Recent content by mikepol

Hi, I ran into conflicting definitions of integral domain. Herstein defines a ring where existence of unity for multiplication is NOT assumed. His definition of integral domain is: "A commutative ring R is an integral domain if ab=0 in R implies a=0 or b=0" I looked in 3 other books and...

Hi Petr, Wow!... This is a type of solution I was looking for but couldn't get myself. I don't think I could have come up with this idea of subtracting k/n^2, such that g(k,n) can be bounded by g(n,n), but still being o(1/n), so that their total contribution goes to zero. What sort of...

Oh! I've spent quite a lot of time on this problem, and right after I posted this I got a solution, but it's ugly, I'm not sure if Knopp would have liked it :) Basically I expressed this as a product and then divided it into d groups, where d is some integer held constant for now. Each group...

Hi, I've been skimming through Knopp's book "Theory and Applications of Inifnite Series", mostly to get some practice with sequences/series. The problems there are pretty hard, I've been trying to do this one without much success. It is from Chapter 2, 15(b): show that the following sequence...

g_edgar thanks for your hints, I think I have a solution now. By extreme points you mean the points on L1 unit ball that maximize L2 norm. By using that and drawing a rhombus, I finally saw how to produce a contradiction by showing that if transformation is not a complex permutation matrix, then...

Ok I think I finally got it. I will post my solution later today.

g_edgar thanks for your answer. I'm not sure what you mean by "extreme points" of a unit ball. For example, in real 2D space, unit ball for L1 norm is a rhombus. You are saying that its extreme points are plus/minus the x and y unit vectors. But I don't see why the image of a unit vector...

Hi, I've been trying to show that the set of matrices that preserve L1 norm (sum of absolute values of each coordinate) are the complex permutation matrices. Complex permutation matrix is defined as permutation of the columns of complex diagonal matrix with magnitude of each diagonal element...

Hi JG89, This is probably not the answer you were looking for. What you are describing is usually referred as Abel's test for convergence of the series in the form $\sum a_n b_n$. The formula used in the proof is called Abel's partial summation formula. Every text that I've seen proves this...

Hi, First of all, sorry if my original post confused anyone, I should have thought more before I wrote. The problem I was trying to solve wasn't given in context of Dirichlet's theorem, so I didn't put much thought after generalizing it. I know that proving that there are infinite number...

Hi, It might be that the above statement is not any weaker after all. Please correct me if I'm wrong: Suppose we proved that arithmetic progression a+bk, k=1,2,... contains at least one prime. Suppose that prime p is formed when k=q, so p = a+bq. Then form another progression a+(k+q)b, where...

Hi, Dirichlet's theorem states that any arithmetic progression a+kb, where k is a natural number and a and b are relatively prime, contains infinite number of primes. I'm wondering if there is an easy proof of a much weaker statement: every arithmetic progression a+kb where gcd(a,b)=1...

Hi SixNein, It is not clear what you mean here. You don't have probability of switching, you either switch or you don't, with probability of 1. The confusion with this problem stems with people analyzing the strategy of switching and not switching at the same time. The probability that you...

Hi Focus, thanks for pointing this out. Sorry for the confusion, I didn't use this term in a prcise way. What I want is get rid of the sum. However, factorials are still allowed. Is this possible for this problem? Thanks!

Thanks CRGreathouse. The formula for all b bins having at most 1 ball after k tossings that you wrote is: \frac{(b)_k}{b^k} So the probability that at least one collision occurs in k throwings (that at least one bin will contain at least two balls) is: 1-\frac{(b)_k}{b^k} But that...