Search Results

$\newcommand\R{\Bbb R}\newcommand\la{\lambda}$Remark 1: Let \begin{equation*} g(s):=q^T(P+sI)^{-2}q \end{equation*} for real $s$ such that $P+sI$ is invertible. The function $g$ is nonnegative an …

Note that $$I_n^*\le E(X_1+\cdots+X_n)=n/2.$$ This upper bound $n/2$ on $I_n^*$ is attained if $n=2m$ is even and $X_{2i}=1-X_{2i-1}$ for $i=1,\dots,m$. So, $$I_n^*=n/2$$ for even $n$. It also follows …

Note that the constant function $f=c$ satisfies your conditions for any real $c\le-1$. So, $$\inf\int_0^1 f=-\infty.$$ Consider now the same problem but with the additional condition $f\ge0$. Then, …

Under the conditions specified in your post, the best lower bound on $g(a_1,\dots,a_n)$ is $\sum'_{i,j}|f(a_i-a_j)-1/2|$, where $\sum'_{i,j}$ denotes the summation over all pairs $(i,j)$ with $x_i ||x …

Rewrite the constraint as $$x_n \le f_n(\rho_1,\dots,\rho_n):=\ln\Big(B\log_2\Big(1+\frac{e^{\rho_n} g_n^2}{\sum_{i=1}^{n-1} e^{\rho_i} g_i^2+\sigma^2}\Big)\Big).$$ The problem is then to prove the co …

$\newcommand\si\sigma\newcommand\Si\Sigma$If $x$ is centered, then $u\cdot x\sim N(0,\si_u^2)$, where $\si_u^2:=u\cdot\Si u$ and $\Si$ is the covariance matrix of $x$. So, $$\frac{E(u\cdot x)^4}{E(u\c …

$\newcommand\R{\mathbb R}\newcommand\b{\hat\rho(x)}\newcommand\a{\tilde\rho(x)}$This inequality is false in general. For instance, if $\varepsilon=1/10$, $M=1/10$, and for some $x\in\R^d$ we have $\a …

A counterexample is given by $n=2$, $[a_1,b_1]=[-2,1]$, $[a_2,b_2]=[-1,2]$. (Make a picture.) Even if the $n$-box $S$ is required to be a subset of $[0,\infty)^n$, the answer will still be no. E.g., …

$\newcommand\R{\mathbb R}$The answer is no. E.g., let $A=B=\R$, $B_a=[1-a^2,2]$ for $a\in A$, and $\ell(b)=b^3-3b$ for $b\in B$. Then $\ell(b)$ and $B_a$ are perfectly smooth, but $L$ is not different …

Of course, not. Indeed, suppose that $U_k=X$ for $k=2,\dots,p$. Suppose also that $U_k(i,j)=X(i,j)$ for $k=0,1$ if $(i,j)\notin\{(0,0),(0,1)\}$. Then the question becomes the following: Is it true th …

We only need to consider $n\ge5$. Let us move each of the $n$ "equidistant points" on the equator slightly towards one of the two poles of the globe, so that after such a movement the points on the sp …

Your objective function is already convex. This follows because the Hessian matrix $$\frac2{y^3}\,\begin{bmatrix} y^2&-xy\\-xy&x^2 \end{bmatrix}$$ of the map $\mathbb R\times(0,\infty)\ni(x,y)\mapsto …

$\newcommand{\al}{\alpha}\newcommand\la\lambda\newcommand\R{\mathbb R}$Such a construction of $M$ and $\al$ is always possible. Indeed, take any complex $\la$. Rearranging columns and rows of the matr …

$\newcommand{\al}{\alpha}\newcommand{\be}{\beta}\newcommand{\vpi}{\varphi}\newcommand{\thh}{\theta}\newcommand{\I}{\mathscr I}\newcommand{\J}{\mathscr J}$The conjunction of your conditions, \begin{equ …

We shall assume that $l\in(0,\infty)$, so that for any $h\in H_0^1$ we have $$\int_0^l|h|^2=\int_0^l dx\,\Big|\int_0^x h'\Big|^2 \le\int_0^l dx\,\Big(\int_0^l|h'|\Big)^2 \\ \le\int_0^l dx\,l\,\int_0 …