mm-4359 === Subject: Choosing the choice relation I have become rather disillusioned and confused with the Axiom of Choice today. I have previously understood the reason for the axiom of choice as follows: sometimes, you need to choose elements for a definition and there is no deterministic way to do this in general. The classical example is the proof that any infinite set has a countable subset. We take an infinite set X and choose some x_0 in X. Then we choose an x_1 in Xsetminus{x_0}. Then we choose an x_2 in Xsetminus{x_0, x_1}. We continue like this; at stage k we choose x_k in Xsetminus{x_0, x_1, ..., x_{k-1}}, and this is always possible otherwise X would be finite with size less than or equal to k. The supposed problem is that the arbitrary choice of x_k at each stage is not allowed by the normal axioms of set theory, and so we must introduce the axiom of choice. The usual formulation of the axiom of choice I have seen (c.f. Godel 1940 p7) is that there exists a relation (choice relation) C such that for any non empty set x, there is some y in x, such that (y,x) is an element of C; also, if (z,x) is an element of C then y=z. We can perform the above proof by setting x = X, Xsetminus{x_0} and so on. Therefore we get a unique sequence and we do not have to worry about arbitrary choice. My problem now that I have just realised today is this: the axiom of choice postulates the existence of the choice relation. However, there may be more than one choice relation; in fact intuitively there are many. How do we know which choice relation to choose when we perform the above actions? Surely the non-arbitrary choice of sequence is only dependent on the arbitrary choice of choice relation in the first place? When we performed the countable subset proof above, we should have begun: choose a choice relation C. Then let x_0 be the unique element with (x_0, X) in C. Let x_1 be the unique element with (x_1, Xsetminus{x_0}) in C. However the fact that we have chosen a choice relation seems to negate having one in the first place. Is my logic flawed, or is there some way around this? Help would be much appreciated. === Subject: Re: Choosing the choice relation > I have become rather disillusioned and confused with the Axiom of Choice > today. > > I have previously understood the reason for the axiom of choice as > follows: sometimes, you need to choose elements for a definition and there > is no deterministic way to do this in general. The classical example is > the proof that any infinite set has a countable subset. We take an > infinite set X and choose some x_0 in X. Then we choose an x_1 in > Xsetminus{x_0}. Then we choose an x_2 in Xsetminus{x_0, x_1}. We > continue like this; at stage k we choose x_k in Xsetminus{x_0, x_1, ..., > x_{k-1}}, and this is always possible otherwise X would be finite with > size less than or equal to k. The supposed problem is that the arbitrary > choice of x_k at each stage is not allowed by the normal axioms of set > theory, and so we must introduce the axiom of choice. > > The usual formulation of the axiom of choice I have seen (c.f. Godel 1940 > p7) is that there exists a relation (choice relation) C such that for > any non empty set x, there is some y in x, such that (y,x) is an element > of C; also, if (z,x) is an element of C then y=z. We can perform the above > proof by setting x = X, Xsetminus{x_0} and so on. Therefore we get a > unique sequence and we do not have to worry about arbitrary choice. > > My problem now that I have just realised today is this: the axiom of > choice postulates the existence of the choice relation. However, there may > be more than one choice relation; in fact intuitively there are many. How > do we know which choice relation to choose when we perform the above > actions? Surely the non-arbitrary choice of sequence is only dependent on > the arbitrary choice of choice relation in the first place? > > When we performed the countable subset proof above, we should have begun: > choose a choice relation C. Then let x_0 be the unique element with (x_0, > X) in C. Let x_1 be the unique element with (x_1, Xsetminus{x_0}) in > C. However the fact that we have chosen a choice relation seems to negate > having one in the first place. Is my logic flawed, or is there some way > around this? Help would be much appreciated. The thing about the axiom of choice is that it is really only necessary for infinite sets; finite choice IS part of logic. Any time you have a statement There exists x such that P(x) for some proposition P, you are allowed to specify x and eliminate the quantifier. You can do this for finitely many such statments, since they can all be conjoined into a single statement. What the axiom of choice gives you is the ability to specify an arbitrary set of existentials. This axiom is, itself, an existential statement; the usual formulation is: For any set X whose elements are nonempty sets, there exists a subset C of X x Union(X) such that: 1. Every element S of X has some (S, t) in C; 2. If (S, t) is in C, then t is in S; 3. If (S, t) and (S, u) are in C, then t = u. C can also be regarded as a choice function because of property 3. To use the axiom, one employs ordinary specification to get some choice function C, and then uses C to dole out choices for every element in the family X. It is true that the choice of C is arbitrary, but it is but a single choice, which is okay. In fact, this arbitrariness is what makes the use of the axiom of choice nonconstructive. It's not that the axiom of choice eliminates arbitrary choices of elements; it actually enables that choice in the first place by reducing very large enterprises to a single arbitrary choice which is allowed under the rules of logical inference. To connect with your specific example: the problem with constructing a sequence x_0, x_1 in X setminus {x_0}, x_2, ... is not that at each stage the choice is arbitrary; the problem is that you have infinitely many stages, and ALL of them are arbitrary. It would be okay, for example, if you just wanted ONE x_0 in X, to say pick any one. That isn't the axiom of choice; it's just the fact that X is nonempty. Just like you don't worry, when picking an arbitrary element of a single set, that it isn't deterministic, you don't worry when using the axiom of choice that the particular choice function is arbitrary. It's the same thing, in fact. Note that the version of the axiom that you've stated is called global choice and brings in some set-theoretic technicalities having to do with proper classes (C can't be a SET, for example). In practice one always works in some kind of a universe so this isn't an issue. -- Ryan Reich ryan.reich@gmail.com === Subject: Re: Choosing the choice relation > >> I have become rather disillusioned and confused with the Axiom of Choice >> today. makes the same point I did. -- Ryan Reich ryan.reich@gmail.com === Subject: An elementary example of deny... i received some emails of people claiming there is no integer polynomial that has a deniel in terms of other integer polynomials... ( apart from the trivial 2n and 2n+1 ) i will give another elementary example : deny 3n = (3a+1)(3b+2)(3c+2) and vice versa of course. tommy1729 === Subject: Re: An elementary example of deny... >i received some emails of people claiming there is no integer polynomial that has a deniel in terms of other integer polynomials... ( apart from the trivial 2n and 2n+1 ) i will give another elementary example : deny 3n = (3a+1)(3b+2)(3c+2) That doesn't work. The right side is always congruent to 1 mod 3, so misses all the numbers congruent to 2 mod 3. quasi === Subject: Re: An elementary example of deny... >i received some emails of people claiming there is no integer polynomial that has a deniel in terms of other integer polynomials... >>( apart from the trivial 2n and 2n+1 ) >>i will give another elementary example : >>deny 3n = (3a+1)(3b+2)(3c+2) That doesn't work. The right side is always congruent to 1 mod 3, so misses all the >numbers congruent to 2 mod 3. But the following function is valid version of deny 3n ... f(x,y) = x^2 + 3y + 1 Stated more precisely: If f(x,y) = x^2 + 3y + 1, then f(Z^2) = {a in Z | a = 1 or 2 mod 3}. proof: Let A = {a in Z | a = 1 or 2 mod 3}. First we show f(Z^2) is a subset of A. Let x,y be integers. Then x^2 = 0 or 1 mod 3, hence f(x,y) = 1 or 2 mod 3, so f(Z^2) is a subset of A. Next we show A is a subset of f(Z^2). Let a be an element of A. If a = 1 mod 3, then a = 3b + 1 for some integer b, hence f(0,b) = a. If a = 2 mod 3, then a = 3b + 2 for some integer b, hence f(1,b) = a. Either way, a is in f(Z^2), hence, A is a subset of f(Z^2). Therefore f(Z^2) = A, as claimed. Equivalently, in tommy-speak, x^2 + 3y + 1 is deny 3n. quasi === <46cdb987.6203293@news.telus.net> <46d07264$0$31919$4c368faf@roadrunner.com> <46d0b87f.1988497@news.telus.net> As Milty Friedman might reason, if monied interests pay outspoken market economists to dodge issues fundamental to markets, next thing you know, you have a lot of outspoken market economists dodging issues fundamental to markets. > Peter, please don't feel our little election cycle troll. Feel? You Repugverts are _disgusting_ and need to go to alt.restroom. Anyway, here's the GOP shill tank letter offer one more time: This is _not_ a wager. It's an offer of a free market free trade: I'll pay $200 for the first letter from an outspoken market economist from Hoover, Heritage, Am. Enterprise, the Chicago School, Cato, von Mises, etc., who answers this question: Does free speech precede each and every free trade. Just get all this on hardcopy: 1. letterhead of GOP think tank 2. the question 3. an answer -- it can be _any_ text whatsoever. 4. the name of the outspoken market economist 5. the signature of the outspoken market economist Scan the letter and email it to BretCahill@aol.com with a mailing address. You'll get a M. O. for $200 in one week. Bret Cahill === Subject: Semidistributivity in complete orthocomplemented pseudocomplemented lattices I'm working on some representation theorems for a structure that seems to be isomorphic to complete orthocomplemented pseudocomplemented lattices. Does anyone know of results on the combination of orthocomplemented and pseudocomplemented lattice (explicitly neither orthomodular nor uniquely complemented)? Orthocomplementation seems to be mainly of interest for quantum mechanics In particular, I'm interested in semidistributivity of those lattices? Are there such lattices that are not semidistributive (obviously they are not distributive). If necessary I can provide more details on the background. Torsten Hahmann === Subject: HELP! Could you please send me the solutions to, Principles and Applications of Electrical Engineering 5th edition please! Will === Subject: Re: help! <24584278.1188570474196.JavaMail.jakarta@nitrogen.mathf orum.org>, > Greg, Michael, et al., > > > message > .com... > How about a new group called > sci.math.solutionmanuals? It seems that we are > getting quite a large number of requests these > days... > > Yes ... welcome to the new school year. > > With any luck some big publisher will take note > of this, track him down, and have him hauled into > court. > > Several have. > > Or rather, several publishers have taken the first step, at least. > > Or rather, anti-piracy lawyers from several publishers -- and a few textbook authors, themselves -- have noticed discussion archives such as the Forum's, and asked us to confirm that we do not condone the activity captured on our sites. > Now that I have your attention please modify your nntp injection program to set line lengths to less than eighty characters, and to include more than one item in -- Michael Press === Subject: Re: help! Hi again, Michael, > Now that I have your attention please modify your nntp > injection program to set line lengths to less than > eighty characters, and to include more than one item Appreciatively, -- Richard Tchen The Math Forum @ Drexel 3210 Cherry Street Philadelphia, PA 19104-2713 tel1: (215)895-1787 tel2: (800)756-7823 facs: (215)895-2964 The Math Forum @ Drexel (http://mathforum.org/) is a research and educational enterprise of the Drexel School of Education (http://drexel.edu/soe/). === Subject: Re: HELP! > Could you please send me the solutions to, Principles and > Applications of Electrical Engineering 5th edition please! Will > no. === Subject: Re: Solving this equation - what a headache! <6192347.1188939857212.JavaMail.jakarta@nitrogen.mathforum.org>, > does anyone know how to solve the following equation to derive x: > > e = x * (y + (z / (1 + (k * x))) > x = ?????? > > i'm having such a big headache over this. > > i have the values of the variables e/k/y/z, but need to formulate it into x = > ???? > > any help is appreciated! If you multiply both sides of the equation by '1 + k * y' and simplify, you will get an equations which is quadratic in x. === Subject: Re: syllogism in set (builder) notation Just curious... Are classical syllogisms still widely taught in > philosophy and other university departments? > I don't know. But you will find a short treatment of syllogisms in many textbooks about (modern) formal logic. Or have they been largely replaced by predicate calculus (FOL) > and set theory? > Well, you can treat (the formal theory of) syllogisms quite nicely in nothing but a small subset of FOPL.) See E. J. Lemmon, Beginning Logic for example. F. -- E-mail: infosimple-linede === Subject: Re: No Identity Bijection for Omega > There is no identity bijection for Omega. When it comes to the various so-called cranks, I usually try to find some axiomatization that may suit their intuitions better. But the standard mathematicians have criticized me for trying to find someone like the OP (Russell Easterly or RE), who doesn't believe in the existence of bijections between any given set and itself. Perhaps what RE wants is for something other than bijections to be the basis of determining set size or cardinality. Then he would be better served by arguing that way, rather than disprove the existence of an identity bijection. The remainder of RE's post exploits the fact that omega, unlike the finite nonzero ordinals, is not the successor of any ordinal, and so the successor function fails to be onto. Of course, this doesn't actually prove what RE wants it to prove at all. Notice that in the hyperreals, it turns out that every positive hypernatural has a predecessor, so perhaps RE's problem is omega's lack of a predecessor, not its lack of an identity bijection. === Subject: Re: No Identity Bijection for Omega > When it comes to the various so-called cranks, > I usually try to find some axiomatization that > may suit their intuitions better. What axiomatizations have you found? MoeBlee === Subject: Re: No Identity Bijection for Omega > When it comes to the various so-called cranks, >> I usually try to find some axiomatization that >> may suit their intuitions better. What axiomatizations have you found? MoeBlee > ZF minus the Axiom of Infinity seems to work for most of them. I have never seen a crank post on finite sets. === Subject: Re: No Identity Bijection for Omega >> When it comes to the various so-called cranks, >> I usually try to find some axiomatization that >> may suit their intuitions better. > What axiomatizations have you found? > Moreover: Did they (!) agree that this is what they (!) want? (Anyone of them?) F. -- E-mail: infosimple-linede === Subject: Re: No Identity Bijection for Omega > Let B be the identity bijection of w+1 = w U {w} . > B is an ordered set of ordered pairs. B = < <0,0>, <1,1>, <2,2>, <3,3>, ..., No, if B is the identity function on wu{w}, then B = {<0 0> <1 1> ... }. I.e., curly braces on the outside. > For each element, B_i, let C_i be the set of > all B_i that appear in the bijection before B_i. Then C_i = i = B_i. > Let C_i also includes B_i. (Skeptics - this is the one to attack.) Then C_i is not as you first described, but rather: C_i = i+1. > Let D_i be the set that C_i is a bijection for. > (D_i is the set of all natural numbers that appear in C_i). What does that appear in mean? If you mean 'that are elements of', then D_i = C_i. > B_0 = <0,0 B_1 = <1,1 B_2 = <2,2 ... B_w = That is NOT the identity function on wu{w}. So, now we see that B is not as you first described, but rather B = {<0 <0 0>> <1 <1 1>> ... >}. Wow, why are you still not competent to just say what you mean from the start? You've been doing your little refute set theory act for how many years now? And you still don't know how to make utterly simple formulations. > C_0 = < <0,0 C_1 = < <0,0>, <1,1 C_2 = < <0,0>, <1,1>, <2,2 ... > C_w = < <0,0>, <1,1>, <2,2>, ..., Again, you're using '< >' on the outside rather than curly braces. That would be okay, except when you get to C_w, we have to know what you mean by an infinite < >. Ordinarily, it would be a function as opposed to a plain n-tuple. Or maybe you mean curly braces: C_0 = {<0 0>} etc. So, we have C_i = range(B restricted to i+1). > D_0 = {0} > D_1 = {0,1} > D_2 = {0,1,2} > ... > D_w = {0,1,2,...,w} Then D_i = iu{i}. You've taken us from B to C to D, just to get the successor function! That's ridiculous! > Assume D_k = w for some k. That's an assumption that contradicts Z set theory. So what is your point? By assuming something that contradicts Z set theory, you're going to derive a contradiction with Z set theory? Wow, that's deep. > D_k = {0,1,2,...,k} and k must be the largest natural number. Well, since you assumed a contradiction. Anything follows. > Therefore, no D_k = w. You didn't need to tell us that, but now that you have, are we still under the assumption that for some k, we have D_k = w? Looking down below, I don't see anything that came from your assumption. It was pointless. > Assume there exists a C_k that is a bijection for w. Again, that assumption contradicts Z set theory. So what's the point? > Then there exists D_k. There exists D_k is not even a statement. There exists a D_k SUCH THAT WHAT? Anyway, since you've taken another assumption that contradicts Z set theory, you can, with your assumption and Z set theory, derive any statement whatsoever in the language of Z set theory. > Assume there exists an identity bijection, A, for w. Do you mean a bijection A from w onto w? In Z set theory we prove there does exist such a bijection. > A must be an proper subset of B. No. NO member of bijection from w onto w is a member of B. A bijection from w onto w looks like: {<0 0> <1 1> ... } No member of that is of the form >. So maybe you have in mind a bijection A such that A is from w onto { | i in w} and such that for each i in w, we have A_i = . Then, yes, A is a proper subset of B. However, if we are still under either of your previous assumptions that contradict Z set theory, then anything follows with Z set theory and that assumption. > A must also be an initial segment of B. Okay, with 'initial segment' suitably defined. > If A is an initial segment of B then A = C_k for some k. That's a complete non sequitur. You've shown no logic, no basis, whatsoever. However, if we are still under either of your previous assumptions that contradict Z set theory, then anything follows with Z set theory and that assumption. And you're formulations of C and A AGAIN don't match themselves. > There is no identity bijection for Omega. You've not shown any such thing. What you did is this: (1) Take a long, unnecessary windup just to get to D, which is the successor function on wu{w}. (2) Make two separate assumptions that are contradictions with Z set theory. Neither of those assumptions leading to anything at all. (3) Make self-contradictory definitions. (4) End by drawing a complete non sequitur. MoeBlee === Subject: Concavity of functions of vectors Let u and v be two vectors belonging to R^n, and define f(u,v) as f:R^n x R^n --> R. My question is: can the strict concavity of f(u,v) be analyzed through the Hessian, treating u and v as scalars (in spite of the fact that they are vectors) in the calculation of the Hessian? For instance, how would you analyze the strict concavity of f(u,v) := -u^4 - v^4, u,v in R^n through the Hessian? Paul === Subject: Re: Concavity of functions of vectors > > Let u and v be two vectors belonging to R^n, and define f(u,v) as > f:R^n x R^n --> R. My question is: > > can the strict concavity of f(u,v) be analyzed through the Hessian, > treating u and v as scalars (in spite of the fact that they are > vectors) in the calculation of the Hessian? > > For instance, how would you analyze the strict concavity of > > f(u,v) := -u^4 - v^4, u,v in R^n > > through the Hessian? > What do you mean by u^4 if u is in R^n? -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Concavity of functions of vectors On Sep 5, 3:15 am, Robert Israel Let u and v be two vectors belonging to R^n, and define f(u,v) as > f:R^n x R^n --> R. My question is: can the strict concavity of f(u,v) be analyzed through the Hessian, > treating u and v as scalars (in spite of the fact that they are > vectors) in the calculation of the Hessian? For instance, how would you analyze the strict concavity of f(u,v) := -u^4 - v^4, u,v in R^n through the Hessian? What do you mean by u^4 if u is in R^n? Sorry, Robert. I should have written f(u,v) := -|u|^4 - |v|^4, u,v in R^n, where |.| means the Euclidean norm. Paul === Subject: online business education Children normally start primary school at the age of four or five, but many schools now have a reception year for four year olds. Children normally leave at the age of 11, moving on to secondary school. Most state schools admit both boys and girls, though some are single-sex. ------------------------------------------------------ online business education http://www.education.tohelpyou.info/education.aspx ------------------------------------------------------ === Subject: Re: #28F perhaps an easy path to a proof that (pi) and (e) are the only two independent Transcendental Numbers Re: ATOM TOTALITY (Atom Universe) THEORY REPLACES BIG BANG THEORY IN PHYSICS > One last time. Above, along time ago, you said: > > K [a constant] can be chosen so that the difference between the > boiling point and freezing point of water is 100 degree. > > The question I have been asking all along is: Why would you choose K > in this particular way? Why not choose K so that the difference > between the boiling point and freezing point of water is 10 degrees? Because temperature had been defined in terms of the material properties of water. After that temperature was defined independently of material properties of any particular substance, but it is convenient to have a rough correspondence with the previous definition. What is the use of making it 10 degrees? -- Michael Press === Subject: Re: #28F perhaps an easy path to a proof that (pi) and (e) are the only two independent Transcendental Numbers Re: ATOM TOTALITY (Atom Universe) THEORY REPLACES BIG BANG THEORY IN PHYSICS One last time. Above, along time ago, you said: K [a constant] can be chosen so that the difference between the > boiling point and freezing point of water is 100 degree. The question I have been asking all along is: Why would you choose K > in this particular way? Why not choose K so that the difference > between the boiling point and freezing point of water is 10 degrees? Because temperature had been defined in terms of the > material properties of water. After that temperature > was defined independently of material properties of any > particular substance, but it is convenient to have a > rough correspondence with the previous definition. It's convenient, but not required. --- Christopher Heckman > What is the use of making it 10 degrees? === Subject: need help regarding string algorithm hello everybody i am trying to solve this problem http://acm.uva.es/p/v9/902.html but getting TLE (time limit exceed ) becoz the input string length could be 10^6 . now what i did to solve this problem . for each substring of given lengh find all possible occurence of substring in the string . i used KMP algroithm but still getting . i think my algorithm is order of n^2. plz some body suggest me how to solve this problem. thnkx . === Subject: Re: need help regarding string algorithm > hello everybody i am trying to solve this problem http://acm.uva.es/p/v9/902.html > but getting TLE (time limit exceed ) becoz the input string length > could be 10^6 . > now what i did to solve this problem . > for each substring of given lengh > find all possible occurence of substring in the string . > > i used KMP algroithm but still getting . i think my algorithm is > order of n^2. > plz some body suggest me how to solve this problem. > thnkx . > IIUC, the KMP algorithm is used to find the occurrences of a known word in the source text, not tabulate the frequencies of unknown words. (P.S. are you allowed help on ACM problems? Then again, the age suggests that this is only practice...) Here is the sketch of an O(N*M) method (N is password length, M is source text length): For each possible starting point, add one to the frequency of the N-substring at that point. The frequency table might be best stored as a hash table. BTW, your algorithm is roughly O[(26^N)*M], not very efficient at all... === Subject: lookin for a book i am looking for solution manual for differential equations and linear algebra by stephen w.goode and scott a.annin with isbn 0130457949. If u have this then e mail me or call me asap9722590593. === Subject: Re: math typesetting freeware for windows > Are there such thing? Latex seems to be popular choice, except I don't > know how to use it. > > The windows office word has an equation editor, but after a while it > keeps asking if I want to buy some stupid program. > Several years ago I toyed with the idea of LaTeX but chickened out and bought the payware program that M$ Equation Editor is based on. I can't remember what it is called, and I have long ago moved the manual from the bookshelf by my PC, so I can't look at that to remind me. A couple of years after that I learnt LaTeX and have used that ever since. Bite the bullet! It's a bit of a pain to start with but I think most people who genuinely need to typeset mathematics will pick up the essentials very quickly. There are excellent online tutorials and books, and an active Usenet group. It's worth buying a proper book though. Mine is sitting where the MathType - I knew it would come back to me! - manual used to live. Only one warning - the guy who persuaded me that LaTeX isn't that difficult next had me writing PostScript. It's a slippery slope that probably ends in APL. :-) Good luck! Mark Atherton === Subject: Re: Algebra with factor ring... >> Hello sir~ >> >> For field Z_7, >> Find |Z_7[x] / |. (number of elements) >> >> ------------------------------------------- >> If x^5 + x + 1 is irreducible, easy by simple extension. >> >> But this is not irreducible over Z_7. > > So Z_7[x] / is just a ring, not a field. > Doesn't matter. > >> >> Does there exist how to solve this problem ? >> or is this mistake of problem maker ? >> >> so, I need your advice. > > Even with a reducible polynomial such as x^5 + x +1, > the elements of |Z_7[x] / | > can be enumerated by representatives of the from > a x^4 + b x^3 + c x^2 + d x + e > for a,b,c,d,e in Z_7. > That makes 7^5 elements unless two of them are equivalent. > But *can* the difference of two such representatives be > a multiple of x^5 + x +1? No by degree argument. So it seems that only the term of the highest power matters. -- -kira === Subject: Re: JSH: What is surrogate factoring? Once more. <329qd3dsesavmfj6oiii947h07k3gsrs5o@4ax.comCould you walk me through, step by step, >how to factor 400 using your method? >Please. Disclaimer - I am not James. To factor 400 target T = 400 The next step is to pick k and n. I shall pick k = 1, > n = 6, 7, 8, 9 ... k remains fixed while n increments if needed. Using the current k and n I calculate the surrogate S: S = 2 * k^2 + n * T so S = 2 * 1^2 + 6 * 400 = 24002. Now factorise S to find all its factors (not just positive, prime or > proper factors): Factors of S = 1, -1, 2, -2, 11, -11, 1091, -1091, 12001, -12001, > 24002, -24002. For each factor f of S we try GCD(f - k, T) to see if any useful > factor of T emerges. GCD( 1 - 1, 400) = 0 > GCD(-1 - 1, 400) = 2. Bingo. We have found that 2 is a factor of 400. If all the factors of the first S had failed to produce a factor of T > then we would have incremented n to 7 and calculated a new value for > S. To derive his method James starts by expressing T as the difference of > two squares: x^2 - y^2 = 0 mod T (1) he then introduces the non-zero integer k such that: k = 2x mod T (2) combining these two equations gets him to: (x+k)^2 - y^2 = 2 * k^2 mod T which is another difference of squares. Introducing n (which is also not zero) to remove the mod gives: (x+k)^2 - y^2 = 2 * k^2 + nT The right hand side of this equation is the surrogate S = 2*k^2 + nT We can now factor the surrogate into f1 and f2, S = f1 * f2, giving: f1 = (x+k) + y > f2 = (x+k) - y from these two we can get: x + y = f1 - k > x - y = f2 - k going back to (1) the original difference of squares allows us to say: (x + y)(x - y) = 0 mod T or (f1 - k)(f2 - k) = 0 mod T Hence for each factor f of S GCD(f - k, T) might show up a proper > factor of T. James is currently working on limiting the range of choices allowed > for k and n. In its current form the method allows too many choices > for k and n so there are too many failed GCDs before finding a proper > factor. With realistic sized numbers, most of the GCDs give either 1 > or T which do not find a factor. I am sure that James will correct me if I have made an error. rossum I think that was a good overview. I didn't notice any problems with it. Readers should note that implementations of surrogate factoring have been done by me and others so we have watched the method factor, and also watched it NOT factor, a lot. I say that to remove the trivial objections that reduce down to--that does not work. As it does work but to my knowledge no one has used it against really big numbers with success. If you think you're a hotshot with mathematics, then analyze the equations and give a mathematical argument predicting the likelihood of success, and I don't care if you think you prove that it does not work. They are simple as you have the congruence of squares: x^2 = y^2 mod T and just one more relation which is k = 2x mod T, so the start should be easy, eh? I have done my own analysis and put much of it on my math blog, so I don't expect anything new from what you might do, but who knows? Currently I am one of the world's experts on surrogate factoring, but hey, I invented it, so I should be. James Harris === Subject: Re: JSH: What is surrogate factoring? Once more. <329qd3dsesavmfj6oiii947h07k3gsrs5o@4ax.com >Could you walk me through, step by step, >how to factor 400 using your method? >Please. Disclaimer - I am not James. To factor 400 target T = 400 The next step is to pick k and n. I shall pick k = 1, > n = 6, 7, 8, 9 ... k remains fixed while n increments if needed. Using the current k and n I calculate the surrogate S: S = 2 * k^2 + n * T so S = 2 * 1^2 + 6 * 400 = 24002. Now factorise S to find all its factors (not just positive, prime or > proper factors): Factors of S = 1, -1, 2, -2, 11, -11, 1091, -1091, 12001, -12001, > 24002, -24002. For each factor f of S we try GCD(f - k, T) to see if any useful > factor of T emerges. GCD( 1 - 1, 400) = 0 > GCD(-1 - 1, 400) = 2. Bingo. We have found that 2 is a factor of 400. If all the factors of the first S had failed to produce a factor of T > then we would have incremented n to 7 and calculated a new value for > S. To derive his method James starts by expressing T as the difference of > two squares: x^2 - y^2 = 0 mod T (1) he then introduces the non-zero integer k such that: k = 2x mod T (2) combining these two equations gets him to: (x+k)^2 - y^2 = 2 * k^2 mod T which is another difference of squares. Introducing n (which is also not zero) to remove the mod gives: (x+k)^2 - y^2 = 2 * k^2 + nT The right hand side of this equation is the surrogate S = 2*k^2 + nT We can now factor the surrogate into f1 and f2, S = f1 * f2, giving: f1 = (x+k) + y > f2 = (x+k) - y from these two we can get: x + y = f1 - k > x - y = f2 - k going back to (1) the original difference of squares allows us to say: (x + y)(x - y) = 0 mod T or (f1 - k)(f2 - k) = 0 mod T Hence for each factor f of S GCD(f - k, T) might show up a proper > factor of T. James is currently working on limiting the range of choices allowed > for k and n. In its current form the method allows too many choices > for k and n so there are too many failed GCDs before finding a proper > factor. With realistic sized numbers, most of the GCDs give either 1 > or T which do not find a factor. I am sure that James will correct me if I have made an error. rossum I think that was a good overview. I didn't notice any problems with > it. Readers should note that implementations of surrogate factoring have > been done by me and others so we have watched the method factor, and > also watched it NOT factor, a lot. I say that to remove the trivial objections that reduce down to--that > does not work. As it does work but to my knowledge no one has used it against really > big numbers with success. If you think you're a hotshot with mathematics, then analyze the > equations and give a mathematical argument predicting the likelihood > of success, and I don't care if you think you prove that it does not > work. They are simple as you have the congruence of squares: x^2 = y^2 mod T and just one more relation which is k = 2x mod T, so the start should > be easy, eh? I have done my own analysis and put much of it on my math blog, so I > don't expect anything new from what you might do, but who knows? Currently I am one of the world's experts on surrogate factoring, but > hey, I invented it, so I should be. James Harris What are X and Y, and how do you use them in your method? Could you please answer my question, as to how to factor 400, step by step, using your method? Or how about even something like 4302394? === Subject: Re: JSH: What is surrogate factoring? Once more. <329qd3dsesavmfj6oiii947h07k3gsrs5o@4ax.com >Could you walk me through, step by step, >how to factor 400 using your method? >Please. Disclaimer - I am not James. To factor 400 target T = 400 The next step is to pick k and n. I shall pick k = 1, > n = 6, 7, 8, 9 ... k remains fixed while n increments if needed. Using the current k and n I calculate the surrogate S: S = 2 * k^2 + n * T so S = 2 * 1^2 + 6 * 400 = 24002. Now factorise S to find all its factors (not just positive, prime or > proper factors): Factors of S = 1, -1, 2, -2, 11, -11, 1091, -1091, 12001, -12001, > 24002, -24002. For each factor f of S we try GCD(f - k, T) to see if any useful > factor of T emerges. GCD( 1 - 1, 400) = 0 > GCD(-1 - 1, 400) = 2. Bingo. We have found that 2 is a factor of 400. If all the factors of the first S had failed to produce a factor of T > then we would have incremented n to 7 and calculated a new value for > S. To derive his method James starts by expressing T as the difference of > two squares: x^2 - y^2 = 0 mod T (1) he then introduces the non-zero integer k such that: k = 2x mod T (2) combining these two equations gets him to: (x+k)^2 - y^2 = 2 * k^2 mod T which is another difference of squares. Introducing n (which is also not zero) to remove the mod gives: (x+k)^2 - y^2 = 2 * k^2 + nT The right hand side of this equation is the surrogate S = 2*k^2 + nT We can now factor the surrogate into f1 and f2, S = f1 * f2, giving: f1 = (x+k) + y > f2 = (x+k) - y from these two we can get: x + y = f1 - k > x - y = f2 - k going back to (1) the original difference of squares allows us to say: (x + y)(x - y) = 0 mod T or (f1 - k)(f2 - k) = 0 mod T Hence for each factor f of S GCD(f - k, T) might show up a proper > factor of T. James is currently working on limiting the range of choices allowed > for k and n. In its current form the method allows too many choices > for k and n so there are too many failed GCDs before finding a proper > factor. With realistic sized numbers, most of the GCDs give either 1 > or T which do not find a factor. I am sure that James will correct me if I have made an error. rossum I think that was a good overview. I didn't notice any problems with > it. Readers should note that implementations of surrogate factoring have > been done by me and others so we have watched the method factor, and > also watched it NOT factor, a lot. I say that to remove the trivial objections that reduce down to--that > does not work. As it does work but to my knowledge no one has used it against really > big numbers with success. If you think you're a hotshot with mathematics, then analyze the > equations and give a mathematical argument predicting the likelihood > of success, and I don't care if you think you prove that it does not > work. They are simple as you have the congruence of squares: x^2 = y^2 mod T and just one more relation which is k = 2x mod T, so the start should > be easy, eh? I have done my own analysis and put much of it on my math blog, so I > don't expect anything new from what you might do, but who knows? Currently I am one of the world's experts on surrogate factoring, but > hey, I invented it, so I should be. James Harris- Hide quoted text - - Show quoted text - I have a souped up version of Surrogate Factoring working. It still thinks that a very small class of composites are prime, but I think I can fix that. Also needed is some more output information about k, n, S, and the factors of S tried. Programming language is VB Script running under Windows XP. speed < 1 second Random Example T=7333679 = 1777 * 4127 Number of GCD's = 73 Random Example T = 4264147 = 853 * 4999 Number of GDC's = 356 I've got this pig flying. Now I just need to put some lipstick on it and it'll be ready to go. At that point, I'll post the code. Bass Ackwards into the Unknown, Enrico === Subject: Re: JSH: What is surrogate factoring? Once more. <329qd3dsesavmfj6oiii947h07k3gsrs5o@4ax.com >Could you walk me through, step by step, >how to factor 400 using your method? >Please. > Disclaimer - I am not James. > To factor 400 > target T = 400 > The next step is to pick k and n. I shall pick > k = 1, > n = 6, 7, 8, 9 ... > k remains fixed while n increments if needed. > Using the current k and n I calculate the surrogate S: > S = 2 * k^2 + n * T > so S = 2 * 1^2 + 6 * 400 = 24002. > Now factorise S to find all its factors (not just positive, prime or > proper factors): > Factors of S = 1, -1, 2, -2, 11, -11, 1091, -1091, 12001, -12001, > 24002, -24002. > For each factor f of S we try GCD(f - k, T) to see if any useful > factor of T emerges. > GCD( 1 - 1, 400) = 0 > GCD(-1 - 1, 400) = 2. Bingo. > We have found that 2 is a factor of 400. > If all the factors of the first S had failed to produce a factor of T > then we would have incremented n to 7 and calculated a new value for > S. > To derive his method James starts by expressing T as the difference of > two squares: > x^2 - y^2 = 0 mod T (1) > he then introduces the non-zero integer k such that: > k = 2x mod T (2) > combining these two equations gets him to: > (x+k)^2 - y^2 = 2 * k^2 mod T > which is another difference of squares. > Introducing n (which is also not zero) to remove the mod gives: > (x+k)^2 - y^2 = 2 * k^2 + nT > The right hand side of this equation is the surrogate S = 2*k^2 + nT > We can now factor the surrogate into f1 and f2, S = f1 * f2, giving: > f1 = (x+k) + y > f2 = (x+k) - y > from these two we can get: > x + y = f1 - k > x - y = f2 - k > going back to (1) the original difference of squares allows us to say: > (x + y)(x - y) = 0 mod T > or > (f1 - k)(f2 - k) = 0 mod T > Hence for each factor f of S GCD(f - k, T) might show up a proper > factor of T. > James is currently working on limiting the range of choices allowed > for k and n. In its current form the method allows too many choices > for k and n so there are too many failed GCDs before finding a proper > factor. With realistic sized numbers, most of the GCDs give either 1 > or T which do not find a factor. > I am sure that James will correct me if I have made an error. > rossum I think that was a good overview. I didn't notice any problems with > it. Readers should note that implementations of surrogate factoring have > been done by me and others so we have watched the method factor, and > also watched it NOT factor, a lot. I say that to remove the trivial objections that reduce down to--that > does not work. As it does work but to my knowledge no one has used it against really > big numbers with success. If you think you're a hotshot with mathematics, then analyze the > equations and give a mathematical argument predicting the likelihood > of success, and I don't care if you think you prove that it does not > work. They are simple as you have the congruence of squares: x^2 = y^2 mod T and just one more relation which is k = 2x mod T, so the start should > be easy, eh? I have done my own analysis and put much of it on my math blog, so I > don't expect anything new from what you might do, but who knows? Currently I am one of the world's experts on surrogate factoring, but > hey, I invented it, so I should be. James Harris- Hide quoted text - - Show quoted text - I have a souped up version of Surrogate Factoring > working. It still thinks that a very small class > of composites are prime, but I think I can fix that. > Also needed is some more output information about > k, n, S, and the factors of S tried. Programming language is VB Script > running under Windows XP. > speed < 1 second > Random Example T=7333679 = 1777 * 4127 > Number of GCD's = 73 > Random Example T = 4264147 = 853 * 4999 > Number of GDC's = 356 I've got this pig flying. Now I just need to > put some lipstick on it and it'll be ready to go. > At that point, I'll post the code. Bass Ackwards into the Unknown, > Enrico Sounds like a good idea. Then James's claims will either stand or fall, *objectively*. === Subject: Re: JSH: What is surrogate factoring? Once more. >> hey, I invented it, so I should be. My dog invents a new theorm on the lawn every day, right out of his arss. your surrogate factoring is dead, but you always knew that. James Harris > === Subject: Re: JSH: What is surrogate factoring? Once more. > On Sep 4, 11:12 am, riderofgiraffes I think you are overlooking what is maybe the >> central unavoidable problem with the Harris idea. >> As with Fermat's algorithm (and Dixon's, and the >> quadratic sieve), he wants >> X^2 = Y^2 mod T, i.e., >> (X + Y)*(X - Y) = 0 mod T. >> So say T = 77. Let k = 1 and n = 2. >> You can't do this. He assumes that k=2x (T) >> *THAT'S* the central, unavoidable problem. I think we may be saying the same thing or closely > related things. He cannot let k = 2X mod T until he > specifies X. What he does in fact is the following. > He FIRST chooses k and n, and lets S = 2k^2 + nT. > Then he factors S as S = F1 * F2, and AFTER THAT, > he chooses X and Y. In the process, he forgets that > he originally wanted X^2 = Y^2 mod T, and in general > the algorithm he specifies does not give him this > crucial equality. So as a rule he does not end up > with a difference of squares which equals a multiple > of T. This largely explains why his method, unlike > that of Dixon or the quadratic sieve method, fails > most of the time. Marcus. a product of a weak mind. JSH may talk big, but is weak intellectually, and a troll. > === Subject: Re: MULTIPLE NOMINATIONS: Re: China freer than USA? :-) pR9MNF=QZrhF$HA44sG?n(RnC{0!e>5EKJK{?&5QI$HfYWdP*zNGgP79xb%0Sch26o G8|k?UC[~)lu]WnopcGdzzVoCsDY?/{MgR9eDs>Y~X},^2Ay|9n9~~1&UUCoDQ|O| eYokL{N-%%x6C_4lKt2,LOOY#A6rAjZ5~[1-s,It[DTBZa1v_2D7;0Yq1rQ{LcW@J2* )f > [malicious crossposts trimmed] > In message <46d791f8$0$16365$88260...@free.teranews.com>, Tom Potter >>> In message <46d58c95$0$16334$88260...@free.teranews.com>, Tom Potter > [...] >>> It's amazing that lying plagiarist Potter's family don't slap some >>> kind >>> of >>> restraining order on him to stop him spreading their pictures and >>> personal information all over his websites. (Apart from the privacy >>> aspect, it might even be libellous - what sane person of good >>> character >>> would want their relationship with TP to be made public? ;-) >>As can be seen by his personal attacks on my family, >>> And once again lying plagiarist Potter demonstrates his inability to >>> comprehend even the simplest of postings. >>> *What* personal attacks on [his] family ? > [snip Potter boiler plate] >>NoteRichard Herring'sstatement: >>what sane person of good character >>would want their relationship with TP to be made public? >>asserts that anyone who >>would want their relationship with TP to be made public >>is not a sane person of good character. > Correct. (nearly - it doesn't assert, it implies.) > Potter appears to be trying to construct a syllogism: the missing premise > (which I neither assert nor imply) is that the people referred to would in > fact want their relationship with TP made public. > So by asserting that premise, it appears to be *Potter* who is making > personal attacks on his own family. Draw your own conclusions. > [snip remaining potterplate] > -- >Richard Herring >>I suggest that anyone who associates withRichard Herring, >>or has a supportive family or business relationship withRichard Herring, >>is a NOT a sane person of good character, >>because as can be seen by many of his posts, >>Richard Herringis a bigoted racist and a dishonest person. >>It's amazing that lying, bigoted, racistRichard Herring'sfamily >>doesn't slap some kind of restraining order on him to stop him >>from disgracing his family, >>and ifRichard Herringis employed, >>it's amazing that lying, bigoted, racistRichard Herring'semployer >>doesn't slap some kind of restraining order on him to stop him >>from disgracing his employer. >>As can be seen, >>by displaying his racist bigotry and dishonesty on the Internet >>it appears to be *Richard Herring* who is making >>a statement about his own family, his own race, and his own religion. >>Of course it may be thatRichard Herring >>is a victim of his religion, and has been brainwashed to >>Jimmy Carter and Mel Gibson folks >>who express views that his religion has brainwashed him suppress. >>I urge all folks to take a look at the following web site >>to see how racist bigots likeRichard Herringand Art Deco operate. >> *ding* >> Good to see you are still trained, TommY. >>Draw your own conclusions. >> Will do. >> Bolo Bullis Foam Duck >> The spoonerism for dumb , for those who've lost more marbles >> than a Chinese Checker factory will ever make. >> Yep, that sounds like Crackpotter, so nominated. >> Seconds? >> Busted Urinal Award >> The reward for the Lamest of the Lame on Usenet. One who clogs and >> stinks up the joint like a busted urinal, you could say. >> Oh yes, that is Crackpotter, too. >> Seconds? >> Tony Sidaway Drama Queen Award >> For those logically consistent persons who leave Usenet for ever, >> rather more often than the Cabal's rules require. It isn't necessary to >> leave Usenet for ever three times in a single week to win this award, >> but it helps. Operatic flouncing of other kinds, including unprovoked >> meltdowns and ridiculous ranting may also lead to a nomination for this >> award. Named after Drama Empress Tony Sidaway. >> Note the ridiculous ranting part -- no one rants like Crackpotter. >> Seconds? >> Joseph Bartlo Pathetic Anal Pineapple Award >> The Jojo Pineapple is awarded every month to the Usenet poster who >> best establishes a pattern of stupidity, idiocy, moronism, silliness, >> cretinism and general kookery in a way exemplified by the eponymous >> founder of this award, Joseph Foamboi Bartlo. >> Anyone who points out TommY's nonsense is automatically a lying, >> bigoted, racist, brainwashed, needs a restraining order, etc. Art Deco raises a good point! Anyone who points out >that Art Deco is a full time, professional, bigoted racist is deserving of the following awards: >kook of the month >Bolo Bullis Foam Duck >Busted Urinal Award >Tony Sidaway Drama Queen Award >Joseph Bartlo Pathetic Anal Pineapple Award >some kind of trained dog award that Art brags about >and various anti science awards. Here is a video of Art Deco in action: >http://www.kooklinks.com/ and here is a web site that lists some of the victims >of Art Deco's handlers: >http://www.kooklinks.com/ > Get some new rants, Crackpotter, these are getting quite worn. -- Official Overseer of Kooks and Saucerheads for alt.astronomy Wee Davie Tholen is a grade-school lamer Trainer and leash holder of: Honest Clockbrain John nightbat fro0tbat of alt.astronomy Tom TommY Crackpotter Potter Classic erroneous presupposition. Others developed websites so that they could have the Last Word, Deco. In the newsgroups, I could counter their lies. --David Tholen === Subject: Re: MULTIPLE NOMINATIONS: Re: China freer than USA? :-) Expires: 6/30/2007 Keywords: x >> [malicious crossposts trimmed] >> In message <46d791f8$0$16365$88260bb3@free.teranews.com>, Tom Potter >>>> In message <46d58c95$0$16334$88260bb3@free.teranews.com>, Tom Potter >>>> [...] >>>> It's amazing that lying plagiarist Potter's family don't slap some >>> kind >>> of >>> restraining order on him to stop him spreading their pictures and >>> personal information all over his websites. (Apart from the privacy >>> aspect, it might even be libellous - what sane person of good >>> character >>> would want their relationship with TP to be made public? ;-) >>>>As can be seen by his personal attacks on my family, >>>> And once again lying plagiarist Potter demonstrates his inability to >>> comprehend even the simplest of postings. >>>> *What* personal attacks on [his] family ? >> [snip Potter boiler plate] >>Note Richard Herring's statement: >>what sane person of good character >>would want their relationship with TP to be made public? >>asserts that anyone who >>would want their relationship with TP to be made public >>is not a sane person of good character. >> Correct. (nearly - it doesn't assert, it implies.) >> Potter appears to be trying to construct a syllogism: the missing premise > (which I neither assert nor imply) is that the people referred to would in > fact want their relationship with TP made public. >> So by asserting that premise, it appears to be *Potter* who is making > personal attacks on his own family. Draw your own conclusions. >> [snip remaining potterplate] >> -- > Richard Herring >>I suggest that anyone who associates with Richard Herring, >>or has a supportive family or business relationship with Richard Herring, >>is a NOT a sane person of good character, >>because as can be seen by many of his posts, >>Richard Herring is a bigoted racist and a dishonest person. >>It's amazing that lying, bigoted, racist Richard Herring's family >>doesn't slap some kind of restraining order on him to stop him >>from disgracing his family, >>and if Richard Herring is employed, >>it's amazing that lying, bigoted, racist Richard Herring's employer >>doesn't slap some kind of restraining order on him to stop him >>from disgracing his employer. >>As can be seen, >>by displaying his racist bigotry and dishonesty on the Internet >>it appears to be *Richard Herring* who is making >>a statement about his own family, his own race, and his own religion. >>Of course it may be that Richard Herring >>is a victim of his religion, and has been brainwashed to >>Jimmy Carter and Mel Gibson folks >>who express views that his religion has brainwashed him suppress. >>I urge all folks to take a look at the following web site >>to see how racist bigots like Richard Herring and Art Deco operate. *ding* Good to see you are still trained, TommY. >>Draw your own conclusions. Will do. Bolo Bullis Foam Duck The spoonerism for dumb , for those who've lost more marbles >than a Chinese Checker factory will ever make. Yep, that sounds like Crackpotter, so nominated. Seconds? >Busted Urinal Award The reward for the Lamest of the Lame on Usenet. One who clogs and >stinks up the joint like a busted urinal, you could say. Oh yes, that is Crackpotter, too. Seconds? >Tony Sidaway Drama Queen Award For those logically consistent persons who leave Usenet for ever, >rather more often than the Cabal's rules require. It isn't necessary to >leave Usenet for ever three times in a single week to win this award, >but it helps. Operatic flouncing of other kinds, including unprovoked >meltdowns and ridiculous ranting may also lead to a nomination for this >award. Named after Drama Empress Tony Sidaway. Note the ridiculous ranting part -- no one rants like Crackpotter. Seconds? >Joseph Bartlo Pathetic Anal Pineapple Award The Jojo Pineapple is awarded every month to the Usenet poster who >best establishes a pattern of stupidity, idiocy, moronism, silliness, >cretinism and general kookery in a way exemplified by the eponymous >founder of this award, Joseph Foamboi Bartlo. Anyone who points out TommY's nonsense is automatically a lying, >bigoted, racist, brainwashed, needs a restraining order, etc. Seconds? I don't see one yet. I so second. 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I'm not so sure your claim is true, but if it is true, my feeling is >that continuity should be enough. >In other words, for every smooth map F from S^1 to R^3 there exists a >>vector v in R^3 such that g(x) = F(x) * v has both 2 global maxima and >>2 global minima. Of course you intended to require the vector v to be nonzero. Also, >it's clear that the vector v can be assumed to be a unit vector. >Here I define a maxima as a point x on S^1 such that >>there is a neighborhood of x where the function is no greater than >>g(x). So g(x) need not be strictly larger then g for other points in >>the neighborhood. I think it should be easy to prove that there's a vector v which gives >2 global minima, and also a vector v which gives 2 global maxima. It's >not clear to me that there must exist a vector v which simultaneously >gives both 2 global minima and 2 global maxima. I just noticed -- you need at least one other restriction. You don't > want a constant map (which would collapse the S^1 to a point). Maybe this is why smoothness is required? --- Christopher Heckman > Also, you didn't insist on a homeomorphic image, so then is any > nonconstant continuous image acceptable? If so, maybe you can dispense > with the image of S^1 and simply allow any compact path connected set > with more than one point. In any case, I would start with the just assuming a continuous image, > rather than requiring smoothness. Alternatively, allow the image to be > piecewise smooth. This allows you to consider the special cases where the image is > piecewise linear (for example, a polygon). Of course, if the image > lies in a plane, then the result is immediate. Thus, the result > automatically holds for triangles. For quadrilaterals, it might hold > -- I can't quite visualize a counterexample, but I'm not sure. If > there is a counterexample, and if a quadrilateral is not enough, then > I suspect a pentagon should suffice, or at worst, a hexagon. On the other hand, if you can prove the claim for all piecewise linear > images, then perhaps a convergence argument could allow a > generalization to all compact path connected sets with more than one > point. quasi === Subject: Re: S^1 in R^3 Can anyone help me prove that every smooth (continuous?) map of S^1 >into R^3 can be rigidly rotated so as to simultaneously have 2 or more >global minima points and 2 or more global maxima points? In other words, for every smooth map F from S^1 to R^3 there exists a >vector v in R^3 such that g(x) = F(x) * v has both 2 global maxima and >2 global minima. Here I define a maxima as a point x on S^1 such that >there is a neighborhood of x where the function is no greater than >g(x). So g(x) need not be strictly larger then g for other points in >the neighborhood. > If you only require continuity, then any continuous surjection > F:S^1 -> S^2 gives a counterexample. (This also shows that it is > possible to have only one maximum in every direction.) For a less abstract example, let g be a bump function supported > on [2/3, 5/6] and then let F:[0,1] -> R^3 be given by F(t) = (cos(4 pi t), sin(4 pi t), g(t)). So F traces the unit circle in the X-Y-plane twice, the second time > with a little bump. Here you can certainly have two maxima, but never > two maxima and two minina simultaneously. RS Hello. extremely helpful, but I think they are not true counter examples. I hope I am not misunderstanding. By the way, your example of the sphere-filling curve was a perfect argument for why continuous map F would be insufficient for my Proposed counter-example (from Robert Sheskey): > For a less abstract example, let g be a bump function supported > on [2/3, 5/6] and then let F:[0,1] -> R^3 be given by F(t) = (cos(4 pi t), sin(4 pi t), g(t)). Let me clarify my question. Remember I am looking for distinct global maxima on the domain of F, S^1 (not the image), so here we can just choose v = [1,0,0] and get that t=0 and t=1/2 both map to [1,0,0] maximizing the dot product with v, and t=1/4 and t=3/4 both map to [-1,0,0] and [-1,0,1] minimizing the dot product with v. So, 2 distinct minima and 2 distinct maxima. Proposed counter-example 2 (from Robert Israel): > ... > But on the first circle, x.[1,0,2] = cos(t)+4 has its maximum at t=0 > at the point [1,0,2], and on the second circle x.[1,0,2] = 1 + 4 cos(t) > has its maximum at t=0 which corresponds to the same point [1,0,2]. Again, I am not asking that the image points be distinct, only that there exist 2 distinct points in the domain (S^1) which map to that maximum. I am not persuaded that there is a smooth function F as you describe which does not map 2 distinct points to [1,0,2]. This really explains the source of my question. I am imagining a wire loop in R^3 and imagining balancing it on the table (2 points of contact). Among the set of such balance positions, I believe one of them has 2 maxima. continues to keep your interest. -GoBee === Subject: Re: S^1 in R^3 > gob...@gmail.com says... >Can anyone help me prove that every smooth (continuous?) map of S^1 >into R^3 can be rigidly rotated so as to simultaneously have 2 or more >global minima points and 2 or more global maxima points? >In other words, for every smooth map F from S^1 to R^3 there exists a >vector v in R^3 such that g(x) = F(x) * v has both 2 global maxima and >2 global minima. Here I define a maxima as a point x on S^1 such that >there is a neighborhood of x where the function is no greater than >g(x). So g(x) need not be strictly larger then g for other points in >the neighborhood. > If you only require continuity, then any continuous surjection > F:S^1 -> S^2 gives a counterexample. (This also shows that it is > possible to have only one maximum in every direction.) For a less abstract example, let g be a bump function supported > on [2/3, 5/6] and then let F:[0,1] -> R^3 be given by F(t) = (cos(4 pi t), sin(4 pi t), g(t)). So F traces the unit circle in the X-Y-plane twice, the second time > with a little bump. Here you can certainly have two maxima, but never > two maxima and two minina simultaneously. RS > > > Hello. > > extremely helpful, but I think they are not true counter examples. I > hope I am not misunderstanding. > > By the way, your example of the sphere-filling curve was a perfect > argument for why continuous map F would be insufficient for my Since a sphere-filling curve can't be one-to-one, it's not so clear that there are counterexamples here. You'd want a curve such that for every x on the sphere, either f^(-1)(x) or f^(-1)(-x) has cardinality 1. It's not at all clear that such a thing can exist. > Proposed counter-example (from Robert Sheskey): > > For a less abstract example, let g be a bump function supported > on [2/3, 5/6] and then let F:[0,1] -> R^3 be given by F(t) = (cos(4 pi t), sin(4 pi t), g(t)). > > Let me clarify my question. Remember I am looking for distinct global > maxima on the domain of F, S^1 (not the image), so here we can just > choose v = [1,0,0] and get that t=0 and t=1/2 both map to [1,0,0] > maximizing the dot product with v, and t=1/4 and t=3/4 both map to > [-1,0,0] and [-1,0,1] minimizing the dot product with v. So, 2 > distinct minima and 2 distinct maxima. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Digital Signal Processing: A Computer Based Approach, 3rd Ed., Mitra I need the solutions manual for Digital Signal Processing: A Computer Based Approach, 3rd Ed., by S. Mitra. Anyone have it? I'll pay. === Subject: Re: JSH: Contradictory behavior, issue of math fraud > > Your math is nearly 100% wrong, the exception being your > prime counting algorithm. This has nothing to do with any > genetic argument. > JSH: the I hate whitey cause they know Math troll === Subject: Re: JSH: Contradictory behavior, issue of math fraud THUS JSH THE RACIAL BIGOT SPEAKS: > And I think that you resist all evidence for class reasons, and a need > to try and hold onto a nasty worldview that some people in the United > States want because it justifies dominating other races based on the > idea that they are inherently genetically inferior. If you acknowledge that mathematicians made mistakes that I > discovered, or missed simple mathematical ideas that I've found, you > lose the genetic argument. You can no longer claim that whites are born to rule, and that > claims otherwise are about being politically correct, or to keep > minorities from rioting. > The terrible thing is that the potential for surrogate factoring may >>be with very large numbers where that number of probes is significant >>and far better than random, but with small numbers and an inability to >>pick k and n properly it can vastly inflate that number of checks to >>support opinions against surrogate factoring. > Easy enough to show - tell us how to pick k and n for large numbers. > Unless there is a quick way to get to the right choices for k and n > then your method is likely to be slower than existing methods for > large numbers. The problem with your method is that it cannot get quickly to the > right choices. Currently all we seem to be able to do is to guess > starting points and go from there. I have seen versions where you > pick n and vary k as well as vice-versa. I have seen you use k=1, > k=2, k=30, k=floor(T/30) and others. You need to concentrate on how to pick k and n from a much narrower > range than currently - it is the wide range of possibilities to be > tried that is slowing your method. rossum > Surrogate Factoring is DEAD. === Subject: help please hi there , hope yourr doing great I wish if you could help me locate the solution manual and or test bank for this book Managerial Accounting: Tools for Business Decision Making, 3rd Edition Jerry J. Weygandt, Univ. of Wisconsin, Madison Donald E. Kieso, Northern Illinois Univ. publisher : wiley your cooperation with this matter is greately appreciated === Subject: help please Cc: sbooks4s...@hotmail.com hi there , hope yourr doing great I wish if you could help me locate the solution manual and or test bank for this book Managerial Accounting: Tools for Business Decision Making, 3rd Edition Jerry J. Weygandt, Univ. of Wisconsin, Madison Donald E. Kieso, Northern Illinois Univ. publisher : wiley your cooperation with this matter is greately appreciated. please reply to mazinmosman@yahoo.com === Subject: help please Cc: bookstoday...@gmail.com hi there , hope yourr doing great I wish if you could help me locate the solution manual and or test bank for this book Managerial Accounting: Tools for Business Decision Making, 3rd Edition Jerry J. Weygandt, Univ. of Wisconsin, Madison Donald E. Kieso, Northern Illinois Univ. publisher : wiley your cooperation with this matter is greately appreciated === Subject: Re: Solution Manuals Available now in PDF! > Contact bwen...@hotmail.com! A Course in Algebraic Number Theory by Henry Cohen > A Course in Game Theory by Martin J. 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John Proakis with ebook > Digital Image Processing 2nd Ed. by Gonzalez (Instructor's Manual) > Digital Integrated Circuits by Rabaey 2nd Ed. (selected problems) > Digital signal processing A Computer-Based Approach 1st Ed. by Sanjit > K. Mitra > Digital Signal Processing by Proakis & Manolakis > Digital Signal Processing by Thomas J. Cavicchi > Discrete-Time Signal Processing 2nd Ed. by Oppenheim, R.W. Schafer and > J. R. Buck > Econometric Analysis 5th Ed. by William H. 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Lin > Linear Systems and Signals by B P Lathi > Logic Computer Design Fundamentals 2th Ed. by Morris Mano (selected > problems) > Managerial Accounting 11th Ed. > Material Science and Engineering an Introduction 6th Ed. by Callister > Material Science and Engineering an Introduction 7th Ed. by Callister > Materials & Processing in Manufacturing 9th Ed. by Demargo > Mathematical Methods for Physics and Engineering 3rd Ed. by > K.F.RileyandM.P.Hobson > Mathematics for Economics by Simon and Blume > Mathematical Physics by Arfken > Matrix Analysis And Applied Linear Algebra by Carl D. Meyer > Mechanical Engineering Design 7th Ed. Shigley > Mechanics of Fluids 8th Ed. by Bernard Massey > Mechanics of materials Hibbeler 6th > Mechanics Of Materials(3rd Ed , read more ?... I would like to have a copy of the solution manuel to Fundamentals of Fluid Mechanics 5th Ed. by Munson, thank you. ben === Subject: Re: Solution Manuals Available now in PDF! > Contact bwen...@hotmail.com! A Course in Algebraic Number Theory by Henry Cohen > A Course in Game Theory by Martin J. Osborne and Ariel Rubinstein > A First Course In Probability 7th Ed by Sheldon Ross > Adaptive Control 2nd Ed. by Karl.J.Astrom > Advanced Engineering Marhematics ERWIN KREYSZIG 9E > Advanced Engineering Mathematics 8Ed Erwin Kreyszig > Advanced Engineering Mathematics 8Ed Erwin Kreyszig in korean (all > even and odd no.) > Advanced Engineering Mathematics 2nd Ed. by Zill & Cullen > Advanced Engineering Mathematics 3RD ED. by ZILL & CULLEN > Advanced Macroeconomics by Jeffery Rohaly > Advanced Modern Engineering Mathematics 3rd Ed. by Glyn James > Analog Integrated Circuit Design by David Johns, Ken Martin > Analysis and Design of Analog Intergrated Circuits 4th Ed. by Paul R. > Gray > Analytical Dynamics by Barcuh (selected problems) > Analytical Mechanics 7th Ed. by Fowles > Antenna for all application 3rd Ed.by John D. Kraus (Instructor's > Manual) > Antenna Theory 2nd Ed. by balanis > Applied Linear Statistical Models 5th Ed. by MichaelH.Kutner > Applied Numerical Analysis 7th Ed. by Curtis F. Gerald, Patrick O. > Wheatley > Applied Numerical Methods with Matlab for Engineers and Scientists by > Chapra > Applied Statistics Probability for Engineers 3rd ed Montogomery > (selected problems) > Applied Strength of Materials 4th Ed. by Robert L. Mott > Automatic Control Systems 8th Ed. Kuo and Golnaraghi > Basic Engineering Circuit Analysis 7th Ed. by Irwin > Biochemistry 2nd Ed. by Philip W. Kuchel, Gregory B. Rakston > Bioprocess Engineering Principles by Doran, 1995 rare > C++ How to Program 3rd ed. Deitel, Deitel & Nieto (Instructor's > Manual) > Calculus 5th Ed. by James Stewart (Complete Instructors Solutions) > Calculus by Jerrold Marsden & Alan Weinstein Vol 1 > Calculus Early Transcendentals 5th Ed. by James Stewart > Calculus of Variations with solutions by Russak 2002 > Chemical And Engineering Thermodynamics 3Ed by Wiley > Chemical Engineering Solutions to the Problems in Volume 1 by J R > Backhurst > Classical Dynamics 5th Ed. by Thornton > Classical Dynamics by Marion > Classical Electrodynamics 3rd Ed. by J.D. Jackon > CMOS Analog Circuit Design 2nd Ed. by P.E. Allen > Combinatorics by Merris > Communication Systems 4th Ed. by Simon Haykin > Communication Systems Engineering 2nd Ed. by Proakis J. > Communication Systems: An Introduction to Signals and Noise in > Electrical Communication 4th. Ed. by Bruce Carlson > Computational Techniques for Fluid Dynamics by K. Srinivas and > C.A.J.Fletcher > Computer Networking: A Top-Down Approach Featuring the Internet 2nd > ed. by Jim Kurose and Keith Ross > Computer Networks 4th Ed. by Andrew.S.Tanenbaum > Computer Networks: A Systems Approach 3rd Ed. by larry L. Peterson and > Bruce S. Davie > Computer Organization And Design - The Hardware Software Interface 3rd > Ed. > Control Systems Engineering 3rd Ed. by Norman S. Nise > Control Systems Engineering by Nise, 4th > Design and Analysis of Experiments 6th Ed. by Montgomery > Design Of Analog Cmos Integrated Circuits by Razavi > Design with Operational Amplifiers and Analog Integrated Circuits 3rd > Ed. by Franco, Sergio > Device Electronics for Integrated Circuits 3rd Ed. by Muller and > Kamins > Differential Equations with Boundary-Value Problems 5th Ed. by Zill > and Cullen > Digital communication: Fundamentals and Applications 2nd Ed. by > Bernard Sklar > Digital communications 4th Ed. John Proakis with ebook > Digital Image Processing 2nd Ed. by Gonzalez (Instructor's Manual) > Digital Integrated Circuits by Rabaey 2nd Ed. (selected problems) > Digital signal processing A Computer-Based Approach 1st Ed. by Sanjit > K. Mitra > Digital Signal Processing by Proakis & Manolakis > Digital Signal Processing by Thomas J. Cavicchi > Discrete-Time Signal Processing 2nd Ed. by Oppenheim, R.W. Schafer and > J. R. Buck > Econometric Analysis 5th Ed. by William H. Greene > Econometric Analysis of Cross Section and Panel Data by Jeffrey M. > Wooldridge > Electric Circuits 7th Ed. by Nilsson > Electric Machinery 6th Ed. by Fitzgerald, Kingsley, Uman > Electric Machinery and Power System Fundamentals 1st Ed. by Stephen J. > Chapman (Instructor's Manual) > Electric Machinery Fundamentals 4th Ed. by Stephen J Chapma > (Instructor's Manual) > Electrical Engineering Principles And Applications 4th Edition by > Giorgio Rizzoni > Electrical Engineering Principles And Applications by Hambley > Electronic Circuit Analysis and Design 2nd Ed. by Donald A. Neamen > Electronic Physics by Strabman > Electronics 2nd Ed. by Allan R. Hambley > Elementary Differential Equations and Boundary Value problems 7th Ed. > by Boyce, DiPrima (book and solutions) > Elementary Differential Equations and Boundary Value Problems 8th Ed. > Elementary Linear Algebra by K. R. Matthews (book and solution > manuals) > Elementary Mechanics and Thermodynamics by John W. Norbury > Elements of Chemical Reaction Engineering 3rd Ed. by Timothy Hubbard, > Jessica hamman and David Johnson > Elements of Information Theory by Thomas M. Cover > Engineering Biomechanics: Statics by Angela Matos, Eladio Pereira, > Juan Uribe and Elisandra Valentin > Engineering Economic Analysis 9E by Newman (selection problems) > Engineering Circuit Analysis 6th Ed. by Hayt > Engineering Electromagnetics 6th Ed. by William H. Hayt, Jr. and Hohn > A. Buck > Engineering Fluid Mechanics 7th Ed. by Crowe > Engineering Fluid Mechanics 8th Ed. by Crowe > Engineering Mathematics 4th Ed. by John Bird > Engineering Mechanics Dynamics 10th edition by Hibbeler > Engineering Mechanics Dynamics 11th edition by Hibbeler > Engineering Mechanics Dynamics 5th Ed. by Meriam > Engineering Mechanics Dynamics by Bedford and Fowler > Engineering Mechanics statics 10th edition by Hibbeler > Engineering Mechanics statics 11th edition by Hibbeler > Engineering Mechanics Statics 4th Ed. Bedford > Engineering Mechanics Statics by Meriam > Feedback Control of Dynamic Systems 4th Ed. by G. F. Franklin, J. D. > Powell, A. Emami > Foundations of Materials Science and Engineering, Ed. 3 by Smith and > Hashemi > Foundations of Materials Science and Engineering Ed. 4 by Smith and > Hashemi > Fractal Geometry: Mathematical Foundations and Approach by Kenneth > Falconer > Fluid Mechanics 5th Ed. by White > Fluid Mechanics by Shames > Fluid Mechanics and Thermodynamics of Turbomachinary, 4E & 5E by > Dixon > Foundations of Materials Science and Engineering 3E & 4E by Smith > Fundamentals of Machine Component Design 4th Ed. by Juvinall > Fundamentals of Aerodynamics 2nd Ed. by John D. Anderson > Fundamentals Of Aerodynamics 3rd Ed. by Anderson (Instructor's > Solutions Manual) > Fundamentals of Aerodyanmics 5 th Ed. by John D. Anderson (Instructor > Manual) > Fundamentals of Aircraft Structural Analysis by Howard Curtis > (Instructor's solutions) > Fundamentals of Applied Electromagenetics 5th Ed+ > Fundamentals of Differential Equations by Nagle Saff Snider > Fundamentals of Digital Logic with Verilog Design 1st Ed. by S. Brown, > Z. Vranesic > Fundamentals of Digital Logic with VHDL Design 1st Ed. by S. Brown, Z. > Vranesic > Fundamentals of Electric Circuits 2nd Ed. by C.K.Alexander > M.N.O.Sadiku > Fundamentals of Engineering Thermodynamics 5th ed (M. J. Moran & H. N. > Shapiro) > Fundamentals of Financial Management With Infotrac Concise 4th Ed. by > Eugene Brigham > Fundamentals of Fluid Mechanics 3th Ed. by Munson (Student solution > manual) > Fundamentals of Fluid Mechanics 4th Ed. by Munson > Fundamentals of Fluid Mechanics 5th Ed. by Munson > Fundamentals of Heat and Mass Transfer 5th Ed. Incropera > Fundamentals of Logic Design 5th Ed. by Charles Roth > Fundamentals of Machine Component Design 3rd Ed. by Juvinall and > Marshek > Fundamentals of phsics 7th ed. by David Halliday, Robert Resnick, and > Jearl Walker (instructor's manual) > Fundamentals of Thermal-Fluid Sciences by Cengal > Fundamentals of Thermodynamics 6th Ed. Richard E. Sonntag, Claus > Borgnakke, Gordon J. Van Wylen > Gas Dynamics 2E by James John > Guide to Energy Management 5th Ed. by Klaus-Dieter E. Pawlik > Heat Transfer 2nd Ed. by Cengel > Hydraulics in Civil and Environmental Engineering 4th Ed. by Andrew > Chawick > Introduction To Algorithms 2nd Ed. by Philip Bille (selected > problems) > Introduction To Algorithms 2nd Ed. by Thomas H. Cormen, Charles > E.Leiserson, Ronald L. Rivest, Clifford Stein > Introduction to Antenna Theory by Palacios > Introduction to electric circuits 6th Ed. by Dorf-Svaboda > Introduction to electrodynamics 3rd Ed. by David Griffiths > Introduction to Fluid Mechanics 5ed. by Fox > Introduction to Fluid Mechanics 6th Ed. by Fox > Introduction to Linear Algebra 3th Ed. by Gilbert Strang > Introduction to Quantum Mechanics 2ed Griffiths, David J. > Linear Algebra and its Applications 3rd Ed. by Lay > Linear Algebra by Jim Hefferon > Linear circuit analysis by R. A. DeCarlo and P. Lin > Linear Systems and Signals by B P Lathi > Logic Computer Design Fundamentals 2th Ed. by Morris Mano (selected > problems) > Managerial Accounting 11th Ed. > Material Science and Engineering an Introduction 6th Ed. by Callister > Material Science and Engineering an Introduction 7th Ed. by Callister > Materials & Processing in Manufacturing 9th Ed. by Demargo > Mathematical Methods for Physics and Engineering 3rd Ed. by > K.F.RileyandM.P.Hobson > Mathematics for Economics by Simon and Blume > Mathematical Physics by Arfken > Matrix Analysis And Applied Linear Algebra by Carl D. Meyer > Mechanical Engineering Design 7th Ed. Shigley > Mechanics of Fluids 8th Ed. by Bernard Massey > Mechanics of materials Hibbeler 6th > Mechanics Of Materials(3rd Ed , read more ?... I would like to have a copy of the solution manual to Fundamentals of Fluid Mechanics 5th Ed. by Munson, thank you very much. Ben === Subject: Re: Solution Manuals Available now in PDF! Hi i would like to know if you have the Engineering Circuit Analysis, 7th Edition by Hayt. or maybe if you dont i would like the Engineering Circuit Analysis, 6th Edition contact me at luisgarcia87@gmail.com please i need it as soon as possible === Subject: Re: Complete electronic solution manual in pdf ! Get it in hours! I need solution manual for these books advanced engineering mathematics (9/e) by ERWIN KREYSZIG Electric Circuits (8th Edition) by James W Nilsson. I need it asap. also wanna know how to make payment? momo > I need Engineering Mechanics: Dynamics, 10th Ed., by Russell C. > Complete electronic solution manual in pdf ! Get it in hours! > I have the complete electronic SOLUTION MANUAL in PDF format > containing ALL (odd and even - except where noted) solutions for the > books listed below. These are not paper books, they are ebooks and > most are only $12.00 each - paypal accepted. > Books for which I have electronic solution manual: > 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, 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 > 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, Sixth 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: Priciples, 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 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 > Electromagnetic Fields and Waves by Iskander (...) > Electronic Circuit Analysis, 2nd Ed., by Donald Neamen > Electronics, 2nd Ed., by Allan R. Hambley > Elementary Differential Equations, 8th Edition, by Boyce, DiPrima > (some odd/even) > 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 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 > Engineering Mechanics: Statics, 6th Ed., Meriam > Engineering Mechanics: Statics, 10th Ed., by Russell C. Hibbeler > Engineering Mechanics: Statics 11th Ed. by Hibbeler > Experiments with Economic Principles by Bergstrom, Miller > 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] > Essentials of Fluid Mechanics: Fundamentals and Applications, 1st Ed., > by Cengel & Cimbala > Fluid Mechanics with Engineering Applications, 10th Edition, by > Finnemore > Fundamentals of Aerodynamics, 3rd Edition, by J. D. Anderson, Jr. > Fundamentals of Aerodynamics, 4th Edition, by Anderson > Fundamentals of Applied Electromagnetics, 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 Digital Logic with VHDL Design by Stephen Brown, > Zvonko 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, Schmid > 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 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 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 Quantum Mechanics, 2nd Ed. by Griffiths > Introdution to Solid State Physics, 8th Edition by Kittel > Intro to Thermal Systems Engineering: Thermodynamics,... > read more E If the calc 5e by stewart has chapter 7 as being inverse functions i > need it. === Subject: simple question... It's 20 years ago, that I was in school. I don't know, how to do it... I've got a pencil about 30 cm long, r = 3, 525. It stucks 14 cm in a pencil sharpener, r = 3,54. How wide you can swing the pencil at the lower end (x)? What is the angle? Can you help me? === Subject: Re: Prolems in Mathematics Education > My high school courses were honors courses [...] I wonder if I'm the only person here who has never > taken an honors, a gifted, or a talented course in > high school or college, in any subject? I only mentioned it at the end. I would not expect the same > depth in non-honors courses, BUT: Perhaps I expressed > myself badly. I was trying to convey the idea that with good > teachers and motivated students that it is possible to learn > some rigorous mathematics in high school -- even in non-honors > courses. Indeed, I suspect that the difference between students in > honors vs. non-honors courses lies more in motivation than it does > talent. The problem we have today is that (1) We don't have good teachers (because public schools pay > diddly squat so that good people get jobs elsewhere) (2) We have too many students who don't CARE. Both are a result of public and parental apathy. I agree about the motivation (with selection for to eliminate students who can't/don't do the work), but don't necessarily agree with (1). The teachers in the Akron Public School system start with a BS/BA at about $35,000, and get about $73,000 after 18 years with a Master's +30. That is better than the average professor at the university where I am employed. Salaries in many of the suburbs are substantially better than that. === Subject: Re: Prolems in Mathematics Education >We don't have good teachers (because public schools pay >diddly squat so that good people get jobs elsewhere) > nonacademic issuesone one has to endure. E.g., bureaucratic requirements, unions, parents, testing, student behavior,... I have been told that that is generally the difference between public and private schools: Public schools pay more but are really quite unbearable, whereas you have tend to have more leeway in a private school, where the pay is much less. -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan === Subject: Re: Prolems in Mathematics Education >> I wonder if I'm the only person here who has never > taken an honors, a gifted, or a talented course in > high school or college, in any subject? > as much as I was curious about how many people fall >into my category of never having taken a special >grouping type of class, especially among those who >have a graduate degree. Times certainly change. For one thing, my generation learned our multiplication tables in primary school, and how to deal with signed numbers and fractions in Junior High. When I was in High School in 1953 -- 1955 in Rapid City, SD, population 25,000, we could take the usual sequence of algebra courses, a year of plane geometry (which was entirely devoted to Euclidean proofs) as a sophomore, a semester or year of trigonometry (I don't recall which) and a semester of solid geometry. There were no honors courses nor the possibility of earning a grade point average greater than 4.0 on a 4.0 scale by taking the hard courses. And we could take a year each of Biology, Chemistry, and Physics, complete with labs. No calculus nor analytic geometry. Then as a freshman at SD School of Mines & Tech, *everyone* started with trigonometry followed by analytic geometry. Calculus was for sophomores. This was a bit repetitious with the trigonometry but what we didn't have in our Sophomore Calculus classes was the endless stream of questions about the steps the teacher was doing because students in the class couldn't factor or keep signs and parentheses straight. Perhaps it's just a romantic view of long ago, but I think we were better off educationally then than now. As exhibit A I present the many posts we see here from undergraduates who can't properly phrase their questions, let alone type them with correct use of parentheses. Not to mention all the remedial algebra courses offered in the Universities. --Lynn === Subject: Twelve Proofs that Muhammad is a True Prophet My brothers and sisters everywhere! With this essay, I am not singling out the adherents of Islam - to which I ascribe - but rather I am writing this essay to every man and woman throughout the whole world. I ask Allah that He facilitates tat this essay reaches every ear, falls under the sight of every eye, and is understood by every heart... Muhammad the son of `Abdullah is Allah's Prophet and the Final Messenger Sent by Allah to the Inhabitants of Earth. My brothers and sisters everywhere! You should know that the Messenger, Muhammad the son of `Abdullah (may Allah's blessings and peace be upon him) is Allah's Messenger in reality and truth. The evidences that show his veracity are abundant. None but an infidel, who out of arrogance alone, could deny these signs. Among these proofs: 1. Muhammad (may Allah's blessings and peace be upon him) was raised illiterate, unable to read or write, and remained like that till his death. Among all his people, he was known as being truthful and trustworthy. Before receiving revelation, he had no prior knowledge of Religion or any previously sent Message. He remained like that for his first forty years. Revelation then came to Muhammad with the Koran that we now have between our hands. This Koran mentioned most of the accounts found in the previous scriptures, telling us about these events in the greatest detail as if he witnessed them. These accounts came precisely as they were found in the Torah sent down to Moses and in the Gospel sent down to Jesus. Neither the Jews or Christians were able to belie him regarding anything that he said. 2. Muhammad (may Allah's blessings and peace be upon him) also foretold of everything that would occur to him and his community after him, pertaining to victory, the removal of the tyrannical kingdoms of Chosroes [the royal title for the Zoroastrian kings of Persia] and Caesar, and the establishment of the religion of Islam throughout the earth. These events occurred exactly as Muhammad foretold, as if he was reading the future from an open book. 3. Muhammad (may Allah's blessings and peace be upon him) also brought an Arabic Koran that is the peak of eloquence and clarity. The Koran challenged those eloquent and fluent Arabs of his time, who initially belied him, to bring forth a single chapter like the Koran. The eloquent Arabs of his day were unable to contest this Koran. Indeed, till our day, none has ever dared to claim that he has been able to compose words that equal-or even approach-the order, grace, beauty, and splendor of this Glorious Koran. 4. The life history of this Noble Prophet was a perfect example of being upright, merciful, compassionate, truthful, brave, generous, distant from all evil character, and ascetic in all worldly matters, while striving solely for the reward of the Hereafter. Moreover, in all his actions and dealings, he was ever mindful and fearful of Allah. 5. Allah instilled great love for Muhammad (may Allah's blessings and peace be upon him) in the hearts of all who believed in and met him. This love reached such a degree that any of his companions would willingly sacrifice his (or her) self, mother or father for him. Till today, those who believe in Muhammad honor and love him. Anyone of those who believe in him would ransom his own family and wealth to see him, even if but once. 6. All of history has not preserved the biography of any person in the manner it has preserved the life of Muhammad, who is the most influential human in history. Nor has the entire earth known of anyone whom every morning and evening, and many times thereafter throughout the day, is thought of by those who believe in him. Upon remembering Muhammad, the believers in him will greet him and ask Allah to bless him. They do such with full hearts and true love for him. 7. Nor has there every been a man on earth whom is still followed in all his doings by those who believe in him. Those who believe in Muhammad, sleep in the manner he slept; purify themselves (through ablution and ritual washing) in the manner he purified himself; and adhere to his practice in the way they eat, drink, and clothe themselves. Indeed in all aspects of their lives, the believers in Muhammad adhere to the teachings he spread among them and the path that he traveled upon during his life. During every generation, from his day till our time, the believers in this Noble Prophet have fully adhered to his teachings. With some, this has reached the degree that they desire to follow and adhere to the Prophet's way in his personal matters regarding which Allah has not sought of them to adhere to in worship. For example, some will only eat those specific foods or only wear those specific garments that the Messenger liked. Let alone all that, all those who believe in Muhammad repeat those praises of Allah, special prayers, and invocations that he would say during each of his actions during day and night, like: what he would say when he greeted people, upon entering and leaving the house, entering and leaving the mosque, entering and leaving the bathroom, going to sleep and awaking from sleep, observing the new crescent, observing the new fruit on trees, eating, drinking, dressing, riding, traveling and returning from travel, etc. Let alone all that, all those who believe in Muhammad fully perform- even to the minute detail-every act of worship-like prayer, fasting, charity, and pilgrimage-as this Noble Messenger (may Allah's blessings and peace be upon him) taught and as he himself performed. All of this allows those who believe in him, to live their lives in all aspects with this Noble Messenger as their example, as if he was standing before them, for them to follow in all their doings. 8. There has never been nor will there ever be a man anywhere upon this earth who has received such love, respect, honor, and obedience in all matters-small and large alike-as has this Noble Prophet. 9. Since his day, in every region of the earth and during every period, this Noble Prophet has been followed by individuals from all races, colors and peoples. Many of those who followed him were previously Christians, Jews, pagans, idolaters, or without any religion. Among those who chose to follow him, were those who were known for their sound judgment, wisdom, reflection, and foresight. They chose to follow this Noble Prophet after they witnessed the signs of his truthfulness and the evidences of his miracles. They did not choose to follow Muhammad out of compulsion or coercion or because they had adopted the ways of their fathers and mothers. Indeed many of the followers of this Prophet (may Allah's blessings peace be upon him), chose to follow him during the time when Islam was weak, when there were few Muslims, and when there was severe persecution of his followers on earth. Most people who have followed this Prophet (may Allah's blessings and peace be upon him) have done so not to acquire some material benefits. Indeed many of his followers have suffered the greatest forms of harm and persecution as a result of following this Prophet. Despite all this harm and persecution, this did not turn them back from his religion. My brethren! All of this clearly indicates to anyone possessing any sense, that this Prophet was truly and really Allah's messenger and that he was not just a man who claimed prophethood or spoke about Allah without knowledge. 10. With all this, Muhammad came with a great religion in its credal and legal make-up. Muhammad described Allah with qualities of complete perfection, and at the same time in a manner that is free of ascribing to Him any imperfection. Neither the philosophers or the wise could ever describe Allah like such. Indeed it is impossible to imagine that any human mind could conceive of an existing being that possesses such complete ability, knowledge, and greatness; Who has subdued the creation; Who has encompassed everything in the universe, small or large; and Who possesses such perfect mercy. Nor is it in the ability of any human being to place a perfect law based upon justice, equality, mercy and objectivity for all human activity on earth like the laws that Muhammad brought for all spheres of human activity - like buying and selling, marriage and divorce, renting, testimony, custody, and all other contracts that are necessary to uphold life and civilization on earth. 11. It is impossible that any person conceive wisdom,, morals, good manners, nobleness of characters as what this honorable Prophet (may Allah's blessings and peace be upon him) brought. In a full and complete manner, Muhammad spread a teaching regarding character and manners toward one' parents, relatives, fiends, family, humanity, animals, plants and inanimate objects. It is impossible for the human mind alone to grasp all of that teaching or come with a similar teaching. All of that unequivocally indicates that this Messenger did not bring an) of this religion from his own accord, but that it was rather a teaching and inspiration that he received from the One Who created the earth and the high heavens above and created this universe in its miraculous architecture and perfection. 12. The legal and credal make-up of the religion that the Messenger, Muhammad, (may Allah's blessings and peace be upon him) brought resembles the engineering of the heavens and the earth. All of that indicates that He who created the heavens and the earth is the One Who sent down this great law and upright religion. The degree of inimitability of the Divine law that was sent down upon Muhammad is to the same degree of inimitability of the Divine creation of the heavens and earth. For just as humanity cannot create this universe, in the same manner humanity cannot bring forth a law like Allah's law that He sent down upon His servant and messenger Muhammad (may Allah's blessings and peace be upon him). === Subject: least squares superimposition of two similar tetrahedra now, but so far cannot figure out how. Given two similar tetrahedra, ABCD and A'B'C'D' (the letters represent the corners of the 2 tetrahedra), and their coordinates (x,y,z) are all known. Also, note that it is known that A corresponds to A', B to B', and so on. Does anyone know a formula that can superipose one onto the other? Assume one of them stays stationary while the other is moved (translated and rotated) to superimpose onto it. I need a 3x3 transformation matrix, right? How do I solve for those 9 numbers? Jason K. Please send answers or ideas to: jasonmkawasaki@gmail.com === Subject: Re: least squares superimposition of two similar tetrahedra > now, but so far cannot > figure out how. Given two similar tetrahedra, ABCD and > A'B'C'D' (the letters represent > the corners of the 2 tetrahedra), and their coordinates (x,y,z) are > all known. Also, note > that it is known that A corresponds to A', B to B', and so on. Does > anyone know a > formula that can superipose one onto the other? Assume one of them > stays stationary > while the other is moved (translated and rotated) to superimpose onto > it. I need a 3x3 > transformation matrix, right? How do I solve for those 9 numbers? Jason K. Please send answers or ideas to: > jasonmkawas...@gmail.com There is a rotation (3 Euler angles) matrix,translation (vector x,y,z difference, 3 coordinate differences) matrix and 1 magnification/ similitude ratio of the corresponding sides as a single multiplication scalar factor. Narasimham === Subject: Re: least squares superimposition of two similar tetrahedra > now, but so far cannot > figure out how. Given two similar tetrahedra, ABCD and > A'B'C'D' (the letters represent > the corners of the 2 tetrahedra), and their coordinates (x,y,z) are > all known. Also, note > that it is known that A corresponds to A', B to B', and so on. Does > anyone know a > formula that can superipose one onto the other? Assume one of them > stays stationary > while the other is moved (translated and rotated) to superimpose onto > it. I need a 3x3 > transformation matrix, right? How do I solve for those 9 numbers? Actually a 3x3 matrix for the rotation plus a 3x1 vector for the translation, which can be put together to form a 3x4 matrix. In any case, you have 12 numbers to solve for. And you have 12 equations that say you map A -> A', B -> B', C -> C', D -> D' (3 coordinates for each). Of course the transformation won't be a rotation unless the tetrahedra are actually congruent. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: math lingo sharp I have a dumb question. Suppose $X$ some function space and I say the following inequality is sharp F(u) <= C G(u) for u in X. Does this just mean the $C$ is the best constant ? craig === Subject: Re: math lingo sharp > I have a dumb question. > > > Suppose $X$ some function space and I say the following inequality is sharp > > F(u) <= C G(u) for u in X. > > > Does this just mean the $C$ is the best constant ? > > > > > craig It can mean that. It typically means that the inequality is somehow as good as it can be, where the notion of how to define best has to be understood from the context, and preferably should be made explicit. Without a proper context, I would tend to think it means what you suggest. === Subject: Re: math lingo sharp > I have a dumb question. > > > Suppose $X$ some function space and I say the > following inequality is sharp > > F(u) <= C G(u) for u in X. > > > Does this just mean the $C$ is the best constant ? > > > > > craig > > It can mean that. It typically means that the > inequality is somehow as > good as it can be, where the notion of how to define > best has to be > understood from the context, and preferably should be > made explicit. > > Without a proper context, I would tend to think it > means what you suggest. > An example (say Classic Hardy's inequality) So (depending what side you put $C$ ) on we know the best constant is given by C= (n-2)^2 /4 where u in H_0^1(Omega). So with this $C$ I can say inequality is sharp? Does it have anything to do whether the inequality obtains the best constant in the respective function space? craig === Subject: Re: math lingo sharp > I have a dumb question. >>> Suppose $X$ some function space and I say the >> following inequality is sharp > F(u) <= C G(u) for u in X. >>> Does this just mean the $C$ is the best constant ? >>> craig >> It can mean that. It typically means that the >> inequality is somehow as >> good as it can be, where the notion of how to define >> best has to be >> understood from the context, and preferably should be >> made explicit. >> Without a proper context, I would tend to think it >> means what you suggest. > > An example (say Classic Hardy's inequality) > > So (depending what side you put $C$ ) on we know the best constant is given by > > C= (n-2)^2 /4 where u in H_0^1(Omega). > > So with this $C$ I can say inequality is sharp? > > Does it have anything to do whether the inequality obtains the best constant in the respective function space? Well, for me 'Hardy's inequality' is http://en.wikipedia.org/wiki/Hardy's_inequality I have heard of the other inequality also called Hardy's inequality but to be honest I don't know enough about it to comment on what you said in a meaningful fashion. === Subject: Re: math lingo sharp > I have a dumb question. >>> Suppose $X$ some function space and I say the >> following inequality is sharp > F(u) <= C G(u) for u in X. >>> Does this just mean the $C$ is the best constant > ? >>> craig >> It can mean that. It typically means that the >> inequality is somehow as >> good as it can be, where the notion of how to > define >> best has to be >> understood from the context, and preferably should > be >> made explicit. >> Without a proper context, I would tend to think it >> means what you suggest. > > An example (say Classic Hardy's inequality) > > So (depending what side you put $C$ ) on we know > the best constant is given by > > C= (n-2)^2 /4 where u in H_0^1(Omega). > > So with this $C$ I can say inequality is sharp? > > Does it have anything to do whether the inequality > obtains the best constant in the respective function > space? > > Well, for me 'Hardy's inequality' is > > http://en.wikipedia.org/wiki/Hardy's_inequality > > I have heard of the other inequality also called > Hardy's inequality > but to be honest I don't know enough about it to > comment on what you > said in a meaningful fashion. > craig === Subject: Re: Electronic Solution Manuals in .pdf/doc - Get it Within 30 Minutes Garrison Please send it to me at sal.nuinui@gmail.com thank you === Subject: Re: Poor teaching of econ > On Sep 3, 3:51 pm, professorchaos > [ Illiterate babble - deleted. ] > >> You will find the model collapses empirical when something structural it >> can not predict changes. Like the supply shocks of the 1970's when >> something Keynes thought to be impossible occurred that is high >> inflation and high unemployment simultaneously. > > By 1962, Keynes' collaborators at Cambridge had developed a > theory of stagflation: > > http://robertvienneau.blogspot.com/2006/10/explanation-of-stagflation.html > > I, of course, do not expect John to not pass on to his > suffering students the misrepresentations of the > literature his professors taught him. > Still sniping and dodging my question Rob. What does the CCC have to do with any New Keynesian model? Why are menu cost not microeconomic foundations for New Keynesian models? Staggered wage contracts? Implicit contracts? Efficiency wages? What does the CCC have to do with any of these? Why are none of these microeconomic foundations? I might have read somewhere that if somebody offers a theory that explains a phenomenon before it is observed, scientists will tend to take that theory up for closer examination. But that would have not been written about economics. Robinson may not have gotten too make attention with this statements but this is just flat out deceptive. Friedman and Phelps predicted that systematic policy would change the way prices and wages were set before the 1970's. The correctly predicted the fall of the Keynesian model as Americans saw it before it happened. There were explanations before it occurred and people did take those theories up for closer examination. Although Rational expectations is seen as Lucas' revolution. The ideas were in his mentor Friedman's work before Lucas came along. This were studied extensively and yet the jury is still sort of out on them. I think there is little doubt about RE during stagflation and implications for permenant versus temporary tax cuts. Yet other than the Fed being able to reduce inflation in the 1990's without increasing unemployment it is not clear how useful RE is to your average recession where monetary policy is to be used. I can not say why Robinson's statement was not looked at and why Friedman's was. I wasn't even alive in the 1960's and did not even begin my studies of economics until the 90's. So I can't speak for the climate or the reasons behind it. However, tust because an economist you seem to like did not get the attention while others did does not mean economics ignores theories. For instance, Schumpter gave an intriguing and interesting explanation of the great depression. One I am inclined to believe was better than Keynes. My opinion. I have no evidence for this. In Schumpter's work recession happened due to adoption cost and creative destruction from new technology. It caused structural unemployment in which tax and spend policy as well as monetary doesn't address very well. Why Schumpter was ignored until the late 1990's when endogenous growth writers picked up his ideas and offered up new ideas that were very different from Romer? Why is his book on business cycles somewhat hard to find and Keynes easy to find? Why did the profession choose Keynes over Schumpter? I can not say. This was a time when econometrics was just getting its legs. Some writers seem to have more influence than others. Having been a student of Marshall one would expect Robinson had more pull but that was not the case. something very similar to Keynes get ignored. Perhaps in this case it was due to a limited audience since he did write in English. === Subject: Re: Poor teaching of econ <46cdb987.6203293@news.telus.net> <46d07264$0$31919$4c368faf@roadrunner.com> <46d0b87f.1988497@news.telus.net Requiring economics would just ensure that evryone > was equally brainwashed with lies. > Research by psychologists has shown > that economics students are the most > amoral on campus, even beating out law students. > OK, I'll bite. > How does one do 'research' into amorality? By watching outspoken economists dodge questions fundamental to free marketry: I'll pay $200 for the first letter from an outspoken market economist from Hoover, Heritage, Am. Enterprise, the Chicago School, Cato, von Mises, etc., who answers this question: Does free speech precede each and every free trade. Just get all this on hardcopy: 1. letterhead of GOP think tank 2. the question, Does free speech precede each and every free trade? 3. an answer -- it can be _any_ text whatsoever. 4. the name of the outspoken market economist 5. the signature of the outspoken market economist Scan the letter and email it to BretCahill@aol.com with a mailing address. You'll get a M. O. for $200 in one week. Bret Cahill === Subject: Re: Poor teaching of econ > And how is it that Martin Feldstein, President of NBER, didn't violate this >> standard when he argued (incorrectly, as it turned out---but data never >> stops these handmaidens to the powerful, does it?) that Clinton's 1993 >> income tax hike would create a recession? > He didn't because he was using standard macroeconomic theory that > raising taxes decreases aggregate demand. >> >> And that theory wouldn't happen to favor the interests of those who >> should rightly be paying more tax, would it? No it would because the same textbook will tell you that some Keynesians >advocated raising taxes to mitigate inflation. >Standard macro >theory does not say raising taxes is bad at all times. Only doing so >during a recession. A case can be made for raising taxes when the >economy is experiencing inflation. However, the case for tax and spend >policy is poor due to lags. The same textbook would also tell you >cutting taxes is likely not to have an effect until the recession is >over and increase inflation because it hits during the expansionary phase. All of which neatly elides the little matter of _whom_ should be taxed, and how, and why. > No major media columnist is immune to the political whims of whom they > work for. >> >> But somehow economists are....? You snipped the bit where I stated only university professor who have >academic freedom and the possibility of tenure come CLOSE to being >apolitical. Hehe. How do they GET tenure? Aye, there's the rub. >Academic freedom says that if an university fires you >because they disagree with a political statement you can sue them. Fires you? Why would they hire you in the first place if they were not certain you were not going to serve their rich masters? It's not like there aren't lots of candidates willing to chant the approved lies for the rest of their lives. And politics is hardly the point. You can be a communist, for all they care. Those lies serve the interests of the privileged just as well as the neoclassical lies, after all. But if you tell the truth about economics, you have no chance of being hired, because you have no chance of being published in a peer-reviewed journal. >Tenure, if you get it, means the university can't find some silly reason >to fire you if they disagree with your politics. To get tenure, you first have to agree with _their_ economics. -- Roy L === Subject: Re: Poor teaching of econ > Fires you? Why would they hire you in the first place if they were > not certain you were not going to serve their rich masters? If this were how it works how did Ward Churchill get tenure? Go ahead and feed your paranoid fantasies I need some rest. I have a new project to start tomorrow. > But if you tell the truth > about economics, you have no chance of being hired, because you have > no chance of being published in a peer-reviewed journal. > Well no if it the truth according to Roy where supply is a carpenter's definition of how much is on hand and not how much people are willing to sell at different prices. Not if you use Roy's definition where supply and quantity supplied are interchangeable. Not if you use Roy's definition where some efficiency is measured in degrees instead of used as a base line for comparison. If you live in Roy's world you can't get published. Not because you are telling the truth but because you have muddled the terms so much an economist dismisses you outright because if you can't get the terms right it is unlikely that you got the theory right. What do you think would happen if you tried to publish a physics paper when momentum and acceleration were used as interchangeable terms? Where you defined gravity as not being constant because gravity meant the force of the wind pushing on you? Do you think you would get very far? > To get tenure, you first have to agree with _their_ economics. > On the contrary. If that were true Samuelson, Lucas, nor Sergeant would have ever been hired much less got tenure. There were very in your face to the Keynesian ideas of the day (Lucas and Sergeant). Samuelson took the use of mathematics to a whole new level. Marginalist professors would have never gotten tenure in universities if there were true. Go feed your paranoia. === Subject: solutions manual Soltion manual needed for: Introduction To Chemical Engineering Thermodynamics, 7th Ed., by Van Ness, Smith, Abbott ASAP === Subject: Re: cardinality of multivariate polynomial preimages > Once again, I thank tommy1729, whose off-beat questions inspired some > questions of my own. > > I'll make a few conjectures ... > > Let f be an n-variate polynomial, n>1, with integer coefficients. > Regard f as a function from Z^n to Z. > > Conjecture 1: > > sup( {card(f^(-1)(y)) | y in Z} ) = infinity. > > Conjecture 2: > > If f is surjective, then card(f^(-1)(y)) = infinity for some y in Z. Erdos asked whether there is a polynomial (with integer coefficients) P(x) such that all the sums P(a) + P(b), -1 < a < b, are distinct. It is widely believed (but as yet unproved) that P(x) = x^5 will do. If so, then the polynomial f(x, y) = x^5 + y^5 takes on no positive value more than twice. Presumably f(x, y) = x^6 + y^6 takes on no integer value more than twice. A reference is problem F30 of Guy's Unsolved Problems in Number Theory. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: cardinality of multivariate polynomial preimages > Once again, I thank tommy1729, whose off-beat questions inspired some >> questions of my own. >> >> I'll make a few conjectures ... >> >> Let f be an n-variate polynomial, n>1, with integer coefficients. >> Regard f as a function from Z^n to Z. >> >> Conjecture 1: >> >> sup( {card(f^(-1)(y)) | y in Z} ) = infinity. >> >> Conjecture 2: >> >> If f is surjective, then card(f^(-1)(y)) = infinity for some y in Z. Erdos asked whether there is a polynomial (with integer coefficients) >P(x) such that all the sums P(a) + P(b), -1 < a < b, are distinct. It is >widely believed (but as yet unproved) that P(x) = x^5 will do. If so, >then the polynomial f(x, y) = x^5 + y^5 takes on no positive value >more than twice. Presumably f(x, y) = x^6 + y^6 takes on no integer value more >than twice. If so, this would contradict my conjecture (1), but as far as I can see, conjecture (2) would still be alive. quasi === Subject: Re: cardinality of multivariate polynomial preimages >> Once again, I thank tommy1729, whose off-beat questions inspired some > questions of my own. > > I'll make a few conjectures ... > > Let f be an n-variate polynomial, n>1, with integer coefficients. > Regard f as a function from Z^n to Z. > > Conjecture 1: > > sup( {card(f^(-1)(y)) | y in Z} ) = infinity. > > Conjecture 2: > > If f is surjective, then card(f^(-1)(y)) = infinity for some y in Z. >>Erdos asked whether there is a polynomial (with integer coefficients) >>P(x) such that all the sums P(a) + P(b), -1 < a < b, are distinct. It is >>widely believed (but as yet unproved) that P(x) = x^5 will do. If so, >>then the polynomial f(x, y) = x^5 + y^5 takes on no positive value >>more than twice. >>Presumably f(x, y) = x^6 + y^6 takes on no integer value more >>than twice. If so, this would contradict my conjecture (1), but as far as I can >see, conjecture (2) would still be alive. In fact, for conjecture (2), I think the hypothesis can be substantially weakened, thus yielding a stronger conjecture ... Let f be an n-variate polynomial, n>1, with integer coefficients. Regard f as a function from Z^n to Z. Conjecture (3): If range(f) is unbounded both above and below, then card(f^(-1)(y)) = infinity for some y in Z. quasi === Subject: Re: cardinality of multivariate polynomial preimages >> Once again, I thank tommy1729, whose off-beat questions inspired some >> questions of my own. >> >> I'll make a few conjectures ... >> >> Let f be an n-variate polynomial, n>1, with integer coefficients. >> Regard f as a function from Z^n to Z. >> >> Conjecture 1: >> >> sup( {card(f^(-1)(y)) | y in Z} ) = infinity. >> >> Conjecture 2: >> >> If f is surjective, then card(f^(-1)(y)) = infinity for some y in Z. >>Erdos asked whether there is a polynomial (with integer coefficients) >P(x) such that all the sums P(a) + P(b), -1 < a < b, are distinct. It is >widely believed (but as yet unproved) that P(x) = x^5 will do. If so, >then the polynomial f(x, y) = x^5 + y^5 takes on no positive value >more than twice. >>Presumably f(x, y) = x^6 + y^6 takes on no integer value more >than twice. >>If so, this would contradict my conjecture (1), but as far as I can >>see, conjecture (2) would still be alive. In fact, for conjecture (2), I think the hypothesis can be >substantially weakened, thus yielding a stronger conjecture ... Let f be an n-variate polynomial, n>1, with integer coefficients. >Regard f as a function from Z^n to Z. Conjecture (3): If range(f) is unbounded both above and below, then >card(f^(-1)(y)) = infinity for some y in Z. > How about f(v,w,x,y,z) = v(w^2 + x^2 + y^2 + z^2) range(f) = Z, but f^-1(k) is finite, all k. Maybe you should try Conjecture (4): if range(f) = Z then card(f^-1(k)) is unbounded. Who knows, it might even be true! RS === Subject: help please Hi there, I quite have difficuty to show that function f:R^2->R that is continuously differentiable function is not 1-1, anyone have idea how to show that and how to generalize it to function that is continuously differentiable f:R^n->R^m mR that is continuously differentiable function is not 1-1, anyone have idea how to show that and how to generalize it to function that is continuously differentiable f:R^n->R^m m Hi there, I quite have difficuty to show that function f:R^2->R that is > continuously differentiable function is not 1-1, > anyone have idea how to show that and how to generalize it to > function that is continuously differentiable f:R^n->R^m m Think about the image under f of [-1,1]x{0} and {0}x[-1,1]. === Subject: Re: Pi in non-euclidiean spaces : : : > Rabbi Albert Einstein >: : : > The Androclitty doctrine: Protect confused angry ignorance against >: > achievement through technical incompetence exercising intellectual >: > irresponsibility. >: : > Hey Androclitty, Hab SoSlI' Quch! >: : >: So then he obviously does not want to fill his ignorance >: with knowledge (otherwise why guard it?) and gain any sort >: of technical compentecy. So then why bother arguing? He's >: _never_ going to concede a single point! Never! >I 'd leki ot avhe omse compentecy fi I wekn hawt ti aws. >What the is it? The Uncle Stooopid doctrine: > http://sound.westhost.com/counterfeit.jpg Androcles, answer the question in the sig. w. -- An Empirical Question for the anti-relativists: What is the GPS carrier modulation signal frequency? [ ] 1.023000000000 MHz (theor. unaffected) [ ] 1.022999999543 MHz (rel. corrected) [x ] (example for Androcles and hanson) === Subject: Re: Pi in non-euclidiean spaces : : : > : >: > : > Rabbi Albert Einstein >: : > [snip crap] >: : > head Jew. >: : >: Insults are not a real argument. Of course not, but Uncle Stooopid is only artificially intelligent. >: >: Insults prove nothing but the weakness of one's case. >: If you must resort to insult to argue a case, then >: you have lost the debate, right there, period, end >: of story. >: >: If *anyone* and I do mean ANYone, YOU included, >: resort to insults, you have shown you are unable to >: defend your position through rational argument. >: >: Concession accepted. I'd agree with you if there were a debate to win, but the blatant >denial of verifiable facts isn't recognisably debate in anyone's >book, it's bigotry. Bigots should be insulted vociferously just >for the fun of it. Which fact do you wish to debate? Androcles, answer the question in the sig. w. -- An Empirical Question for the anti-relativists: What is the GPS carrier modulation signal frequency? [ ] 1.023000000000 MHz (theor. unaffected) [ ] 1.022999999543 MHz (rel. corrected) [x ] (example for Androcles and hanson) === Subject: Re: Pi in non-euclidiean spaces : [snip prolix crap] >: >: > Rabbi Albert Einstein >: [snip rest of 61 lines of prolix crap] >: >: The Androclitty doctrine: Protect confused angry ignorance against >: achievement through technical incompetence exercising intellectual >: irresponsibility. >: >: Hey Androclitty, Hab SoSlI' Quch! >The Uncle Stooopid doctrine: > http://sound.westhost.com/counterfeit.jpg Androcles, answer the question in the sig. w. -- An Empirical Question for the anti-relativists: What is the GPS carrier modulation signal frequency? [ ] 1.023000000000 MHz (theor. unaffected) [ ] 1.022999999543 MHz (rel. corrected) [x ] (example for Androcles and hanson) === Subject: Re: Pi in non-euclidiean spaces : [snip prolix crap] >: >: > Rabbi Albert Einstein >: [snip rest of 38 lines of prolix crap] >: >: The Androclitty doctrine: Protect confused angry ignorance against >: achievement through technical incompetence exercising intellectual >: irresponsibility. >: >: Hey Androclitty, Hab SoSlI' Quch! >The Uncle Stooopid doctrine: > http://sound.westhost.com/counterfeit.jpg Androcles, answer the question in the sig. w. -- An Empirical Question for the anti-relativists: What is the GPS carrier modulation signal frequency? [ ] 1.023000000000 MHz (theor. unaffected) [ ] 1.022999999543 MHz (rel. corrected) [x ] (example for Androcles and hanson) === Subject: Re: Pi in non-euclidiean spaces : >: : >Helmut - Wabnig >: >: >: >: : >: >Helmut Wabnig , <.... .-- .- -... -. .. --. @ .- --- -. dot .- - //: >: >: >: >: >: : >: >: >: [snip crap] >: >: >: >: >: >: >: >: > Rabbi Albert Einstein >: >: >: >: [snip rest of crap] >: >: >: >: >: >: >: >: The Androclitty doctrine: Protect confused angry ignorance >against >: >: >: >: achievement through technical incompetence exercising >intellectual >: >: >: >: irresponsibility. >: >: >: : >: >: >The Uncle Stooopid doctrine: >: >: >: > http://sound.westhost.com/counterfeit.jpg >: >: >: : >: >: >: >: >: >: >: >: >'we establish by definition that the time required by >: >: >: >light to travel from A to B equals the time it requires >: >: >: >to travel from B to A' >: >: >: >: >: >: What's wrong with that? >: >: : >: >The explanation is in pictures for imbeciles who can't read, think >: >: >or calculate... idiots just like you, Wabnigga: >: >: > http://www.androcles01.pwp.blueyonder.co.uk/Smart/tAB=tBA.gif >: >: : >: : >: : >: : >: I already have told you that your little graphic is wrong. >: : >I've already plonked you before and don't bother reading your . >: > off, imbecile. Delete it ahahahaaaa....Andro, you don't like counterarguments, but you like hanson, hahahaha...(laugh=copyright hanson) >: >: >: >'we establish by definition that the time required by >: >: >: >light to travel from A to B equals the time it requires >: >: >: >to travel from B to A' Wrong? BECAUSE ANDROCLES SAYS SO......ahahahahahaha... Go raise sheep, Androcles, or answer the question in the sig. w. -- An Empirical Question for the anti-relativists: What is the GPS carrier modulation signal frequency? [ ] 1.023000000000 MHz (theor. unaffected) [ ] 1.022999999543 MHz (rel. corrected) [x ] (example for Androcles and hanson) === Subject: Re: Pi in non-euclidiean spaces ahahaha... AHAHAHAHA... GOOD one!... ahahaha... > [Andro] The rect-Al Uncle Stooopid doctrine: http://sound.westhost.com/counterfeit.jpg > : > : [Andro quoting Einstein] 'we establish by definition that the time required by light to travel from A to B equals the time it requires to travel from B to A' ..... because I, Einstein, SAY SO and you have to agree because I'm the great genius, STOOOPID, don't you dare question it. -- Rabbi Albert Einstein > : [Someone] What's wrong with that? > [Andro] The explanation is in pictures for imbeciles who can't read, think or calculate... idiots just like you, Wabnigga: http://www.androcles01.pwp.blueyonder.co.uk/Smart/tAB=tBA.gif [D.9alj Banana story] > I am quite rational > 1. I realise that a banana can not incapsulate time, no matter how > fast and hard a monkey throw it. > 2. And even if we suppose the monkeys actually could encapsulate time > in bananas, they would be useless for measuring the SSPS staines > per second. > 3. Now the SSPS is said to be an invariant every monkey who sticks a > banana in another monkeys ass will get the same number of staines on > it. That's why it is beleived that when a banana comes out with less > staines than normal the banana actually shrunk..... and time go very > sloooow for the monkey with banana in ass. > 4. But hey didn't I just say that the number of staines on it would be > invariant/constant, yes but since the banana just shrunk and the > monkey feel time goes very slow. The bananas is monkey variant > depending on the monkey who get the ass job. But now you probably will > protest say if the bananas have different size in different monkeys, > how can i really know that the number of staines on it really is > invariant. > Well just do like Einstein smell the banana and say Hey that smells > just like invariant staines > And from this follow of course that there will be very slow monkeys > playing with different sizes of bananas. > This all to me make perfect sense but maybe not in the way the > monkeys, would like it. > [JT] I really need this as a GIF androcles... > [hanson] .... ahahahaha... you like to see Andro make a Banana clock for you? The Einstein Dingleberries will be distraught about you for not having asked for a Dingleberry clock so that they can see for how long they'll have to hang onto Albert's sphincter. === Subject: Re: Pi in non-euclidiean spaces <20070902173431.681$ho@newsreader.com> <5IGCi.5800$xp6.4742@fe3.news.blueyonder.co.uk> : > : > : >: [snip crap] > : >: > : >: > Rabbi Albert Einstein > : >: [snip rest of crap] > : >: > : >: The Androclitty doctrine: Protect confused angry ignorance against > : >: achievement through technical incompetence exercising intellectual > : >: irresponsibility. > : > : >The Uncle Stooopid doctrine: > : > http://sound.westhost.com/counterfeit.jpg > : > : > : > : >'we establish by definition that the time required by > : >light to travel from A to B equals the time it requires > : >to travel from B to A' > : > : What's wrong with that? > The explanation is in pictures for imbeciles who can't read, think > or calculate... idiots just like you, Wabnigga: > http://www.androcles01.pwp.blueyonder.co.uk/Smart/tAB=tBA.gif > Ok i admit i don't like to read but i love your gif's, so simple and > yet beyond the relativists understanding. You...don't like to read...and you think relativity is wrong. I think > it's safe to say that you are an idiot.- D?lj citerad text - - Visa citerad text - I don't think relativity is wrong, i know it is. It is a monkey theory > with BANANA meters, where each monkey push up s banana in the ass of > the other, take it out looking at the staines to see how long the > banana is. Is a theory where the intervall of the staines determine the > length of the banana. END OF STORY support SR, but since you don't like to read I won't bother. Since you don't like to read, you should probably not participate on this newsgroup. === Subject: Re: Pi in non-euclidiean spaces <20070902173431.681$ho@newsreader.com> <5IGCi.5800$xp6.4742@fe3.news.blueyonder.co.uk> : > : > : >: [snip crap] > : >: > : >: > Rabbi Albert Einstein > : >: [snip rest of crap] > : >: > : >: The Androclitty doctrine: Protect confused angry ignorance against > : >: achievement through technical incompetence exercising intellectual > : >: irresponsibility. > : > : >The Uncle Stooopid doctrine: > : > http://sound.westhost.com/counterfeit.jpg > : > : > : > : >'we establish by definition that the time required by > : >light to travel from A to B equals the time it requires > : >to travel from B to A' > : > : What's wrong with that? > The explanation is in pictures for imbeciles who can't read, think > or calculate... idiots just like you, Wabnigga: > http://www.androcles01.pwp.blueyonder.co.uk/Smart/tAB=tBA.gif > Ok i admit i don't like to read but i love your gif's, so simple and > yet beyond the relativists understanding. > You...don't like to read...and you think relativity is wrong. I think > it's safe to say that you are an idiot.- D?lj citerad text - > - Visa citerad text - I don't think relativity is wrong, i know it is. It is a monkey theory > with BANANA meters, where each monkey push up s banana in the ass of > the other, take it out looking at the staines to see how long the > banana is. Is a theory where the intervall of the staines determine the > length of the banana. END OF STORY support SR, but since you don't like to read I won't bother. Since you don't like to read, you should probably not participate on > this newsgroup.- D?lj citerad text - - Visa citerad text - That is really not up to you === Subject: Re: Pi in non-euclidiean spaces <20070902173431.681$ho@newsreader.com> <5IGCi.5800$xp6.4742@fe3.news.blueyonder.co.uk> : > : > : >: [snip crap] > : >: > : >: > Rabbi Albert Einstein > : >: [snip rest of crap] > : >: > : >: The Androclitty doctrine: Protect confused angry ignorance against > : >: achievement through technical incompetence exercising intellectual > : >: irresponsibility. > : > : >The Uncle Stooopid doctrine: > : > http://sound.westhost.com/counterfeit.jpg > : > : > : > : >'we establish by definition that the time required by > : >light to travel from A to B equals the time it requires > : >to travel from B to A' > : > : What's wrong with that? > The explanation is in pictures for imbeciles who can't read, think > or calculate... idiots just like you, Wabnigga: > http://www.androcles01.pwp.blueyonder.co.uk/Smart/tAB=tBA.gif > Ok i admit i don't like to read but i love your gif's, so simple and > yet beyond the relativists understanding. > You...don't like to read...and you think relativity is wrong. I think > it's safe to say that you are an idiot.- D.9alj citerad text - > - Visa citerad text - > I don't think relativity is wrong, i know it is. It is a monkey theory > with BANANA meters, where each monkey push up s banana in the ass of > the other, take it out looking at the staines to see how long the > banana is. > Is a theory where the intervall of the staines determine the > length of the banana. > END OF STORY support SR, but since you don't like to read I won't bother. Since you don't like to read, you should probably not participate on > this newsgroup.- D.9alj citerad text - - Visa citerad text - That is really not up to you You sure do like to bitch and cry about physics a lot for someone who doesn't like to read. Get attention from somewhere else. === Subject: distance function / viscosity solution Suppose one wants to solve | grad u | =1 in Omega u=0 on bdry- Omega then there will be a bunch of solutions but the viscosity solution is given by u(x) = dist(x, bdry-Omega). I was curious about whether one can extend the above ideas to the standard uniformly elliptic operator: Let A(x) denote a symmetric , uniformly positive definite matrix and define L(u) = - sum_{i,j} a^{i,j}(x) u_{x_i} u_{x_j} so $L$ is just the standard uniformly elliptic operator one always sees. Let B(u,v) denote the associated bi-linear form and define the associated norm | grad u |_A^2:= B(u,u). QUESTION: I want to define a A- distance function so the obvious choice would be to let $u$ denote a viscosity solution to | grad u |_A = 1 in Omega u=0 on bdry-Omega. (I don't even know what a viscosity solution is but suspect the existence of the above $u$ isn't a problem.) Will $u$ behave similar to the distance function ? Clearly it would be Lipz but what about other properties of the distance function ? (I know I haven't asked a proper question but any comments would be appreciated) craig === Subject: Matrix multiplication A^T * A is there a simple solution for the following problem? Given is a sparse matrix A (n x m). Is it possible, and if yes how, to determine all nonzero postions in the solution for the matrix multiplication A^T * A if those for A are known? Rico === Subject: wanted - Elementary Differential Equations, 8th Edition, by Boyce, DiPrima solution manual Elementary Differential Equations, 8th Edition, by Boyce, DiPrima solution manual === Subject: Re: NASA exposes Apollo Hoax On Aug 29, 6:23 pm, Major Quaternion Dirt Quantum > hte first thing that I found for sims on JPL's site was this, > which seems to show that Venus is very close to Earth (also Mars), so > that > you might not be able to see it, in the actual sky, at all, > whether morming or evening, without special equipment, > if there is any thing like that. maybe not even fromMoon, > even if you were on the limb at sunset or sunrise. Still no interactive 3D orbital simulator. How downright Yid pathetic. - Brad Guth - === Subject: Re: NASA exposes Apollo Hoax On Aug 24, 3:54 pm, Major Quaternion Dirt Quantum anyway, since theApollolandings were not near lunar sunrise or > sunset, how could Venus be visible, without occultation > of Sun byMoon-- > *why* is it called the morningstar? We see that you still haven't got that fully interactive 3D orbital simulator up and running. What's the excuse this time around? - Brad Guth - === Subject: Re: Mathematics: art or science? >One has to be careful of what one is talking about >here. Evolution is a fact in evidence, observable >and observed. The explanation for evolution is Darwin's >theory of common ancestry. >What we actually _observe_ is kind of common design. >No. we actually observe evolution in action; e.g., >the mutation of bacteria and virii that keep the drug >companies in business. >>That's too naive. Your so called mutation of >>bacteria and virii will never change one species >>into another. It will never mutate virii into >>bacteria. And you know it. > > Nah. There is no barrier to species branching, save > in the minds of creationist believers. I am not going to > respond to this any more, as it is off topic, and there > are plenty other forums to debate this non-issue. Nah. At least your naive vision upon the world is quite consistent. Quoted from another poster in this thread: > Others pick their good axioms just like you pick > yours. So what? Empirical support is not and never > was a requirement for a good mathematics result. So I'll take your evolution theory as a good mathematics result, with no empirical support whatsoever. Discussion closed. Han de Bruijn === Subject: Analysis with subsequence. Hello sir~ (x_n) is bounded. (x_n) does not converge. Show that there exist two different subsequences that converge to two different limits. (hint : Bolzano - Weierstrass theorem) ----------------------------------------------------------------------- I can make two subsequences with lim sup and lim inf. Namely, If lim sup x_n = a, there exist infinity many x_n such that a-1 < x_n. Choose x_(n_1) among these. there exist infinity many x_n such that a-(1/2) < x_n. Choose x_(n_2) among these such that n_1 < n_2. ...... If I used the fact that there is N>0 such that n>=N ==> x_n < a + e for each e>0. then x_(n_k) --> a. Similarly, lim inf. ------------------------------------------------------------------ But this problem appear before lim sup section. so, I want to show by B-W theorem. Since (x_n) is bounded, there (x_n) has a convergent subsequence (x_(n_k)) converges to some c. 1) Consider (y1_n) = (x_n) - (x_(n_k)). (=/= empty, of course, infinity because (x_n) does not converge.) If y1_n converges to some d_1 , and d_1 =/= c, then End. If d_1 = c, then (x_n) --> c. so contradiction. So, suppose that y1_n does not converge. Since (y1_n) is bounded, there (y1_n) has a convergent subsequence (y1_(n_k)) converges to some c_1. If c_1 =/= c, End. 2) If c_1 = c, Consider (y2_n) = (y1_n) - (y1_(n_k)). (=/= empty, of course, infinity because (y1_n) does not converge.) If y2_n converges to some d_2, and d_2 =/= c, then End. If d_2 = c, then (x_n) --> c. so contradiction. So, suppose that y2_n does not converge. Since (y2_n) is bounded, there (y2_n) has a convergent subsequence (y2_(n_k)) converges to some c_2. If c_2 =/= c, End. 3) If c_2 = c, Consider (y3_n) = (y2_n) - (y2_(n_k)). ......... iteration. If all convergent subsequence (yi_(n_k)) converges to c, then (x_n) --> c. so, contradiction. so, there is convergent subsequence (yi_(n_k)) does not converge to c. End. -------------------------------------------------------------------- is this no problem ?? and Let (A_n), (B_n) are subsequences of (x_n). If {A_n | n in N} U {B_n | n in N} = {x_n | n in N} and {A_n | n in N} U {B_n | n in N} = empty, and (A_n) --> c, (B_n) --> c. Then (x_n) --> c. is this right ? Maybe, this is a key in my solution. === Subject: Proving an inequality I'm having trouble proving an inequality. Suppose the function f(t,x) is locally Lipschitz on the domain G in R^2, that is, |f(t,x_1)-f(t,x_2)| <= k(t) |x_1 - x_2| for all (t, x_1), (t,x_2) in G. Define I = (a,b) and phi_1(t) and phi_2(t) are 2 continuous functions on I. Assume that, if (t, phi_i(t)) is in G, then the function f(t, phi_i(t)) is an integrable function on I for i = 1, 2. Suppose that for i = 1,2 and t in I, phi_i(t) = phi_i(t_0) + INTEGRATE (from t_0 to t){ f(s, phi_i(t)) }ds + E_i(t) and (phi_1(t_0) - phi_2(t_0)| <= d for some constant d. Show that for all t in (t_0, b) we have |phi_1(t) - phi_2(t)| <= d exp(INTEGRATE(from t_0 to t){ k(s) }ds) + E(t) + INTEGRATE(from t_0 to t){ E(s)k(s)exp[INTEGRATE(from t_0 to s) { k(r) } dr] }ds where E(t) = |E_1(t)| + |E_2(t)| I managed to get d exp(INTEGRATE(from t_0 to t) k(s) ds) using triangle inequality and Gronwall's inequality, but I cannot seem to get the last 2 terms in the inequality. Here's what I did: |phi_1(t) - phi_2(t)| = |phi_1(t_0) + INTEGRATE (from t_0 to t){ f(s, phi_1(s)) }ds + E_1(t) - phi_2(t_0) - INTEGRATE (from t_0 to t){ f(s, phi_2(s)) }ds - E_2(t)| = |phi_1(t_0) - phi_2(t_0) + INTEGRATE (from t_0 to t){ f(s, phi_1(s)) }ds - INTEGRATE (from t_0 to t){ f(s, phi_2(s)) }ds + E_1(t) - E_2(t)| <= |phi_1(t_0) - phi_2(t_0)| + |INTEGRATE (from t_0 to t){ f(s, phi_1(s)) }ds - INTEGRATE (from t_0 to t){ f(s, phi_2(s)) }ds + E_1(t) - E_2(t)| <= d + |E_1(t) - E_2(t)| + |INTEGRATE (from t_0 to t){ f(s, phi_1(s)) - f(s, phi_2(s)) }ds| <= d + |E_1(t) - E_2(t)| + INTEGRATE (from t_0 to t){ |f(s, phi_1(s)) - f(s, phi_2(s))| }ds <= d + |E_1(t) - E_2(t)| + INTEGRATE (from t_0 to t){ k(s)|phi_1(s) - phi_2(s)| } ds <= (d + |E_1(t) - E_2(t)|) exp(INTEGRATE (from t_0 to t){ k(s) } ds) = d exp(INTEGRATE (from t_0 to t){ k(s) } ds) + (|E_1(t) - E_2(t)|) exp(INTEGRATE (from t_0 to t){ k(s) } ds) This is where I got stuck. Please help. Nott === Subject: Re: the use of 'iff' and 'if and only if' <040920070836338123%edgar@math.ohio-state.edu.invalid As I understand it... Traditionally, in English-language mathematical writing, definitions > were stated with if: A triangle is called equilateral if all three sides are congruent. But of course everyone had to understand that in a definition, if is > actually used to mean if and only if. Instructors of mathematics > found this to be sometimes confusing to students. J. L. Kelley, > in his 1955 textbook, decided to do something about it. So he invented > a word iff, still pronounced if, but to be used in definitions, and > meaning if and only if. P. R. Halmos, liked that idea and adopted > it in his writing, too. When others saw this, they started using iff > as a replacement for if and only if in all locations, not just in > definitions. Some purists reject that broadened use, but many find it > very convenient. > My style avoids the confusion of logical connectives and definitions. A triangle is equilateral when all sides are equal. If x /= 0, then x/x = 1. Never, when x /= 0, x/x = 1. === Subject: Re: the use of 'iff' and 'if and only if' > 4ax.com: > > > IFF. On the last page of his autobiography, Paul R. Halmos (1916-2006) > > My most nearly immortal contributions are an abbreviation and a > typographical symbol. I invented iff, for if and only if -- but I > could never believe that I was really its first inventor. > > And we frequent travelers to Alberta are grateful for the > ingenious soul who shortened Banf and Only Banf to Banff. > It makes reservations soooo much quicker. John Conway has been known to write onne for one and only one, twwo for two and only two, etc. I remember Hugh Montgomery once including in a lecture, ...if and only if, and *only* if and only if... -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: the use of 'iff' and 'if and only if' <040920070836338123%edgar@math.ohio-state.edu.invalid> John Conway has been known to write onne for one and only one, > twwo for two and only two, etc. > Truuthh. The truth, the whole truth and nothing but the truth. > I remember Hugh Montgomery once including in a lecture, > ...if and only if, and *only* if and only if... > Wow, iff and only iff. Sufficiently necessary and necessarily sufficient? === Subject: Regularity of Schemes Suppose X is a scheme such that it is regular at every *closed* points. Is this sufficient to say that X is also regular at every *generic points as well? If not, then it would imply that there exists smooth varieties (the maximum spectrum) that are not smooth as a scheme (the prime spectrum including the generic points). Is this possible? -- -kira === Subject: Re: Regularity of Schemes > Suppose X is a scheme such that it is regular at every *closed* points. > Is this sufficient to say that X is also regular at every *generic > points as well? What definition of regularity do you have in mind? When is a field (e.g. the stalk of the structure sheaf at the generic point of an irreducible reduced scheme) regular? Maybe you mean smooth ... ? J. === Subject: Re: Regularity of Schemes > Suppose X is a scheme such that it is regular at > every *closed* points. > Is this sufficient to say that X is also regular at > t every *generic > points as well? Yes, since regularity is stable under localization, that is if O is a regular, local ring, then for every prime p of O the local ring O_p is regular too. > If not, then it would imply that there exists smooth > varieties (the > maximum spectrum) that are not smooth as a scheme > (the prime spectrum > including the generic points). Be careful here: regularity and smoothness are different notions. For varieties over a field k they are equivalent only if k is perfect. However: smoothness as defined e.g. in EGA is an open concept (with respect to the Zariski topology). More concretely a variety X over a ring R (R suitable ...) is smooth at the point x if there exists an open neighborhood U of x satisfying some properties. Thus if y specializes into x, then U is also an open neighborhood of y with the required properties, so that X is smooth at y too. H === Subject: Re: Homogenizing coordinates? On 2007-09-04 07:00:42 -0400, desktop said: > I am currently reading J. Foley, A. van Dam, S. Feiner, J. Hughes, > C Edition, Interactive Computer Graphics: Principles and Practice. > > In the appendix page 1090 he gives an example of homogenizing a vector (x,y,h). > > The result is (x/h, y/h, 1) which is where the line from (x,y,h) to the > origin intersects with the affine plane. This sounds more like dehomogenizing a vector to me. You started out with a homogeneous vector (x,y,h) in a projective space P^2. Then you consider its embedding (x/h, y/h, 1) in the affine space H<>0. > > He calls this the homogenizing of (x,y,h). > > But I have learned that if you homogenize a point P = (x,y) in R^2 the > result is a point in R^3 which is defined as: P'= (ax, ay, a). > > How does this fit with the homogenization of the point (x,y,h) to (x/h, > y/h, 1)? Usually, homogenizing a vector means to embed the vector into a projective space. So, when you say homogenize P=(x,y) in R^2, you're saying embedding P in a projective space, which in this case is P^2=(R^3)/~ where a~b if a=kb for some invertible constant k. -- -kira === Subject: Re: Homogenizing coordinates? > In the appendix page 1090 he gives an example of homogenizing a vector > (x,y,h). The result is (x/h, y/h, 1) which is where the line from (x,y,h) to the > origin intersects with the affine plane. He calls this the homogenizing of (x,y,h). But I have learned that if you homogenize a point P = (x,y) in R^2 the > result is a point in R^3 which is defined as: P'= (ax, ay, a). How does this fit with the homogenization of the point (x,y,h) to (x/h, > y/h, 1)? > In homogeneous coordinates, (x,y,z) is the same point as (ax,ay,az) in the projective or affine plane when a /= 0. The line at infinite, the line in which all parallel lines intersect, is { (a,b,0) | a,b in R }. === Subject: Re: Many Solutions Manuals and Ebooks in Electronic (PDF)Format! can you please send me Solution Manual for Electrical Circuits (7/ e) please??????? knash2@hotmail.com === Subject: Re: Basic solution for the one dimensional Laplace/Poisson equation Hello everyone: My question is as follows: I want to apply boundary element method to one dimensional > Poisson equation. Therefore I need to obtain the basic solution of the > Laplace/Poisson equation: frac{d^2 U}{d x^2} + delta(x - x_0) = 0 I've used Fourier transformation to solve this problem, but > run across difficulties in the following integration: U = frac{1}{2pi} int_{-infty}^{+infty} frac{1} > {w^2}exp^{iwx} dw where, w is the variable in fourier frequency domain. I've also used Maple to solve the above integration, but > failed. Would you please help me about this problem, thank you very > much! If all you want is a solution (the Green's function) and don't care > whether you use Fourier transforms or not, the problem is easy; I had > this stuff in undergraduate mathematical physics. In the two regions > {x < x0} and {x > x0}, the equation reads as d^2U/dx^2 = 0, so in each > region the solution is of the form A*x + B: U = A1*x + B1 for x < x0 > and U = A2*x + B2 for x > x0. Integrating the DE from x0-epsilon to > x0+epsilon, we have that U'(x0+) - U'(x0-) = -1, so A2 - A1 = -1. > Also, we want continuity: U(x0-) = U(x0+), so A1*x0 + B1 = A2*x0 + B2. > We still need two more conditions, which must be supplied, somehow, > from outside the DE; they may be problem-context dependent. For > example, we could take A1 = B1 = 0. Interestingly, the Fourier method > does not seem to need any extra conditions; it is not clear to me > why this is. R.G. Vickson Firstly, the piecewise function should be continuous at x=x0, secondly this function should be symmetric about x=x0 because it is the response of an impulse excitation. In this way, I got the right answer. === Subject: Re: Basic solution for the one dimensional Laplace/Poisson equation Hello everyone: My question is as follows: I want to apply boundary element method to one dimensional > Poisson equation. Therefore I need to obtain the basic solution of the > Laplace/Poisson equation: frac{d^2 U}{d x^2} + delta(x - x_0) = 0 Notice that d^2 U / dx^2 = 0 when x =/= x_0, so U will be a piecewise > linear function. One suggestion you might try is U(x) = |x-x_0| I've used Fourier transformation to solve this problem, but > run across difficulties in the following integration: U = frac{1}{2pi} int_{-infty}^{+infty} frac{1} > {w^2}exp^{iwx} dw where, w is the variable in fourier frequency domain. I've also used Maple to solve the above integration, but > failed. Would you please help me about this problem, thank you very > much! That way could be made to work using a principle value What was the > difficulty? are right, this function has to a piecewise function. Previously, I thought this function was to be: U(x) = const, x> tommy1729 a .8ecrit : >>> tommy1729 a .8ecrit : >> where do the zero's lie of integral zeta ? >>> Not to mention the problem of Riemann surfaces, >> what do you take as >>> integration constant? >>>>> Always getinf your questions out of your hat, do >> you? >> I though that he got them from somewhere > else....... >> haha very funny guys >> *rolleyes* >> how about actually answering a question and doing > or explaining math , apart from just making fun of > the OP ?? >> tommy1729 >>> But this was an answer : what about that constant? > (and Riemann surfaces) >>a primitive integral ... >> If you'd think about what you're saying instead >> of trying so hard to be a smartass you'd understand >> when people explain why the question makes no sense. >> we understood that by integral zeta you meant a >> primitive integral. There are several problems with >> that. First, the fact that zeta has a pole with >> non-vanishing residue at 1 means that it does not >> _have_ a primitive integral in various sets; >> the integral is multivalued, just like the >> integral of 1/z. (That's a simple-minded way >> of expressing what Denis is getting at when he >> mentions Riemann surfaces). >> Second, even if we ignore that problem, the question >> still makes no sense, because a given function >> has many different integrals. What are the zeroes >> of integral (x) on the line? Well, integral(x) >> could be x^2 + 1, in which case it has no zeroes. >> Or it could be x^2 - 4, which has two zeroes. Etc. >> ************************ >> David C. Ullrich > David, you just gave me a _wonderful_ idea: I came up with >something I call complex numbers! Cool! Ilias PS. Waiddaminute...It's (1/2)x^2, not x^2 ! Dammit...back to the >drawing board :-( Yeah, someone else already pointed out that I got that wrong. Invalidates the entire argument. ************************ David C. Ullrich === Subject: Re: integral zeta > On 2 Sep., 00:41, tommy1729 On 31 Aug., 12:35, David C. Ullrich > > To tommy1729: > If you insist that primitives are > uniquely >> determined > simply by the > concept of just set C=0, you are going > to > run >> into > some problems. > For example, let f(x )= e^x - x. > Let's integrate f(x) two different ways > .... > method (1) [standard Calc 1 approach]: > int f(x) dx = e^x - x^2/2 + C > Setting C=0 yields e^x - x^2/2 > method (2) [using power series]: > e^x = 1 + x + x^2/2 + x^3/6 + x^4/24 + > ... > f(x) = e^x - x = 1 + x^2/2 + x^3/6 + > x^4/24 + >> ... > int f(x) dx = (x + x^3/6 + x^4/24 + > x^5/120 + > ... > ... ) + C > Setting C=0 yields x + x^3/6 + x^4/24 + > x^5/120 + > 0 + ... > which equals e^x - x^2/2 - 1. > So which one is the unique primitive? > quasi >>intresting argument. :-) >>first note : i have already defined > integral > zeta >> by a series. >>although perhaps there might be discussion > about the >> analytic continuation or riemann surface > .... >>but anyways integral zeta is already > defined... >> First, it _had not_ been defined at the > point > where >> you started insulting people for not knowing > what you >> meant. Second, that series does not converge > except >> for Re(z) > 1, and it has many _different_ >> continuations >> to larger regions - hence that series does > _not_ >> specify where the zeroes in the plane are - > where the >> zeroes in the plane are depends on what > particular >> continuations we're talking about. Saying > that > there >> might be discussion about analytic > continuation >> is progress, but it shows you still have no > idea >> what you're talking about - the question of >> non-uniqueness of analytic continuation is > crucial >> here. >>in your argument, perhaps a good way to > clarify > is >> this >>f(0) = INTEGRAL f(0)dx if possible by > changing > C. >>so f(x) = exp(x) - x >>f(0) = 1 = integral f(0) >>therefore the integral is method(1). >>another reason for avoiding series is the > taylor >> expansion of exp(x^2) at 0 ... >>you know what happens there dont you ? >>a bit off topic perhaps but can you give > nice >> examples of an integral which can take 3 or > more >> values ... >>just for fun :-) >>tommy1729 >> ************************ >> David C. Ullrich >ok , so you FINALLY accepted the series. > Huh? What does it mean to accept a series? >the continuation is arguable yes. >i admit that. >but there are zero's that are in common with > any > possible continuation ; critical ones. >correct me if im wrong about that >( would surprise me ; the wrong , not the > correction ) >( critical is not the same here as usual ; > not > referring to a line but the zero' s that are in > common to all continuations ) >i never claimed to be an expert on > continuation > though ... > In fact you know nothing whatever about it, but > that > doesn't stop you from making statements about > it. > I'll give you a hint, so you can realize > exactly > how absurd you're being here: > Hint: What you're claiming is this: If f is a > function > defined in a certain region and g is defined in > the > same region by g = f + 2 pi i then f and g have > certain zeroes in common. > Indeed. > And to allow tommy1729 to learn something: > zeta(s) has one pole, of residuum 1. > Loosely speaking, this allows us to look at > integral > zeta mod 2pi. > In other words, the research question is: > DEFINITION1 : > f:U->C is called a tommy-function if > - U is a region in C > - 0 in U, 1 not in U > - f holomorphic on U > - f(0)=0 > - f'(s)=zeta(s) fr all s in U > DEFINITION 2: > If f:U->C is a tommy-function, a point z in C is > called > a tommy-point for f if z in U and f(z) is an > integer > multiple of 2pi. > tommy's objective is to study tommy-points. > Trivial LEMMA: tommy-pointnesss does not depend > on > the choice of f (and/or U). > Which allows > DEFINITION 3: > z in C{1} is called a tommy-point if there > exists > a tommy-function of which z is a tommy-point > So, now there is something defined. > Next question: What is interesting about > tommy-points? > hagman hahaha hagman tommy-points lol your making stuff up i never said , just like > david. > > I merely try to make some sense of your original > post. > And this is as close to a definition of > zero of integal zeta as I can get. > i assume its a parody :-) if not , well then you made that up ( tommy points) > ... > > I needed to coin a word for it and (ab?)used your > name - thus acknowledging your first contribution > to the subject. > You would have prefered to speak of zeroes of > integeral > zeta but I invented tommy-points as your > descriptive > term is downright misleading (similarly one should > never > call something a vacuum cleaner if it is unable to > clean > the vacuum). > and yes zeta(1) has a pole since it is the harmonic > series 1+1/2+1/3+... > > zeta(1) has no pole. > zeta has a pole at 1. > Moreover, the fact that > sum 1/n^s > does not converge does not prove that there is a > pole, > otherwise zeta would have a pole at 1/2 as well. > you have outlined a clearer similar version of the > topic , yet ullrich ignores it ... just as he ignored the series i gave him ... instead he invents stuff ala tommy-points :-) > > No, he tries to make sense of your statements as well > and tries (again and again) to convince you that > the statement in the form you gave it does not make > sense. > Simply because being zero is not well-defined for > integral zeta > for two reaons: > - integration constant > - nonzero residue of zeta at 1 > During this thread, You have appeared quite unwilling > to > consider either of these two problems. > > hagman > im sorry hagman i assumed you were joking... what i meant by the integral is the integral term by terms considering the n^x as the terms... that gave the series i have given before ... this should solve the constant problem, not ? as for different continuations , consider the domain where no continuations are neccesary and there are still zero's perhaps ? tommy1729 === Subject: Re: Solving a Recursive Function > I'm having a little difficulty solving this >> equation, maybe someone >> could help? >> a[0]=i, >> a[n]=(1+((1+r/100/c)^c-1)*(1-t/100))*a[n-1] >> i,r,c and t are constants. I need to solve for >> a[n] where n is any >> integer >= 0. >> >> FYI, >> >> For i=20000, r=10, c=12 and t=21.2 it should be >> >> a[0]=20,000 >> a[1]=21,650.28 >> a[2]=23,436.73 >> a[3]=25,370.58 >> ... >> >> I think >> >> what makes you think there is a closed form solution ? That's a good question. Hmm. I wonder if the problem was instead a[0] = 1, a[n+1] = 2*a[n], could we find a closed-form solution for that? Pretty tough... >tommy1729 ************************ David C. Ullrich === Subject: Re: Solving a Recursive Function > > I'm having a little difficulty solving this > equation, maybe someone > could help? > > a[0]=i, > a[n]=(1+((1+r/100/c)^c-1)*(1-t/100))*a[n-1] > > i,r,c and t are constants. I need to solve for > a[n] where n is any > integer >= 0. > > FYI, > > For i=20000, r=10, c=12 and t=21.2 it should be > > a[0]=20,000 > a[1]=21,650.28 > a[2]=23,436.73 > a[3]=25,370.58 > ... > > I think > > > > what makes you think there is a closed form solution ? Maybe the fact that it's a constant-coefficient linear equation? -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Question about logarithms > Opened my math book and it says you cannot takle the >> logarithm of >> negative numbers. Assuming, we forget the complex >> numbers stuff, I can >> clearly see: >> >> -2 ^ 3 = -8, so log of -8 to the base -2 is 3. >> >> Why is this wrong? Is the book assuming that the base >> is always >> positive. Of course, I understand that a positive >> number cannot be >> raised by a power operation to a negative number. >> >> >> Anja >> -2 ^ 3 = -8, so log of -8 to the base -2 is 3. nothing wrong with that at all :-) if the book does not work with complex numbers , it usually does not work with base -2 either ... since log of -8 to the base -2 is ln(-8)/ln(-2) and logaritms of negative numbers are not considered since they are complex ... to avoid complex numbers one can use ln(a*b)=ln(a)+ln(b) so -> ln(-8)/ln(-2) = [ln(-1)*ln(8)]/[ln(-1)*ln(2)] For heaven's sake, you should really try to avoid saying things about subjects you know nothing about. >and we can reduce ln(-1) -> ln(8)/ln(2) = 3 ..as expected... hope this clarifies it for you. tommy1729 ************************ David C. Ullrich === Subject: Re: Question about logarithms > Opened my math book and it says you cannot takle the logarithm of > negative numbers. Assuming, we forget the complex numbers stuff, I can > clearly see: > > -2 ^ 3 = -8, so log of -8 to the base -2 is 3. > > Why is this wrong? Is the book assuming that the base is always > positive. Of course, I understand that a positive number cannot be > raised by a power operation to a negative number. > > > Anja > David Cantrell is the only person you really got it right. In order to understand logarithms of negative numbers, you need to understand complex numbers. And then you find that the logarithm of a number is not uniquely defined. Now if you stick to positive numbers, well, one of the logarithms of a positive number is a real number, and with that proviso it is unique. But when you start taking logs of negative numbers, there is no natural choice for which of the many logarithms you should choose. The choice is called a branch. So, the short answer is that 3 is definitely a logarithm of log(-8) to base (-3), but there are many other answers, and there really isn't any good reason why you should pick the answer 3 over any other answer. Stephen === Subject: Re: Question about logarithms > Opened my math book and it says you cannot takle the logarithm of > negative numbers. Assuming, we forget the complex numbers stuff, I can > clearly see: > > -2 ^ 3 = -8, so log of -8 to the base -2 is 3. > > Why is this wrong? Is the book assuming that the base is always > positive. Of course, I understand that a positive number cannot be > raised by a power operation to a negative number. > Yes, you can define (-2)^3. But how would you define (-2)^(1/2)? With positive numbers there are no restrictions. If a > 0, there is a natural way of defining a^b for any real number _b_. But then a^b will always be positive and therefore it will make no sense to speak of log_a x when _x_ is negative or zero. Jose Carlos Santos === Subject: Re: Question about logarithms Opened my math book and it says you cannot takle the logarithm of > negative numbers. Assuming, we forget the complex numbers stuff, I can > clearly see: -2 ^ 3 = -8, so log of -8 to the base -2 is 3. Why is this wrong? Is the book assuming that the base is always > positive. Of course, I understand that a positive number cannot be > raised by a power operation to a negative number. > Anja > -2 ^ 3 = -8, so log of -8 to the base -2 is 3. nothing wrong with that at all :-) if the book does not work with complex numbers , it usually does not work > with base -2 either ... since log of -8 to the base -2 is ln(-8)/ln(-2) and logaritms of negative > numbers are not considered since they are complex ... to avoid complex numbers one can use ln(a*b)=ln(a)+ln(b) so -> ln(-8)/ln(-2) = [ln(-1)*ln(8)]/[ln(-1)*ln(2)] No! > and we can reduce ln(-1) -> ln(8)/ln(2) = 3 ..as expected... Your method might be called wishful thinking. If we use the principal branch of the (complex) logarithm, then log(-8)/log(-2) = (log(-1) + log(8))/(log(-1) + log(2)) does _not_ reduce to 3, as you expected. Note the + signs, instead of your * signs. But 3 is one of the possible values. To get 3 for log(-8)/log(-2), if you use the principal branch in the denominator, for example, then you have to use a different branch in the numerator: log(-8)/log(-2) = (3*pi*i + ln(8))/(pi*i + ln(2)) = 3. 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Autodesk Raster Design 2006 Autodesk Building Systems 2006 Autodesk Revit Building 8.1 AUTODESK REVIT STRUCTURE 2 VERICUT 5.4 Artist Drums The Denny Jaeger Master Violin Library Steinberg The Grand 2 Steinberg Hypersonic 2 formZ 5.5 PCDJ FX VRM v7.2.1 Logic Express 7.1 Spectrasonics Atmosphere Natural Motion endorphin 2.5.2 Maya Unlimited 7.01 Liquid Instrument Electric Bass With Electric Spectrasonics Trilogy TotalBass Steinberg WaveLab v5.01b VMware Workstation v5.5.1.19175 Linux Geomagic Qualify 8.0 STYLUS RMX VST RTAS FL Studio 6 XXL Edition LabVIEW 8.0 Full Nero 7 Premium Steinberg Virtual Guitarist Electric Edition Cakewalk Project 5 v2 Steinberg Virtual Guitarist VSTi v1.0 ANSYS DesignSpace 10.0 FL Studio 6 Producer Edition Quark XPress 6.5 Passport Swift3D Version 4.5 SmartSound Sonicfire Pro 3 InstallShield 11.5 Toast 7 Titanium Premium Edition ANSYS Workbench V10.0 Yamaha Vocaloid LOLA Real Guitar 1.5 work GarageBand Jam Pack4 GarageBand Jam Pack3 GarageBand Jam Pack2 GarageBand Jam Pack1 Native Instruments Absynth 3.0 DVD Studio Pro 4 Sonic DVDit Pro v6.0 Ansys CFX 10 Ansys CFX-TASCflow V2.12.2 MixMeister Pro 6 Geomagic Studio 7 (with tutorial) Aperture Native Instruments Reaktor 5 Final Cut Pro 5 HKS ABAQUS 6.5 Solid Edge v17 CiberCut 5.6 Max Mathcad 13 Apple Shake 4.0 Eovia Carrara 5.0 Pro Ableton LIVE V5 for MAC 3D Weather Globe GlobalCAD LandARCH 2005 Professional Edition 1.2 PGMusic Band-in-a-Box MegaPAK 2006 for Windows Cakewalk Sonar v5.0 Producer.Edition Logic Pro 7 Adobe Acrobat 7.0 Pro COMSOL Multiphysics 3.2 Ultiboard 8 Cadence SPB 15.2 MovieDV6 Alias PortfolioWall 3.0 Sony ACID Music Studio v5.0a Sony DVD Architect 3.0 Alias Imagestudio 3.0 Unigraphics NX 4.0 (UG NX4.0) Native Instruments Guitar Rig 2 HYBRID MaxonForm v9.103 For Archicad Corel Paint Shop Pro X 10.0 Digital Fusion 5.0 Ulead MediaStudio Pro 8.0 STAR-CD 3.24 Alias MotionBuilder 7 Maple 10 Pro Amadis DVD Ripper v 1.0.5 SIMATIC STEP 7Step7 v5.3 +SP3 Pro Pro/ENGINEER Wildfire 2.0 M150 Maya Unlimited 7.0 FL.Studio.v5.0.0.XXL.Producer.Edition Exceed PowerSuite 2006 ProgeCAD 2006 Pro Steinberg Nuendo v3.1.1.944 Think3 Thinkdesign 2005.1 Boris Red 3GL v3.04 HipHop eJay 6.0 Soundtrack Pro Mac osx Hollywood Edge The Car Chase Scene Set CD1-CD5 Sun StarOffice 8.0 Luxology MODO V102 Luxology MODO V103 Sony Vegas 6.0c Sony Sound Forge 8.0 Geomagic Studio 8.0 Alias PortfolioWall 2.2 Mathematica 5.1.1 Ulead VideoStudio 9 MatheMatica 5.2 CSI Sap2000 9.03 EWQL Symphonic Orchestra Gold Edition Autodesk VIZ 2006 Cimatron E7 Symantec Norton Systemworks 2006 Premier FileMaker Pro v8.0 Advanced Borland Together Architect 2006 FileMaker Pro 8 Painter IX Alias StudioTools 13 Borland JBuilder 2006 Enterprise Autodesk AutoCAD 2006 Avid Xpress Pro HD v5.20 Autodesk 3d Studio Max v8.0 Reason 3.0 Vue 5 Infinite 5.05 Magic Bullet Editors Bundle MAC Ulead DVD Workshop 2.0 Adobe After Effects 6.5 Pro STAAD Pro 2005 ANSYS 10.0 Borland Developer Studio 2006 (Architect Edition) Moldflow MPI 5.0 Autodesk Inventor 9.0 Professional Solidworks 2006 AUTODESK.REVIT.BUILDING.V8.0 Microsoft Windows XP Professional with SP2 Adobe CS2 Premium Plus DVD Adobe Creative Suite 2 Premium Edition(7CD) SOFTIMAGE|XSI v.5.01 Advanced Steinberg Hypersonic Modules XXL Steinberg Virtual Bassist Sibelius 4 Steinberg Groove Agent 2 Guitar Pro5 Cubase SX 3.1 Traktor DJ Studio 3 3D TOTAL TEXTURES 1-15CDs VectorWorks 11.5 (Advanced architectural design) Artcam Pro v8.0 UGS IMAGEWARE NX V12.1 MakeMusic Finale 2006 MATLAB 7.1 LightTools 5.1 +SP1 OrCAD PSpice 10.5 ArchiCAD 9 architectural design Freeway 4 Express Mac Freeway 4 Pro Mac AutoCAD Electrical 2006 Macromedia Studio 8 TurboCAD Professional 11.2 UGS I-deas 11 NX Series Autodesk Architectural Desktop2006 MicroGDS 9.0 Pro ( Architectural Design ) Unigraphics NX 3.0(UG NX3.0)