divergence of split sequences with $\sum a_i + \sum b_i = \infty \Rightarrow...
A given series $(x_i)_{i=1}^{\infty}$, is split into $(a_i)$ and $(b_i)$ with $(x_i) = (a_i) \cup (b_i) ,$ such that $\lim a_i = \infty,$ and $\lim b_i = 0.$If then $\sum_{i=1}^{\infty} a_i +...
View Articleshow that $x\longmapsto \arg \min\|x-y\| $ is continuous
Let $\varnothing\ne M\subset\mathbb R^n$ be a convex compact set, suppose $M\subseteq B_r(u)$ (closed ball).Then the map\begin{align}f:\ B_r(u)&\longrightarrow M\newline x&\longmapsto f(x) =...
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Proving legendre transform exists locally if $f$ twice differentiable,...
BackgroundsLegendre transform is defined as follows:And the problem I am trying to solve follows:My thoughtsI searched up to acquired that the matrix concerned is a hessian matrix.Also, the previous...
View ArticleGeometric Interpretation of the Jacobian Matrix and Its Eigenvectors
I understand that for scalar-valued functions $g: \mathbb{R}^n \to \mathbb{R}$, the gradient represents the direction of maximum ascent. Similarly, for vector-valued functions $f: \mathbb{R}^n \to...
View ArticleWhat does it mean that the complement of the rationals has empty interior?
This question refers to How to prove closure of $\mathbb{Q}$ is $\mathbb{R}$I want to prove that the closure of $\mathbb{Q}$ is $\mathbb{R}$. I am trying to understand the accepted answer, but when it...
View ArticleInterpretation of a sigma-algebra not generated by a random variable.
I am wondering what is the interpretation of a sigma-algebra that is not generated by a random variable?For example if we have a random variable $X$ on a probability space $(\Omega, \mathcal{A},P)$...
View ArticleSeeking Analytical Solution for Constrained Optimization Problem with...
I am struggling to solve the following optimization problem\begin{aligned} \min_{\{u_{1},u_{2},u_{3}\}} & \quad10u_{1}+15u_{2}+5u_{3} \\ \text{s.t.} \quad & \frac{1}{u_{1}}+\frac{1}{u_{2}}+...
View ArticleProve that $\lim_{n \to \infty} \frac {1}{\sqrt n}=0$.
Prove that $\lim_{n \to \infty} \frac {1}{\sqrt n}=0$. Attempy: $\forall \varepsilon>0 $, we have to find $M\in N$ such that $|\frac {1}{\sqrt n}-0|<\varepsilon$ for $n \ge M$. Let $\varepsilon...
View Articlean infinite set S to itself is one-to-one if and only if it is onto [closed]
prove that a function from a finite set S to itself is one-to-one if and onlyif it is onto. Is this true when S is infinite?the foregoing question is from contemporary abst algebra by joseph gallian...
View ArticleContinuity and Surjectivity of Distance-Expanding Functions
Let $K\subseteq \mathbb R$ be non-empty and $f: K\to K$ be continuous such that $$|x-y|\leq |f(x)-f(y)|~\forall x,y\in K.$$ Which of the following statements are true?$1.$$f$ need not...
View ArticleA continuous function can be bounded from above by a smooth 'adapted' function?
Consider a continuous function $f:\mathbb{R}\to\mathbb{R}$. Is there a standard strategy that produces a function $g\in C^{\infty}(\mathbb{R},\mathbb{R})$, such that$f(t)\le g(t)$$\forall t$;$g(t)$ can...
View ArticleAsymptotic analysis of the finite product
I have the following product$$\prod _{n=1}^{\frac{L}{2}} \frac{e^{-\frac{2 \cos \left(\frac{\pi n}{L+1}\right)}{T}}+1}{e^{-\frac{2 \cos \left(\frac{\pi n}{L}\right)}{T}}+1}$$I am interested in...
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Question:Find $\;\displaystyle\lim_{t\to0^+}\,t\!\!\int_{t^2}^t\frac{\cos(x)}{x^{3/2}}\,\mathrm{d}x$.Attempted solution:Let: $$f(x)=\begin{cases}\frac{\cos x-1}{x^{3/2}}&\text{if...
View ArticleContinuity of optimal value of a functional on a Hilbert space
Let $f:X\times Y\to \mathbb{R}$ be a bounded continuous function on the product topological space $X\times Y$. It is well known that if $Y$ is compact, then the infimum function $g(x) = \inf_{y\in...
View Articlewhether lipschitz continuous in one variable and continuous in another...
I am considering a two-variable function\begin{align*}f:X\times Y&\to\mathbb{R}.\end{align*}Assume that $f(\cdot,y)$ is Lipschitz continuous for any $y\in Y$ and $f(x,\cdot)$ is continuous for any...
View ArticleFor any measurable space $(X,\mathscr{M})$, are all weighted counting...
I have been given the following definitions:A measure $\mu$ is a weighted counting measure on $(X,\mathscr{M})$ if there exists a function $f:X\rightarrow[0,\infty]$ such that $$\forall...
View ArticleUnconditional / Conditional Convergence of Fourier Series
Let $f \in L^1[0, 1]$, then we know we can write $$f = \sum_{n \in \mathbb{Z}} \hat{f}(n) e^{2 \pi i n t},$$ where we have convergence in $L^1$-norm. My question is, do we have unconditional...
View ArticleShowing a convolution is well defined
Let $f: \Bbb{R}\to \Bbb{R}$ and $g: \Bbb{R}\to \Bbb{R}$ be continuous functions, where $g(x) = 0$ for all $x \notin [a,b]$ for some interval $[a,b]$.a) Show that the convolution...
View ArticleShow that $2xy+\frac{1}{x}+\frac{1}{y}$ attains global minimum
Let be $f:]0,\infty[\times]0,\infty[\to\mathbb{R}$ where $f(x,y):=2xy+\frac{1}{x}+\frac{1}{y}$. We already know that $f$ has only one local minimum at...
View ArticleCharacterizing the asymptotic properties of $f(k)>\frac{ak^2}{k-1}$
Context: Let $a>0$ be some given constant. Let $f:\{2,3,\text{...}\}\to\mathbb{R}_+$ be some increasing function. Consider the following inequality:$$\qquad f(k)> a\frac{k^2}{k-1}. \tag{$*$} $$I...
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