mm-3239 
 === 
 Subject: Re: Continuous Injections 
 In a small neighborhhod of (x_0,0) we have 0<f_2(x,y)<1.So if we choose  
 x=x_0 and y_0<0 such that |y_0| is sufficiently small, both the inequalities  
 
 hold: 
 0<f_1(x_0,y_0)<1, 0<f_2(x_0,y_0)<1. 
 Unfortunately, these reasonings do not apply to the cases of C^0 and C^1  
 extensions. 
 ======= 
 Subject: MathWorld complaint 
 The MathWorld entry concerning odd perfect numbers starts with this 
 sentence: 
     In Book IX of The Elements, Euclid gave a method for constructing 
     perfect numbers (Dickson 2005, p. 3), although this method applies 
     only to even perfect numbers. 
 I used the COMMENT on this page link in order to say: 
    The second half of the first sentence of this entry is rather silly. 
    It makes no sense to say that Euclid's method *applies* only to 
    even perfect numbers. What happens is that Euclid's method allows us 
    to *create* only even perfect numbers. 
 Then I got a rather rude and moronic reply by a guy named Ed Pegg Jr. 
 Has something like this happened to someone else? I exchanged some 
 e-mails with Eric W. Weisstein in the past, and he always had a nice 
 behaviour. 
 Jose Carlos Santos 
 Subject: Re: MathWorld complaint 
 This guy, Ed Pegg Jr., is a totally rude asshole. That's a well established 
 fact, reported to me by at least two more independent sources. The more you 
 stay away from him, the better you are. You want to correct or add a 
 Mathworld entry? Tough. It's got to go through this asshole. 
 See: 
 http://tinyurl.com/bltj6 
 Eric should kick his useless ass out of Mathworld. 
 --  
 I. N. Galidakis 
 http://users.forthnet.gr/ath/jgal/ 
 Eventually, _everything_ is understandable 
 Subject: Re: mag5 stuff. 
 Let's improve the notation and make crisp the definition. 
 For real t, let r(t) be the real root of x^5 + x - t = 0. 
 Also let's clear the air, there's nothing magic about mag_5. 
 In fact it's much lacking, for what's r(t) when x^5 + x - t = 0 
 has more that one real solution?  It will have 1,3 or 5 real roots. 
 Which one is r(t)? 
 Clearly 
         r(t)^5 + r(t) = t 
         r(t^5 + t) = t 
 Not clear is 
         if x real and x^5 + x = t, then x = r(t) 
 Trival results of the first now obviously 
 r(ab)^5 + r(ab) = ab 
 r(a)^5 + r(a) = a 
 r(b)^5 + r(a) = b 
 r(ab) = r((r(a)^5 + r(a)) (r(b)^5 + r(b))) 
 depending upon the yet to be made precise definition of r(t). 
 Subject: Re: mag5 stuff. 
 Yep. 
 Well, a more rigorous definition can be found by the series formula, 
 mag_5(z) = sum_(n=0...infinity) ((-1)^n z^(4n+1))/(4n+1) C(5n, n), 
 where C(p, q) is the binomial coefficient p choose q. The series, 
 however, only converges for z within a very small disc. 
 Actually, that is clear. 
 Subject: Re: mag5 stuff. 
 This is not a more rigorous definition because it's not defined for all 
 real t.  What the radius of convergence?  1? 
 Prove it.  Prove it for t when the series doesn't converge. 
 Subject: Re: Is the set N of natural numbers well defined? 
 First you'll need to define what you mean by balanced, and then make 
 sure that what you mean to say actually makes sense for infinite sets. 
 Only after you've done that can you even hope to start proving the 
 assertion the braces are balanced for every set. 
 Before you do that, the rest of your reasoning is not even meaningful, 
 let alone correct. 
 If I guess correctly what you mean by written out in the list form, 
 then this is true. 
 - Tim 
 Subject: Re: Is the set N of natural numbers well defined? 
 By S(n) do you mean the successor of n?  If so, then two of your 
 earlier limits fail this criterion. 
 What makes you think that limits are distributive over set 
 difference?  Surely you'd have to prove that first. 
 No set belongs to itself. 
 What makes you think taking the limit is distributive over set union? 
 Another thing to be proved... 
 Exactly.  That's why it's so important to properly define it before 
 trying to prove anything about it. 
 - Tim 
 Subject: Renewal theorem (Feller) AND/OR Number Theory 
 I have had troubles with one theorem for some time, and could not find a 
 solution. 
 Indeed I have two problems. 
 1) There is an elementary theorem in Feller (Ch. XIII) which states 
 Let u_0=1 and define u_n as follows: 
         u_n = f_1*u_{n-1}+f_2 u_{n-2}+....+f_n*u_0 
 The problem is that I understand proof except one thing, Feller states that 
 following theorem is a result of some elementary (!) 
 number theory: 
 Theorem 1.2: Let a_1, a_2,...,a_m be any positive integers such that 
 representation: 
         k=a_1*x_1 + a_2*x_2 + ... + a_m*x_m 
 where x_i are positive integers. 
 The question is - where can I find a proof of this ?! 
 There is similar theorem which says that for above a_i's any number k (not 
 x_i's can be any integers (i.e. positive or negative or 0)) - and I am not 
 able to conclude Theorem 1.2 from this. 
 I also tried to make a proof by myself by induction, it is ok for m=2, but 
 later I have troubles (because gcd{a1,...,a_{m+1}}=1 does 
 not imply gcd{a1,...,am}=1 what I would need for induction) 
 Please give me some solution to this problem, I cannot cope with it:) 
 2) Maybe there is somewhere another proof of Theorem 1.1 ? It is often 
 cited,  but I want it proved. 
 I am looking forward for any clues 
 Pavello 
 Subject: Re: Renewal theorem (Feller) AND/OR Number Theory 
 In your induction step, let k = gcm(a1,..., a_m). For sufficiently large  
 n, you can represent k*n. Then work this in with a_(m+1). 
 --  
 Stephen J. Herschkorn                        sjherschko@netscape.net 
 Math Tutor in Central New Jersey and Manhattan 
 Subject: generalized eigenvalues of wishart matrices 
 Is anyone aware of works dealing with the distribution of the max of 
 the generalized eigenvalues of (A,B) where A and B are both Wishart, 
 Results are known for the non-generalized case (Khatri) but I can't 
 find nothing about the generalized case ??? 
 jonagold 
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 Subject: How can i derive the lognormal & Rayleigh Probability density  
 function ? 
 Hello 
 I want to derive the lognormal pdf & Rayleigh pdf. 
 Anyone can show me that or give me useful URL ? 
 :) 
 Subject: Re: strategy for an instant, elementary proof of FLT 
 (for relatively prime polynomials with rational coefficients). 
 My approach assumes only that f, g, h are relatively prime 
 as polynomials, although not necessarily coprime (relatively 
 prime in pairs), and that their coefficients are rational. 
 The nature of k (e.g. whether or not it is a p-th power) 
 isn't involved. 
 For odd p, given your supposed identity, in the plane x + y = 0 
 the surface f^p + g^p + h^p = 0  [1] has a p-fold root at z = 0. 
 (where @ denotes a Greek alpha). 
 Hereafter f, @f/@z, .., evaluated at z = 0 are polynomials in 
 one variable, say x, to which the deg function refers. 
 (As polynomials in x, z, deg(@f/@z){z=0} _equals_ deg(f){z=0}. 
 [1] and [2], considered as a pair of linear homogenous 
 equations in f^(p-1), g^(p-1), h^(p-1), then imply for 
 some pair of relatively prime polynomials u, v: 
 either: 
    u.f, u.g, u.h = v.@f/@z, v.@g/@z, v.@h/@z 
 or: 
    u.f^(p-1) = v.(g.@h/@z - h.@g/@z) 
    u.g^(p-1) = v.(h.@f/@z - f.@h/@z) 
    u.h^(p-1) = v.(f.@g/@z - g.@f/@z) 
 In either case, because f, g, h are relatively prime, we can 
 assume that v is constant. 
 Denoting d_f := deg(f), the first case requires d_u + d_f = d_f, 
 so that u is constant and consequently f must be a multiple (by 
 a polynomial in x) of e^(v.z/u), which is impossible. 
 In the second case, we have: 
    d_u + (p-1).d_f <= d_g + d_h 
    d_u + (p-1).d_g <= d_h + d_f 
    d_u + (p-1).d_h <= d_f + d_g 
 Intuitively, the symmetric differences on the RHSs can reduce 
 their overall degree by cancellation down from (typically for 
 the first) d_g + d_h to the corresponding LHS degree; but the 
 RHS degrees cannot exceed the maximum of the degrees of their 
 individual terms (whose degrees as it happens are equal). 
 Adding these gives the following, with equality if and only if 
 equality holds separately for each of the above: 
   3.d_u + (p-1).(d_f + d_g + d_h) <= 2.(d_f + d_g + d_h) 
 which forces d_u = 0 (u constant) and p = 3, and equality 
 so that the triplet above gives: 
    d_f = d_g = d_h   = d say 
 f, g, h must all be non-zero integers, thereby giving a non-zero 
 integer solution to f^3 + g^3 + h^3 = 0, which is impossible. 
 Otherwise f, g, h are constant, and hence their _original_ 
 equivalents in your identity are each polynomials in x + y 
 and z alone. 
 But the whole proof can be repeated on the plane x + z = 0, 
 at y = 0, so we conclude that the original f, g, h are also 
 polynomials in x + z and y alone, and consequently they are 
 each polynomials in x + y + z alone (exercise for the reader!) 
 render the LHS of your supposed identity constant, whereas 
 the factor x^p + y^p + z^p of its RHS could become arbitrary 
 large which, since all factors are integers, is impossible. 
 Note that for p = 2 one can obtain forms of your identity, 
 by using the quaternion norm formula: 
   f^2 + g^2 + h^2 + k^2 = 
     (p^2 + q^2 + r^2 + s^2).(x^2 + y^2 + z^2 + 0^2) 
 where p, q, r, s are polynomials in x, y, z chosen to render 
 s as a suitably signed inner product of them with x, y, z, 0 
 as zero, e.g. for any polynomial F(x,y,z): 
    p, q, r, s  =  2yz, - xz, - xy, F(x,y,z) 
 whereupon say: 
   k = p.x - q.y - r.z - s.0 = 0 
 [See http://mathworld.wolfram.com/Quaternion.html ] 
 John R Ramsden 
 Subject: Re: strategy for an instant, elementary proof of FLT 
 I haven't had time to review your argument but if it's right -- bravo! 
 quasi 
 Subject: Analysis question 
  Hello everybody, 
  I look for an answer to the following question. Let U(x) be continous 
 bounded function on (0,1) and 
 it true that U(0+)exists? 
 mladensavov@yahoo.com 
 Subject: Re: Analysis question 
 i.e. h=1/2.  by U(0+) I assume you mean lim from above of U(x).  It  
 doesn't exist here. 
 Subject: Re: Analysis question 
 How do you get this? It isn't at all obvious to me that for small 
 s, int_0^s sin(1/x) dx is anything like s/2. 
 Subject: Re: Analysis question 
 i should have used |sin(1/x)|.   My reasoning is that the oscillations 
 become so tight that adjacent peaks and valleys have nearly the same  
 area so that half the area is filled.  Maybe I'm wrong. 
 Subject: Re: Analysis question 
 If you use sin^2(1/x), the reasoning is correct. 
 For |sin(1/x)|, the value is like 2s/pi, and for sin(1/x),  
 the integral is O(s^2). 
 --  
 This address is for information only.  I do not claim that these views 
 are those of the Statistics Department or of Purdue University. 
 Herman Rubin, Department of Statistics, Purdue University 
 hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558 
 Subject: Re: Analysis question 
 Anyway I think that (Int_0^s sin(1/x))/s converges to 0 .I asked the 
 question whether the statement is true for U positive ,but if that with 
 see that what I have asked David Ulrich is clear. 
 Subject: Re: Analysis question 
 No. 
 ************************ 
 David C. Ullrich 
 Subject: Re: Analysis question 
  And what about if U is positive? 
 Subject: Stanley Tennenbaum 
 At the website http://www.cs.ru.nl/~freek/comparison/comparison.ps.gz 
 is an essay with a preface by Dana Scott in which he refers to the late 
 Stanley Tennenbaum. Has anyone published a biography of Stanley  
 Tennenbaum? 
 --  
 Ignorantly, 
 * Disclaimer: I am a guest and *not* a member of the MIT CSAIL. My actions  
 and 
 * comments do not reflect in any way on MIT. Also, I am nowhere near  
 Boston. 
 Here is a (loosely) related question this reminds me of... 
 f(x) = x+1 and g(x)=x^3 generate a free group? 
 If not, we would have a previously unknown identity of real numbers. 
 If so, we would get a simple example of something in logic (I forget 
 what). 
 At one point Zassenhaus published a paper claiming to prove this is 
 free.  But George Bergman said the paper is, at best, incomplete. 
 (My own knowledge of Galois theory is not good enough to judge.) 
 So, this may still be an open problem... 
 --  
 G. A. Edgar                                
 http://www.math.ohio-state.edu/~edgar/ 
 Subject: Re: Representation of Functions 
 On Tue, 20 Sep 2005 12:25:00 -0700, The World Wide Wade 
 Yeah, I got that far, which is why I changed my yes to no. 
 Well duh, why didn't I think of that. 
 (For anyone wondering why a compact set is nowhere dense: 
 a compact set is (locally) equicontinuous, and it's 
 clear that an open set in C(R) is not equicontinuous.) 
 ************************ 
 David C. Ullrich 
 Subject: Re: Representation of Functions 
 Maury 
 Subject: Re: Representation of Functions 
 s(x) = k p(x-k) for x in [k,k+1), k in Z. 
 Subject: Coset leader of a coset code of RS codes 
 I need some help on a problem regarding coset codes of Reed-Solomon 
 codes (using GF(2^m)). I will quickly describe the problem. 
 Define the same_symbol_weight of a codeword c as the maximum number 
 of same symbols contained in c. 
 Define the same_symbol_weight of a codebook C as the number of same 
 symbols of c_s, where c_s is a codeword in C containing the maximum 
 number of same symbols. 
 As an example, the same_symbol_weight of the all-zero codeword of an 
 (n,k) RS code is equal to n.  Also, the same_symbol_weight of the 
 all e codeword (e in GF(2^m)) is equal to n.  Thus the 
 same_symbol_weight of an (n,k) RS codebook is equal to n. 
 Through simulation, I have found that the minimum same_symbol_weight of 
 (C + h) is equal to k.  I also found that coset leaders that will give 
 a same_symbol_weight of k can be constructed by either (x - alpha^1)(x 
 - alpha^2)...(x-alpha^{n-k-1}) or (x - alpha^2)(x - alpha^3)...(x - 
 alpha^{n-k}), given that the RS generator polynomial is (x - 
 alpha^1)(x - alpha^2)...(x - alpha^{n-k}).  [I assume that if the 
 generator polynomial is constructed with the first factor as (x - 
 alpha^{i+1}), the coset leaders should be adjusted accordingly 
 (leaving out the first or last factor) and having the same results, 
 although I haven't test it yet...] 
 How can I prove that the above coset leaders will always give a 
 same_symbol_weight(C+h) of k?  (C is the RS code, h is one of the coset 
 leaders shown, and same_symbol_weight(C+h) is the same symbol weight of 
 the coset code (C+h)). 
 Any help will be greatly appreciated, 
 Jaco Versfeld 
 Subject: Re: PDE book recommendation 
 I do like 
 Partial Differential Equations : An Introduction  by Walter A. Strauss 
 Some people dont like it as it will not spell out every single step for  
 you. 
 If you are willing to put in the work using this book will pay off. But  
 that 
 is not you dont buy it get it out of the lib. 
 Analytic Methods for Partial Differential Equations by G. Evans, J. 
 Blackledge, P. Yardley. 
 In some ways it kinda throws all types of Method of solution methods at  
 you. 
 It also starts of with a botany of the different types of PDE not specfic 
 examples. 
 Applied Partial Differential Equations by J. David Logan 
 This does give you a more application oritated look at some PDE's it could 
 be hard to get into if you are just starting PDEs and is not that extensive 
 in the coverage. It is good to have a look at. 
 You may want to have a look at some of the Dover books i am sure they have 
 many titles on PDE's (They are also cheap). 
 If you just want solutions and examples to look at i think there should be  
 a 
 Schaums book on the subject. 
 Be sure to see what books are sitting in the Maths lib at uni. Sure its not 
 to have a couple of good books but nothing beats doing problems esp the 
 non-trivial questions. 
 Hape that helps. 
 Subject: Efficient algorithms for multidimensional patterns 
 Hi all, 
 I'm interested in finding out what are the most efficient algorithms 
 (in terms of time) for the following two problems: 
 1.      determining the convex hull for a set of points 
 2.      determining if two convex sets intersect 
 but I'm interested in the theoretical results (O, Omega, Theta 
 complexities) for larger number of dimensions. 
 How about incremental versions? If someone has links to related 
 Lucian.