Let $\xi$ be a complex line bundle over a CW-complex $B$. I want to prove that $\xi$ is trivial if and only if $c_1(\xi)=0$. My attempt: Suppose $c_1(\xi)=0$. Then the Euler class $e(\xi)=0$. Since $e(\xi)=o_2(\xi)$, there exists a nonvanishing cross section of $\xi|_{sk^2 (B)}$, denoted as $X\...

Let $X,Y,Z$ be random variables on $\Bbb{R}$ such that $(X,Z)\stackrel{d}{=}(Y,Z)$. I want to show that $X\stackrel{d}{=}Y$. My idea was the following. Let me take any $A\subset \Bbb{R}$ then $$\Bbb{P}(X\in A)=\Bbb{P}((X,Z)\in A\times \Bbb{R})\stackrel{*}{=}\Bbb{P}((Y,Z)\in A\times \Bbb{R})=\Bb...

My question pertains to Ted Shifrin's Multivariable Mathematics Section 8-2.3 Pullbacks. We are to use the following pieces, given in the definition, to build the pullback $\mathbf{g}^{*}\omega\in\mathcal{A}^{k}\left(U\right)$ of $\omega$ by $\mathbf{g}.$ The symbol $\mathbb{I}$ is a multi-index....

Let $s$ be a complex number. $$T(s)=\sum_{n>0} (n^s + n^{-s})^{-1} $$ This is well defined for $Re(s) > 1$. It seems $T(s) = T(-s)$ but then again we have it only defined for $Re(s)>1$ for now. We try analytic continuation by $$\sum_{n>0} (n^s + n^{-s})^{-1} - n^{-s} = - \sum_{n>0} (n^{3s} + n^{...

I think $f(x) = \sum_{n=1}^{\infty}(-1)^n \frac{x^{2n-1}}{(2n)! \ln 2n}$ is totally monotone. Bernstein's theorem and his integral transform convinced me. Am I correct ? Probably related or useful is a function like $f(x,k) = \sum_{n=1}^{\infty}(-1)^n \frac{x^{2n-1}}{(2n)! \ln (2n+k)}$ which foll...

Suppose $A$ and $B$ are diagonalizable matrices. Prove or disprove that $A$ is similar to $B$ iff $A$ and $B$ are unitarily equivalent. Is there a way to disprove this without using the idea of showing: $\Sigma_{i,j}^n|A_{i,j}|^2 = \Sigma_{i,j}^n|B_{i,j}|^2$? I ask because the question that esta...