Please help me with the partial differentiation of a matrix elementwise
BackgroundHelp me calculate the triple summationProblemWe want to show that$$\frac{\partial}{\partial\xi_i}\left[\sum_{i=1}^k\sum_{j=1}^k a_{ij}(\bar{x}_i-\xi_i)(\bar{x}_j-\xi_j)\right] =...
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Prove that $|z^2+1|\le 2$ implies $|z^3+3z+2|\le 6$
Show that $$\{z \in \mathbb{C}: |z^2+1|\le 2 \} \subseteq \{z \in \mathbb{C} : |z^3+3z+2|\le 6 \} \tag{*}$$(In other words: Let $z\in \mathbb{C}$ satisfy $|z^2+1|\le 2$. Prove that $|z^3+3z+2|\le...
View ArticleAny hints on how to compute the given limit manually?
I want to compute the given limit when $x$ tends to $n\pi$ from right and $n\in\mathbb{N}$; all the parameters involved $g,u,w,z$ are real.$$ \lim_{x\to (n\pi)^+} \frac{-x}{\sin x}\sqrt{\left(...
View ArticleSequential divergence criterion for functional limit, diverging function...
According to Abbott 4.2.5 "Divergence Criterion for Functional Limits",Let $f$ be a function defined on $A$, and $c$ be a limit point of $A$. If there exist two sequences $(x_n)$, $(y_n)$ with $x_n...
View ArticleIntegrability of derivatives to deduce integrability of the function
$\varphi:[0,1] \to [0,1],$ is an absolutely continuous and increasing homeomorphism. If we have $$\int_0^1 \frac{\log ^2 \varphi^{\prime}(x)}{|x-1|^2} d x<\infty,$$then do we have $$\int_0^1...
View ArticleIs this true: $\left| \|A\|_{\infty} \|x\|_{\infty} - \|B\|_{\infty}...
I am trying to lower bound the following norm in the Euclidean space\begin{align}\left \|Ax + By \right\|_{\infty},\end{align}where $A, B \in \mathbb{C}^{m \times n}$ and $x, y \in \mathbb{C}^n$.I made...
View ArticleA closed subset of separable normed space $X$ is separable.
Let $(X,||\cdot||_X)$ be a normed separable space and let $F$ be a closed subset of $X$, then $F$ is separable.My attemptLet $D\subseteq X$ countable and dense, that is $cl_X(D)=X$, then $D\cap...
View ArticleQuestions about functions with nonzero derivatives almost everywhere
I have two questions about the properties of the functions with nonzero derivatives almost everywhere.Let $f(x)$ and $g(x)$ be two functions where $f'(x)\neq 0$ and $g'(x)\neq 0$ a.e.Would $f(x)g(x)$...
View ArticleFinding the power series of a rational function
In many combinatorial enumeration problems it is possible to find a rational generating function (i.e. the quotient of two polynomials) for the sequence in question. The question is - given the...
View ArticleAny hint on how to show that the given limit (from left and right) tends to...
I have this limit for $x\to n\pi$ with $n\in\mathbb{N}$; the parameter $g$ is real. Numerically, I see that, in general, the limit for $x\to n\pi$ is indeterminate since the left and right limits are...
View ArticleFind a bounded real sequence, whose range has exactly one accumulation point,...
Example of bounded sequence on $ \mathbb{R}$ s.t. the set $\{X_n : n \in \mathbb{N}\}$ has exactly one accumulation point but $X_n$ is not convergentMy thoughts: Since Xn is bounded it has convergent...
View ArticleMaximum point of a function about binomial coefficients
For a fixed positive integer $m$, define $$g(d)=\sum_{j=d+1}^m\binom{m}{j}\binom{j-1}{d},\ \ 1\leq d\leq m-1.$$I used MATLAB to calculate the value of $g$ and it seems that $g(d)$ is maximal iff...
View ArticleContinuity on the torus
Let $\mathbb{T}^d$ be the $d$-dimensional torus, for all $r >0,\eta_r(x):=e^{-r|x|^2},x \in \mathbb{T}^d,$ we denote by $\mathscr{F}^{-1}\eta_r$ the inverse Fourier transform defined by...
View ArticleA certain distance-related map is well-defined
I am trying to understand this answer.Context: we want to show that there exists a retract $C\to A$ where $C$ is the Cantor set and $A\subseteq C$ is a nonempty closed subset.They suggest to take a...
View Article2-D Real-analytic function (power series) bounded above and below by a...
Let $B_{\delta}\in \Bbb{R}^2$ be a closed ball of radius $\delta < 1$ be centered at $(0,0)$. Let $f:B_{\delta} \to \Bbb{R}_{\geq 0}$ be real-analytic. Moreover assume that the only zero of $f(x,y)$...
View ArticleMinimums of 2-D function form a continuous function itself.
Let $B_{\delta}\subseteq \Bbb{R}^2$ be a closed ball of radius $\delta < 1$ centered at $(0,0)$. Let $f:B_{\delta}\to \Bbb{R}_{\geq 0}$ be real-analytic and have only one zero in $B_{\delta}$,...
View ArticleSufficient condition for moving limit?
If we are told that the integral$$\int f(x,y)dx$$converges, then can we say that$$\lim_{y\to0}\int f(x,y)dx = \int \lim_{y\to0}f(x,y)dx$$?If this isn't enough information, what do we need to know in...
View ArticleProof check that lim sup $(c_n^{\frac{1}{n}})_{n=m}^{\infty} \leq$ lim sup...
Let $(c_n)_{n=m}^{\infty}$ be a sequence of positive numbers.$L$ := lim sup $(\frac{c_{n+1}}{c_n})_{n=m}^{\infty}$. Choose any $\epsilon > 0$, this implies that for some $N \in \mathbb{N}$ we have...
View ArticleInjective continuous function [closed]
I recently came across this question.Let $\gamma : [0,1) \longrightarrow \mathbb{R}^k, k \geq 2$ be a injective and continuous function such that $Im(\gamma) \cap \omega(\gamma) = \emptyset$,...
View ArticleAbel's theorem infinite case
I would like to prove the second remark in the Wikipedia article of Abel's theorem:Let $a_k $ be real numbers. If $\sum_{k=0}^{\infty} a_k = +\infty $ then $$\lim_{z\to 1^-} \sum_{k=0}^{\infty} a_k z^k...
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