How to prove $(F\ast\sin)(x)=-\sin(x)$, where $F(x)=\frac{1}{2}|x|$?
Wikipedia states in this article about fundamental solutions that if $F\left( x \right) = \tfrac{1}{2} \left| x \right|$, then$$\left( F \ast \sin \right)\left( x \right) :=...
View ArticleAverage cluster size of a nxn matrix
I asked a question about the cluster size inside a vector here. As a result, I finally used the expression $\frac{n}{-k+n+1}$ as the average cluster size, although it´s not proved correct for every...
View ArticleExample of the function $F(x)$ that has non zero total variation as $x$ goes...
I am trying to find an example of the function $F(x)$ that has non zero total variation as $x$ goes to $-\infty$.Let $T_F(x) := sup\{ \sum_{i=1}^n |F(x_j) - F(x_{j-1})| :n \in \mathbb{N},...
View ArticleIntegral of $\int_{\Re\lambda=\gamma_0} \frac{e^{\lambda...
I want to deform the contour to the imaginary axis but avoiding the poles $\lambda=\pm ic$. To avoid the poles one can use half-circles $C_\pm=\{\lambda\in\mathbb C:\lambda = \pm ic+\epsilon...
View ArticleTrying to prove monotone-sequences property ⇒ Archimedean property
Monotone-sequences property ⇒ Archimedean propertyToday I just started learning this and having trouble understanding parts of the proof. Sorry if these are easy to some, I just couldn't find this...
View ArticleProve ${b_n} = \sup \{ {a_k}:k \geqslant n\} $for $n \geqslant 1$. Prove that...
Let $\left(a_n\right)_{n = 1}^\infty $ be bounded sequence. Define the sequence $b_n=\sup \{ a_k:k \geqslant n\}$ for $n \geqslant 1$. Prove that $(b_n)$ converges.Proof.Define$$A_k=\{a_k,a_{k +...
View ArticleConditions on coefficients for a family of multivariate trigonometric...
Let $f$ be a real trigonometric polynomial of $d$ variables that is bounded $|f|\le 1$. Further assume that the maximum degree for each variable is 1. Then we can write it as the sum of monomials, each...
View ArticleIs...
I need to calculate the limit (if it exist) $$\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$$ where $\{x\}$ denotes the fractional part of $x$, $n!$ is the factorial of $n$...
View ArticleHow to prove $\int_0^{\infty}e^{-sx}\frac{\sin(x)^2}{x}dx = \frac{1}{4}\ln(1-...
I need some help with in trying to prove that$$\int_{0}^{\infty}{\rm e}^{-sx}\,\frac{\sin^{2}(x)}{x}\,{\rm d}x = \frac{1}{4}\ln\left(1- \frac{4}{s^{2}}\right)\quad\mbox{using}\...
View ArticleTheorem 7.48 in Apostol's MATHEMATICAL ANALYSIS, 2nd ed: Lebesgue's Criterion...
Here is Theorem 7.48 (Lebesgue's Criterion for Riemann Integrability) in the book Mathematical Analysis - A Modern Approach to Advanced Calculus by Tom M. Apostol, 2nd edition:Let $f$ be defined and...
View ArticleGlobal existence of solution of ODE - Gray-Scott model
I am dealing with the ODE version of Gray-Scott model:\begin{equation}\begin{split}\dot{x} &= -xy^2 + F(1-x)\\\dot{y} &= xy^2 - (F + k)y,\end{split}\end{equation}where $F>0$ and $k>0$ are...
View ArticleApplication of Gronwall's lemma to the expectation
Let $X_t$, $Y_t$ be real-valued continuous stochastic processes with finite second moments such that$$E |X_t-Y_t|^2 \leq \int_0^t K(s) \left( E |X_s - Y_s |^2 + W_2^2( \mu_s, \nu_s) \right) ds, \quad t...
View ArticleQuestion on the formulation of Stokes' theorem
In the book I'm reading, the Stokes' Theorem is stated as follows:Let $\Omega\subset\mathbb{R}^2$ be open, bounded and of class $C^1$. Let $V\subset\mathbb{R}^3$ be an open set. Let...
View ArticleDouble integral of projectile problem
Given the projectile problem$$x^{\prime \prime}=-\frac{1}{(1+\epsilon x)^{2}}$$$$x(0) = 0 \quad x^{\prime}(0) = 1$$By integrating twice, the projectile problem can be rewritten as the integral...
View ArticleWe get $L^2$ convergence, but do we get a.s. convergence?
Assume you have a sequence of independent Bernoulli random variables$X_i$ each with probability $p_i$. Let $c_i$ be a sequence of real numbers and $ m,M$ be a real numbers such that$0 < m...
View ArticleConvergence of linear functionals
Let $(\ell_n)_{n\in\mathbb{N}}$ be a sequence of linear functionals in $\mathrm{BV}^*$, namely the dual of the space of functions with bounded variation. Suppose that $\ell_n$ converges to $\ell$ as $n...
View ArticleSolving for an Exercise from Zorich's Mathematical Analysis I Chapter 5
Let $f\in C^{(n)}((-1,1))$ and $\sup_{-1<x<1}|f(x)|\le 1.$ Let $m_k(I)=\inf_{x\in I}|f^{(k)}(x)|$ where $I$ is an interval contained in $(-1,1)$. Show that:a) if $I$ is partitioned into three...
View ArticleHow to understand that both the lim inf and lim sup of the sets of real...
I am reading the Real Analyais (4th edition) by Royden, H. L., & Fitzpatrick, P. In Section 1.4, it is said thatboth the lim inf and lim sup of a countable collection of sets of real numbers, each...
View ArticleComposition of functions, once for all
I need certainties. Consider two functions $f: A \to B$ and $g: C \to D$. For what I am going to ask, it's not a loss of generality if we consider multivariable functions with domains and or codomains...
View ArticleLebesgue outer measure in $\mathbb{R}^2$ in terms of a grid of $h$-squares
For a set $D\subseteq\mathbb{R}^2$, the Lebesgue outer measure of $D$ is defined by$$\lambda^\ast(D)=\inf\bigg\{\sum_i\lambda(I_i)\mid D\subseteq\bigcup_iI_i\bigg\},$$where $\{I_i\}$ is a sequence of...
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