All Questions

Please help me to prove $$ \sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad \...

Let $A$ be a continuous $2\times 2$ matrix-valued function on $[0,1]$. Define $X_A$ as the solution of $$ X_A'(t) = A(t) X_A(t), \qquad X(0) = I. $$ In other words $X_A$ is the ordered exponential of $...

Given bounded positive functions $f, g, h \in L^1$, with $g, h$ finitely supported, I would like to compare the following two expressions: $$\begin{aligned} a_1 &= \int_{0}^{\infty} \! dx \, f(x) \...

$\newcommand\norm[1]{\lVert#1\rVert} \newcommand\opnorm[1]{\norm{#1}_{\text{op}}} \newcommand\Frnorm[1]{\norm{#1}_{\text F}}$Define \begin{align*} S_t=\{ (A,B)\in\mathbb{R}^{n\times n}\times\...