Negativity of a function with differential inequality as condition
The question comes from Tao's book on dispersive PDE. In the appendix B, he shows uniqueness of ground state. In the second part of lemma B.10, he proves that$$\partial_t^2 u_y(t) + \frac{d-1}{t}...
View ArticleCauchy sequences are convergent, part of the proof verification
I just want to confirm that one particular reasoning in the proof of the theorem is correctLet $x_n$ be a Cauchy sequence, and let $k,N \in \mathbb N, m=N$ and $k \geq N$. Then since $x_n$ is Cauchy we...
View ArticleWhat is the value of $\int_{3}^\infty \frac{1}{x\ln(x)(\ln(\ln(x))^2)}dx$?
If I calculate the value of $\int\limits_{3}^\infty \frac{1}{x\ln(x)(\ln(\ln(x))^2)}dx$ by wolfram alpha it says $\approx 10.663$. However, If I do it by hand via a substitution and integration by...
View ArticleProving that the expansion of a convex set is convex
I am trying to solve problem 2.14 from Stephen Boyd's 'Convex Optimization', which is as follows.Given a convex set $S \subset \mathbb{R}^n$, a norm $\lVert \cdot \rVert$ on $\mathbb{R}^n$, and a...
View ArticleSolving $y'=\cos x \int \cos x^2 dx$
This seems a very hard ODE that I couldn't solve.It is from Zorich Mathematical Analysis so I guess there must be a somewhat-enlightening solution.Though my half-attempt is here:$$y'(x)=\cos x\int \cos...
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Contraction Mapping Theorem. Prove $\{ y_{1},f(y_{1}),f(f(y_{1})),\ \ldots)...
Let $f$ be a function defined on all of $R$. Assume there is a constant $c$ such that $0< c <1$ and $ |f(x)\ -f(y)\leq c|x-y|$ for all $x,\ y\in R$.(a) Show that $f$ is continuous on $R$ for all...
View ArticleHow to decode an irrational number into 2 rationals
Context:A friend chooses 2 rational numbers to use, R1 and R2.The friend chooses an irrational number, I1The friend uses this equation to get I2R1 + (R2 * I1) = I2The friend gives I1 and I2 to you, and...
View ArticleProof that a Function is Uniformly Continuous
Let A and B be intervals. Suppose we have two uniformly continuous functions $f:A\to\mathbb{R}$ and $g:B\to\mathbb{R}$ such that for all $x\in{A\cap{B}}$, $f(x)=g(x)$. Also suppose that $A\cap{B}$ is...
View ArticleProve $\lim\limits_{x \to +\infty } \frac{{f(x)}}{x} = \lim\limits_{x \to...
Let $f:\Bbb R \to \Bbb R$ be differentiable, and $\mathop {\lim }\limits_{x \to \infty } \frac{{f(x)}}{x}$ (the slope of some asymptote) exists and the limit of the derivative $\mathop {\lim...
View Article$n$-th moment of the generalized pareto distribution
I need urgent help in calculating the $n$-th moment of the generalized pareto distribution. This distribution is characterized by a shape paramter $\xi\in\mathbb{R}$ und scale parameter $\sigma>0$....
View ArticleSolve an integral analytically
I'm trying to show that$$\int_{0}^{1} \sqrt{-\ln (x)}\frac{x}{1-x}dx>\int_{0}^{1} \sqrt{-\ln (x)}dx.\quad (E1)$$But $(E1)$ is equivalent to$$\int_{0}^{1} \frac{\sqrt{-\ln...
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Showing $S(f;P)-S(f;\tilde{P})\le\omega(f;[a,b])\cdot (\Delta...
ContextThe problem I am trying to solve is the following:Attempt for the first partI have successfully showed the left part of inequality for both...
View ArticleProof verification: Showing a measurable set that is translation invariant...
Suppose that $E$ is measurable and $E+q \subseteq E$ for all $q \in \mathbb Q$. Show that either $m(E)=0$ or $m(E^c)=0$.I wanted to try proving the statement using the Lebesgue Density theorem. This is...
View Articleinequality related to roots of $(x-1)\log(x)=m$
Let $f(x) := (x-1)\log {x}$. Suppose $f(x_1)=f(x_2)=m$ for some $0<x_1<x_2$.Show that $\frac{9}{5}+\log{(1+m)}<x_1+x_2<2+\frac{m}{2}$.If we apply Hermite-Hadamard inequality, it's easy to...
View ArticleHeine-Borel for Finite Dimensional Normed Vector Spaces
I would like to show that finite-dimensional normed vector spaces have the Heine-Borel property (any subset is compact if and only if it is closed and bounded). I have decided to take the following...
View ArticleWhy a metric on a set can not be $\infty$? [duplicate]
I ask this question because I came acorss a problem (b) in exercise 3 in page 93 in section 13 in chapter 2 in the second edition of Elementary Analysis by Ross. The question is described below:Let $B$...
View ArticleProof of fundamental lemma of calculus of variation.
Suppose $\Omega$ is an open subset of $\mathbb{R}^n$ and let $L^1_\text{Loc}(\Omega)$ denote all locally integrable functions on $\Omega$ and $C^{\infty}_0(\Omega)$ for smooth functions whose support...
View Articleright-sided/left sided differentiability of $2\pi$-periodic extension
Let be $f:\mathbb{R}\to\mathbb{R}$ a differentiable function and $g:\mathbb{R}\to\mathbb{R}$ with $g(x):=\begin{cases}f(x),&x\in~\!\!]-\pi,\pi]\\f(\pi),&x=-\pi.\end{cases}$ its $2\pi$-periodic...
View ArticleUnion of a finite or countable number of uncountable sets
Prove that the union of a finite or countable number of setseach of power $c$ (continuum power) is itself of power $c$.To prove this I would start considering two sets $A$ and $B$ of power $c$. I would...
View ArticleProve that $Sup(A + B) = Sup(A) + Sup(B)$
Earlier on in the book it showed that to prove $a = b$ it is often best to show that $a \leq b$ and that $b \leq a$. This is the way I want to go about the proof. I am sure there is an easier way but I...
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