Curl of a vector field belongs to the tangent space of a manifold
While studying Analysis on Manifolds in Elon Lages book, in the Chapter of Stokes Theorem there was this problem:Let U $\subseteq$$\mathbb{R}^3$ be a open set and consider $F: U \rightarrow...
View Articlehow to find explicit formula for $f_{n+1}=af_n+\frac{b}{f_n}$?
I tried to find an explicit formula for the recurrence relation$$f_{n+1}=af_n+\frac{b}{f_n} , f_0 \ne0$$I will show what I got in five casesCase1for $f_0=-\frac{\sqrt{b}}{\sqrt{1-a}}\ne0$ I got that...
View ArticleIs $L^p$ linear for $0
The following is an exercise from Bruckner's Real Analysis:Show that for all $0 <p< \infty$ the collections $L^p$ of measurablefunctions defined on a measure space $(X, \mathcal{M},μ)$ such that...
View ArticleClik here to view.
Solution Review: Evaluating Monotonicity of Solutions to IVP
I'm studying a problem from an old ODE exam and I have some general ideas on how to solve it but I feel as though these arguments are kind of heuristically clear but wanting in formality. We're given...
View ArticleIf the slope of a measurable function is always irrational, is the function...
If $f : \mathbb{R} \rightarrow \mathbb{R}$ is a Lebesgue measurable function, such that all distinct $x,y \in \mathbb{R}$ we have that:$$\frac{f(x)-f(y)}{x-y} \not\in \mathbb{Q}$$Can we conclude that...
View Articlewhat is Q-regular Banach space? [closed]
https://ideas.repec.org/a/hin/jjmath/5938029.html in this article about The Quasisymmetric Mappings under the QuasihyperbolicMetric in Real Banach Spaces they mentioned about Q-regular Banach space. I...
View ArticleWhy does $C=\{x\in\mathbb{R}:\nu(\{x\})\neq0\}$ belong to...
Let $\nu$ be a finite measure on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$. Let $C=\{x\in\mathbb{R}:\nu(\{x\})\neq0\}$. I want to show that $\nu$ can be decomposed into the sum of a discrete meausre, a...
View ArticleInterchanging differentiation and double integral
Consider the double integral $$I(a)=\int_0^1\int_0^1 f(x,y,a)dxdy$$What are the conditions needed so that we can differentiate under the integral sign of this double integral? That is when do we have...
View ArticleShow a parametric function is convex
Let $0<u<\sin x<x<\frac{\pi}{2}$, and$$f(x)=\frac{x}{u}\left(\frac{1}{(\sin x) - u}-\frac{1}{(\sin x) + u}\right)-\frac{1}{x-u}-\frac{1}{x+u}.$$Show that $f(x)$ as a function of $x$ with...
View ArticleDoes this theorem hold under specific conditions and not in general?
In baby rudin, Thm 2.34, states:Compact subsets of metric spaces are closed.The proof goes as :Proof Let $K$ be a compact subset of a metric space $X$. We shall provethat the complement of $K$ is an...
View ArticleProve using epsilon - delta
let $f(x) = \dfrac {x-\sqrt x} {5x-4}$prove using $\epsilon-\delta$ that $\lim_{x\to1} f(x) = 0$I got it to -let $\epsilon >0$.choose $\delta = min(0.1 , \dfrac \epsilon 2)$Than for all $\lvert x-...
View Articleconvergence of $\sum_{n=1}^{\infty}\left(\frac{1}{n}-\sin\frac{1}{n}\right)^p$
I want to find all $p\in \mathbb{R}$ such that the following series converges:$$\sum_{n=1}^{\infty}\left(\frac{1}{n}-\sin\frac{1}{n}\right)^p.$$We know $$\frac{1}{n}-\sin\frac{1}{n}\leq...
View ArticlePositive linear functional: If there is one, there are infinitely many
Let $\mathbf{y}=\{Y_{n}\}_{n=1}^{\infty}$ be a sequence in aninfinite-dimensional Lebesgue space $L^{p}(\mathsf{\Omega},\mathcal{F},\mathrm{P})$, $p\geq1$, on some probability space...
View ArticleClosed form: show that $\sum_{n=1}^\infty\frac{a_n}{(n+1)4^n}=\zeta(2) $
Let $a_n$ be the sequence defined via$$ a_n=\sum_{k=1}^n{2n \choose {n-k}}\frac{1+(-1)^{k+1}}{k^2}$$then prove that$$\sum_{n=1}^\infty\frac{a_n}{(n+1)4^n}=\frac{\pi^2}{6} $$I tried simplifying $a_n$...
View ArticleWhy does the triangle inequality for space $L^p$ with $0
It is said that triangle inequality for the space $L^p(\mathbb{R})$ space doesn't hold if $0<p<1$.Does anyone know an example for this?Also, what we can say, for example, about the quantity like...
View ArticleFractional part of a sum
Define for $n\in\mathbb{N}$$$S_n=\sum_{r=0}^{n}\binom{n}{r}^2\left(\sum_{k=1}^{n+r}\frac{1}{k^5}\right)$$I need to find $\{S_n\}$ for $n$ large where $\{x\}$ denotes the fractional part of...
View ArticleDoes there exist a set containing only rational numbers with no isolated point?
My answer:Construct a set E that consists of all the rational numbers between 0 and 1, and choose an arbitrary element a $\in$ E. By the density of Q in R, all $\epsilon$-neighborhood of a contains...
View ArticleComputing an integral with limit
Let $$I_n=\int_{\log~n}^{e^n} x^{2022}e^{-x^9}dx$$Then compute $\lim_{n\to\infty} I_n$how i can use lebesgue dominated convergence theorem?
View ArticleClik here to view.
Definite integral involving K Bessel function and a square root
I have recently been trying to evaluate some integrals involving the modified Bessel function $K_0(x)$. The specific integrals are$$L(x,u) = \int_0^{1} K_0\left( 2x \sqrt{r(1-r)} \right) \exp(2ixur) dr...
View ArticleContinuity of this function and well definedness
Consider $f: (0, 2] \to [-5, 7]$ defined as $f(x) = \begin{cases} x & 0 < x\leq 1 \\ 1 & 1 < x \leq 2 \end{cases}$Is this function well defined? Is it continuous?AttemptsI believe $f$ is...
View Article