Recent content by acarchau

Let 1, z_1, \dots z_{n-1} be the roots of unity. We want to find : \prod_{i=1}^{n-1} \frac{1}{2i} (z_i - \frac{1}{z_i}), or \left(\frac{1}{2i}\right)^{n-1} \times (-1)^{n-1} \prod_{i=1}^{n-1} \frac{(1 -z_i)(1+z_i)}{z_i} . Now, if P(z)=z^{n} -1 , then, for z \neq 1...

Let me try, \frac{1}{\Gamma(z)} is entire and has zeroes at -1,-2, \dots , so \frac{1}{\Gamma(\frac{-1}{1-z})} would have 0's at 1-\frac{1}{n} and since the pole closest to 0 is at 1, the radius of convergence is 1. Does that look right? [Added later] f(\frac{1}{1-z}) where f is an entire...

Suppose I have an analytic function, f, with radius of convergence 1. From http://planetmath.org/encyclopedia/ZerosOfAnalyticFunctionsAreIsolated.html" it follows that any 0 of f in \{ z : |z| < 1 \} is isolated. But is it possible that the zeroes have a limit point on the boundary of...

I have a proof which uses the first two derivatives, btw the polynomial consists of the first few three terms of the taylor expansion of (x+1)^{x+1}. I was curious if someone could come out with a better proof.

Prove/disprove the following: For x \geq 0, (1+x)^{1+x} \geq 1+x+x^2 .

You are correct about the above matrix. Try to figure out why (hint: determinants). What do you think about the following matrix: A=\begin{pmatrix}1&10\\ 10&-1\end{pmatrix}

I think this also implies that the eigenvalues of A dominate the eigenvalues of B in the following way: if \lambda_i(A) is the i^{th} largest eigenvalue of A and \lambda_i(B) is similarly defined. Then \sum_{i=1}^{k} \lambda_i(A) \geq \sum_{i=1}^{k} \lambda_i(B) for k=1,\dots,n. This...

My linear algebra is very rusty so be careful of the argument below. A-B is p.s.d . I am assuming the matrices are real. Now from the given condition we have for any orthogonal matrix U, the diagonal elements of U^T (A-B) U are non negative. In particular choose U to be the orthogonal...

I meant an eigenvector. My problem was with the claim Ax was an eigenvector of B when x was an eigenvector of B, even though it was not obvious to me why Ax was not the zero vector.

Right. That is a simple counterexample.

I can't follow an argument in Horn and Johnson's Matrix analysis in a suggestion (actually an outline of a proof) that follows problem 8 following section 1.3 (pg 55 in my copy). They argue that if A and B are complex square matrices of order n which commute, and if all eigenvalues of B are...

My take is this: 1. X_n is constructed by dividing [0,1] into 2^n equal intervals and dropping alternate ones, the idea being that only half of any "good" set lies in X_n, upto a resolution. This makes X_n to be the set of points having their nth bit 0, it is a disjoint union of 2^{n-1}...

Is X_n the set of all x in [0,1] whose nth bit, after the "decimal" point, is 0 in the binary representation?

the approach appears sound

You are correct, thanks!