I'd suggest Hausdorff dimension. > Simply put, if a metric space X has dimension d, then the number of > balls > of radius r needed to cover X is proportional to (1/r)^d. > But then a negative dimension would imply that one needs > less small balls to cover what can be covered by more big balls. > This is of course impossible. > The empty metric space has no balls. === Subject: Re: The negative dimension <472c1464$0$21145$7a628cd7@news.club-internet.fr> <472c55bb$0$21146$7a628cd7@news.club-internet.frWilliam Elliot a .8ecrit : Give an example of a space which has -1 dimension. > But since {} x V = {} for all V, should not dim {} = -infty > so that dim (UxV) = dim U + dim V can hold? > The result dim (UxV) = dim U + dim V holds for many things which are > not vector spaces (manifolds, to name one, or self-similar fractals) >It does not for empty space, what ever dim nulspaced is. > dim nulspaced = dim (nulspaced x somespace) > t= dim nulspaced + dim somespace dim nulspaced is infinite, you say? Yes (-oo) > dim nulspaced = oo, will also satisfy oo + n = oo > Let dim nulspaced = Aleph small > dim somespace = Aleph big Do you believe the only infinites are the cardinal ones? We are > calculating in the extended real line IR U {-oo,+oo} here... As it's homeomorphic to [0,1], it's one dimensional. Even the long line and the longer lines are one dimensional. Do you believe that for all r in [0,1], there is a subset A r of R, or if you must, of extended R with dimension A r = r? > Moreover, even if your reasoning was correct, it is not obvious > arbitrary infinite dimensional spaces exists... > R^(Aleph alpha) > Then > tAleph small = Aleph small + Aleph big = Aleph big Small is big and big is tall and tall is small. > === Subject: Re: The negative dimension > William Elliot a .8ecrit : ' Are there any spaces which are more negatively > dimensional? > A new notion of fractal dimension is defined. > When it is positive, it > effectively falls back on known definitions. But > its motivating virtue > is that it can take negative values, which measure > usefully the degree > of emptiness of empty sets.[...] And negative > fractals dimensions [...] > are latent (= hidden, but present). (Benoit B. > Mandelbrot, 1990) Indeed. Have you a fractal of dimension -1/2 or -1 > to present? ' I dont, but Mandelbrot looks like he have ; see > http://www.math.yale.edu/mandelbrot/web_pdfs/123negati > veFractalDimensions.pdf See also the recent preprint http://www.math.yale.edu/mandelbrot/web_pdfs/emptiness_of_latent_sets.pdf and the references therein. Here Mandelbrot introduces a new notion D_t, called the test dimension, or average-box-counting dimension, and gave examples of processes yielding almost surely empty sets for which D_t=-1 and D_t=-2. But these are not sets, of course. === Subject: uniformly continuous related to integral Suppose that H=[a,b]x[c,d] is a rectangle, f:H->R is continuous, and g:[a,b]->R is integrable. Prove that F(y)=integral (from a to b)g(x)f(x,y)dx is uniformly continuous on [c,d]. === Subject: Re: uniformly continuous related to integral > Suppose that H=[a,b]x[c,d] is a rectangle, f:H->R is continuous, > and g:[a,b]->R is integrable. Prove that ' F(y)=integral (from a to b)g(x)f(x,y)dx ' is uniformly continuous on [c,d]. Use the fact that _f_ is uniformly continuous. Jose Carlos Santos === Subject: Equivalency between the undeciability of the halting problem and the 2nd law of thermodynamics. For al long time I've suspected but never been able to prove in any way that there exists an equivalency betwen the 2nd law of therodynamics and the undeciability of the halting problem. That is if you had a halting oracle you could use it to build a free energy device and visa versa. Of cause both devices are impossable. But if this equiverlence could be shown it would strongly suggest the correctness of the strong Church-Turing Thesis. What would be the best way to go about proving this? === Subject: Re: Equivalency between the undeciability of the halting problem and the 2nd law of thermodynamics. On Nov 4, 5:20 am, David Formosa (aka ? the Platypus) > For al long time I've suspected but never been able to prove in any > way that there exists an equivalency betwen the 2nd law of > therodynamics and the undeciability of the halting problem. That is > if you had a halting oracle you could use it to build a free energy > device and visa versa. Of cause both devices are impossable. But if this equiverlence could > be shown it would strongly suggest the correctness of the strong > Church-Turing Thesis. What would be the best way to go about proving > this? Hmm... for a long time I had a hunch that the impossibilty of obtaining polynomial time solutions to certain problems was equivalent to the existence of irrational numbers, and of programs which do not halt. Unfortunately, however appealing our hunches our, we must eventually reduce them to concreteness! Off hand I don't see a strong family resemblance with the second law of thermodynamics in this collection. Programs which don't halt are something like irrational sequences are something like problems which cannot be solved significantly faster than by exhaustive numeration: they all involve some non-iterative explosion of complexity. But the second law of thermodynamics is, AFAIK, essentially a tautology, which states that class of macroscopically similar states with overwhelmingly greater total probability will appear with overwhelmingly greater total probability. We may invite it over for === Subject: Re: Equivalency between the undeciability of the halting problem and the 2nd law of thermodynamics. > For al long time I've suspected but never been able to prove in any > way that there exists an equivalency betwen the 2nd law of > therodynamics and the undeciability of the halting problem. That is > if you had a halting oracle you could use it to build a free energy > device and visa versa. Of cause both devices are impossable. But if this equiverlence could > be shown it would strongly suggest the correctness of the strong > Church-Turing Thesis. What would be the best way to go about proving > this? That depends on exactly what you mean by the undecidability of the halting problem and the second law of thermodynamics. And the strong Church-Turing Thesis, of course -- I take it you have in mind the thesis that all mechanical devices that are physically constructible, in some idealised sense, compute recursive functions. If by undecidability of the halting problem you mean the purely mathematical result there is no hope of any kind of equivalence proof; mathematical results are simply not equivalent to empirical, contingent questions like the validity of (any of) the laws of thermodynamics. But perhaps you have in mind some other formulation of either the second law of thermodynamics -- as a purely mathematical result that just happens to pertain to mathematical structures we use as models of physical things -- or of the halting problem -- reformulated, in some manner, as a physical claim? As to supporting the strong Church-Turing thesis, obviously if we established that provided the second law of thermodynamics holds all physical devices compute recursive functions that would hugely increase our confidence in the strong thesis; we should then be as confident of its validity as we are of that of the second law of thermodynamics, which is quite confident indeed. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Equivalency between the undeciability of the halting problem and the 2nd law of thermodynamics. > If by undecidability of the halting problem you mean the purely mathematical > result there is no hope of any kind of equivalence proof; mathematical > results are simply not equivalent to empirical, contingent questions like > the validity of (any of) the laws of thermodynamics. But perhaps you have in > mind some other formulation of either the second law of thermodynamics -- as > a purely mathematical result that just happens to pertain to mathematical > structures we use as models of physical things -- or of the halting problem > -- reformulated, in some manner, as a physical claim? Or perhaps he means to interpret the second law of thermodynamics by using the information theoretic definition of entropy. -- Jesse F. Hughes Disney has now succeeded in preventing anyone from doing to Mickey Mouse what Disney did to Quasimodo. -- Randolph Rackovitz, on Eldred vs. Ashcroft === Subject: Re: Is a line segment composed of points? <472b44c1$0$6689$afc38c87@news.optusnet.com.au> conditions for showing a set is larger than the other is to show that some elements in the first set don't have corresponding elements in the second set. I realize this. We can't really prove that lines have different number of points. However I have a few more thoughts. Two lines have equal number of points on them? I would say a line has as many ways of dividing into two parts as any other line. I'm not talking about the contents of the line, but ways of dividing its contents. Ways of dividing contents are the same as content? If you keep cutting a line at different places, will it be cut into pieces of zero length? It is self contradictory, because when the piece if of zero length, then it is not a result of cutting at different places. So, this means we can't keep cutting at different places. We'll start cutting at the same place when the number of cuts are approaching infinity? Is it a blind belief, or common sense? It seems to me that we are tying two unfathomable things (infinite number of cuts and piece length becoming zero) together and hoping that they somehow fit with each other. This works for all practical purposes, but I doubt its validity for pure mathematics. You might say that the cardinality of the set of all possible cuts of a line is the same as some infinite number set. But which number set? Integers, real numbers? As Dave said, some rules stop being valid at limiting conditions. So when I keep one end of the line fixed and keep moving second end towards the first, the length keeps decreasing and becomes zero and again increases while point moves to other side (or is it negative length?) So it suddenly stopped being a line for a moment when the length was zero. And the rules became invalid for that moment. May be this is a problem is due to defining a line as a set of points? - venkat > A lot of this stuff seems paradoxical at first. It is known to be correct, > because it can be proved from set theory. A big part of this is defining exactly what you mean by the number of > points in a set. The paradox you have below stems from this. A set is not > assumed to be bigger than another just because you can pair the elements off > together and still have some left over. Its a little more subtle; one set is > said to be larger than another if you can pair all of the elements of the > smaller set with ones from the larger set, and also prove that there is no > possible way to do it in reverse. For finite sets, this new definition works > exactly the same; for infinite sets it gives the only real definition of the > number of points in a set that produces sensible mathematics. === Subject: Re: Is a line segment composed of points? <472b44c1$0$6689$afc38c87@news.optusnet.com.auIf you keep cutting a line at different places, will it be cut into > pieces of zero length? It is self contradictory, because when the > piece if of zero length, then it is not a result of cutting at > different places. Incorrect. Start with [0,1]; at step n cut at 1/n; Do this for every n. You end up with an infinite number of pieces, one of which is a single point of length zero. (Note there is no last cut.) - William Hughes === Subject: Re: Is a line segment composed of points? ' If you keep cutting a line at different places, will it be cut into > pieces of zero length? It is self contradictory, because when the > piece if of zero length, then it is not a result of cutting at > different places. ' Incorrect. Start with [0,1]; at step n cut at 1/n; > Do this for every n. You end up with an infinite number of pieces, > one of which is a single point of length zero. (Note there > is no last cut.) end up with clashes with no last. Phil -- -- Microsoft voice recognition live demonstration === Subject: Re: Is a line segment composed of points? <472b44c1$0$6689$afc38c87@news.optusnet.com.au> <878x5ejbll.fsf@nonospaz.fatphil.org> On Nov 4, 10:04 am, Phil Carmody pieces of zero length? It is self contradictory, because when the > piece if of zero length, then it is not a result of cutting at > different places. Incorrect. Start with [0,1]; at step n cut at 1/n; > Do this for every n. You end up with an infinite number of pieces, > one of which is a single point of length zero. (Note there > is no last cut.) end up with clashes with no last. As Dave Seaman pointed out, end up with means after all the cuts. This does not imply a last cut. - William Hughes === Subject: Re: Is a line segment composed of points? ' If you keep cutting a line at different places, will it be cut into > pieces of zero length? It is self contradictory, because when the > piece if of zero length, then it is not a result of cutting at > different places. ' Incorrect. Start with [0,1]; at step n cut at 1/n; > Do this for every n. You end up with an infinite number of pieces, > one of which is a single point of length zero. (Note there > is no last cut.) > end up with clashes with no last. Not at all; end up with means after all the cuts, which does not imply a last cut. -- Dave Seaman Oral Arguments in Mumia Abu-Jamal Case heard May 17 U.S. Court of Appeals, Third Circuit === Subject: Re: Is a line segment composed of points? ' If you keep cutting a line at different places, > will it be cut into > pieces of zero length? It is self contradictory, > because when the > piece if of zero length, then it is not a result > of cutting at > different places. ' Incorrect. Start with [0,1]; at step n cut at > 1/n; > Do this for every n. You end up with an infinite > number of pieces, > one of which is a single point of length zero. > (Note there > is no last cut.) ' end up with clashes with no last. ' Phil > -- Not if you interpret end up with as in the limit which is certainly what was intended. === Subject: Re: Is a line segment composed of points? > If you keep cutting a line at different places, will it be cut into > pieces of zero length? It is self contradictory, because when the > piece if of zero length, then it is not a result of cutting at > different places. Incorrect. Start with [0,1]; at step n cut at 1/n; > Do this for every n. You end up with an infinite number of pieces, > one of which is a single point of length zero. (Note there > is no last cut.) Well, hold on a minute. If you do this in the normal way you certainly don't end up with a piece of single length zero. There is no n corresponding to a piece of length zero. A piece of length zero would correspond to the ordinal w, which is not part of N. === Subject: Re: Is a line segment composed of points? <472b44c1$0$6689$afc38c87@news.optusnet.com.au> <472ddd3a$0$19803$afc38c87@news.optusnet.com.au > If you keep cutting a line at different places, will it be cut into > pieces of zero length? It is self contradictory, because when the > piece if of zero length, then it is not a result of cutting at > different places. Incorrect. Start with [0,1]; at step n cut at 1/n; > Do this for every n. You end up with an infinite number of pieces, > one of which is a single point of length zero. (Note there > is no last cut.) Well, hold on a minute. If you do this in the normal way you certainly don't end up with a piece of > single length zero. Of course you do. After all N cuts are done there are N line segements and one point, the point 0. Note each of the line segments correpsonds to a cut, but the point 0 does not. If you want, you can look at it as each cut removing the line segment corresponding to that cut. The point 0 is what remains after all the line segments are cut off. > There is no n corresponding to a piece of length zero. Correct. The piece left over does not correspond to a cut. A piece of length zero would correspond to the ordinal w, which is not part > of N. There is a cut for each element of N. There is no cut omega. There are omega cuts, however there is no omega'th cut. - William Hughes === Subject: Re: Is a line segment composed of points? >If you keep cutting a line at different places, will it be cut into > pieces of zero length? It is self contradictory, because when the > piece if of zero length, then it is not a result of cutting at > different places. > Incorrect. Start with [0,1]; at step n cut at 1/n; > Do this for every n. You end up with an infinite number of pieces, > one of which is a single point of length zero. (Note there > is no last cut.) > Well, hold on a minute. If you do this in the normal way you certainly don't end up with a > piece of single length zero. There is no n corresponding to a piece > of length zero. I think he means the piece containing the endpoint 0 has zero length, not any of the pieces [1/n+1,1/n]. -- Jesse F. Hughes There's a thrill that's gone that I'll probably not have in quite the same way again. After all, FLT was a unique animal, and we had a great dance. -J.S. Harris on proving Fermat's last theorem === Subject: algebraic problem in geometry Let $mathfrak{g}$ be a real Lie algebra of dimension $n$. We assume that $mathfrak{g}$ can be decomposed into orthogonal sum of two commutatif ideals $mathfrak{g}=Koplus K^perp$. To any $rinwedge^2mathfrak{g}$ we associate the linear map $r_{sharp}$ : [begin{array}{cccc} r_{sharp} : & mathfrak{g}^* & longrightarrow & mathfrak{g} & alpha & longmapsto & r_{sharp}(alpha) end{array},qquadbetaleft(r_{sharp}(alpha)right)=r(alpha, beta) ] we associate to $r$, the element $[r,r]inwedge^3mathfrak{g}$, defined by : [[r,r](alpha,beta,gamma)=alpha(r_{sharp}(beta),r_{sharp} (gamma))+ beta(r_{sharp}(gamma),r_{sharp}(alpha))+gamma(r_{sharp} (alpha),r_{sharp}(beta))] To any $r$ solution of the ``classical Yang-Baxter equation'' $ [r,r]=0$ we have a Lie bracket on the dual $mathfrak{g}^*$ of $ mathfrak{g}$ defined by [[alpha,beta]_r=mathrm{ad}_{r_sharp(beta)}^*alpha- mathrm{ad}_{r_sharp(alpha)}^*beta] where $mathrm{ad}_{r_sharp(alpha)}^*$ is the co-adjointe representation. We can easily see that $ker r_sharp$ est a commutatif ideal, and we have an exact sequence : [0longrightarrowker r_sharphookrightarrowmathfrak{g}^* longrightarrowmathrm{im}, rhookrightarrowmathfrak{g}] textbf{Question :} How to situate $mathrm{im},r$ with respect to $K $ and $K^perp$ ? in particular, is $mathrm{im},r$ a commutatif sub- algebra, or ideal, of $mathfrak{g}$ ? === Subject: Re: algebraic problem in geometry > Let $mathfrak{g}$ be a real Lie algebra of dimension $n$. We assume > that $mathfrak{g}$ can be decomposed into orthogonal sum of two > commutatif ideals $mathfrak{g}=Koplus K^perp$. > To any $rinwedge^2mathfrak{g}$ we associate the linear map > $r_{sharp}$ : > [begin{array}{cccc} > r_{sharp} : & mathfrak{g}^* & longrightarrow & mathfrak{g} > & alpha & longmapsto & r_{sharp}(alpha) > end{array},qquadbetaleft(r_{sharp}(alpha)right)=r(alpha, > beta) ] > we associate to $r$, the element $[r,r]inwedge^3mathfrak{g}$, > defined by : > [[r,r](alpha,beta,gamma)=alpha(r_{sharp}(beta),r_{sharp} > (gamma))+ > beta(r_{sharp}(gamma),r_{sharp}(alpha))+gamma(r_{sharp} > (alpha),r_{sharp}(beta))] > To any $r$ solution of the ``classical Yang-Baxter equation'' $ > [r,r]=0$ we have a Lie bracket on the dual $mathfrak{g}^*$ of $ > mathfrak{g}$ defined by > [[alpha,beta]_r=mathrm{ad}_{r_sharp(beta)}^*alpha- > mathrm{ad}_{r_sharp(alpha)}^*beta] > where $mathrm{ad}_{r_sharp(alpha)}^*$ is the co-adjointe > representation. > We can easily see that $ker r_sharp$ est a commutatif ideal, and we > have an exact sequence : > [0longrightarrowker r_sharphookrightarrowmathfrak{g}^* > longrightarrowmathrm{im}, rhookrightarrowmathfrak{g}] > textbf{Question :} How to situate $mathrm{im},r$ with respect to $K > $ and $K^perp$ ? in particular, is $mathrm{im},r$ a commutatif sub- > algebra, or ideal, of $mathfrak{g}$ ? ****************************************************************** Try again: the above is practically impossible to read without going nuts. Tonio === Subject: x^n + y^n = k (mod m) Does there exist an integer n > 2, and an integer k, such that (1) x^n + y^n = k has no integer solutions. (2) For all integers m > 1, the congruence x^n + y^n = k (mod m) has integer solutions. ? quasi === Subject: Re: Different values of the dot product ' suppose we have the n different vectors x_1, x_2, > ..., x_n in the R^m. No two vectors are the same and > no vector is the zero vector. Suppose we now calculate > for every 1<=i to get, regardless of the choice of x_1, x_2, ..., x_n? ' Certainly, if n<=m, we can set x_1 = (1,0,0...,0)^T, > x_2 = (0,1,0...)^T, ... - each dot product will be > zero, we therefore only get one number as a result. > This, of course, also works for any orthogonal > basis of R^m. > For any m, you can get at least m+1 vectors with only a single dot product. As an example, take the vertices of a regular simplex centred at the origin. This is not the only solution, though. Obviously, for m=1 any two non-zero numbers will do. For m=2, a general solution, up to similarity, is (1, 0), (a, b), (a, (a-a^2)/b) for any non-zero a and b. Note that this will give you any inner product you want, except 0. For these two cases it's also fairly easy to prove that m+1 is the maximum. I would guess that this holds for all m, but I don't have a proof. For m=1 I think that generally the least number of products is obtained by 1, -1, 2, -2, 4, -4, 8, -8, ... This gives for odd n >= 3 a total of 2n-3 products, and for even n >= 4 a total of 2n-4 products. products is at most linear in n. For m=2 you can take the vertices of a regular n-gon centred at the origin. This gives a total of floor(n/2) products. For n <= 5 this is an optimal solution. For higher n, I don't see how to prove that yet. -- Niels Diepeveen === Subject: Re: Different values of the dot product On 2 Nov, 09:34, stefan.steinerber...@gmail.com suppose we have the n different vectors x_1, x_2, > ..., x_n in the R^m. No two vectors are the same and > no vector is the zero vector. Suppose we now calculate > for every 1<=i to get, regardless of the choice of x_1, x_2, ..., x_n? Certainly, if n<=m, we can set x_1 = (1,0,0...,0)^T, > x_2 = (0,1,0...)^T, ... - each dot product will be > zero, we therefore only get one number as a result. > This, of course, also works for any orthogonal > basis of R^m. But what if n>m? Is anything known about this problem? > (It might very well have a trivial solution I overlooked). Best wishes, > Stefan With n = m(m+1)/2, we can get only two different numbers by choosing the vectors with 1 or 2 1's and 0's otherwise. But for m = 1, 2 and 5 this is not optimal: For m = 1 we can choose any two distinct non- zero vectors, for m = 2 we can choose the five vectors from the centre of a regular pentagon to its vertices, and for m = 5 we can choose the 16 vectors (+/-1, +/-1, ...) with an even number of negative entries. Can we also do better than m(m+1)/2 for m = 3 and m = 4? Are my examples for m = 2 and 5 optimal? --- J K Haugland http://home.no.net/zamunda === Subject: Re: Different values of the dot product Is it true that the maximum number of vectors in the R^3 resulting in three different numbers is 12, so that any 13 vectors in the R^3 determine at least four different numbers? 12 vectors with 3 different scalar product results are {1, p, 0}, {-1, p, 0}, {1, -p, 0}, {-1, -p, 0}, {p, 0, 1}, {-p, 0, 1}, {p, 0, -1}, {-p, 0, -1}, {0, 1, p}, {0, -1, p}, {0, 1, -p}, {0, -1, - p} where p = (1+sqrt(5))/2. (Those are the coordinates of the twelve vertices of a regular icosahedron). Is it true that any 21 vectors in the R^3 determine at least six different scalar product results? 20 vectors with five results are given by the vertices of the regular dodecahedron (p = (1+sqrt(5))/2)). {-1, -1, -1}, {-1, -1, 1}, {-1, 1, -1}, {-1, 1, 1}, {1, -1, -1}, {1, -1, 1}, {1, 1, -1}, {1, 1, 1}, {0, 1/p, p}, {0, 1/p, -p}, {0, -1/p, p}, {0, -1/p, -p}, {1/p, p, 0}, {-1/p, p, 0}, {1/p, -p, 0}, {-1/p, -p, 0}, {p, 0, 1/p}, {-p, 0, 1/p}, {p, 0, -1/p}, {- p, 0, -1/p} === Subject: Re: Different values of the dot product On 2 Nov, 09:34, stefan.steinerber...@gmail.com suppose we have the n different vectors x_1, x_2, > ..., x_n in the R^m. No two vectors are the same and > no vector is the zero vector. Suppose we now calculate > for every 1<=i to get, regardless of the choice of x_1, x_2, ..., x_n? Certainly, if n<=m, we can set x_1 = (1,0,0...,0)^T, > x_2 = (0,1,0...)^T, ... - each dot product will be > zero, we therefore only get one number as a result. > This, of course, also works for any orthogonal > basis of R^m. But what if n>m? Is anything known about this problem? > (It might very well have a trivial solution I overlooked). Best wishes, > Stefan With n = m(m+1)/2, we can get only two different numbers by choosing the vectors with 1 or 2 1's and 0's otherwise. But for m = 1, 2 and 5 this is not optimal: For m = 1 we can choose any two distinct non-zero vectors, for m = 2 we can choose the five vectors from the centre of a regular pentagon to its vertices, and for m = 5 we can choose the 16 vectors (+/-1, +/-1, ...) with an even number of negative entries. Can we also do better than m(m+1)/2 for n = 3 and n = 4? Are my examples for m = 2 and 5 optimal? --- J K Haugland http://home.no.net/zamunda === Subject: Putnam problem I'll skip the preliminaries and cut to the chase: f(x) = 2x - 3x^3 c is a constant, unknown, c > 0 K and L are the positive roots of the equation f(x) - c = 0; K < L I [P .. Q] {g(x) dx} is defined as the integral of g, over P < x < Q Your mission, Jim, is to find the c which satisfies: I [0 .. K] { (c - f(x)) dx} = I [K .. L] { (f(x) - c) dx} Good luck, Jim. If you know the formula for the roots of a cubic polynomial, it's straightforward. But if not... is there a trick, whereby one might solve it, without determining those roots? If anyone wants the text of the full problem, I'll post it. Mark === Subject: Re: Putnam problem > I'll skip the preliminaries and cut to the chase: f(x) = 2x - 3x^3 > c is a constant, unknown, c > 0 K and L are the positive roots of > the equation f(x) - c = 0; K < L I [P .. Q] {g(x) dx} is defined as the > integral of g, over P < x < Q Your mission, Jim, is to find the c which satisfies: I [0 .. K] { (c - f(x)) dx} = I [K .. L] { (f(x) - c) dx} Add I [0 .. K] { (f(x) - c) dx} to both sides: 0 = I [0 .. L] { (f(x) - c) dx} 0 = -3/4 L^4 + L^2 - c L (1) and we know 0 = -3L^3 + 2L - c (2) Subtract L/4*(2) from (1): 0 = 1/2 L^2 - 3/4 c L hence (L=0 can be excluded) L = 3/2 c Substitute this into (1)/L^2: 0 = -3/4 L^2 + 1/3 = -27/16 c^2 + 1/3 Thus c = 4/9 as the negative solution ca be excluded. I hope I didn't mistype anything, but the way should work anyhow. hagman Good luck, Jim. If you know the formula for the roots of a cubic > polynomial, it's straightforward. But if not... is > there a trick, whereby one might solve it, > without determining those roots? If anyone wants the text of the full problem, > I'll post it. Mark === Subject: Central Limit Theorem for Mode Central limit theorem is for the mean value of distribution. However not only the mean value but also the mode might follow similar limit theorem. I tried to prove it in the following site with experimental verification using Java applet; http://hecoaustralia.fortunecity.com/mode1/clt-mode.htm === Subject: Re: Central Limit Theorem for Mode ' Central limit theorem is for the mean value of distribution. > However not only the mean value but also the mode might follow > similar limit theorem. I tried to prove it in the following > site with experimental verification using Java applet; ' http://hecoaustralia.fortunecity.com/mode1/clt-mode.htm If I'm not mistaken, the Central Limit Theorem is not only for the mean value of the distribution, but it shows that the distribution approaches the normal distribution. That would mean that the mode (or the kurtosis or ...) of the sum will approach the mode (or the kurtosis or ...) of a normal distribution with a mean equal to the sum of means and a variance equal to the sum of variances. Refreshing my memory at Wikipedia, I see there are a family of similar convergence results. The one I'm most familiar with requires that the summed random variables be independent and have finite variances. Jim Burns === Subject: Re: Confirmation of Shannon's Mistake about Perfect Secrecy of One-time-pad > [BTW, could you *plase* switch to a more standard way of quoting? > Because of your current unusual method, I always get terribly lost > about the authorship of text fragments a few levels inside] Without looking at the encrypted text (i.e. a priori) the > keys K are uniform. > After looking at the encrypted text (i.e. a posteriori) > teh situation is different as I have quantified above. Exactly. > And the keys are not uniform a posteriori. > And the plaitext distribution is absolutely unchanged. > And the latter is what i scalled perfect secrecy. > Thus OPT is perfectly secret. If you throw a fair die, the probaility of getting a 6 is 1/6. > As soon as you observe the result, the a posteriori probability that > you just got a 6 is either 1 or 0, depending on what you observe. > ==yes, the condition changed. But you seem to be unable to pin down a single step that is > invalid. You don't. You merely say that you point out the problem (see, you > just did). > So I ask again: Which of my small handful of steps is not backed by > standard probability theory? Alternatively, which result from > standard probability theory that I used is wrong? Which conditions cannot coexist? coexist, So the three items that cannot coexist are a) the prior, > b) the key uniform cyphertext fixed and > c) OTP ? > OTP clearly exists. > The prior (if it means an a priori distribution for plaintexts?) does > exist. > The sequence of words the key uniform cyphertext fixed does not > make sense to me anyway, so I don't care if that doesn't exist. look at the papers carefully, do not just ask. I've taen a swicft look over one of your papers (they are all the same > anyway) > and have already pointed out one error of yours (something like > mixing up P(A|B) with P(B|A)) in another post. While it is correct that the theorem cannot be proved by example, ==yes you admit your problem. read the sentence to its end. > I don't have to prove Shannon by example anyway. > I am satisfied with the fact that the example *disproves* > your gross deviation. it is clear that IF you were right and observing the encrypted > message made the a-posteriori probabilities of the plaintext > message differ from the a-priori probabilities THEN you > should be able to make a guess of the plaintext that > would agree better than expected. > Additionally, one CAN make (as predicted by my statements) > a good guess (about 90% digits correct) of the key > used in my example - I just verified it from the raw data > I still have on my disk. > However, your result that the plaintext suddenly has > a different distribution can be checked by statistical > tests applied to my example and thus looks highly unlikely. > The possible conclusions are: > What all mathematicians except wangyong consider correct > arguments in probability theory is logically incorrect > but exactly describes the phenomena for the description > of which probability theory was developed in the first place. > What wangyong considers correct produces wrong descriptions > of simple testable statistical phenomena. > uncertainty increase dont predicate incorrect. you just ask me to > guess a fixed value, or possibel value, but not a random variable. You may view my 1000 character plaintext as a sequence of 1000 > single character plaintexts. > Then 1000 is big enough a number that you are allowed to > guess a distribution instead. > But the distribution for the plaintext is still > P(M=0)=0.9 both for cyphertext 0 and for cypertext 1. > Do you want to come up with a significantly different distribution > or finally admit that no knwoledge (not even a changed distribution) > about the plaintext could be gained (an effect aka. perfect security)? you take a mistake like shannon and OPT. Being only as wrong as Shannon and the perfect secrecy of OTP > makes me quite satisfied. > OTOH, I see that you are totally incapable of pointing out > one single error in a short derivation and only resort to > your Shannon made a mistake lamenting, running logically > in circles again and again (as you do with writing essentially > the same paper again and again). > I take this as a sign that ma presence in this thread > is of no use whatsoever and shall therfore leave it. > Good luck to all others trying to succeed in this struggle. hagman- - - - hagman Exactly. And the keys are not uniform a posteriori. And the plaitext distribution is absolutely unchanged. You don't. You merely say that you point out the problem (see, you just did). the answer is like the following. see the condition that the ciphertext y is a fixed value is never considered when computing P(M = x C = y). We can get that result by reductio ad absurdum. Suppose for fixed y, if P (K = (x y))=2-n (that is used in the proof, but indeed it is wrong. It is used just to get wrong conclusion), we can get P(M = x C = y)= 2-n because there is a one-to-one correspondence between all the plaintexts and keys for the fixed ciphertext in OTP. But it is obviously wrong, for the prior probabilities of all plaintexts are seldom equally likely. So P(M = x)?P (K = (x y)) stand for the joint probability of x and y when y is not fixed. But Shannon thought of the posterior probability as the probability of plaintext when ciphertext had been intercepted, we can see that there is a presupposition in P(M = x C = y) that y is fixed, but in P(M = x), P (K = (x y)) and P(C=y), y is not fixed, otherwise we can get obviously wrong results. In such way, the Bayes's formula was misused for the probability was not on the same presupposition and the equation does not come into existence. In OTP there are complex and crytic conditions that influence the probability of plaintext, key and ciphertext, so it is essential to cognize all the conditions and carefully use probability theory. The proof did not realize the crytic condition that ciphertext was a fixed value (even though unknown) rather than a random variable. So I ask again: Which of my small handful of steps is not backed by standard probability theory? Alternatively, which result from standard probability theory that I used is wrong? conditional probability. just misuse it. > Which conditions cannot coexist? > coexist, So the three items that cannot coexist are a) the prior, b) the key uniform cyphertext fixed and c) OTP ? OTP clearly exists. The prior (if it means an a priori distribution for plaintexts?) does exist. The sequence of words the key uniform cyphertext fixed does not make sense to me anyway, so I don't care if that doesn't exist. > look at the papers carefully, do not just ask. I've taen a swicft look over one of your papers (they are all the same anyway) and have already pointed out one error of yours (something like mixing up P(A|B) with P(B|A)) in another post. read the sentence to its end. I don't have to prove Shannon by example anyway. I am satisfied with the fact that the example *disproves* your gross deviation. You may view my 1000 character plaintext as a sequence of 1000 single character plaintexts. Then 1000 is big enough a number that you are allowed to guess a distribution instead. But the distribution for the plaintext is still P(M=0)=0.9 both for cyphertext 0 and for cypertext 1. Do you want to come up with a significantly different distribution or finally admit that no knwoledge (not even a changed distribution) about the plaintext could be gained (an effect aka. perfect security)? discuss.displacing the problem to cover up your mistake seems to be your strongpoint > you take a mistake like shannon and OPT. Being only as wrong as Shannon and the perfect secrecy of OTP makes me quite satisfied. OTOH, I see that you are totally incapable of pointing out one single error in a short derivation and only resort to your Shannon made a mistake lamenting, running logically in circles again and again (as you do with writing essentially the same paper again and again). I take this as a sign that ma presence in this thread is of no use whatsoever and shall therfore leave it. Good luck to all others trying to succeed in this struggle. ---------------I see. your All tricks have been exhausted. dicussion just exposed your mistakes. === Subject: Re: Confirmation of Shannon's Mistake about Perfect Secrecy of One-time-pad [BTW, could you *plase* switch to a more standard way of quoting? > Because of your current unusual method, I always get terribly lost > about the authorship of text fragments a few levels inside] Without looking at the encrypted text (i.e. a priori) the > keys K are uniform. > After looking at the encrypted text (i.e. a posteriori) > teh situation is different as I have quantified above. Exactly. > And the keys are not uniform a posteriori. > And the plaitext distribution is absolutely unchanged. > And the latter is what i scalled perfect secrecy. > Thus OPT is perfectly secret. If you throw a fair die, the probaility of getting a 6 is 1/6. > As soon as you observe the result, the a posteriori probability that > you just got a 6 is either 1 or 0, depending on what you observe. > ==yes, the condition changed. But you seem to be unable to pin down a single step that is > invalid. You don't. You merely say that you point out the problem (see, you > just did). > So I ask again: Which of my small handful of steps is not backed by > standard probability theory? Alternatively, which result from > standard probability theory that I used is wrong? Which conditions cannot coexist? coexist, So the three items that cannot coexist are a) the prior, > b) the key uniform cyphertext fixed and > c) OTP ? > OTP clearly exists. > The prior (if it means an a priori distribution for plaintexts?) does > exist. > The sequence of words the key uniform cyphertext fixed does not > make sense to me anyway, so I don't care if that doesn't exist. look at the papers carefully, do not just ask. I've taen a swicft look over one of your papers (they are all the same > anyway) > and have already pointed out one error of yours (something like > mixing up P(A|B) with P(B|A)) in another post. While it is correct that the theorem cannot be proved by example, ==yes you admit your problem. read the sentence to its end. > I don't have to prove Shannon by example anyway. > I am satisfied with the fact that the example *disproves* > your gross deviation. it is clear that IF you were right and observing the encrypted > message made the a-posteriori probabilities of the plaintext > message differ from the a-priori probabilities THEN you > should be able to make a guess of the plaintext that > would agree better than expected. > Additionally, one CAN make (as predicted by my statements) > a good guess (about 90% digits correct) of the key > used in my example - I just verified it from the raw data > I still have on my disk. > However, your result that the plaintext suddenly has > a different distribution can be checked by statistical > tests applied to my example and thus looks highly unlikely. > The possible conclusions are: > What all mathematicians except wangyong consider correct > arguments in probability theory is logically incorrect > but exactly describes the phenomena for the description > of which probability theory was developed in the first place. > What wangyong considers correct produces wrong descriptions > of simple testable statistical phenomena. > uncertainty increase dont predicate incorrect. you just ask me to > guess a fixed value, or possibel value, but not a random variable. You may view my 1000 character plaintext as a sequence of 1000 > single character plaintexts. > Then 1000 is big enough a number that you are allowed to > guess a distribution instead. > But the distribution for the plaintext is still > P(M=0)=0.9 both for cyphertext 0 and for cypertext 1. > Do you want to come up with a significantly different distribution > or finally admit that no knwoledge (not even a changed distribution) > about the plaintext could be gained (an effect aka. perfect security)? you take a mistake like shannon and OPT. Being only as wrong as Shannon and the perfect secrecy of OTP > makes me quite satisfied. > OTOH, I see that you are totally incapable of pointing out > one single error in a short derivation and only resort to > your Shannon made a mistake lamenting, running logically > in circles again and again (as you do with writing essentially > the same paper again and again). > I take this as a sign that ma presence in this thread > is of no use whatsoever and shall therfore leave it. > Good luck to all others trying to succeed in this struggle. hagman- - - - hagman Exactly. > And the keys are not uniform a posteriori. > And the plaitext distribution is absolutely unchanged. > You don't. You merely say that you point out the problem (see, you > just did). the answer is like the following. > see the condition that the ciphertext y is a fixed value is never > considered when computing P(M = x C = y). We can get that result by > reductio ad absurdum. Suppose for fixed y, if P (K = (x y))=2-n (that > is used in the proof, but indeed it is wrong. It is used just to get > wrong conclusion), we can get P(M = x C = y)= 2-n because there is a > one-to-one correspondence between all the plaintexts and keys for the > fixed ciphertext in OTP. But it is obviously wrong, for the prior > probabilities of all plaintexts are seldom equally likely. So P(M = > x)?P (K = (x y)) stand for the joint probability of x and y when y is > not fixed. But Shannon thought of the posterior probability as the > probability of plaintext when ciphertext had been intercepted, we can > see that there is a presupposition in P(M = x C = y) that y is fixed, > but in P(M = x), P (K = (x y)) and P(C=y), y is not fixed, otherwise > we can get obviously wrong results. In such way, the Bayes's formula > was misused for the probability was not on the same presupposition > and > the equation does not come into existence. > In OTP there are complex and crytic conditions that influence the > probability of plaintext, key and ciphertext, so it is essential to > cognize all the conditions and carefully use probability theory. The > proof did not realize the crytic condition that ciphertext was a > fixed > value (even though unknown) rather than a random variable. So I ask again: Which of my small handful of steps is not backed by > standard probability theory? Alternatively, which result from > standard probability theory that I used is wrong? > conditional probability. just misuse it. Which conditions cannot coexist? > coexist, So the three items that cannot coexist are a) the prior, > b) the key uniform cyphertext fixed and > c) OTP ? > OTP clearly exists. > The prior (if it means an a priori distribution for plaintexts?) does > exist. > The sequence of words the key uniform cyphertext fixed does not > make sense to me anyway, so I don't care if that doesn't exist. look at the papers carefully, do not just ask. I've taen a swicft look over one of your papers (they are all the > same > anyway) > and have already pointed out one error of yours (something like > mixing up P(A|B) with P(B|A)) in another post. read the sentence to its end. > I don't have to prove Shannon by example anyway. > I am satisfied with the fact that the example *disproves* > your gross deviation. You may view my 1000 character plaintext as a sequence of 1000 > single character plaintexts. > Then 1000 is big enough a number that you are allowed to > guess a distribution instead. > But the distribution for the plaintext is still > P(M=0)=0.9 both for cyphertext 0 and for cypertext 1. > Do you want to come up with a significantly different distribution > or finally admit that no knwoledge (not even a changed distribution) > about the plaintext could be gained (an effect aka. perfect > security)? discuss.displacing the problem to cover up your mistake seems to be > your strongpoint you take a mistake like shannon and OPT. Being only as wrong as Shannon and the perfect secrecy of OTP > makes me quite satisfied. > OTOH, I see that you are totally incapable of pointing out > one single error in a short derivation and only resort to > your Shannon made a mistake lamenting, running logically > in circles again and again (as you do with writing essentially > the same paper again and again). I take this as a sign that ma presence in this thread > is of no use whatsoever and shall therfore leave it. > Good luck to all others trying to succeed in this struggle. ---------------I see. your All tricks have been exhausted. > dicussion just exposed your mistakes. Please send me the Chinese version if you have one. I cannot be convinced here, even after I read your paper in detail and all your replies. Jiang Bian jxbian@ualr.edu Univ. Of AR at LIT === Subject: Re: Confirmation of Shannon's Mistake about Perfect Secrecy of One-time-pad If the ciphertext is fixed, something is going on to block > the ciphertexts that are different from the first > ciphertext. What mechanism does this? > four feet table In my paper. What is the compromise? I don't understand. You're not alone. Despite the large quantities of verbiage generated, > NOBODY has any clue what wangyong is on about. All attempts to elicit > clearer explanations lead round in circles, or to even greater > confusion. Initially I thought there might be some coherent (albeit > probably wrong) idea behind his theories that was potentially worth > discussing but wasn't getting across clearly because of language > difficulties. Now I'm not so sure... (Ooops... I said I was giving up on this thread, didn't I? Oh > well...!) when your All tricks have been exhausted. but he said that beforehand. your afterhand excuse is useful to give your grace to cover up your mistakes and foolish. You're not alone. Despite the large quantities of verbiage generated, NOBODY has any clue what wangyong is on about. All attempts to elicit clearer explanations lead round in circles, or to even greater confusion. Initially I thought there might be some coherent (albeit probably wrong) idea behind his theories that was potentially worth discussing but wasn't getting across clearly because of language difficulties. Now I'm not so sure... mistakes.these are the sticking points. === Subject: Re: Confirmation of Shannon's Mistake about Perfect Secrecy of One-time-pad So assuming that the key is uniformly distributed is equivalent to > assuming that > the plaintext is uniformly distributed. > ----yes. Thus the condition that the key is uniformly distributed is > equivalent to the condition that the plaintext is uniformly > distributed. If you want to impose the condition that the key > is uniformly distributed you must abandon your > assumption that the plaintext might not be uniformly > distributed. - William Hughes and different probability . This does not matter. You have agreed assuming that the key is uniformly distributed is equivalent to > assuming that the plaintext is uniformly distributed So whatever conditions you use you cannot have both > the key uniformly distributed and the plaintext not > uniformly distributed at the same time. > If you start with a situation where the plaintext might > not be uniformly distributed, then you the key cannot > be uniformly distributed for a fixed cyphertext. - William Hughes- - - - This does not matter. You have agreed assuming that the key is uniformly distributed is equivalent to assuming that the plaintext is uniformly distributed So whatever conditions you use you cannot have both the key uniformly distributed and the plaintext not uniformly distributed at the same time. If you start with a situation where the plaintext might not be uniformly distributed, then you the key cannot be uniformly distributed for a fixed cyphertext. the key uniformly distributed and the plaintext not uniformly distributed at the same time. so you are sophistic by ignoring the conditions. === Subject: Re: Confirmation of Shannon's Mistake about Perfect Secrecy of One-time-pad If you start with a situation where the plaintext might > not be uniformly distributed, then you the key cannot > be uniformly distributed for a fixed cyphertext. > Note *for a fixed cyphertext*. The reply is irrlevant. The condition (repeated many times by you), is that the cyphertext is fixed). - William Hughes === Subject: Re: Confirmation of Shannon's Mistake about Perfect Secrecy of One-time-pad [...] In practice, when using OTP, the ciphertext can be anything. > If the ciphertext is fixed, something is going on to block > the ciphertexts that are different from the first > ciphertext. What mechanism does this? > four feet table In my paper. What is the compromise? I don't understand. > Are you calculating a conditional probability? > Something like: > Suppose event 'X' is observed. What is the chance that event 'Y' is > observed? David Bernier We give a simple example of OPT to discuss the problem, plaintext space is M {0,1}, ciphertext space is C {0,1} and key space is K {0,1}. According to the information that cryptanalysts got beforehand, they can get the prior probability of plaintext as P(M=0) = 0.9 and P(M=1) = 0.1. Later the ciphertext C=0 is intercepted. When only considering C=0 and the cryptosystem (regardless of the prior probability of plaintext), we can educe that the plaintexts are equally likely, for there is a one-to-one correspondence between all the plaintexts and keys for C=0. The prior probabilities of plaintexts are seldom the same, so the two probability distributions of conflicting. compromise of the two probability distributions is indispensable. The compromised posterior probability of the plaintext would be between the two corresponding probabilities of the two sectional conditions. When C=0 is intercepted, the posterior probability P(M=0) is between 0.9 and 0.5, and P(M=1) is between 0.1 and 0.5. The compromised posterior probability of the plaintext isn't equal to the prior probability, so OTP is not perfectly secure. === Subject: Re: Confirmation of Shannon's Mistake about Perfect Secrecy of One-time-pad If the ciphertext is fixed, something is going on to block > the ciphertexts that are different from the first > ciphertext. What mechanism does this? > four feet table In my paper. What is the compromise? I don't understand. > Are you calculating a conditional probability? > Something like: > Suppose event 'X' is observed. What is the chance that event 'Y' is > observed? David Bernier We give a simple example of OPT to discuss the problem, plaintext > space is M {0,1}, ciphertext space is C {0,1} and key space is K > {0,1}. According to the information that cryptanalysts got > beforehand, > they can get the prior probability of plaintext as P(M=0) = 0.9 and > P(M=1) = 0.1. Later the ciphertext C=0 is intercepted. When only > considering C=0 and the cryptosystem (regardless of the prior Note we are explicitely condidering a fixed cyphertext. > probability of plaintext), we can educe that the plaintexts are > equally likely, for there is a one-to-one correspondence between all > the plaintexts and keys for C=0. However, the key probabilities are not equal for a fixed cyphertext so the correxpondence tells you nothing about the probability distribution on the plaintext. - William Hughes === Subject: Re: Confirmation of Shannon's Mistake about Perfect Secrecy of One-time-pad [...] In practice, when using OTP, the ciphertext can be anything. > If the ciphertext is fixed, something is going on to block > the ciphertexts that are different from the first > ciphertext. What mechanism does this? > four feet table In my paper. What is the compromise? I don't understand. > Are you calculating a conditional probability? > Something like: > Suppose event 'X' is observed. What is the chance that event 'Y' is > observed? David Bernier What is the compromise? I don't understand. Are you calculating a conditional probability? Something like: Suppose event 'X' is observed. What is the chance that event 'Y' is observed? conditions are conflicting. === Subject: Re: 2000 availables Solutions manual I'm interested in downlowding the follwing items, Fluid Mechanics (5th Ed., White) + Ebook Heat Tranfer (2nd Ed., Cengel) + original Ebook > Heat and Mass Transfer: A Practical Approach (3rd. Ed., Cengel) + > original Ebook thankyou very much, Edan > My List of Solutions Manual > contact me to : newbergh123yahoo.com > newbergh123(at)yahoo.com ot to : mattos...@gmail.com > mattosbw1(at)gmail.com If your wanted solutions manual ins't on this list, also can ask me if > is available . These are some only. This same list of tites (not links) is available from : http://rapidshare.com/files/64945514/List of solutions manual.txt - Mechanics, Mechanical Engineering & Aerospace Engineering: > Classical mechanics (2nd Ed., Goldstein) > Classical Mechanics (Douglas Gregory) + original Ebook > Advanced Dynamics (Greenwood) + original Ebook > Advanced Engineering Dynamics (2nd Ed., Jerry Ginsberg) + Ebook > Classical Dynamics (Jorge V. Jos.8e) + Ebook > Impact Mechanics (W.J. Stronge) > Introduction to Mechanical Engineering (Rizza) > Mechanical Engineering Principles (Bird & Ross) + original Ebook > Engineering Fluid Mechanics (William Graebel) > Advanced Fluid Mechanics (William Graebel) + original Ebook > Mechanics of Fluids (8th Ed., Massey) + original Ebook > Fluid Mechanics (5th Ed., White) + Ebook > Fluid Mechanics (6th Ed., White) > Viscous Fluid Flow (3rd Ed., White) + Ebook > Fundamentals of Thermal-Fluid Sciences (1st Ed., Cengel) + original > Ebook > Fundamentals of Thermal-Fluid Sciences (2nd Ed., Cengel) + original > Ebook > Fundamentals of Thermal-Fluid Sciences with Student Resource CD (3rd > Ed., Cengel & Turner) > Thermodynamics: An Engineering Approach (5th Ed., Cengel) + original > Ebook > Thermodynamics: An Engineering Approach (6th Ed., Cengel) + original > Ebook > Essentials of Fluid Mechanics: Fundamentals and Applications (1st Ed., > Cengel) + original > Fluid Mechanics (1st Ed., Cengel) + original Ebook > Heat Tranfer (2nd Ed., Cengel) + original Ebook > Heat and Mass Transfer: A Practical Approach (3rd. Ed., Cengel) + > original Ebook > Design and Simulation of Thermal Systems (Suryanarayana & Arici) > Introduction to Fluid Mechanics (6th Ed., Robert Fox, Alan McDonald & > Philip Pritchard) > Fluid Mechanics (5th Ed., Douglas) > Fluid Mechanics (3rd Ed., Kundu) > Fluid Mechanics with Engineering Applications (Finnemore) > Fundamentals of Fluid Mechanics, 4th Ed (Bruce R. Munson, Donald F. > Young, Theodore H. Okiishi) + original ebook > Fundamentals of Fluid Mechanics, 5th Ed (Bruce R. Munson, Donald F. > Young, Theodore H. Okiishi) > A Brief Introduction to Fluid Mechanics, 3rd Ed (Donald F. Young, > Bruce R. Munson, Theodore H. Okiishi) > A Brief Introduction to Fluid Mechanics, 4th Ed (Donald F. Young, > Bruce R. Munson, Theodore H. Okiishi, Wade W.) > Engineering Fluid Mechanics, 7th Ed (Clayton T. Crowe, Donald F. > Elger, John A. Roberson) > Engineering Fluid Mechanics, 8th Ed (Clayton T. Crowe, Donald F. > Elger, John A. Roberson) > Mechanics of Fluids (3rd Ed., Potter) > Mechanics of Fluids (4th Ed., Shames) > Extended Irreversible Thermodynamics (3rd Ed., D. Jou, J. Casas- > Vazquez & G. Lebon) > Thermodynamics: An Integrated Learning System (Schmidt, Ezekoye, > Howell & Baker) > Introduction to Thermal and Fluids Engineering (Kaminski & Jensen) > Heating, Ventilating and Air Conditioning Analysis and Design (6th > Ed., McQuiston) > An Introduction to Fluid Dynamics: Principles of Analysis and Design > (Middleman) > Introduction to Mass and Heat Transfer: Principles of Analysis and > Design (Middleman) > Heat Transfer (2nd Ed., Mills) > Convective Heat and Mass Transfer (4th Ed., Kays & Crawford) > Advanced Engineering Thermodynamics (3rd Ed., Bejan) > Convection Heat Transfer (2nd Ed., Bejan) > Convection Heat Transfer (3rd Ed., Bejan) > Thermal Design and Optimization (Bejan) > Shape and Structure, from Engineering to Nature (Bejan) > An Introduction to Combustion: Concepts and Applications (2nd Ed., > Turns) > Thermodynamics: Concepts and Applications (Stephen Turns) > Thermal-Fluid Sciences: An Integrated Approach (Stephen Turns) > Principles of Heat Transfer (Kaviany) > Heat Convection (Latif M. Jiji) + original Ebook > Heat Transfer (9th Ed., Holman) > Fundamentals of Momentum, Heat and Mass Transfer (4th Ed., Welty) > Fundamentals of Momentum, Heat and Mass Transfer (5th Ed., Welty) > Momentum, Heat, and Mass Transfer Fundamentals (Kessler) + original > Ebook > Analytical Methods for Heat Transfer and Fluid Flow Problems (Bernhard > Weigand) > Heat Tranfer (Rao) > Heat Conduction (kakac) > Heat Exchanges (Kakac) > Convective Heat Transfer (kakac) > Heat Exchangers: Selection, Rating and Thermal Design (2nd Ed. Sadik > Kakac & Hongtan Liu) > Fundamentals of Engineering Thermodynamics, 5th Ed (Michael J. Moran, > Howard N. Shapiro) + original Ebook > Fundamentals of Engineering Thermodynamics, 6th Ed (Michael J. Moran, > Howard N. Shapiro) > Fundamentals of Heat and Mass Transfer (5th Ed., Incropera, DeWitt) > Fundamentals of Heat and Mass Transfer (6th Ed., Incropera, DeWitt) > Introduction to Heat Transfer (4th Ed., Incropera, DeWitt) > Introduction to Heat Transfer (5th Ed., Incropera, DeWitt) > Radiation Detection and Measurement (3rd Ed., Glenn Knoll) > Radiative Heat Transfer (2nd Ed., Michael Modest) > Engineering Heat Transfer (2nd Ed., Janna) > Engineering Thermodynamics: Work and Heat Transfer (4th Ed., G.F.C. > Rogers & Y.R. Mayhew) > Elements of Heat Transfer (Yildiz Bayazitoglu and M. Necati Ozisik) > Inverse Heat Transfer: Fundamentals and Applications (M.N. Ozisik & > Helcio R.B. Orlande) > Thermal Radiation Heat Transfer (4th Ed.,Robert Siegel & John R. > Howell) > Computational Heat Transfer (2nd Ed., Jaluria) > Principles of Combustion (2nd Ed., Kenneth Kuan-yun Kuo) > Incompressible Flow (3rd Ed., Panton) > Modern Compressible Flow: With Historical Perspective (3rd Ed., John > D. Anderson) > Non-Newtonian Flow : Fundamentals and Engineering Applications (R P > Chhabra & J F Richardson) + original Ebook > Computational Techniques for Fluid Dynamics (Srinivas, K., Fletcher, > C.A.J.) > Ebook > Theory of Applied Robotics: Kinematics, Dynamics and Control (Reza N. > Jazar) > Kinematic Chains and Machine Components Design (Dan B. Marghitu) + > original Ebook > Kinematics and Dynamics of Machinery (3rd Ed., Wilson & Sadler) > Kinematics, Dynamics, and Design of Machinery (2nd Ed., Waldron & > Kinzel) > Mechanism Design: Analysis and Synthesis-Volume 1 (4th Ed., Erdman & > Sandor) > Machines and Mechanisms: Applied Kinematic Analysis (3rd Ed., > Myszka) > Mechanical Design: A Components Approach (Peter Childs) > Mechanical Design of Machine Elements and Machines: A Failure > Prevention Perspective (Collins) > Fundamentals of Machine Component Design (3rd Ed., Juvinall) > Fundamentals of Machine Component Design (4th Ed., Juvinall) > Design of Machine Elements (8th Ed., Spotts) > Machine Design (Wentzell) > Solutions Manual to the text : Problems on the Design of Machine > Elements (Faires) > Machine Elements in Mechanical Design (4th Ed., Mott) > Mechanical Design: An Integrated Approach (1st Ed., Ugural) > Design of Machinery (3rd Ed., Norton) > Design of Machinery (4th Ed., Norton) > Machine Design (2nd Ed., Norton) > Machine Design : An Integrated Approach (3rd Ed., Norton) > Mechanical Engineering Design (6th Ed., Shigley) > Mechanical Engineering Design (7th Ed., Shigley) > Shigley's Mechanical Engineering Design (8th Ed., Budynas) > Fundamentals of Machine Elements (1st Ed., Hamrock) > Fundamentals of Machine Elements (2nd Ed., Hamrock) > Mechanics of Materials: A Modern Integration of Mechanics and > Materials in Structural Design (Christopher Jenkins & Sanjeev Khanna) > Mechanics of Materials (3th Ed., Beer) > Mechanics of Materials (5th Ed., Gere) > Mechanics of Materials (6th Ed., Gere) > Mechanics of Materials (Ugural) > Simplified Mechanics and Strength of Materials (6th Ed., James > Ambrose) > Engineering Mechanics, Statics, 2nd Ed (William F. Riley, Leroy D. > Sturges) > Engineering Mechanics, Dynamics, 2nd Ed (William F. Riley, Leroy D. > Sturges) > Engineering Mechanics - Statics, 5th Ed (J. L. Meriam, L. G. Kraige) + > Ebook > Engineering Mechanics - Statics, 6th Ed (J. L. Meriam, L. G. Kraige) > Engineering Mechanics - Dynamics, 5th Ed (J. L. Meriam, L. G. Kraige) > Engineering Mechanics - Dynamics, 6th Ed (J. L. Meriam, L. G. Kraige) > Vector Mechanics for Engineers: Statics (7th Ed., Ferdinand P. Beer) > Vector Mechanics for Engineers: Statics (8th Ed., Ferdinand P. Beer) > Vector Mechanics for Engineers: Dynamics (7th Ed., Ferdinand P. Beer) > Vector Mechanics for Engineers: Dynamics (8th Ed., Ferdinand P. Beer) > Statics: Analysis and Design of Systems in Equilibrium (Sheppard & > Tongue) > Dynamics: Analysis and Design of Systems in Motion (Sheppard & Tongue) > Statics and Mechanics of Materials: An Integrated Approach (2nd Ed., > Riley, Sturges & Morris) > Mechanics of Materials (6th Ed., Riley, Sturges & Morris) > Deformable Bodies and Their Material Behavior (Haslach & Armstrong) > Strength of Materials - Volume 1 : Elementary Theory and Problems > (Timoshenko) > Intermediate Mechanics of Materials, (1st Ed., Barber) > Elasticity (2nd Ed., J.R. Barber) + original Ebook > Elasticity: Theory, Applications, and Numerics (Martin Sadd) + > original Ebook > Elasticity in Engineering Mechanics (2nd Ed., Boresi) > Advanced Mechanics of Materials (6th Ed., Boresi) + Ebook > Engineering Mechanics: Dynamics (Boresi) > Metal Fatigue in Engineering (2nd Ed., Stephens, Fatemi & Fuchs) > Applied Mechanics for Engineering Technology (8th Ed., Keith M. > Walker) > Applied Fluid Mechanics (6th Ed., Mott) > Applied Strength of Materials (4th Ed., Mott) > Applied Strength of Materials (5th Ed., Mott) > Intermediate Dynamics for Engineers (Marcelo R.M & Crespo da Silva) > Engineering Mechanics - Statics (4th Ed., Anthony Bedford & Wallace > Fowler) > Engineering Mechanics - Statics (5th Ed., Anthony Bedford & Wallace > Fowler) > Engineering Mechanics - Dynamics (4th Ed., Anthony Bedford & Wallace > Fowler) > Engineering Mechanics - Dynamics (5th Ed., Anthony Bedford & Wallace > Fowler) > Engineering Mechanics: Statics (2nd Ed., Pytel) > Engineering Mechanics: Dynamics (2nd Ed., Pytel) > Engineering Mechanics: Statics (2nd Ed., Shames) > Engineering Mechanics: Statics (4th Ed., Shames) read more E... === Subject: Re: Complete Electronic (.pdf/doc) Solution Manuals. Get witihn 30 Minutes! I have the comprehensive solution manual, solutions manual, solutions > manuals, in electronic format for the following textbooks. They > include complete solutions to all the problems in the text, except > where noted below in the listing. Payment is through Paypal. Email me > bookstoday777[at]gmail.com but please DO NOT POST HERE because I will > not be able to help you, but instead email and ask me for the solution > that you need. Downloads emailed immediately - within 30 minutes! A Course in Game Theory by Osborne, Rubinstein > A Course in Algebraic Number Theory by Cohen > Adaptive Filter Theory, 4th Edition, by Haykin > Adaptive Control, 2nd. Ed., by Astrom, Wittenmark > Advanced Engineering Mathematics, 8th Editoin, by Erwin Kreyszig (even > solutions) > Advanced Engineering Mathematics, 9th Edition, by Erwin Kreyszig (even > solutions) > Advanced Macroeconomics, 1st Ed., by David Romer > Advanced Mathematical Concepts Precalculus With Applications by > Holliday [ISBN: 0028341759] > Advanced Modern Engineering Mathematics, 3rd Ed., by G. James > A First Course In Differential Equations, 7th Edition, by Zill, Cullen > Analog Integrated Circuit Design, 1st Ed., by Johns, Martin (text > ebook and solution manual) > Analysis and Design of Analog Integrated Circuits, 4th Ed., by Gray, > Hurst, Lewis, Meyer > Analytical Mechanics, 7th Edition, by Fowels, Cassiday > An Interactive Introduction to Mathematical Analysis, by Jonathan > Lewin > An Introduction to the Mathematics of Financial Derivatives, 2nd Ed., > by Neftci [ISBN: 0125153929] > Antenna Theory, 2nd Ed., by Balanis > Antennas for all Applications, 3rd Edition, Kraus, Marhefka > Applied Linear Statistical Models, 5th Ed., by Neter (Selected Sol.) > Applied Numerical Analysis, 6th Edition, by Gerald, Wheatley > Applied Numerical Methods with MATLAB for Engineers and Scientists, > 1st Ed,. by Chapra > Applied Statistics and Probability for Engineers, 3rd Ed., by > Montgomery, Runger (Selected Solutions) > Applied Strength of Materials, 4th Edition, by Mott > A Transition to Advanced Mathematics, 5th Edition, by Smith, Eggen, > Andre > Automatic Control Systems, 8th Edition, by Kuo, Golnaraghi Basic Business Statistics: Concepts and Applications, 10th Ed., by > Berenson, Krehbiel, Levine (chap1-18) > Basic Engineering Circuit Analysis, 7th Ed., by J. David Irwin > Basic Engineering Circuit Analysis, 8th Ed., by J. David Irwin, Nelms > (Missing a chapter or 2) > Bioprocess Engineering Principles by Doran Calculus Early Transcendental, 5th Ed., by James Stewart > Calculus - Early Transcendentals, 7th Ed., by Anton, Bivens, Davis > Calculus: Graphical, Numerical, Algebraic, 3rd Ed., Waits, Finney, > Demana, Kennedy > Calculus: Multivariable, 5th Edition, by James Stewart > Calculus: Single Variable, Early Transcendental, 5th Edition, by James > Stewart > Calculus, Single and Multivariable, 3rd Ed., by Hughes-Hallett, > McCallum > Calculus: Study and Solutions Guide, Vol. 1, 7th Ed., by Larson, > Hostetler, Edwards > Chemical and Engineering Thermodynamics, 3rd Ed., Stanley I. Sandler > Chemical Engineering Volume 1, 6th Edition, by Richardson, Coulson, > Backhurst, Harker > Thornton > College Physics, Volume 1: 7th Edition, by Serway, Faugh > College Physics, Volume 2: 7th Edition, by Serway, Faughn > Communications Systems, 4th Ed., by Haykin > Communications Systems Engineering, 2nd Edition, by Proakis > Computational Techniques for Fluid Dynamics by Srinivas, Fletcher > Computer Networks, 4th Ed., by Andrew S. Tanenbaum > Computer Networks: A Systems Approach, 3rd Edition, by Davie > Control Systems Engineering, 4th Ed., by Norman Nise > Corporate Finance, 6th Edition, by Ross > C++ How to Program: Intro Object-Oriented Design with the UML, 3rd > Ed., by Deitel, Nieto Data and Computer Communications, 8th Edition by Stallings > Database Management Systems, 3rd Ed., by Ramakrishnan, Gehrke (Sol. > for Chapters 2-21, odd only) > Design of Analog CMOS Integrated Circuits, 1st Edition, by Razavi > Design of Analysis of Experiments, 6th Edition, Montgomery (missing > chapter 6-8) > Design of Machinery, 3rd Ed by Robert L. Norton > Design With Operational Amplifiers and Analog Integrated Circuits, 2nd > Ed., by Sergio Franco > Design With Operational Amplifiers and Analog Integrated Circuits, 3rd > Ed., by Sergio Franco > Device Electronics for Integrated Circuits 3rd Edition by Muller > Differential Equations with Boundary Value Problems, 2nd Ed., by > Polking, Arnold > Digital And Analog Communication Systems 7th Ed., Leon > W. Couch > Digital Communications, 4th Edition, by Proakis > Digital Communications: Fundamentals and Applications, 2nd Ed, Skylar > Digital Design, 4th Edition, by Mano, Ciletti > Digital Image Processing, 2nd Edition, by Gonzalez, Woods > Digital Integrated Circuits, 2nd Ed., by Rabaey (Solutions ONLY for > Chapters 3, 5, 6, 10) > Digital Signal Processing: A Computer Based Approach, 1st Ed., by > Mitra > Digital Signal Processing: A Computer Based Approach, 2nd Ed., by S. > Mitra > Digital Signal Processing: A Computer Based Approach, 3rd Ed., by S. > Mitra > Digital Signal Processing: Principles, Algorithms and Applications, > 3rd Edition, by Proakis > Discrete Time Signal Processing, 2nd Edition, Oppenheim > Dynamics of Mechanical Systems by C.T.F. Ross Econometric Analysis, 5th Edition, by Greene > Wooldridge > Econometrics of Financial Markets, by Adamek, Cambell, Lo, MacKinlay, > Viceira > Electrical Engineering Principles and Applications, 3rd Ed., by > Hambley > Electrical Properties of Materials, 7th Ed., by D. Walsh, L. Solymar > Electric Circuits 6th Ed. by Nilsson > Electric Circuits 7th Ed. by Nilsson > Electric Machinery, 6th Ed., Fitzgerald, Kingsley, Umans > Electric Machinery Fundamentals, 4th Ed by Chapman > Electric Machinery and Power System Fundamentals, 1st Ed., by S. > Chapman > Electronic Circuit Anlaysis, 2nd Ed., by Donald Neamen > Electronic Fields and Waves by Iskander > Electronics, 2nd Ed., by Allan R. Hambley > Elementary Differential Equations, 8th Edition, by Boyce, DiPrima > (some odd/even) > Elementary Principles of Chemical Processes, 3rd Ed., by Felder, > Rousseau > Elements of Chemical Reaction Engineering, 3rd Ed., by H. Scott Fogler > Engineering and Chemical Thermodynamics, by Koretsky [ISBN: > 0471385867] (No sol. for chapt 6) > Engineering Circuit Analysis, 6th Edition, Hyat > Engineering Electromagnetics, 6th Ed W. Hayt, J. Buck > Engineering Electromagnetics, 7th Ed., Hayt, Buck > Engineering Fluids Mechanics 7th Edition by Crowe > Engineering Fluids Mechanics 8th Edition by Crowe > Engineering Mathematics, 4th Ed., by John Bird > Engineer Mechanics: Dynamics, 4th Ed., by Bedford > Engineering Mechanics: Dynamics, 10th Ed., by Russell C. Hibbeler > Engineering Mechanics: Dynamics 11th Ed. by Hibbeler > Engineering Mechanics: Dynamics 5th Ed. by Meriam, Kraige > Engineering Mechanics: Statics, 4th Edition - A. Bedford, Wallace > Fowler > Engineering Mechanics: Statics, 5th Ed., Meriam, Kraige > Engineering Mechanics: Statics, 6th Ed., Meriam, Kraige > Engineering Mechanics: Statics, 10th Ed., by Russell C. Hibbeler > Engineering Mechanics: Statics 11th Ed. by Hibbeler > Essentials of Fluid Mechanics: Fundamentals and Applications, 1st Ed., > by Cengel & Cimbala > Experiments with Economic Principles by Bergstrom, Miller Facilities Planning, 3rd Edition, by Tompkins, White, Bozer, Tanchoco > Feedback Control of Dynamic Systems, 4th Edition, by Powell, Emami- > Naeini > Financial Accounting, 4th Ed., by Libby, Short (Chap1-14) > Financial Accounting: An International Introduction, 2nd Ed., by > Alexander, Nobes > Finite Element Techniques in Structural Mechanics by Ross > Fluid Mechanics - 5th Edition by Frank M. White > Fluid Mechanics and Thermodynamics of Turbomachinery, 5th Ed., by S. > L. Dixon [ISBN: 0750678704] > Fluid Mechanics with Engineering Applications, 10th Edition, by > Finnemore > Fundamentals of Aerodynamics, 3rd Edition, by J. D. Anderson, Jr. > Fundamentals of Applied Electromagnetics, 1st Ed., 2001 Media Edition, > by Ulaby > Fundamentals of Applied Electromagnetics, 5th Ed., 2008 Media Edition, > by Ulaby > Fundamentals of Digital Logic with Verilog Design, 1st Edition, by > Brown, Vranesic > Fundamentals of Electric Circuits, 2nd Edition, by Alexander > Fundamentals of Electromagnetics with Engineering Appls by Wentworth > Fundamentals of Fluid Mechanics, 5th Ed. by Munson, Young.. > Fundamentals of Heat and Mass Transfer, 4th Ed by Incropera... > Fundamentals of Heat and Mass Transfer, 5th Ed by Incropera... > Fundamentals of Heat and Mass Transfer, 6th Ed by Incropera... > Fundamentals of Logic Design, 5th Ed., by Roth Jr. > Fundamentals of Machine Component Design, 3rd Ed., by Juvinall > Fundamentals of Machine Component Design, 4th Ed., by Juvinall > Fundamentals of Machine Elements, 2nd Ed., Hamrock, Jacobson, Schmidt > Fundamentals of Physics by Halliday, 7th Ed., Walker, Resnick > Fundamentals of Semiconductor Devices, 1st Edition by Anderson > Fundamentals of Structural Analysis, 2nd Ed., Chia-Ming Uang, Kenneth > Leet > Fundamentals of Thermal-Fluid Sciences, 2nd Ed. by Cengel > Fundamentals of Thermal-fluid Sciences, Int'l 2nd Ed. by Cengel > Fundamentals of Engineering Thermodynamics, 5th Ed. by Shapiro > Fundamentals of Engineering Thermodynamics, 6th Ed., by Shapiro > Fundamentals of Thermodynamics, 5th Ed., by Sonntag, Borgnakke... > Fundamentals of Thermodynamics, 6th Ed., by Sonntag Geometry, 04 Edition, by McGraw-Hill [ISBN: 0078296374] > Guide to Energy Management, 5th Edition, by Pawlik Heat Transfer: A Practical Approach - 2nd Edition by Cengel > Hydraulics in Civil and Environmental Engineering, 4th Ed., by Andrew > Chadwick Introduction to Algorithms, 2nd Ed by Cormen, Leiserson (Selected > Sol.) > Introduction To Chemical Engineering Thermodynamics, 7th Ed., by Van > Ness, Smith, Abbott > Introduction to Electric Circuits, 6th Ed., by Dorf, Svoboda > Introduction to Electric Circuits, 7th Ed., by Dorf, Svoboda > Introduction to Electrodynamics, 3rd Ed. by David Griffiths > Introduction to Fluid Mechanics - 5th Ed. by Fox.. > Introduction to Fluid Mechanics - 6th Ed by Fox, McDonald... > Introduction to Linear Algebra, 3rd Ed., by Gilbert Strang > Introduction to Linear Algebra, 5th Ed., Arnold, Johnson, Riess > Introduction to Probability by Grinstead, Snell (odd solutions only, > not just answers but step by step solutions) > Introduction to Quantum Mechanics, 2nd Ed. by Griffiths > Introdution to Solid State Physics, 8th Edition by Kittel > Introduction to Statistical Quality Control, 4th Edition, by > Montgomery > Introduction to Thermal Systems Engineering: Thermodynamics, Fluid > Mechanics, and Heat Transfer by Moran, Shapiro, Munson, > DeWitt Linear Algebra, by J. Hefferon > Linear Algebra And Its Applications, 3rd Ed., by David C. Lay > Linear Algebra with Applications, 2nd Edition - by Otto Bretscher > Linear Algebra with Applications, 3rd Edition - by Otto Bretscher > Linear Circuit Analysis: Time Domain, Phasor and Laplace.., 2nd Ed, > Lin > Linear Systems and Signals, 1st Ed., by B.P. Lathi Machine Design: An Integrated Approach, 2nd Ed., by Robert L. Norton > Machine Design: An Integrated Approach, 3rd Ed., by Robert L. Norton > Managerial Accounting, 11th Ed., by Noreen, Brewer, Garrison > Materials Science and Engineering: An Introduction, 6th Ed. by > Callister > Matrix Analysis and Applied Linear Algebra by Carl Meyer > MC68HC11: An Introduction: Software/Hardware Interf, 2nd Ed, by Huang > Mechanical Engineering Design, 7th Ed. by Mischke, Shigley > Mechanical Vibrations, 3rd Edition, by S. S. Rao (99% same as 4th > Edition, No Solutions for Chapters 6, 9, and 12) > Mechanics of Fluids, 8th Ed., by Bernard Massey > Mechanics of Fluids, 4th Ed., Irving H. Shames > Mechanics of Fluids, 8th Ed., by Bernard Massey > Mechanics of Materials - 3rd Ed. by Beer, Johnston, Dewolf > Mechanics of Materials - 6th Ed. by Hibbeler > Mechanics of Materials, 6th Edition by James M. Gere (missing small > portion, section 8.5) > Mechanics of Materials, 6th Ed., by Sturges, Morris, Riley (part of > Chapt 2 is missing but only #21 thru #60) > Mechanics of Solids by C.T.F. Ross > Microeconomic Analysis, 3rd Ed., by H. Varian (Ans. to Exercises: Ch. > 1- Ch.25) > Microeconomic Theory, by Mas-Colell, Whinston, Green > Microelectronic Circuit Analysis and Design, 3rd Edition, by D. Neamen > Microelctronic Circuits, 5th Ed. by Sedra and Smith > Microelectronic Circuit Design, 2nd Edition by Jaeger, Blalock > Microelectronic Circuit Design, 3rd Edition by Jaeger, Blalock > Microelectronics: Digital and Analog Circuits and Systems by Millman > Microwave and Rf Design of Wireless Systems, 1st Edition, by Pozar > Microwave Engineering, 3rd Ed., by David M. Pozar > Microwave Transistor Amplifiers: Analysis and Design, 2nd Ed., by > Guillermo Gonzalez > Miller & Freund's Probability and Statistics for Engineers, 7th > Edition, Johnson, Miller > Modern Compressible Flow, 3rd Edition, by Anderson > Modern Control Engineering, 3rd Edition, by Ogata > Modern Control Engineering, 4th Edition, by Ogata > Modern Digital and Analog Communication Systems, 3rd Ed., by Lathi > Modern Control Systems, 9th Ed., by Richard C. Dorf, Robert H Bishop > (98% same as the 10th Ed.) > Modern Operating Systems,2nd Ed., by Andrew Tanenbaum > Modern Physics 4th Edition by Tipler > Monetary Theory and Policy, 2nd Edition, by Walsh > Multivariable Calculus, 5th Edition, by James Stewart Numerical Methods, 3rd Ed., by J. Douglas Faires, Richard L. Burden > (Selected Solutions) Operating Systems: Internals and Design Principles, 4th Edition, by > Stallings > Operating System Concepts, 7th Ed., Silberschatz, Galvin, Gagne > Options, Futures and Other Derivatives, 4th Ed., by John Hull > Options, Futures and Other Derivatives, 5th Ed., by John Hull > (Chapters 1 thru 18 ONLY) > Orbital Mechanics: For Engineering Students by Howard Curtis (includes > matlab scripts) > Organic Chemistry, 4th Ed., by Carey, Atkins (Student Study Guide and > Sol. Man.) Partial Differential Equations with Fourier Series and Boundary Value > Problems, 2nd Ed., by Asmar (Student Solutions Manual) > Physical Chemistry - 7th Edition - by Julio de Paula, Peter Atkins > Physics, 6th Edition, by John Cutnell > Physics, 5th Edition, Vol 2 by Halliday, Resnick, Krane (Chap 25-52) > Physics: for Scientists and Engineers with Modern Physics, 3rd Ed., by > Fishbane (Consists of Chapters 1-37) > Physics for Scientist and Engineers by Knight (No Chapt 36-42) > Physics for Scientist and Engineers, 6th Ed., by Serway > Physics for Scientists and Engineers-Vol 1, 5th Edition, Serway, > Beichner (Chap. 1 - 22) > Physics for Scientists and Engineers-Vol 2, 5th Edition, Serway, > Beichner (Chap. 23 - 46) > Physics for Scientists and Engineers, 3rd Ed., by Douglas C. Giancoli > Physics for Scientist and Engineers, 5th Edition, by Tipler, Mosca > Physics: Principles with Applications, 6th Ed. by Giancoli > Power System Analysis and Design, 3rd Ed., by Glover, Sarma > Principles and Applications of Electrical Engineering 4th (Revised) > Edition by Rizzoni > Principles and Practices of Automatic Process Control, 3rd Edition by > Smith, Corripio [ISBN: 0471431907] > Principles of Communication: Systems, Modulation Noise, 5th Ed., > Ziemer > Principles of Physics, 3rd Edition, by Serway > Principles of Physics, 4th Edition, by Serway > Principles of Statics, 10th Ed., by Russell C. Hibbeler [ISBN: > 0131866745] > Probability and Statistics for Engineers and Scientists, 3rd Edition, > Hayter > Probability and Statistics for Engineering and the Sciences, 6th Ed., > by Jay L. Devore > Probability Random Variables, and Stochastic Processes, 4th Ed., by > Papoulis, Pillai > Quantum Mechanics: An Accessible Introduction, 1st Ed., by Robert > Scherrer Recursive Macroeconomic Theory, 1st Ed., by Ljungqvist, Sargent > Recursive Methods in Economic Dynamics, (2002) by Irigoyen, Rossi- > Hansberg, Wright > RF Circuit Design: Theory & Applications, by Bretchko, Ludwig Sears and Zemansky's University Physics 11th Edition by Young.. > Semiconductor Device Fundamentals by Pierret > Semiconductor Devices: Physics and Technology, 2nd Ed., S.M. Sze > Semiconductor Physics And Devices -3rd Ed. by D. Neamen > Separation Process Principles, 2nd Ed., Seader, Henley > Signal Processing and Linear Systems by Lathi > Signals and Systems, 2nd Edition, by Haykin, Van Veen > Signals and Systems, 2nd Edition, Oppenheim, Willsky, Hamid, Nawab > Signals and Systems: Analysis Using Transform Methods and MATLAB, 1st > Ed., by M. J. Roberts > Signals, Systems, and Transforms, 3rd Ed., by Charles L. Phillips, Eve > A. Riskin, John M. Parr > Shigley's Mechanical Engineering Design, 8th Ed. by Budynas, Nisbett > (No Sol. for Chapt 18 & 19) > Simply C#: An Application-Driven Tutorial Approach, by Deitel, Hoey > (Chapters 1-32) > Soil Mechanics: Concepts and Applications, 2nd Ed., by Powrie > Solid State Electronic Devices - 5th Ed by Streetman > Solid State Electronic Devices - 6th Ed by Streetman > Statics and Mechanics of Materials: An Integrated Approach, 2nd Ed., > by Riley, Sturges, Morris > Structural Analysis, 5th Edition, by Hibbeler Theory and Design for Mechanical Measurements, 4th Ed., Beasley, > Figliola > Thermal Physics, 2nd Edition, by Charles Kittel > Thermal Physics, by Ralph Baierlein > Thermodynamics: An Engineering Approach, 5th Ed., by Cengel, Boles > (Missing solutions #118-149 of Chapter 7) > Thermodynamics: An Engineering Approach, 6th Ed., by Cengel, Boles > The Science and Engineering of Materials, 4th Ed., by Donald R. > Askeland, Pradeep P. Phule > Thomas' Calculus, Early Trans., Part 1, 10th Ed. by Thomas, Weir, > Hass, Giordano > Thomas' Calculus: Part 2, 10th Ed. (Multivariable, chs. 8-13), by > Thomas, Weir, Hass, Giordano > Thomas' Calculus, Early Trans., Part 1, 11th Ed. by Thomas, Weir, > Hass, Giordano > Thomas' Calculus: Part 2, 11th Ed. (Multivariable, chs. 11-16), by > Thomas, Weir, Hass, Giordano > Transport Phenomena, 1st Edition, by R. Byron Bird > Transport Phenomena, 2nd Ed., by Bird University Physics 11th Edition by Young.. Vector Mechanics: Statics 7th Edition by Beer > Vector Mechanics: Dynamics, 7th Ed., by Beer, Johnston, Staab, Clausen > Vibrations and Stability: Advanced Theory, Analysis, and Tools, 7th > Ed., by Thomsen Wireless Communications: Principles and Practice, 2nd Ed, by Rappaport Hi! Could you email me Physics 6th Edition by Cutnell unless you have > advance! could you email me the following > Options, Futures and Other Derivatives, 4th Ed., by John Hull > Options, Futures and Other Derivatives, 5th Ed., by John Hull > (Chapters 1 thru 18 ONLY) === Subject: Re: Distortion of a picture with this mapping? >Hi. What happens if you take a picture, consider the coordinates on it to >be complex numbers, then apply the mapping f(z) = z^i (ie. raise each point to a purely imaginary exponent equal to the >imaginary unit) If z = a*exp(-ib) then the remapped coordinates u and v of w=u+jv become:u = exp(-b)(cos(log(a)) ; v = exp(-b)(sin(log(a)) Yours is an excellent suggestion! I checked it out because a similar complex remapping, where f(z) = z^(-1) yields a very interesting remapping of the Mandelbrot: http://home.rochester.rr.com/jbxroads/interests/sci.fractals/Java_Fractals/a ntimbrt/invmnbt.html (or, if that url is too long for your newsreader: http://tinyurl.com/2yn2zr) In this case, the transformation doesn't look too exciting. If you try it (feel free to adapt the source of my inverse mandelbrot as a start) please post the results. One other observation: two applications of mike4's remapping yields john's remapping(?) Hmmm. John === Summary: UDP === This server is temporarily under UDP because of recent massive hipcrime attacks, until the problem is solved by their administrators. We are sorry for the inconvenience. See news.admin.net-abuse.policy and news.admin.net-abuse.usenet for more details. === Subject: Natural numbers and the continuum (13/16). 13.- As a consequence, we have no complete information about natural numbers without the continuum. Fernando. Previous: 1.- The real line, that is a line with a fixed scale so that every real number corresponds to a unique point on the line. The real line works as a coding for the points of a line. 2.- Every bijective function between a half open interval and non negative real numbers changes the coding for the points on a half-line. 3.- Peano Arithmetic is embedded into the half-open interval via bijectivity. 4.-The pairs whose coordinates are numbers of the half-open interval constitute a deformed plane. 5.- We transform the x y = k ( k > 0 ) hyperbolas into curves of the deformed plane. 6.- Geometry does not identify natural number coordinate points of the hyperbolas neither in the plane nor in the deformed plane. 7.- A natural number p is prime iff p > 1 and the only natural coordinate points of the x y = p hyperbola are ( 1, p ) and ( p, 1 ). 8.-It is possible to obtain deformed planes in such a way that we can identify natural coordinate points of the deformed hyperbola of x y = n ( n natural number ). This identification is obtained in terms of differentiability of deformed hyperbolas close to x y = n. 9.-As a consequence we can identify primes via continuum. 10.- Adequate selections of areas between deformed hyperbolas allow to relate two natural numbers and its sum. 11.- In some deformed planes, the second derivative of the mentioned area provide a characterization the Goldbach Conjecture in an infinite set of even numbers. 12.- Deformed planes which characterize Goldbach Conjecture can be continuously deformed in such a way that we keep the Arithmetic but we lose the characterization. === Subject: Re: how to determine sufficient sample size? > I have a problem where I need to determine a minimum > sample size. I took stats a long time ago, so please > bear with me. Suppose I have a bunch of coins, some of which are fairly > weighted, but there are some that are unfairly weighted > (for example, these coins come up heads only 25% of > the time). I want to classify the coins into either one of two classes: > fairly-weighted and unfairly-weighted. I can flip each coin > as many times as I want, but I want to find out the minimum > number of flips I need to determine if a coin is fair or not. I am ok if a coin is fair or not requires a confidence > parameter, which I can give, like 90% or something. > Each flip of the coin can be treated as a binomial random variable, with heads probability p and tails probability q=1-p. Let X be the number of heads obtained in N flips. If Np and Nq are both greater than 5, or N is more than about 30, then the distribution of X can be approximated by a normal distribution with mean Np and variance Npq. What you want to do is a test on the hypothesis that p=0.5, with whatever confidence level you desire. Note that this will be able to tell you if you can reject the hypothesis of a fair coin, but not if you can accept it. You can then do the hypothesis tests on p>0.5 and on p<0.5 to test a coin biased towards heads or tails (respectively). If you can reject 2 of the 3 hypotheses (at your given confidence level), then you can say your tests support the claim of a biased/unbiased coin. If you cannot reject two of them, then you need to increase N. Since your hypothesis tests are always dealing with the null hypothesis including p=0.5, by the above you only need 10 flips to use the normal distribution. However, if coins may only be slightly biased, then it is unlikely that 10 flips will allow you to reject two hypotheses (you may not be able to reject ANY of them). If you have a bound on the amount of bias, you can find a minimum number of trials to allow the normal approximation, but you will never know how many trials will be needed to be able to draw a conclusion, and for slight biases (such as 51/49) you many need hundreds of trials before you can even reject one of the three possibilities. === Subject: Re: how to determine sufficient sample size? > I have a problem where I need to determine a minimum > sample size. I took stats a long time ago, so please > bear with me. Suppose I have a bunch of coins, some of which are fairly > weighted, but there are some that are unfairly weighted > (for example, these coins come up heads only 25% of > the time). I want to classify the coins into either one of two classes: > fairly-weighted and unfairly-weighted. I can flip each coin > as many times as I want, but I want to find out the minimum > number of flips I need to determine if a coin is fair or not. I am ok if a coin is fair or not requires a confidence > parameter, which I can give, like 90% or something. > It also depends on the size of the bias. A 49-51 bias will take more flips to detect than will a 25-75 bias. === Subject: Re: how to determine sufficient sample size? I have a problem where I need to determine a minimum > sample size. I took stats a long time ago, so please > bear with me. Suppose I have a bunch of coins, some of which are fairly > weighted, but there are some that are unfairly weighted > (for example, these coins come up heads only 25% of > the time). I want to classify the coins into either one of two classes: > fairly-weighted and unfairly-weighted. I can flip each coin > as many times as I want, but I want to find out the minimum > number of flips I need to determine if a coin is fair or not. I am ok if a coin is fair or not requires a confidence > parameter, which I can give, like 90% or something. > Use the negative binomial distribution. For each coin, and results of > a serie of its flippings, it is possible to estimate its fairness from > confidence intervals. > kunzmilan many flippings I would need to discover that it is biased. === Subject: Re: Question regarding QR Factorization 6y59pkR8)L]j8{eQmixxBq{y[uG13Ekt3@`{$F6?Hj%@)Y{V[~BT;SJ}c-,bth[~` ']|Y^+JDq5'>mb.3vRMu91|)ffD;27>$9k]`0H > I have a question regarding QR Factorization using Givens rotations, as > presented in Golub and van Loan... Using the basic algorithm (Algorithm 5.2.2), in addition to solving for > R, it is possible to create a matrix Q^T that is the product of the > individual Givens matrices, G_1, G_2, etc.. That is, Q^T = G_K...G_3 G_2 G_1 The basic algorithm works by introducing the zeros column-by-column. > The authors then mention a modification to the algorithm whereby you can > swap the for statements and zero elements row-by-row. This is > attractive for our implementation because we have to pay a large penalty > for row accesses on our machine and this would allow us to the keep a > pair of rows in local memory for more math operations. However, if one uses the modification to work by rows, you see that they > use a trick where they don't revisit the columns that have been zeroed > out. This is fine in a for loop implementation but, I believe, prevents > you from actually computing a Q^T from the individual G_i matrices. The > resultant R matrix is upper diagonal and one could calculate Q by taking > the inverse of R and combining it with the original A matrix [i.e., > Q=AR^(-1)]. However, that is inefficient. Can any anyone confirm that either (a) I am missing something and it is > indeed possible to construct Q from the G_i, when working row-by-row or > (b) it is not possible to work row-by-row and still create a Q matrix. ' Here is a question for you: Why do you want to form Q? If, for > example, you want to multiply a vector by Q or Q^T, then forming Q > first and multiplying it by the vector will take O(n^3) multiplies and > adds, but if you apply the Given's rotations one by one to the vector, > it will take only O(n^2) multiplies and adds. ' The BLAS Givens' rotation generation routines compute a single number > from which the rotation can be reconstructed. This is described in the > original Blas paper: ' C. L. Lawson , R. J. Hanson , D. R. Kincaid , F. T. Krogh, Basic > Linear Algebra Subprograms for Fortran Usage, ACM Transactions on > Mathematical Software (TOMS), v.5 n.3, p.308-323, Sept. 1979 ' as modified in D.S. Dodson and R.G. Grimes, Remark on Algorithm 539: > Basic Linear Algebra Subprograms for Fortran Usage, ACM Transactions > on Mathematical Software (TOMS) v.8, n.4, p.403-404, Dec. 1982. ' You could store these numbers in the lower triangle of the array. > Then, at the end of the QR factorization, the array would contain > everything you need to, e.g., solve a least squares problem Ax=b given > a right-hand-side vector b. ' Dave We have a fairly specialized processor for which we have been asked to develop a set of library routines and benchmarks. The original requirement was to produce both Q and R but I now see that both BLAS and VSIPL return Q in coded form and also provide routines to multiply an arbitrary matrix with the coded Q. This is certainly possible for the library routine. However, I suspect that we will still need to produce Q for the benchmark result because that is what others before us have done (I know that that sounds inane). Ken Prager P.S. Is you paper available on-line? === Subject: Re: Make 100 by using + - x / and 1~9 <2007102518354216807-kirakun@earthlinknet> <2007102518411575249-kirakun@earthlinknet> <5odnj4Fmaj3aU1@mid.individual.netCongrats, Kira, nice, short and quick code. Here a solution inMathematica: In[1]:= > Select[Flatten[ > Table[ > Inner[StringJoin, {1, 2, 3, 4, 5, 6, 7, 8}, > {+, -, *, /}[[{i1, i2, i3, i4, i5, i6, i7, i8}]], > StringJoin] > <> 9, > {i1, 4}, {i2, 4}, {i3, 4}, {i4, 4}, {i5, 4}, {i6, 4}, {i7, 4}, {i8, > 4}]], > (ToExpression[#] == 100) &] Out[1]= > {1+2+3+4+5+6+7+8*9, 1+2+3-4*5+6*7+8*9, 1+2-3*4+5*6+7+8*9, > 1+2-3*4-5+6*7+8*9, 1+2*3+4*5-6+7+8*9, 1+2*3*4*5/6+7+8*9, > 1-2+3*4*5+6*7+8-9, 1-2+3*4*5-6+7*8-9, 1-2*3+4*5+6+7+8*9, > 1-2*3-4+5*6+7+8*9, 1-2*3-4-5+6*7+8*9, 1*2*3+4+5+6+7+8*9, > 1*2*3-4*5+6*7+8*9, 1*2*3*4+5+6+7*8+9, 1*2*3*4+5+6-7+8*9} Although evaluation time is rather short (6 seconds on a Pentium 4, 3 > elegant code. Alfred In[16]:= Select[ StringJoin@Riffle[ToString /@ Range[9], #] & /@ Tuples[{+, -, *, /}, 8], (ToExpression[#] == 100) &] // Timing Out[16]= {4.328, {1+2+3+4+5+6+7+8*9, 1+2+3-4*5+6*7+8*9, 1+2-3*4+5*6+7+8*9, 1+2-3*4-5+6*7+8*9, 1+2*3+4*5-6+7+8*9, 1+2*3*4*5/6+7+8*9, 1-2+3*4*5+6*7+8-9, 1-2+3*4*5-6+7*8-9, 1-2*3+4*5+6+7+8*9, 1-2*3-4+5*6+7+8*9, 1-2*3-4-5+6*7+8*9, 1*2*3+4+5+6+7+8*9, 1*2*3-4*5+6*7+8*9, 1*2*3*4+5+6+7*8+9, 1*2*3*4+5+6-7+8*9}} B. === Subject: Re: RAF: Rational numbers, irrational numbers: each dense in real numbers Consider the functions from reals to reals, bijective functions. That > is a collection of all collections of ordered pairs such that for any > r in R there exists a distinct and unique ordered pair (r, x) in the > function, and for each x in R there is as well a unique (r,x). Then, in well-ordering those by a well-ordering of the reals, there > are no strictly ascending nor strictly descending chains that are > uncountable, else ZFC is inconsistent. Each strictly ascending or > strictly descending chain is countable. Then, no well-ordering with a particular element r_0 has each of > uncountably many lesser reals, Incorrect. No *strictly decreasing chain* with a particular element > r_0 has each of uncountable many lesser reals. This says > nothing about a *well ordering* which is not strictly decreasing. yet a characteristic of the well- > ordering is that for each of those, there are uncountably many > elements of the reals. subset is equivalent to > restricted to that subset. Maybe it would be reasonable to further consider: So there are c^c > many, including one of them > that is the strictly decreasing antisequence, in the real numbers, No this would not be reasonable. Yes there are a lot of them; > no there is no strictly decreasing one. If you want to discuss a putative uncountable strictly decreasing > sequence I have indicated how this needs to be presented. Note, you have an uncoutable number of critical points, > you need a real value for each, and these real values > must be strictly decreasing. So before you can construct > an uncountable strictly decreasing sequence you need an > uncountable strictly decreasing sequence. - William Hughes The critical points are shown to exist. Yes, there are an uncountable number of them. > They are *well ordered*. They are not *striclty > decreasing*. - William Hughes The strictly monotone sequences are each countable in extent, so there are uncountably many sequences. Between each there is a critical point. After each sequence, there is a critical point. Two or more critical points can be adjacent. Many of the critical points could be adjacent. Between any two sequences there is a critical point. There are various ways to consider the critical points at limit points: for example to have them at each limit point or not. Of the critical points, they are a subset of the reals, equivalent to well-ordered. Then, there can't be an uncountable monotone sequence of those. Again, for any of those there are uncountably many eventual successors less than it. Given a well-ordering of the reals, the composition with a bijection from the reals to the reals is also a well-ordering of the reals, and given the collection of functions between reals, that set is complete in terms of representing each possible well-ordering of the reals, and only representing well-oprderings, that is an item from a c^c space. Each function from the reals to reals takes a well-ordering to a well- ordering, because, every composition of well-ordering of ordinal and real and bijections real to real is a well-ordering. Then, any enumeration strategy is an initial segment of a well-ordering. (An enumeration strategy is the selection of choice functions on the set thus that a particular if not distinct or unique well-ordering results as the output of transfinite induction.) So, the strictly monotone sequences of a type can be concatenated. Yet, there being uncountably many of them, they can't all be concatenated, else there would be an uncountable strictly montone subchain. How would the subchains of a type be concatenated? If a subchain ends it's bounded. Only the initial subchain doesn't have predefined elements to the last and the last subchain doesn't have predefined elements beyond it. So, each strictly increasing subchain is bounded above, and each strictly decreasing subchain bounded below. So, the s.i (strictly increasing/strictly ascending, s.a.) initial subchain of a well-ordering is bounded below. Being well-ordered, each subchain is also bounded below/above respectively. So then to concatenate a later s.i subchain s_alpha to an initial s.i. subchain s_0, there are where the s.i. chain was, its content is replaced by elements from those following s_0, so perhaps alpha should be selected so that more than its bounds are taken into account, i.e., replacing a s.d. subchain with a s.a. subchain of the same length. Beyond the alternation (except about limit ordinals) of strictly increasing and strictly decreasing subchains would it not be natural that there exists a 1-1 function between the increasing and decreasing subchains thus that each was mapped to another of the exact same cardinality and limit crossing character, and perhaps even furthermore magnitude between bounds? The point here is that the transfinite recursion schema to select irrationals via an enumeration strategy enables well-ordering more than countably many of them. It is clearly shown that there are at least as many as omega^omega which is countable. A well-ordering of the reals is random, the probability of each point being a critical point is about 1/2. P(critical) = P(x_2>x_1) | P( x_1x_0) = 1/2 1/2 + 1/2 1/2 = 1/4 + 1/4 = 1/2 As well, for each critical point increasing there is one decreasing, and vice versa, because there are no adjacent reals in the standard reals. A well-ordering of the reals would exist and begin 1, 2, 3, ..., as often as any other initial segment, because there exists for any other x_0, x_1, x_2 a bijection R->R such that f(x_0) = 1, f(x_1) = 1, f(x_2) = 2, etcetera. (That is except where the reals are only well-ordered in certain structure-preserving ways. For example there is the equivalency function ordering elements by magnitude regardless of other algebraic properties, where magnitude is not an intrinsically algebraic quantity, in terms of the real numbers general consideration as a complete ordered field, except for zero. They are as well further prototyped in those algebraic spaces assuming that character, generally through polynomials and the introduction of natural numeric constants, then definition through completion. ) So, between any two limit ordinals, if they are separated by uncountably many elements, up to and including the cardinal of the continuum, there are that many strictly increasing and that many strictly decreasing and that many critical points. There is not necessarily the alternation of the critical points from limit ordinal to limit ordinal of the construction towards large strictly descending sequences. They might maintain direction over limit ordinals. I guess I'm under the (mis)impression that with there being uncountably many elements less than a given critical point, they can be reordered by their natural ordering, and that maintains a well- ordering. There are uncountably many chains in the well-ordering. It's not so simple as to select the subset of the reals corresponding to members of strictly monotone type. Also, the boundaries of the subchain, as each is bounded above and below, must each be considered, thus that the subset of the reals as union of elements corresponding to contiguous subchains that are strictly monotone of a type would have that the pairwise intersection of intervals defined by the bounds of the subchain is empty. Otherwise, then pairwise there would have to be de- and re-composition of the original well-ordering thus that the elements are correctly interleaved, which is simple, in terms of pairwise unions, just not unions over all the monotone subchains. Yet, it can be shown for each countable ordinal at least up to infinity to the infinity'th power that a transfinite recursion schema to enumerate irrationals in a strictly monotone manner proceeds, (as well as for rationals), so with there being more irrationals than rationals that should proceed. If it doesn't, it's because there would thus be too many rationals, yet it does, else there aren't enough irrationals. Ross -- Finlayson Consulting === Subject: Re: RAF: Rational numbers, irrational numbers: each dense in real numbers is a collection of all collections of ordered pairs such that for any > r in R there exists a distinct and unique ordered pair (r, x) in the > function, and for each x in R there is as well a unique (r,x). Then, in well-ordering those by a well-ordering of the reals, there > are no strictly ascending nor strictly descending chains that are > uncountable, else ZFC is inconsistent. Each strictly ascending or > strictly descending chain is countable. Then, no well-ordering with a particular element r_0 has each of > uncountably many lesser reals, Incorrect. No *strictly decreasing chain* with a particular element > r_0 has each of uncountable many lesser reals. This says > nothing about a *well ordering* which is not strictly decreasing. yet a characteristic of the well- > ordering is that for each of those, there are uncountably many > elements of the reals. subset is equivalent to > restricted to that subset. Maybe it would be reasonable to further consider: So there are c^c > many, including one of them > that is the strictly decreasing antisequence, in the real numbers, No this would not be reasonable. Yes there are a lot of them; > no there is no strictly decreasing one. If you want to discuss a putative uncountable strictly decreasing > sequence I have indicated how this needs to be presented. Note, you have an uncoutable number of critical points, > you need a real value for each, and these real values > must be strictly decreasing. So before you can construct > an uncountable strictly decreasing sequence you need an > uncountable strictly decreasing sequence. - William Hughes The critical points are shown to exist. Yes, there are an uncountable number of them. > They are *well ordered*. They are not *striclty > decreasing*. - William Hughes The strictly monotone sequences are each countable in extent, so there > are uncountably many sequences. Between each there is a critical > point. Thus there are uncountable many critical points. These have to form a strictly descending sequence. You have not indicated how you would decide which real number applies to each critical point. - William Hughes === Subject: Re: RAF: Rational numbers, irrational numbers: each dense in real numbers [snipalot] Browsing Metamath, consider theorem elisseti: http://us.metamath.org/mpegif/elisseti.html > (element is set), that a member of a class is a set, with the class of > ordinals member of class V. Sorry, I could not find any theorem in metamath that claimed that the class of ordinals were a member of V (that would be On e V, wouldn't it?). Can you give a reference to an existing metamath proof or at least a hint what such a proof might look like? Be sure to read http://us.metamath.org/mpegif/df-v.html carefully. > That seems quite controversial, to say > that if a class is a member of a class that it's then a set. Maybe > instead it's just a faulty description. I track back through the > developments, eghttp://us.metamath.org/mpegif/elisseti.html , and, > it seems to say that, uh, if a class A is an element of a class B then > it's a set, using class equality instead of set equality. So, in > Metamath, the universal class is the collection of all elements > satisfying identity, and, in Metamath, all classes satisfy identity, > and, in Metamath, an element of a class is a set, and, each class is > an element of the universal class. Thus, in Metamath, each class, for > example the class of Ordinals, is a set. As you notice yourself, onprc claims exaclty the opposite. If you WERE able to produce a proof of On e V in metamath, that WOULD be quite interesting. Then, in onprc, it is shown that no set contains all the ordinal > numbers. So, then On =/= On, WRONG > else it would be an element of V WRONG > and a > set, where V is simply defined as the class of all classes, WRONG > sets, that > are equal to themselves. (Identity, of an object being itself, is > generally assumed to hold.) As well, it is stated that universal > quantification is unrestricted. http://us.metamath.org/mpegif/con0.html http://us.metamath.org/mpegif/df-cleq.html http://us.metamath.org/mpegif/df-v.html Second, then there is to be described an infinite set with the well- > ordering thus that via transfinite induction/recursion it exists and > is uncountable. Then, the property to show that holds for transfinite > induction is that for a given ordinal, either it is less or equal than > the cardinality of the irrationals and thus there exists uncountably > many irrationals left from which to select, or it is greater than the > cardinality of the irrationals. http://us.metamath.org/mpegif/tfi.html That is where the desired property for a given ordinal (that there are > more elements in the interval (0, p_alpha) for ordinal alpha less than > the cardinality of the irrationals) holds for ordinals less than or > equal to the cardinality of the irrationals, where for higher ordinals > the property would not consistently hold, but it is not necessary that > it does. So, in the course of values over all ordinals, for ordinals > less than or equal to the cardinality of the irrationals there are at > least that many remaining in the interval (p_alpha, 0). (Otherwise, > there wouldn't be that many in the interval.) For ordinals greater > than the cardinality of the irrationals, they as well satisfy the > property in being greater than the cardinality of the irrationals. > Then, there are as many elements p_alpha as there are ordinals alpha > that are less than or equal to the cardinality of the irrationals. > Then, that holds for sufficiently many irrationals, for each of which > can be displayed a distinct rational, that theorem contradicts another > in the theory. You have consistently been unable to define p_alpha. You never managed to give a valid recursive definition of p_alpha, i.e. using only the previously defined p_beta for beta interval (p_alpha, 0) into (p_alpha, p_alpha+) and (p_alpha+, 0), each > partition has the same cardinality. As long as p_alpha < p_alpha+ < 0, this at last sounds correct. The rationals and irrationals are each dense in the reals. That's correct as well. So, ZFC is inconsistent. Non sequitur. Ross -- > Finlayson Consulting hagman === Subject: Re: RAF: Rational numbers, irrational numbers: each dense in real numbers (element is set), that a member of a class is a set, with the class of > ordinals member of class V. Sorry, I could not find any theorem in metamath that claimed > that the class of ordinals were a member of V (that would > be On e V, wouldn't it?). > Can you give a reference to an existing metamath proof or at least > a hint what such a proof might look like? > Be sure to readhttp://us.metamath.org/mpegif/df-v.htmlcarefully. That seems quite controversial, to say > that if a class is a member of a class that it's then a set. Maybe > instead it's just a faulty description. I track back through the > developments, eghttp://us.metamath.org/mpegif/elisseti.html, and, > it seems to say that, uh, if a class A is an element of a class B then > it's a set, using class equality instead of set equality. So, in > Metamath, the universal class is the collection of all elements > satisfying identity, and, in Metamath, all classes satisfy identity, > and, in Metamath, an element of a class is a set, and, each class is > an element of the universal class. Thus, in Metamath, each class, for > example the class of Ordinals, is a set. As you notice yourself, onprc claims exaclty the opposite. > If you WERE able to produce a proof of On e V in metamath, > that WOULD be quite interesting. Then, in onprc, it is shown that no set contains all the ordinal > numbers. So, then On =/= On, > WRONG > else it would be an element of V > WRONG > and a > set, where V is simply defined as the class of all classes, > WRONG > sets, that > are equal to themselves. (Identity, of an object being itself, is > generally assumed to hold.) As well, it is stated that universal > quantification is unrestricted. http://us.metamath.org/mpegif/con0.html http://us.metamath.org/mpegif/df-cleq.htmlhttp://us.metamath.org/mpegif/df-v . html Second, then there is to be described an infinite set with the well- > ordering thus that via transfinite induction/recursion it exists and > is uncountable. Then, the property to show that holds for transfinite > induction is that for a given ordinal, either it is less or equal than > the cardinality of the irrationals and thus there exists uncountably > many irrationals left from which to select, or it is greater than the > cardinality of the irrationals. http://us.metamath.org/mpegif/tfi.html That is where the desired property for a given ordinal (that there are > more elements in the interval (0, p_alpha) for ordinal alpha less than > the cardinality of the irrationals) holds for ordinals less than or > equal to the cardinality of the irrationals, where for higher ordinals > the property would not consistently hold, but it is not necessary that > it does. So, in the course of values over all ordinals, for ordinals > less than or equal to the cardinality of the irrationals there are at > least that many remaining in the interval (p_alpha, 0). (Otherwise, > there wouldn't be that many in the interval.) For ordinals greater > than the cardinality of the irrationals, they as well satisfy the > property in being greater than the cardinality of the irrationals. > Then, there are as many elements p_alpha as there are ordinals alpha > that are less than or equal to the cardinality of the irrationals. > Then, that holds for sufficiently many irrationals, for each of which > can be displayed a distinct rational, that theorem contradicts another > in the theory. You have consistently been unable to define p_alpha. > You never managed to give a valid recursive definition of p_alpha, > i.e. using only the previously defined p_beta for beta You only did that for alpha=0 and for successor ordinals. Basically for each partition of the irrationals intersecting the > interval (p_alpha, 0) into (p_alpha, p_alpha+) and (p_alpha+, 0), each > partition has the same cardinality. As long as p_alpha < p_alpha+ < 0, this at last sounds correct. The rationals and irrationals are each dense in the reals. That's correct as well. So, ZFC is inconsistent. Non sequitur. Ross -- > Finlayson Consulting hagman The issue I noted with the definition of the universal class is that it defines its elements by any class, not set but class, that satisfies class identity, not just set identity. Each class satisfies class identity, where the universe is defined as something along the lines of: U (V, L) being {x: x = x}, here casually, and in Metamath explicitly using class identity. Ah, there is a set variable in the class builder, not a class variable. So, by preservation of type, there are only sets, not classes, satisfying the predicate of equalling themselves that comprise V, in Metamath's definition. Maybe it's just that using eventually class equality from upcasted sets gives a ready definition of the collection of classes. So, I see that the definition of V the class of all sets in Metamath is of sets. Yet, using class equality to build, with the class variables and class equality, etcetera, in the class builder, instead of set terms, the same primitives are available. Obviously there is no collection of classes in ZF with classes, yet the classes in Metamath are built as collections of classes in the class builder, class element-of is defined with class variables, from set variables. There are class primitives defined. ZFC's universe is the Russell set, from Russell's well-known paradox, in a theory with theorems decided by less axioms than ZFC's, eg ZFC - R, with unrestricted comprehension. Thus it contains itself. So ZFC is inconsistent. Hagman, I defined p_alpha+ in a variety of different ways, including a recursive definition in a course-of-values transfinite recursion scheme, a general type preserving monotonicity. Here there is the consideration of the mutual denseness of each of the rationals and irrationals in the reals. Between any two irrationals, there lie, lay, infinitely many rationals, and between each two of those, infinitely many irrationals, between each two of those, infinitely many rationals, etcetera ad infinitum, in the standard real numbers. (A particular meaningful difference between the verbs lie and lay may be reasonable to consider as technical.) Then in consideration of strictly monotone sequences vis-a-vis critical points, well there are uncountably many strictly monotone sequences, and correspondingly that many critical points. (The critical points are unfortunately not monotone.) Here the consideration of strictness in monotonicity can be ignored, because the values are already unequal. Between each pair of monotone subchains there is a critical point, but between critical points there aren't necessarily non-zero ascending chains. That gets into whether to divide, to have the boundary, to partition, at the point or between the points. In the normal ordering of the reals the boundary is always at a point. So anyways a well-ordering of the real numbers has the property that it is a bijection between the set of real numbers and some ordinal equivalent to the initial ordinal of c the cardinality of the continuum. If there is an uncountable segment that has preservation of monotonicity in the normal total ordering of the real numbers, then as Hughes showed then via a compactness-type argument that would be contradictory. Yet, the well-ordering is the union of monotone subchains, where these critical points as subchains of length one can be considered, variously removing the initial element of the following monotone subchain/critical point. Defining the critical points as simply initial elements of a monotone subchain just leads to many smaller monotone sequences, if they were adjacent, or larger subchains if they were between subchains of opposite monotonicity. (Again there is a variation depending on whether each limit ordinal is defined as a critical point.) In that sense the well-ordering is a union of monotone subchains, where as a collection of ordered pairs, the well- ordering's subsets for given ordinal ranges are all defined. Currently I'm looking at all these but the necessary countability of a strictly descending uncountable chain ((r_alpha, ..., r_beta), alpha < beta) of irrationals seems to belie what is otherwise the uncountability of the irrationals, given a transfinite course-of- values schema defining them. Otherwise there don't exist choice functions on R, which do exist in ZFC. Then there was further consideration and multiple heuristic examples of why the rationals carry as much weight in the reals as the irrationals. For example, unless the (0,p) U (0, p_i), 0 < p_i < p is non-empty, then for any real number it is defined as the set of either the rationals, or the set of irrationals, less than it. If that set-difference was non-empty then it would contain a rational, and a distinct rational, for p. If it's empty, then it is so for the construction using rationals as well, with there not being non- rational irrationals between a rational and all different rationals (completeness of rationals), which is not considered to hold in the standard real numbers. It takes many less rationals than irrationals to fully specifiy a real number. Another example considered convergent sequence representation of reals, and of how each of infinitely many finite subsequences generated a distinct real, and the infinitely many generated only another. The point of this is to find more analytical reckonings to extend the integral calculus, obivously, in redifferintegrosystems, basically for finding classes of non-real functions with reasonable analytical properties. It's said sometimes that well-orderings of the reals are totally random, but any enumeration of a subset of the reals forms a subchain of a well-ordering of the reals. In the standard reals, there are no ordering-sensitive properties of the elements, thus that is so. Ross -- Finlayson Consulting === === Subject: Re: R.C. Hibbeler Dynamics 10th & 11th Edition Solutions Manual (traditional) I would like to have a set of R.C. Hibbeler Dynamics 11th Edition Solutions Manual (traditional) thank you. === Subject: Re: Is this a well known class of variables ? > Well, can something be discrete or continuous indeterminately? On the > one hand it makes no sense. But on the other hand it might make sense. The very premise that anything might be indeterminate makes one > wonder where else this property might be applied. Indeterminate > geometric structure ? Indeterminate existence ? Indeterminate set > membership ? And on and on. > But I still dont see the foundational explanation for indeterminacy in > the first place. I believe first in existential indeterminacy, and that it is reflexive > so that it may or may not even even exist itself. You may have > indeterminacy, and you may not. It is indeterminate. I think you are mainly in love with the sound of this word or the feeling it creates or the satisfaction you feel with yourself over ideas like the existence of indeterminacy is indeterminate. Part of the art of constructing models is a reliable sense of what sounds like it might mean something definite, and what actually means something definite. There is nothing wrong with the former, so long as we keep it apart from the latter. === Subject: Re: Arab transmission of Greek maths McTutor provides a shallow review of this topic. To actually read the Arab methods of writing Greek versions of Egyptian fraction unit fraction arithmetic, read the Liber Abaci per Wikipedia: http://en.wikipedia.org/wiki/Liber_Abaci and one of my blogs: http://liberabaci.blogspot.com Enjoy, Milo Gardner === Subject: Re: Bias coin One out of 1000 coins always gives heads, you flip a coin and get heads 10 times, what is probability that it is biased? I decided, out of interest, to look at the general case of the problem posed by Yves at the start of this thread. conditional probability, and to Phil Carmody for putting me onto Bayes' Theorem. http://en.wikipedia.org/wiki/Bayes'_theorem#Derivation_from_conditional_prob abilities Notation: P(x) = probablity of event x P(x|y) = probability of event x given the occurrence of y Bayes theorem: P(x|y) = p(y|x)p(x) / p(y) Let: b = proportion of biased coins in coin population B = the event in which a biased coin is selected F = the event in which a fair coin is selected n = number of times the coin is flipped H = event where n heads occur out of n flips of coin Problem: find P(B|H) P(B) = b P(F) = 1 - b P(H|B) = 1 P(H|F) = 2^-n When H occurs then either B or F has occurred so: P(B|H) + P(F|H) = 1 According to Bayes theorem P(B|H) = P(H|B)P(B) / P(H) = P(B)/P(H) = b/P(H), P(H|F) = P(F|H)P(H) / P(F) = (1 - P(B|H))P(H) / P(F) = (1 - b/(P(H)) P(H) / P(F) = (P(H) - b) / P(F) substitute 2^-n for P(H|F), 1 - b for P(F) (P(H) - b) / (1 - b) = 2^-n ==> P(H) - b = 2^-n(1 - b) ==> P(H) = 2^-n(1 - b) + b = (1 - b + b.2^n) / 2^n Hence P(B|H) = b/P(H) = b.2^n / (1 - b + b.2^n) Sanity check: (1) b = 0 ==> P(B|H) = 0 / (1 - 0 + 0) = 0 As one would expect if there are no biased coins (2) b = 1 ==> P(B|H) = 2^n / (1 - 1 + 2^n) = 2^n / 2^n = 1 All coins are biased so we always get heads and only biased coins producing them. (3) We choose a coin and keep getting heads no matter how many times we flip the coin As n --> infinity, b.2^n > b, 1 so (1 - b + b.2^n) --> b.2^n hence P(B|H) --> b.2^n / b.2^n = 1 Showing that if we get a large number of heads the coin is almost certainly biased. (4) A special case As b --> 2^-n, b.2^n --> 1, P(B|H) --> 1/(2 - b) where for small b, P(B|H) ~ 0.5 So if the biased coins are rare, and we flip the coin a number of times such that the probability of flipping all heads with a fair coin is comparable to the chance of selecting a biased one, there is a roughly equal chance that the coin is fair or biased. Quite similar to the original problem where b = 1/1000, n = 10 ==> P(H| T) = 1/1024 calculation using these values in general formula derived above gives P(B|H) = 0.506178942, confirming Michael Press's and Phil Carmody's result Stan === Subject: Re: Bias coin B: we selected the biased coin at the beginning > F: we selected a fair coin at the beginning T: A coin selected from the population gives ten out of ten heads. We want to find Pr(B|T). Are you sure that is correct? Isn't the question really asking for Pr(B)? Compare it with this alternative question: One out of 1000 coins always gives heads, you flip a coin 10 times, what is probability that you will get 10 heads? Do you agree that this (which calls for Pr(T)) is not simply 1/2^10? Stan === Subject: Re: Bias coin > One out of 1000 coins always gives heads, you flip a coin and get heads 10 times, what is probability that it is biased? > Define the events > B: we selected the biased coin at the beginning > F: we selected a fair coin at the beginning > T: A coin selected from the population gives ten out of ten heads. > We want to find Pr(B|T). Are you sure that is correct? Isn't the question really asking for >Pr(B)? Without seeing any flip results, the probability that the selected coin is biased is 1/1000. Thus, P(B) = 1/1000. But we are entitled to use the given information about the flip results. Thus, given that the selected coin yielded 10 heads, the appropriate probability is the conditional probability, P(B|T). For this problem, P(B|T) is not equal to P(B), not even close. To dramatize the issue, suppose the outcome was less than 10 heads. With that information, what would you say about the probability that the selected coin is the biased one? The new probability is zero, right? Thus, the information about the flip results can change one's view of the probability. The new probability is called a conditional probability since it's conditional on the given information. >Compare it with this alternative question: One out of 1000 coins always gives heads, you flip a coin 10 times, >what is probability that you will get 10 heads? Do you agree that this (which calls for Pr(T)) is not simply 1/2^10? No, not exactly. P(T) is not 1/2^10 (although it's close). In fact, P(T) = P(B) + P(F)*(1/2^10) = 1/1000 + (999/1000)*(1/2^10) But in any case, the problem isn't asking for the probability of 10 heads. It's definitely not the same problem. quasi === Subject: Re: Bias coin One out of 1000 coins always gives heads, you flip a coin and get heads 10 times, what is probability that it is biased? > Define the events > B: we selected the biased coin at the beginning > F: we selected a fair coin at the beginning > T: A coin selected from the population gives ten out of ten heads. > We want to find Pr(B|T). Are you sure that is correct? Isn't the question really asking for >Pr(B)? Without seeing any flip results, the probability that the selected > coin is biased is 1/1000. Thus, P(B) = 1/1000. But we are entitled to use the given information about the flip > results. Thus, given that the selected coin yielded 10 heads, the appropriate > probability is the conditional probability, P(B|T). For this problem, P(B|T) is not equal to P(B), not even close. To dramatize the issue, suppose the outcome was less than 10 heads. > With that information, what would you say about the probability that > the selected coin is the biased one? The new probability is zero, > right? Thus, the information about the flip results can change one's view of > the probability. The new probability is called a conditional > probability since it's conditional on the given information. Yes, I see it now. After I had posted the above the thought that a single non-heads result would influence my view of it did cross my mind. I couldn't get past the notion that 10 heads did not materially affect whether or not it was a biased coin. I now picture it as follows: I play a game called find the rogue coin Someone (not me) draws a coin from a large vessel containing a 1000 coins, one of which is known to be double-headed. I'm not allowed to examine it. I have the option of placing a bet that the coin is the biased or I can pass. The coin is flipped ten times after which I have a second chance to bet or pass. If I wish to play again the coin is put back in the vessel, which is then given a shake, and another coin is drawn, and so on. Clearly if I played blind i.e. before seeing any results of coin flipping, my chances of winning would be 1/1000 but if I wait for the second phase and then only placed a bet when 10 heads came up my chances of winning would be much greater. Furthermore I can now see that a figure of around 0.5 is plausible. >Compare it with this alternative question: One out of 1000 coins always gives heads, you flip a coin 10 times, >what is probability that you will get 10 heads? Do you agree that this (which calls for Pr(T)) is not simply 1/2^10? No, not exactly. P(T) is not 1/2^10 (although it's close). In fact, P(T) = P(B) + P(F)*(1/2^10) = 1/1000 + (999/1000)*(1/2^10) But in any case, the problem isn't asking for the probability of 10 > heads. It's definitely not the same problem. quasi Quite so. My reason for suggesting the alternative question was to point out that a more complex analysis more akin to that from Michael Press and Phil Carmody would follow. However I accept that I was completely wrong in thinking the original question was merely P(B). Stan === Subject: Re: Bias coin > One out of 1000 coins always gives heads, you flip a coin and get heads 10 times, what is probability that it is biased? If I understand this question correctly, surely the answer is simply > 1/1000 If I flip a fair coin and get 10 heads, that doesn't make it one of > the biased coins. Put another way, the probability that I choose a biased coin will be, > according to the information given, 1/1000. The event defined to be > the coin is biased does not depend on the result of flipping the > coin. So calculations involving 2^10 are irrelevent. Stan How about this interpretation? It gives the same answer as others, but seems simpler (to my simple mind) Where do runs of 10 heads come from? If we do n trials, we'll pick the biassed coin n/1000 of the time, and get a 10-head run for sure... We'll pick an unbiassed coin n * 999 / 1000 times, and from these we'll get a 10-head run 1 / 2^10 of the time. Ratio of biassed 10-head runs to total 10-head runs, after some cleaning up: Ratio = 1024 / 2023 === Subject: Re: Bias coin , > <15888415.1194017792043.JavaMail.jakarta@nitrogen.mathf > orum.org>, ' > One out of 1000 coins always gives heads, you flip a coin and get heads 10 times, what is probability that it is biased? ' Define the events > B: we selected the biased coin at the beginning > F: we selected a fair coin at the beginning ' T: A coin selected from the population gives ten out of ten heads. ' We want to find Pr(B|T). ' Pr(B & T) = Pr(B|T).Pr(T) = Pr(T|B).Pr(B) > = Pr(B) > so > Pr(B) > Pr(B|T) = --------------------- > Pr(B) + Pr(F).Pr(T|F) This is confusing. Should be Pr(B) Pr(B|T) = ------ Pr(T) > We can get ten heads in two ways. > One way is choosing the biased coin, > and one way is choosing a fair coin. > Pr(T) = Pr(B).Pr(T|B) + Pr(F).Pr(T|F). > = Pr(B) + Pr(F).Pr(T|F). ' Pr(T|F) = 1/2^10 ' Pr(B) > Pr(B|T) = --------------------- > Pr(B) + Pr(F).Pr(T|F) ' 1 > = -------------------------- > 1 + Pr(T|F).Pr(F)/Pr(B) ' 1 > = -------------- > 1 + 999/2^10 ' = 0.5062 -- Michael Press === Subject: Re: Bias coin orum.org>, > One out of 1000 coins always gives heads, you flip a coin and get heads 10 times, what is probability that it is biased? Define the events > B: we selected the biased coin at the beginning > F: we selected a fair coin at the beginning T: A coin selected from the population gives ten out of ten heads. We want to find Pr(B|T). Pr(B & T) = Pr(B|T).Pr(T) = Pr(T|B).Pr(B) > = Pr(B) > so > Pr(B) > Pr(B|T) = --------------------- > Pr(B) + Pr(F).Pr(T|F) We can get ten heads in two ways. > One way is choosing the biased coin, > and one way is choosing a fair coin. > Pr(T) = Pr(B).Pr(T|B) + Pr(F).Pr(T|F). > = Pr(B) + Pr(F).Pr(T|F). Pr(T|F) = 1/2^10 Pr(B) > Pr(B|T) = --------------------- > Pr(B) + Pr(F).Pr(T|F) 1 > = -------------------------- > 1 + Pr(T|F).Pr(F)/Pr(B) 1 > = -------------- > 1 + 999/2^10 = 0.5062 -- > Michael Press Haven't gone through your work, but just looking at your answer, the prob you get seems too low. You're basically saying if you flip a coin 1,000 times and 10 times (or 1/100 of the total throws) you get heads, there's a 50:50 chance it's biased. That seems incorrect to me. M === Subject: Re: Bias coin > <15888415.1194017792043.JavaMail.jaka...@nitrogen.mathf > orum.org>, > One out of 1000 coins always gives heads, you flip a coin and get heads 10 times, what is probability that it is biased? Define the events > B: we selected the biased coin at the beginning > F: we selected a fair coin at the beginning T: A coin selected from the population gives ten out of ten heads. We want to find Pr(B|T). Pr(B & T) = Pr(B|T).Pr(T) = Pr(T|B).Pr(B) > = Pr(B) > so > Pr(B) > Pr(B|T) = --------------------- > Pr(B) + Pr(F).Pr(T|F) > ... > = 0.5062 ' Haven't gone through your work, but just looking at your answer, the > prob you get seems too low. You're basically saying if you flip a coin > 1,000 times and 10 times (or 1/100 of the total throws) you get > heads, there's a 50:50 chance it's biased. That seems incorrect to me. You're forgetting that you've been told a priori that in the population of all coins 1/1000 are biased. Stick a different proportion in there, and you'd get a different answer. Maybe if you'd actually gone through his work you'd have realised this, and spared yourself the embarassment. Phil -- -- Microsoft voice recognition live demonstration === Subject: Re: Bias coin > <15888415.1194017792043.JavaMail.jaka...@nitrogen.mathf > orum.org>, > One out of 1000 coins always gives heads, you flip a coin and get heads 10 times, what is probability that it is biased? Define the events > B: we selected the biased coin at the beginning > F: we selected a fair coin at the beginning T: A coin selected from the population gives ten out of ten heads. We want to find Pr(B|T). Pr(B & T) = Pr(B|T).Pr(T) = Pr(T|B).Pr(B) > = Pr(B) > so > Pr(B) > Pr(B|T) = --------------------- > Pr(B) + Pr(F).Pr(T|F) This is confusing. Should be Pr(B) Pr(B|T) = ------ Pr(T) > We can get ten heads in two ways. > One way is choosing the biased coin, > and one way is choosing a fair coin. > Pr(T) = Pr(B).Pr(T|B) + Pr(F).Pr(T|F). > = Pr(B) + Pr(F).Pr(T|F). Pr(T|F) = 1/2^10 Pr(B) > Pr(B|T) = --------------------- > Pr(B) + Pr(F).Pr(T|F) 1 > = -------------------------- > 1 + Pr(T|F).Pr(F)/Pr(B) 1 > = -------------- > 1 + 999/2^10 = 0.5062 ' Haven't gone through your work, but just looking at your answer, the > prob you get seems too low. You're basically saying if you flip a coin > 1,000 times and 10 times (or 1/100 of the total throws) you get > heads, there's a 50:50 chance it's biased. That seems incorrect to me. Go through the work. There is a true but unclear statement as noted. The derivation uses the definition of conditional probability and resolving a family of events into mutually exclusive subsets. These are elementary probability results. Another way of looking at it. At the beginning the odds for a biased coin against a fair coin are 1:999. The odds for a biased coin against a fair coin giving 10 out of 10 heads are 1:1024. The odds that our coin is biased are 1:999 times 1024:1; which is to say that the probability of a biased coin is about 0.5. In fact when the odds for B against F are B:F, Pr(B) = B/(B+F). In our case B:F = 1024:999 so Pr(B) = 1024/(1024 + 999) = 0.5062. -- Michael Press === Subject: Re: Bias coin > <15888415.1194017792043.JavaMail.jaka...@nitrogen.mathf > orum.org>, > One out of 1000 coins always gives heads, you flip a coin and get heads 10 times, what is probability that it is biased? > Define the events > B: we selected the biased coin at the beginning > F: we selected a fair coin at the beginning > T: A coin selected from the population gives ten out of ten heads. > We want to find Pr(B|T). > Pr(B & T) = Pr(B|T).Pr(T) = Pr(T|B).Pr(B) > = Pr(B) > so > Pr(B) > Pr(B|T) = --------------------- > Pr(B) + Pr(F).Pr(T|F) > We can get ten heads in two ways. > One way is choosing the biased coin, > and one way is choosing a fair coin. > Pr(T) = Pr(B).Pr(T|B) + Pr(F).Pr(T|F). > = Pr(B) + Pr(F).Pr(T|F). > Pr(T|F) = 1/2^10 > Pr(B) > Pr(B|T) = --------------------- > Pr(B) + Pr(F).Pr(T|F) > 1 > = -------------------------- > 1 + Pr(T|F).Pr(F)/Pr(B) > 1 > = -------------- > 1 + 999/2^10 > = 0.5062 > -- > Michael Press Haven't gone through your work, but just looking at your answer, the >prob you get seems too low. You're basically saying if you flip a coin >1,000 times and 10 times (or 1/100 of the total throws) you get >heads, there's a 50:50 chance it's biased. That seems incorrect to me. It's correct. Think of it this way. Let each of the 1000 coins be flipped 10 times. The biased coin will, of course, yield 10 heads. Of the other 999 coins, the chance is approximately (1 - 1/e) that at least one of them will yield 10 heads. But there's also a smaller but non-negligible chance that 2 of them could yield 10 heads, and possibly even 3 of them. Those extras are what compensate for the lower chance of success. Thus, in the competition to be selected for 10 heads, the fair coins have less chance of success, but more potential qualifying candidates. It works out to be almost a fair fight. quasi === Subject: Re: Bias coin Yes. question. M === Subject: #246 Summarizing this book; Imaginary North and South Poles are zero; new textbook: Mathematical-Physics (AP-adics primer) for 6 year olds and onwards <472AB80C.7060709@hotmail.com> I should have spent more time on discussing whether it makes sense for the Poles, although imaginary to AP-adics to be both zero, even though they are given the designation of pi and 2pi. Is there anything else in mathematics that demands that the poles be zero? Yes, indeed. The old Euler Identity of: (e)^(2pi)(i) = 1 and its counterpart since both come from trigonometry origins: (e)^(pi)(i) = (-1) For you see, Euler's Identity comes purely from AP-adics of points and numbers on a sphere or circle with their hollowed out inside surface. The one Euler Identity are Positive AP-adics and so for the Equation to hold means that either the 2pi or the i have to be zero. In AP-adics there are many instances of the square root of -1 in Negative AP-adics, but not in Positive AP-adics because they are all positive. so that means in the first case above of Euler Identity the 2pi is zero to render the equation true. In the second Euler Identity of (-1), these numbers are in the Negative AP-adics and here the (i) value exists but is never zero, but the (pi) value is always zero and hence renders the Euler equation. So here we see the example of where the old math supports and validates the true new math of what the Counting Numbers as AP-adics truly are. But the maximum support and validation comes from the fact that a sphere has two poles and those are special points and thus it makes commonsense that the poles should be both zero since the poles serve for both Elliptic and Hyperbolic geometry all in one single cool Model of a sphere. So to have all lines parallel in Hyperbolic geometry the poles are missing in Negative AP-adics while existing and imaginary in Positive AP-adics in order to have no lines parallel for Elliptic Geometry. So you see, the geometry demands the poles to be both zero points. Now whether the fact that the Poles are both zeroes, or imaginary- zeroes as pi and 2pi, whether that fact alone disqualifies the AP-adics as ever being a Galois Group or Ring or Field is unknown to me. What is known is that the points on a sphere or circle surface can never be a Group, Ring or Field because those points disobey commutative, associative and distributive laws. As I repeat the old harp-- Galois theory is only confined to Reals/Complex and Euclidean geometry and cannot be used for Counting Numbers, for Elliptic and Hyperbolic geometry. Now I was worried about having two distinct type of zeroes because it interfers with my 1990s alleged proof that the infinity can only be of one type of infinity since it is the inverse of zero and that zero comes in only one type. Well, as you can see, Poles are two distinct types and thus zeroes are two distinct types. So here I am in a very sad situation, where I marshalled a alleged proof in the 1990s to discredit Cantor's hierarchy of infinities. That here by 2007, mine own alleged proof is working against me. Because the AP-adics are equinumerous with the Reals as set forth in this book since both are **All Possible Digit Arrangements** except one is rightward while the other is leftward. And since these two number systems are the only existing number systems then the world has only one type of infinity. That infinity of the Counting Numbers is the very same as the infinity of the Reals, since both are all possible digit arrangements and countable. So here I am in the ugly spot of where mine own alleged proof that infinity comes in only one type can be used against me. Since the circle or sphere can have two imaginary zeroes, does that mean that the world can have two different types of infinity? I am going to say that I am saved from that ugly dilemma by saying that since those two zeroes are imaginary in both Positive and Negative AP-adics that because they are imaginary they save my argument. And it is noted that in Complex Numbers that i is appended j and k which are also square roots of (-1) that are different, so why not zeroes being imaginary and different. One thing I have not spent much time on is reflection over this Model and why there is a imaginary line due to the imaginary poles. Why multiplication is closed within a hemisphere yet addition trespasses between the two hemispheres. And resolution over the Equator line. But reflection needs time away to brew and stew over what has been accomplished and discovered, so maybe in the future editions some of those reflections will come forth. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Agent Smith? ' over alt.gossip.celebrities? Is that the same guy who thinks he invented mathematics? Did I say write invented, I meant discovered mathematics. Actually I'm not sure which is the proper term, invented or > discovered... All I know is I heard Nelson Rockefeller died while having sex with a > girl > who wasn't his wife....use it and you can die. That would apply if you used it your wife, wouldn't it? Although I suppose doing it with your secretary probably > leads to an increased state of excitement and is more > dangerous (and more fun). Is it invented or discovered? Discovered. You sound very sure of yourself.. > So you're saying, mathematics was not invented by man, it was put here > already...waiting for > humans to come along and discover it ' Where wast thou when the foundations of the earth were laid? > Declare, if thou hast understanding. ' You're trying to say...the foundation makers installed mathematics before > even people arrived > cause they knew we were coming? Who are these founders that *understand* > mathematics and knew > we would hast understanding? Math transcends makers and us. Math exist without us. Approximately ~16 billion year ago the planet earth was made. X + 1. Math. You really need to get a decent college education. Start at a community college in the LA area that can get you into UCLA, get as close to a 4.0 as you can (above a 3.5), and then go to UCLA. And stop spending all day worrying about losing your virginity. === Subject: Re: Agent Smith? ' On Nov 2, 12:58 pm, The Starmaker Who is this Agent Smith that keeps posting newspaper > Is that the same guy who thinks he invented mathematics? > Did I say write invented, I meant discovered mathematics. > Actually I'm not sure which is the proper term, invented or > discovered... > All I know is I heard Nelson Rockefeller died while having > sex with a girl > who wasn't his wife....use it and you can die. > That would apply if you used it your wife, wouldn't it? > Although I suppose doing it with your secretary probably > leads to an increased state of excitement and is more > dangerous (and more fun). > Is it invented or discovered? > Discovered. > You sound very sure of yourself.. > So you're saying, mathematics was not invented by man, it was put > here already...waiting for > humans to come along and discover it ' Where wast thou when the foundations of the earth were laid? > Declare, if thou hast understanding. ' You're trying to say...the foundation makers installed mathematics > before even people arrived > cause they knew we were coming? Who are these founders that > *understand* mathematics and knew > we would hast understanding? ' Math transcends makers and us. Math exist without us. Approximately > ~16 billion year ago the planet earth was made. X + 1. Math. ' You really need to get a decent college education. Start at a > community college in the LA area that can get you into UCLA, get as > close to a 4.0 as you can (above a 3.5), and then go to UCLA. And > stop spending all day worrying about losing your virginity. > Yeah. No use worrying about something that will never happen. === Subject: Sphere's rate of growth When a sphere is described passing through a plane as in Flatland it is said that a circle appears and at first expands rapidly. Then its growth rate slows and then stops. This I understand is when the circle has reached it's maximum size before it starts to shrink. What puzzles me is why the rate of expansion is faster to start with and then becomes slower as the circle === Subject: Re: Sphere's rate of growth > When a sphere is described passing through a plane as in Flatland it is said > that a circle appears and at first expands rapidly. Then its growth rate > slows and then stops. This I understand is when the circle has reached it's > maximum size before it starts to shrink. What puzzles me is why the rate of > expansion is faster to start with and then becomes slower as the circle If the radius grew linearly up to its maximum, you would get two cones joined together. Think of the 2-dimensional analogy: a circle passing through a line. The height of the intersection grows more rapidly to start with. --- J K Haugland http://home.no.net/zamunda === Subject: Re: Sphere's rate of growth > When a sphere is described passing through a plane as in Flatland it is said > that a circle appears and at first expands rapidly. Then its growth rate > slows and then stops. This I understand is when the circle has reached it's > maximum size before it starts to shrink. What puzzles me is why the rate of > expansion is faster to start with and then becomes slower as the circle If one imagines a sphere passing through a plane in such a way that the velocity of the sphere center perpendicular to the plane is constant, and one works out the rate of change of the radius of the circle of intersection between the sphere and plane, that rate of change will be greater when the circle of intersection is smaller. === Subject: #247 and #1 this textbook spawns another new textbook for Set theory; new textbook: Mathematical-Physics (AP-adics primer) for 6 year olds and onwards <472AB80C.7060709@hotmail.com> I am still wrangling and wrestling with how to organize this textbook of AP-primer for 6 year olds and upward. The trouble is that this textbook went far beyond the single subject of AP-adics and the desire to make the four operations accessible to everyone including bright 6 year olds. As I said often that this book is so revolutionary as to be the most revolutionary book ever written in mathematics for it dispels and falsifies more than 50 percent of mainstream mathematics. I said this textbook is about 3 books in one, at least I can organize it as such: (1) AP-adics Primer showing mostly how to add, subtract, multiply and divide Infinite Integers and other operations (2) What is Elliptic and Hyperbolic Geometry coordinate system (3) the Foundations of Numbers of Mathematics So I really had 3 textbooks all written inside one textbook. Maybe I should split those apart and out of that textbook in future editions. I do not know how I will go from here. But one thing that comes to my mind is that I missed diving deep into Set Theory, although I mentioned it in passing. So I should devote some time to a new textbook that explains Set Theory in the context of this book and more so in the context of the Atom Totality theory of Physics. It is the Atom Totality theory that is forcing me to write and do all this science and math. For without it, none of this would have happened, and Ludwig van Ludvig would have been an average person. So let this post be one of the last posts for the AP-adics Primer textbook and let this be the first post of another new textbook: New Textbook: Set theory as the most primitive foundation of Mathematics as Atomic theory is the most primitive foundation of Physics by Archimedes Plutonium written and copyrighted 2007 in sci.physics, textbook because that audience needs these ideas more than the education system. I mentioned in passing that Set Theory is mostly a sparse and vapid wisdom for it has only one prime concept of membership and which membership is never possible to define with accuracy. But there is problems over in Physics on their foundation for many physicists run around with the belief that atoms are not the lowest form of Matter and that they have existence independent of an atom. So, unlike my writing the AP-adics textbook where the Physics side was true and clear and wave duality and the symmetry and symmetry-breaking. In this textbook, I need to straighten up and clear up the messy Physics side as well as the phony Math side. I never had to introduce the Atom Totality theory in the AP-adics textbook and mentioned it only in connection with the idea that pi and e are growing numbers and have holes in them and based on the Cosmic Clock for their ultimate meaning as transcendental. But in this textbook on Set theory, I am going to have to do alot about talking of the Atom Totality theory for it is the foundation of Set theory. In the AP- adics textbook I could get away with merely Quantum Mechanics, but in this textbook I have to go one step deeper into the Foundation of Physics in order to clear up and clean up Set theory of Math. Basically, Atomic Theory is to Physics as the rock bottom ultimate Foundation that Set theory is to Mathematics. And that foundation of Physics follows this simple and beautiful fact. Since the entire Cosmos is nothing but one big atom itself, containing more atoms inside itself to irrelevant. To give a poignant metaphor, if we are tasked to count all humans would we not count all bodies as a count or would we be silly and capricious by counting hands and fingers and legs and toes and organs and even counting the total number of hairs and counting the total number of cells etc etc? This is insane. Well, in physics for much of the 20th century and going strong into the 21st century, the poor physicists has been so dumb and illiterate and lame in brain reality to them. Because of the Atom Totality theory, Physics stops and ends with the Atom. Anything smaller than the atom such as a photon or electron or proton or neutron or nucleus muon or neutrino or of the Atom. The Atom is the start of Physics and the Atom is the final last of Physics. Physicists were mostly insane and not rational in the 20th and 21st century when their mindset was so steeped One of the reasons they ended up in their stupidity corner wearing a dunce cap is because their overarching theory of the Big Bang never led them out of the darkness and kept them being so dumb and stupid, for are brewed in high energy physics labs. But once physicists realize the Cosmos itself is one big atom, that it immediately dispels their big atom itself, then atoms are the final thing and atoms are the sole and unique first things and primary foundational things. Anything not an atom, is a piece or fragment of an atom. So that when physicists are asked to make an inventory of the World, the present day stupid physicists would start the list with Atom of hydrogen then helium etc etc. So the Foundation of Physics is Atoms and the Foundation of Math is Set Theory. So, now, how does Physics clear up and clarify Set Theory with its pitiful notion of membership. Stay tuned. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: 3-D shape question What would be the term to describe the three dimensional shape of a couch pillow whose cross-section through the largest intersection is a square? I'm perhaps not describing this real well, but just consider the shape of a the square-ish pillows that you always see thrown at the end of a couch, you'll know what I mean. Would this be a type of spheroid, and if so, what type? BS === Subject: Re: 3-D shape question > What would be the term to describe the three dimensional shape of a > couch pillow whose cross-section through the largest intersection is a > square? I'm perhaps not describing this real well, but just consider > the shape of a the square-ish pillows that you always see thrown at the > end of a couch, you'll know what I mean. Would this be a type of > spheroid, and if so, what type? It's not a spheroid, but I don't know what it should be called. Your question reminds me of the tea bag problem. See, for example, . David === Subject: Re: 3-D shape question > What would be the term to describe the three dimensional shape of a > couch pillow whose cross-section through the largest intersection is a > square? I'm perhaps not describing this real well, but just consider > the shape of a the square-ish pillows that you always see thrown at the > end of a couch, you'll know what I mean. Would this be a type of > spheroid, and if so, what type? > BS first you have to consider about pillow's upper cross section which can be considered as distorted hemispher.as a whole pillow can be considered as two distorted hemispheres joined through their bases;so i think this is not type of spheroid it is anything else;do you have any referance about it?if so then can you mail it to me? i think you will....... thank you very much!!!!!!!!!!!!! === Subject: Re: Epistemology 501: Angular Mechanics >Hmmm. You might want to rethink your whole approach here, sport. >What approach is that? You mean I might want to >adopt your definitions? The classical definition of L >defines a thing which is constant in the absence >of torque. I find that convenient. I find no reason >to rethink my whole approach, even if you could >define what you mean by that. > Except the classical definition of angular momentum defines angular > momentum period whether or not in the absence of torque. That's right. You think angular momentum can't be defined in >the absence of torque? Then what sense do you make >of the conservation law angular momentum is constant >in the absence of torque? How could it be constant >if it can't be defined? Of course angular momentum can be defined in the presence or absence of torque. It just might help if it were defined correctly. > So the classical definition of angular momentum is just noise now? 'No, but you didn't write that. You assigned completely >different meanings to the symbols, so that the quantity >you were discussing r x p has nothing to do with >classical angular momentum. L=r x p is the classical definition of angular momentum, sport. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics > I'm really less interested in the macro angular mechanics practiced in > ordinary engineering than in integrating the general approaches to > angular mechanics used in micro atomic, macro engineering, together > with celestial orbital mechanics on a common framework. I'm confident > the complexities of conventional engineering render the problem fairly > intractable. But if we consider the problem purely in the abstract of > how rotation is actually mechanized I suspect that approaches to all > forms of angular mechanics will be considerably simplified. > (By the way while I certainly appreciate the candor and tenor of your > remarks, by conversing so sensibly with me you'll undoubtedly earn the > wrath of most around here who insist their opinions are anything more > than hearsay and guesswork.) > ~v~~ OK Lester. I am open to what you are describing. Have you ever >considered a torque as a possible force mechanism no different than >the standard directed forces? Whereas the standard forces generate a >second differential in position these torque forces are going to do >the same in rotation. Here may be a spin interaction that goes pretty >deep. Thermodynamics in its standard interpretation claims to be vibrating >atoms. However this interpretation cannot be so simple since standard >atomic vibrations propagate at the speed of sound in a solid. Thermal >energies transfer far more slowly. If thermal energy is rotational in >nature such a mechanism will provide (with a loose coupling) slow >propagation of energy. I do not have a full model but enough is here >to see a simplistic mechanism. This area is extremely loaded and >overwhelming to consider such behavoirs, yet the discrepancy between >the speed of sound versus the speed of thermal propagation is not >addressed head on by phonon theory, which is the accepted modern >theory of energy propagation in solids. Depending on how large a can >of worms you are willing to work with I think this area is worth your >consideration. I do not ask you to adopt this theory of rotational >energy and torque interactions, but it could nicely fit. This is a >very abstract formulation I am providing here. The most substantive >part is the critique of existing theory. If we remain in 3D space and >accept that the solid is tightly coupled in translation then there is >little else in mechanics to work with other than this rotational >concept. This will bleed into atomic theory pretty heavily and so this >is no small potato to be handling. The fascinating behaviors that take >place at cold temperature are fundamental physics and so if the >problem weren't overwhelming enough try to take all of this theory in >as well. If we take the lid off to this level we see the existing theory is not >sufficient and in that there is a large accumulation it may be >reasonable to consider a collapse to a cleaner and simpler theory. >Most who have done enough work to be proficient in these areas will >not want to go here yet look at the number of poor answers. You >mention spin and here again I have criticism but I'll save it for >another post. Tim, I think I have some appreciation for where you're going with this. However I'm not so sure we can simplify these problems very much even with an accurate mechanical description for angular mechanics. But before getting into all this let me just add that there is another category of angular interactions you might want to add to your list: magnetic effects. The problem I see entails several considerations. First we have only one established constant velocity in space the velocity of light c and every other velocity must be analyzed in reference to it. But beyond that also we have to recognize that even if we can analyze angular mechanics correctly we're still left with the problem of orientations which may defy simple analysiss. In other words we may be able to explain a comprehensive mechanics without necessarily improving the predictive picture greatly. As far as torque is concerned I see that as a non starter. In fact tomorrow morning I'm going to post an addendum to this thread outlining briefly my own approach to the problem which entails a strictly linear mechanics in Newtonian terms. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics Lester Zick ' Because the mathematics states plainly and simply that for constant > p=mv where there is no rotation that angular momentum is infinite. First of all, you're not right. The axis that you choose does not depend >on the current radius of the movement. I am actually tired of repeating >it. While l. momentum is a property of the body itself, ang. momentum >characterizes two things: 1. The material point >2. The axis of rotation. Thus, saying linear movement necessiates the use of an infinite radius >you're making a mistake. If a point of mass m is moving straight-lineary at speed v, I have a >full right to choose the axis of rotation in 1 metger away from the >point and declare its angular momentum (relative to this axis) equals >1*m*v. So the magnitude of classically defined angular momentum L=r x p does not increase from zero at zero radius to infinite at infinite radius? ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics Clearly this situation is unsatisfactory in mechanical terms and we > must find some alternative definition for angular momentum with > consistent relations among all factors. Oh, Lester, why don't you try preening at alt.discussions.look.at.me >or sci.general.buffoons where there is a greater chance that when you >say something it won't be *immediately* obvious how much of a goofball >you are. If someone intends to impersonate a surgeon, they generally do it in >front of people they think they can fool. They don't go to a surgical >conference and attempt it. You are not only an imposter, you're an incompetent imposter, and one >with poor judgement in selecting targets as well. So L=r x p doesn't imply infinite values of angular momentum for > motion in a straight line, Paul? I didn't say anything of the kind, Lester. > I'd like to believe you but unlike > empirics I can't just bury my mistakes. The fact that you can't bury your mistakes is precisely the reason I suggested you take it elsewhere. > Scientists are required to > explain their mistakes instead. Explain their mistakes? Have you explained your mistake, Lester? > It's why empirics can only practice > empiricism in the hopes that maybe someday they'll guess something > right and noncewise why there's no malpractice insurance for empirics. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics >Clearly this situation is unsatisfactory in mechanical terms and we > must find some alternative definition for angular momentum with > consistent relations among all factors. >Oh, Lester, why don't you try preening at alt.discussions.look.at.me >or sci.general.buffoons where there is a greater chance that when you >say something it won't be *immediately* obvious how much of a goofball >you are. >If someone intends to impersonate a surgeon, they generally do it in >front of people they think they can fool. They don't go to a surgical >conference and attempt it. >You are not only an imposter, you're an incompetent imposter, and one >with poor judgement in selecting targets as well. > So L=r x p doesn't imply infinite values of angular momentum for > motion in a straight line, Paul? I didn't say anything of the kind, Lester. Well getting you to say anything of any kind specifically to do with the subject at hand is rather a nuisance, Paul. > I'd like to believe you but unlike > empirics I can't just bury my mistakes. The fact that you can't bury your mistakes is precisely the reason I >suggested you take it elsewhere. Take what elsewhere? > Scientists are required to > explain their mistakes instead. Explain their mistakes? Have you explained your mistake, Lester? No, only because you haven't explained what mistake I've made. > It's why empirics can only practice > empiricism in the hopes that maybe someday they'll guess something > right and noncewise why there's no malpractice insurance for empirics. ~v~~ === Subject: Re: Epistemology 501: Angular Mechanics > So L=r x p doesn't imply infinite values of angular momentum for > motion in a straight line, Paul? I didn't say anything of the kind, Lester. So L=r x p does imply infinite values of angular momentum for motion in a straight line, Paul? ~v~~ === Subject: Re: Unsolved Quaternion problem is now solved! Content-Length: 2792 Originator: rusin@vesuvius > This is a minor unsolved quaternion problem. Tian, Yongge, The equations ax - xb = c, ax - x*b = c, and x*ax = b > Asian Bulletin of Mathematics 28, 343-362. http://www.scnu.edu.cn/seam-bulletin/vol28no2/21.pdf the equation ax - x*b = c in quaternions, when |a| != |b| has a unique solution, > which Tian gives in his eqn 3.35, but says it's an unsolved > problem how to compose the solution with a formula in terms > of the given quaternions, a,b,c. Yesterday, I solved this and found x = N(a,b,c) / D(a,b) where, N(a,b,c) = (|a|^2 + |b|^2).( |a|^2.a*.c + a*.c*.a.b + b.c.b*.a* + |b| > ^2.b.c* ) > - ( (ab) + (ab)* ).( |b|^2.a*.c* + b.c*.a.b + a*.c.b*.a* + |a| > ^2.b.c ) D(a,b) = (|a|^2 - |b|^2).( (|a|^2 + |b|^2)^2 - ( (ab) + (ab)* )^2 ) the method will be explained in my forthcoming paper > General solutions to linear problems in quaternions. > due Oct, 27, 2007 on my website, > where I solve all linear problems explicitly, and Tian's > problems are just trivial examples used to illustrate the > method. is a long time ago in the world of math. At any rate, keys > to the method are already described in my hexpentaquaternion > paper, http://www.hypercomplex.com/research/emgrav/abs20060129a.html in my forthcoming paper I will delve into depth on how to apply > the two-hand quaternion method I introduced there. pmj > m...@hypercomplex.com Solutions for scalar problems have recently been solved in Turner, J.D., Solving Linear and Quadratic Quaternion Equations, Technical Note, AIAA Journal of Guidance, Dynamics, and Control, Vol. 29, No. 6, November- December 2006, pp 1420-1423. Turner, J.D., An Object-Oriented Operator-Overloaded Quaternion Toolbox, Paper No. AIAA-2006-6160, Proceedings of the AIAA Guidance, Navigation, and Control Conference, Keystone Colorado, Aug. 2006, AIAA Meeting Papers on Disc, CD Code: 1305-349138. The basic idea is the factor the scalar and vector parts of the eqn. as ax+xb+c=0 (a0x0-av*xv,a0xv+avx0+avxxv)+ (b0x0-bv*xv,b0xv+bvx0+xvxbv)+ (c0,cv)=(0,0) OR ( (a0+b0)x0-(av+bv)*xv+c0, (a0+b0)xv+(av+bv)x0+(av-bv)xxv +cv) ) = (0,0) Use vector part to solve for xv: xv=-inv[(a0+b0)I+tilde(av-bv) ](cv+(av+bv)x0), tilde denotes matrix form of cross product Introduce in scalar equation: (a0+b0)x0+(av+bv)*inv[(a0+b0)I+tilde(av-bv) ](cv+(av+bv)x0) + c0 = 0 Solve for x0 x0 = -(c0+(av+bv)*inv[(a0+b0)I+tilde(av-bv) ]cv ------------------------------------------------- (a0+b0)+(av+bv)*inv[(a0+b0)I+tilde(av-bv) ](av+bv) Substitute into xv Done! Many special cases. === Subject: generalised hypergeometric function Content-Length: 298 Originator: rusin@vesuvius For the generalised hypergeometric function denoted by F_pq(a_1,a_2...a_p;b_1,b_2...b_q;z), under what condition(s) does the function converge for z=1? Is this condition some sort of region of convergence in the (p+q)-dimensional space spanned by the p a's and the q b's? If so what exactly is it? === Subject: Re: Equivalent to Axiom of Choice? <472b1208$0$4360$9b4e6d93@newsspool4.arcor-online.net> Content-Length: 1043 Originator: rusin@vesuvius >Theorem: Every partially ordered set can be extended to a total order. The easy proof I've found of this uses the axiom of choice (well, > specifically, Zorn's Lemma.) Is it equivalent to the axiom of choice? no, it does not even imply that everyBooleanalgebra has aprimeideal. See http://citeseer.ist.psu.edu/87611.html -- > Stefan Wehmeier > stef...@math.upb.de I did like the proof in that paper of the Order Extension Theorem from the Well Ordering Principle. If (P,<=) is a partial order, and we have a well-ordering on the set P, then we define: l(x)={z<=x: z in P} And extend <= by defining: xRy if x=y or if the least element of the symmetric difference between l(x) and l(y) (under the well-ordering of P) is a member of y. Now R is a total order and x<=y implies xRy. Much nicer than the Zorn Lemma proof, which feels 'less constructive' on an intuitive level, even if it isn't logically any different. === Subject: Re: Equivalent to Axiom of Choice? <472b1208$0$4360$9b4e6d93@newsspool4.arcor-online.net> Content-Length: 586 Originator: rusin@vesuvius > Theorem: Every partially ordered set can be extended to a total order. The easy proof I've found of this uses the axiom of choice (well, > specifically, Zorn's Lemma.) Is it equivalent to the axiom of choice? no, it does not even imply that every Boolean algebra has a prime ideal. See http://citeseer.ist.psu.edu/87611.html -- > Stefan Wehmeier > stef...@math.upb.de equivalent to AC, but I was curious. === Subject: Re: Equivalent to Axiom of Choice? <472b1208$0$4360$9b4e6d93@newsspool4.arcor-online.net> Content-Length: 392 Originator: rusin@vesuvius > Theorem: Every partially ordered set can be extended to a total order. it does not even imply that every Boolean algebra has a prime ideal. How about the following kind-of-dual variant - where does it fit into the one-way implications above: ### Every partial order can be RESTRICTED to a maximal total order. Maximal here refers to order-preserving inclusions. Bill Taylor === Subject: AIMS can't calculate kids' math prowess http://www.tucsoncitizen.com/daily/opinion/67337.php Published: 10.31.2007 Stanton: AIMS can't calculate kids' math prowess BILLIE STANTON, Tucson Citizen Kudos to students in 111 Arizona schools, including 15 in Pima County, who have improved their math scores on our state test. And a nod to the Rodel Charitable Foundation of Arizona, which has been providing these low-income schools with its MAC-Ro program to rev up math learning. But while higher achievement on AIMS (Arizona's Instrument to Measure Standards) is a terrific accomplishment, the true test will be how these children fare with math years from now. A child needs more than the ability to give correct answers on AIMS; she needs to understand the concept behind that math. Noted intellectual Sheila Tobias, Tucson author of Overcoming Math Anxiety and many other books, says today's students get a process that focuses on memorization, testing for short-term recall and extrinsic rewards. What we have to inculcate, Tobias says, is the intrinsic curiosity and satisfaction and learning of concepts that will motivate them to stick with math when math gets really hard. The rise in test scores may or may not be meaningful, says Alan H. Schoenfeld, education professor at the University of California at Berkeley. In many cases, scores on a state test went up, but scores on other tests, such as the National Assessment of Educational Progress, did not. The kids were able to do precisely what they were drilled on, Schoenfeld says, but there was no transfer and no greater understanding of core mathematical concepts than before. Student understanding of mathematics appears to be shallow, indeed, according to a survey of students graduating from New York high schools. There were three effects they were absolutely sure of, says Florence D. Fasanelli, mathematician in residence at the American Association for the Advancement of Science If you see a decimal point, move it. If you see a sign, change it. And if you see a fraction, invert it, Fasanelli laments. But such misconceptions begin long before high school. Kids should understand that the equal sign (=) means quantities flanking that sign are the same. Most U.S. kids are taught that it means 'the answer goes here,' says Phil Daro, a mathematics educator at the University of California, senior fellow in mathematics at America's Choice and site director of SERP in San Francisco. In kindergarten through third grade, few children are exposed to the number line - which shows numerical progression: -2 -1 0 1 2 3 4 5 and so on. Spending time to deepen conceptual understanding of numbers using the number line might not improve K-3 performance compared to practicing subtraction problems, Daro says, but it will pay off in improved fourth-grade performance. So how can schools infuse a deep understanding of mathematical concepts in their students? Train the teachers, Fasanelli says. A typical elementary school teacher will not have had more than two years of math in high school, she says. Then they get a course in how to teach it. And that's it. That's all they know. Veteran teachers must be given an opportunity to gain a deeper knowledge of content. Mathematics is not just a series of steps; there's some underlying logic to it. A teacher with a strong grasp of mathematical concepts can impart that information more easily to the child. The future depends on these children, Fasanelli says, particularly children from underrepresented groups because there are so many of them. We have to do it right. Tobias says, Somewhere between here and college, they're going to have to develop an intrinsic interest and satisfaction in learning mathematics. Whether MAC-Ro will have lasting effects is uncertain. It is important to follow the students in the program and see how they benefit in higher grades before proclaiming victory, Daro says. If they persist in standing above the crowd, terrific. Fred Stevenson, mathematics professor at the University of Arizona, echoes that sentiment. If Rodel and institutions like Rodel can fund these kinds of ventures for generations and even lifetimes, rather than for the short intervals of time that administrators and legislators serve, then our country can become what our leaders claim they want it to be - a truly mathematically literate country, Stevenson says. Billie Stanton may be reached at bstanton@tucsoncitizen.com and 573-4664. 8 Comments on this Story Noted intellectual Sheila Tobias... Oh, please. Her inane theories on math anxiety have stifled progress on teaching mathematics. Elementary teachers have only had two years of high school math. Add to that, they have only had one college math course; Math for Elementary Teachers. Most of them could not pass College Algebra and many are Math(s) flunkees. That is why they were elementary ed majors to begin with. My experience with people educated using modern techniques has been far from great. I have had employees with Masters degrees that had a difficult time in writing a comprehensive paragraph not to mention that they could not spell. I have had college graduate employees with business degrees that could not even balance their own check books. It wasn't that these people were stupid, far from it, they just had not gotten the idea of practical applications for what they learned. At least testing these students to a standard gives a flicker of an idea of what they have ahead. Rating: 3 Thumbs Up Perhaps an annual AIMS type test for teachers, with mandated remedial education courses for those not achieving acceptable grades would be a step in the right direction. Just a thought,as statistics seem to indicate the national public education system, but particularly in ARIZONA, is deplorable. 5. Comment by shane l. (maranaman) - October 31,2007 @ 12:34PM Rating: 3 Thumbs Up I am sick and tired of always blaming the teachers. My children's education is my responsibility and I appreciate the teachers ASSISTING me. The education system is fine; it is the parents that are deplorable. I know exactly what subjects my children are learning in school and I go over the subjects with them. I assist the mathematic lessons by showing them what a number line is. I make sure all their homework is finished. I make sure they understand the concepts of the lessons. If your children are not learning, look in the mirror. Place the blame where it belongs. 6. Comment by Hugh M. (shuggie) - October 31,2007 @ 1:01PM Rating: 1 Thumb Down If we're talking about numeracy, fine, but school mathematics has little to do with real mathematics, and lack of success with it at school proves very little. The subject bored me stiff at school, but I graduated in it at university and I now pursue it as a hobby. The penny drops eventually or it doesn't. I had a lecturer with a doctorate in math who asked us not to submit the arithmetical bits, since neither we nor would get any better at that particular branch of the subject. True. 7. Comment by Dabil G. (Red Star) - October 31,2007 @ 4:00PM Rating: 1 Thumb Up If the student hasn't learned, the teacher hasn't taught. 8. Comment by J Z. (#4898) - October 31,2007 @ 4:26PM Rating: 1 Thumb Up ... and the parent hasn't cared. === Subject: Math skills lag for LCC freshmen Math skills lag for LCC freshmen By Carrie Pederson / The Daily News Oct 29, 2007 Most of the students who enroll at Lower Columbia College aren't ready for college's lowest-level math courses. Ninety-three percent of first-year LCC students are required to enroll in a pre-college-level math course, and 89 percent of them are straight out of high school. LCC and high school educators are now trying to figure out how to deal with the problem. LCC offers seven courses below College Algebra, the school's lowest- level college math class. First-year student Anna Owen, who graduated from Kelso High School last year, is taking a pre-college course from Tim Trinkle, an adjunct professor. I wish that I had placed in a higher class, that way it wouldn't take as long, she said. Some students have to take four classes to get to the college level, LCC math professor Rick Swee said. A lot of students place in Math 092 (Elementary Algebra), said Janelle Runyon, spokeswoman for LCC. That class is four levels below College Algebra. To complete many of the college's programs, students need to be at the college level, said Dawn Draus, faculty chair of the math department at LCC. For transfer programs, as well as programs in nursing, business technology, early childhood education and welding, College Algebra is not necessarily required, she said. There are more math-intensive programs-- engineering and some science and computer-related programs that require about six college-level math classes, Swee said. Some students think that the college's placement test doesn't accurately reflect their skills. The test is way harder than (Trinkle's) class, but I can't go further without placing higher, said Samantha Petrich, a first-year student who graduated from W.F. West in Chehalis last year. Students don't get to go around the test very often, Swee said. In some cases, they may convince a counselor they can work at the top of their ability in math, he said. Students used to be able to sign up with whatever they want, that created havoc, he said. They would fail the class, retake it and fail again. To tackle the problem of getting students to pass the test and advance in math more quickly at LCC, local math educators went to a conference in Leavenworth last summer called the Transitions Math Project. The project, about two years old, was started through the State Board of Community and Technical Colleges to address the fact that, statewide, 46 percent of first-year community college students just out of high school were below the college math level. Teachers from Kelso, Mark Morris, R.A. Long, Kalama and Castle Rock compared their math courses with the LCC's pre-college course work by looking at a map of the college's curriculum. We say these are our expectations, if you hope students are placed in college algebra, Swee said. I found out about some things kids are being tested on for placement, said Wayne Heuett, math teacher at Castle Rock. I thought we were covering everything, but it turns out we didn't match up on all the curriculum. Some of the stuff the we are teaching in pre-calculus we could teach in algebra two, and vice-versa, he said. Castle Rock's traditional math curriculum, which teaches algebra, geometry and calculus sequentially, more closely aligns with LCC, Swee said. Some local schools use the integrated system which hits equations in a different order, he explained. The group continues to expand its mapping project-- Woodland, Toutle Lake, Clatskanie, Wahkiakum and Rainier high schools have been invited to join. The idea is to get lots of schools to aim for a more universal seamless transfer, Swee said, but any time you get this many institutions and people trying to pull in one direction it's a logistic nightmare. With much to consider, plans are in the infancy stage, Heuett said, though he's learning through the process. I'm learning more now than in 10 years of teaching what's happening around me, he said. It's been a fun opportunity to re-access what I'm doing. In the meantime, students like Aurelia Bell appreciate the chance to strengthen math skills through Trinkle's class and LCC's pre-college program. Math is easier here, said the Castle Rock graduate. The environment is better, it's less rushed to learn. User Comments: Just Opt Out This is a problem in every single college and university in this state. It's called FUZZY math, supported and mandated by the Office of the Superintendent of Public Instruction (OSPI). This reform math does not prepare students for college. WASL rules our educational world, and this is a sad result of teaching our kids new math that meets the standards the state has set. Our state is in the process of revising the math standards, but they have hired The Dana Center in Texas to do this monumental task...the Dana Center is widely known for screwing up Texas math education and they loudly promote this new reform math...the same math that is getting us into trouble. There is not ONE person on the revision committee that is against this math, which is troubling. Contact your legislature and tell them you've had enough. If anyone reading this has had difficulties helping your elementary student with math homework, you know firsthand what I'm speaking of. The WASL must go. I encourage ALL parents to opt your children out of this test. Elementary and middle school students aren't required to pass this nutty test, so there is NOTHING to lose by opting out. It does not effect grades or placement in classes, or promotion to the next grade. There are many other measuring tools our schools use to judge your child's progress. If we, as parents, don't finally take a stand...things will get much worse. Write a letter to your child's principal politely requesting your child not participate in the WASL. If enough parents do this, the state will be forced to listen. Taking pre-college level math courses in college may be nice for those students we've failed in public school, but having our kids prepared in the first place would be a much better path. === Subject: MathPuzzle 201: Snooker in Ruurlo Good morning ! For today's puzzle I have been inspired by BBC's Snooker coverages. For those who do not know how snooker is being played, there is a link to an explanatory website. Not to say that you really need this though to solve the puzzle.. Please have fun with the new puzzle. Quite a few puzzles are still open. Solutions are kindly invited and more than welcome. Peter direct link: http://home.planet.nl/~p.j.hendriks/p201e.htm Please answer by email and not in this newsgroup. === Subject: solution manuals List of Solution manuals: Engineering Circuit Analysis 6Ed - Hayt Solutions Manual Norbury - Solutions manual for mechanics and thermodynamics Physics For Scientists And Engineers - Solution Manual RC Hibbeler statistics 11th edition DigitalSignal Processing; A Computer Based Approach 1st ed intro to C++ Dietel and diitel solution manual Openheims discrete time soln manual Coulson & Richardson's Chemical Engineering, Volume 5, Solutions to the Problems in Chemical Engineering Volume 2 & 3.pdf Solutions Manual] [Instructors] Introduction to Linear Algebra--3rd Edition - Gilbert Strang.pdf Introduction to VLSI Circuits and Systems (2001 draft) - John P Uyemura - Solutions Manual.pdf PAPOULIS & PILLAI, PROB. R.V & STOCHASTIC PROCESSES 4TH EDITION SOLUTION Student Solutions Manual & Study Guide for Hornback's Organic Chemistry (2nd Ed.) - Joseph M. Hornback.pdf Field and wave electromagnetics 2nd edition Java Cookbook Solutions and Examples for Java Developers.pdf An Introduction to the Mathematicis of Financial Derivatives(Neftci)-- Solution Manual.pdf probability and statistical inference hogg and tanis 7th ed Probablility and statistics fundamentals Signals adn SYstems 2nd edition solution manual An intro to database systems 8th edition ELementary differential equations and boundary value problems Introduction to probability introduction to algorithms Microwave engineering 3rd edition Operating systems concepts soln manual [Solutions Manual] Mechanical Engineering Design 7th Ed. Shigley [Solutions Manual] Engineering Mechanic STATICS 10th Ed. R.C. Hibbeler Stallings W - Instructors manual Operating Systems 4ed Digital Signal Processing - Proakis & Manolakis - Solutions Elements of Chemical Reaction Engineering - Solutions Manual Solid State Electronic Devices Streetman Solution Manual [Solutions Manual] Fourier and Laplace Transform - Antwoorden Elements of Chemical Reaction Engineering - Solutions Manual Computational techniques for fluid dynamics - Solutions Manual Networks - Book Solution Stallings W - Instructors manual Operating Systems 4ed solutions Solution Manual For Communication Systems (4th edt) by Simon Haykin Fundamentals Of Logic Design 5Ed - Charles Roth - Solutions Manual Solution Manual For Microelectronic Circuits By Adel Sedra Modern Digital and Analog Communications Systems - B P Lathi Solutions Manual Fundamentals of Heat and Mass Transfer - Solutions Manual [Solutions Manual] [Instructors] Calculus 5Th Ed James Stewart Proakis J. (2002) Communication Systems Engineering - Solutions Manual Solution Manual for Semiconductor Physics and Devices 3ed Neamen serway - physics for scientists and engineers - solution manual Fundamentals Of Logic Design 5Ed - Charles Roth - Solutions Manual [Solutions Manual] Thermodynamics - An Engineering Approach, 5Th Cengal Boles [Solutions Manual] Engineering Mechanic STATICS 10th Ed. R.C. Hibbeler [Solutions Manual] Probability And Statistics For Engineers And Scientists Fundamentals of digital logic with VHDL design solutions manual Sonntag-Borgnakke-Van Wylen -Fundamentals of Thermodynamics Solution Manual Chapters 10-16 Fundamentals of Heat and Mass Transfer [Frank P.Incropera - David P.DeWitt] Solution Manual Introduction to VLSI Circuits and Systems (2001 draft) - John P Uyemura - Solutions Manual [Solutions Manual] [Instructors] Introduction to Linear Algebra--3rd Edition - Gilbert Strang Dorf-Svaboda - Solution Manual For Introduction To Electric Circuits 6th Edition An Introduction to the Mathematicis of Financial Derivatives(Neftci)-- Solution Manual Fiber Optics Technicians Manual (2nd Ed.) DigitalComm Fundamentals App Solution Manual Solution Manual For Communication Systems (4th edt) by Simon Haykin Atkins Solution Manual Microelectronics - Millman Solution Manual Modern Physics-4th Edition Solutions Manual Sakurai - Modern Quantum Mechanics. Solutions to Problems.djvu Modern Digital and Analog Communication Solutions---Funadamentals of Communication Norbury - Solutions manual for mechanics and thermodynamics Engineering Circuit Analysis 6Ed - Hayt Solutions Manual serway - physics for scientists and engineers - solution manual Sonntag-Borgnakke-Van Wylen -Physics - Classical Mechanics - Fundamentals of Thermodynamics Solution Manual Chapters 1-9 Prentice Hall - Solutions Manual; Communication Systems Engineering (McGraw-Hill) (Instructors Manual) Electric Machinery Fundamentals 4th Edition (Stephen J Chapman) Prentice.Hall- Digital image processing - Gonzalez 2Ed- Solutions Manual (2002) Prentice.Hall- Digital image processing - Gonzalez 2Ed- Solutions Manual (2002) [Instructors Solutions Manual] Introduction to Eletrodynamics - 3rd ed. David J. Griffiths [Manual Solution] Mechanics of Materials Hibbeler 4th-Chapter 12 Operating System Concepts 7th ed - Exercises & Solutions [Problemas y Soluciones] 854 Problemas Seleccionados de F.92sica Elemental. (B.B.B.9cjotsev - V. D. Kr.92vehemkov - G. Ya. Mi.87kishev - I. M. Sar.87eva)(1979) [Solu.8d.8bo dos problemas] Redes de Computadores - 4a ed. - ANDREW S. TANENBAUM Chemical and Engineering Thermodynamics- 3rd Edition- Solutions Manual [Soluciones a los problemas] FISICA 1 -2a ed. Luis Rodrigus Valencia [Soluciones a los problemas] Suplemento Calculo Infinitesimal Calculus- Michael Spivak.pdf [Solution Manual] CD Physics - Halliday, Resnick and Walkers - Fundamentals of Physics 1, 2, 3 and 4 (4th ed.)(over 2000pages) [Solutions Manual] Classical Electrodynamics - 2nd Ed. John David Jackson byKasper van Wijk [Solutions Manual] Communication Systems Engineering Proakis J (2002) [Solutions Manual] [Instructors] Advanced Engineering Mathematics 8Ed - Erwin Kreyszig [Solutions Manual] [Instructors] Calculus 5Th Ed James Stewart [Solutions Manual] [Instructors] Introduction to Linear Algebra--3rd Edition - Gilbert Strang [Solutions Manual] [Instructors] Physics by Resnick Halliday Krane, 5th Ed. Vol 2 [Solutions Manual] Anton Bivens Davis CALCULUS early transcendentals 7th edition [Solutions Manual] Applied Statistics and Probability for Engineers 3rd Ed. Douglas C Montgomery, George C. Runger [Solutions Manual] Applied.Statistics.and.Probability.for.Engineers.-.Student.,.3rd.Ed. [Solutions manual] Calculus George Thomas 10th ed Vol 1 [Solutions manual] Calculus George Thomas 10th ed Vol 2 [Solutions Manual] Communication Systems 4Th Edition Simon Haykin [Solutions Manual] Control Systems Engineering, Nise [Solutions Manual] Design of Analog CMOS Integrated Circuits [McGraw Hill].pdf [Solutions Manual] Digital Signal Processing - Proakis & Manolakis [Solutions Manual] Digital Signal Processing; A Computer-Based Approach 1st ed [Solutions Manual] Econometric Analysis - Greene , Williame H. - 5th Ed [Solutions Manual] Electric Machinery 6Ed Fitzgerald, Kingsley, Uman - [Solutions Manual] Elementary Mechanics & Thermodynamics [2000] by Professor Jhon W. Norbury [Solutions Manual] Elementary Mechanics & Thermodynamics [2000] by Professor Jhon W. Norbury [Solutions manual] Engineering - Materials Science, Milton Ohring [Solutions Manual] Engineering Electromagnetics - 6th Edition - William H. Hayt, John A. Buck [Solutions Manual] Engineering Fluid Mechanics, 7th ed. Clayton T. Crowe, Donald F. Elger and John A. Roberson [Solutions Manual] Engineering Mechanic STATICS 10th Ed. R.C. Hibbeler Edition, (2002) - J. L. Meriam and L. G. Kraige [Solutions Manual] Fourier and Laplace Transform - Antwoorden [Solutions Manual] Fundamental os Heat and Mass Transfer [Frank P. Incropera - David P.DeWitt] [Solutions Manual] Fundamentals of Engineering Thermodynamics Moran, M.J. & Shapiro H.N. [Solutions Manual] Fundamentals of Engineering Thermodynamics, M. J. Moran and H. N. Shapiro, 5th edition [Solutions Manual] Fundamentals Of Fluid Mechanics 3Rd And 4Th Edition.pdf [Solutions Manual] Fundamentals of Machine Component Design 3rd Edition by Robert C. Juvinall and Kurt M. Marshek [Solutions Manual] Fundamentals of Thermodynamics 6th Ed Sonntag- Borgnakke-Van Wylen [Solutions Manual] Fundamentals of Thermodynamics [Sonntag-Borgnakke- Van Wylen] [Solutions Manual] Fundamentals.of.Thermodynamics.[Sonntag-Borgnakke- Van.Wylen] [Solutions Manual] Hibbeler 4ed - Resist.90ncia dos Materiais [Solutions Manual] Introduction to Fluid Mechanics (Fox, 5th ed) [Solutions Manual] Introduction to Linear Algebra 3Ed - Gilbert Strang [Solutions Manual] Introduction to VLSI Circuits and Systems (2001 draft) - John P Uyemura [Solutions Manual] Mechanical Engineering Design 7th Ed. Shigley [Solutions Manual] Mechanics Of Materials - (3Rd Ed , By Beer, Johnston, & Dewolf) [Solutions Manual] Mechanics of Materials, 6th Ed. by R. C. Hibbeler [Solutions manual] Oppenheim's Discrete Time Signal Processing text [Solutions Manual] Probability And Statistics For Engineers And Scientists [Solutions manual] Probability and Statistics for Engineers and Scientists Manual HAYLER [Solutions Manual] Signals and Systems 2nd Ed. - Haykin [Solutions Manual] Signals And Systems - 2nd Ed.- Oppenheim & Wilsky.pdf [Solutions Manual] Thermodynamics - An Engineering Approach, 5Th Cengal Boles [Solutions Manual] University Physics - Sears and Zemansky's 11th Ed Manual.pdf Classical Mechanics - Goldstein Solved problems Daniel Shanks - Solved And Unsolved Problems In Number Theory (2Nd Ed), 1978.pdf Electric Machinery Fundamentals (Solutions Manual) Elementary Differential Equations And Boundary Value Problems, 7Th Ed - Boyce And Diprima Student Solutions Manual, Charles W Haines Ode Architect Companion Fundamentals of Logic Design 5Ed - Charles Roth - Solutions Manual Fundamentals of Thermodynamics 6th Ed (Solutions Manual) - Sonntag- Borgnakke-Van Wylen Griffiths, David - Introduction To Electrodynamics Solutions Manual - With Update Halliday, Resnick - Fundamentals Of Physics - 7Th Edition Instructors Solutions Manual Instructors Solution Manual, Static- Meriam and L. G. Kraige Instructor's Solutions Manual - Marion, Thornton - Classical Dynamics Introduction To Algorithms 2Nd Edition Solutions(Instructor's.Manual) Introduction to Probability - Solutions Manual Dorf-Svaboda-Solution manual for Introduction to electric circuits 6th edition Juvinall, Marshek - Fundamentals of Machine Component Design, 3rd ed - Student Solutions Manual McgrawHill - William H. Hayt, John A. Buck - Engineering Electromagnetics, 6th Edition Solutions Manual !!!!!!!!!!!!!! Microwave Engineering 3e - David M Pozar - Solutions Manual Microwave Engineering 3E - David M Pozar - Solutions Manual Munson - Young - Okiishi Operating Systems Concepts 6th SOLUTIONS MANUAL !!! Physical Chemistry 7ed - Peter Atkins - Julio de Paula - instructors solution manual Physics For Scientists And Engineers 6E By Serway And Jewett - Solutions Manual Vol 2 Proakis J. (2002) Communication Systems Engineering - Solutions Manual (299s) Probability and Statistics for Engineering and the Sciences by Jay L. Devore Probability Random Variables and Stochastic Processes Solutions Manual.Papoulis.McGraw Hill.2002 Schaums Mathematical Handbook of Formulas and Tables Signal Processing and Linear Systems - B P Lathi - Solutions Manual Solution Manual to engineering fluid mechanics 7e Solution To Two-Dimensional Incompressible Navier-Stokes Equations- Maciej Matyka Thomas' Calculus, Early Trascendentals 10th ed Instructors Solutions Manual Wankat & Oreovicz - Teaching Engineering Wiley - Pozar - Microwave Engineering 3ed - Solutions Manual Wiley Chemical And Engineering Thermodynamics 3Ed Solutions Manual Zwillinger D. et al - CRC Standard Probability and Statistic Tables and Formulae (1999) [Solutions Manual] Fundamental os Heat and Mass Transfer [Frank P. Incropera - David P.DeWitt] ANOTHER EDITION Halliday, Resnick- Fundamentals Of Physics (7Th Ed)- Solutions DigitalComm Fundamentals App Solution Manual.pdf u can email and tell me which u want at epictheman@yahoo.com .All payments are via paypal only and one solution manual will cost you US $10 if u email the name of the book whose solutions u want i will reply bak to u === Subject: Papoulis Solution Manual can anyone help me? I need the solution manual to the book of randon variables and stochastic of Papoulis... but the manual that I found on the net, the chapters 12,13,15 and 16 are missing... does anyone have the solutions for this chapters?? Marcos === Subject: Re: Int[-oo..oo] e^-|x| ' I hadn't thought of that but what still bothers me is if e^(-|x|) always = e^(-x) then why integrate e^(x)e^(iax) on the interval int[-oo...0]? Clearly I'm missing something. Saying that e^(-|x|)=e^(-x) is the same as saying that -|x| = -x. I feel it is clear why this is not so. xizar === Subject: Re: Int[-oo..oo] e^-|x| I hadn't thought of that but what still bothers me is if e^(-|x|) always = e^(-x) then why integrate e^(x)e^(iax) on the interval int[-oo...0]? Clearly I'm missing something. But e^(-|x|) *doesn't* always = e^(-x), and that is the point. |x| is frequently written as a two-piece formula: |x| = x if x >= 0 and -x if x < 0. So -|x| = -x if x >= 0 and x if x < 0 so e^(-|x|) = e^(-x) if x >= 0 and e^x if x < 0. It doesn't equal e^(-x) if x < 0. Try x = -1 if you still don't see it. You might note this function never gets greater than 1. --Lynn === Subject: Re: List of solutions manual (thousands) Hello. Can you send me Computer Networking: A Top-Down Approach Featuring the Internet (3rd Ed., James F. Kurose & Keith W. Ross) ? please. ArkangelArctico(at)gmail(dot)com === Subject: Re: Probability Random Variables and Stochastic Processes Solutions Manual, Papoulis http://www.ebookee.com/Probability-Random-Variables-and-Stochastic-Processes -4th-Edition-with-Solution-Manual_136898.html === Subject: Need help... It has been several years since I have been in school, and I have recently been give a problem: reduce (a^3+b^3)/(a+b)^3 to its simplest form. The book shows a solution of (a^2-ab+b^2)/(a+b)^2, but does not explain the solution process. Can anyone offer insight? === Subject: Re: Need help... in alt.math.undergrad: > It has been several years since I have been in school, and > I have recently been give a problem: reduce > (a^3+b^3)/(a+b)^3 to its simplest form. The book shows a > solution of (a^2-ab+b^2)/(a+b)^2, but does not explain > the solution process. Can anyone offer insight? One of the standard factorizations is a^3 + b^3 = (a + b)(a^2 - ab + b^2); you can check it by multiplying out the righthand side. Once the numerator has been factored in this way, one factor of a + b in the denominator gives the stated result. Note that you could have deduced this from the answer given by the book. The book's claim is that (a^3 + b^3)/(a + b)^3 = (a^2 - ab + b^2)/(a + b)^2. This means that if you put the fractions over a common denominator, they must have the same numerator. The simplest common denominator is (a + b)^3. The first fraction already has this denominator, and to put the second -- the book's answer -- over it requires multiplying by (a + b)/(a + b). Thus, the book's claim is equivalent to the assertion that a^3 + b^3 = (a^2 - ab + b^2)(a + b), which can, as noted above, be checked by direct multiplication. Brian