Inverse Laplace of $1/(\sqrt{s}+ae^{-\sqrt{s}})$
Is there any known result for a nice expression for the inverse Laplace transform of$$F(s)=\frac{1}{\sqrt{s}+ a e^{-\sqrt s}},$$where $a\geqslant0$.It seems quite difficult to compute. The actual...
View ArticleConnectedness of complement of intersection of two balls
Let $B_r(x_0)\subset \mathcal{R}^n$ be an open ball with radious $r>0$ and centered at $x_0$. Let $y_0\in \partial \overline{B}_r(x_0)$. I want to argue: there exists some $\delta>0$ such that...
View ArticleConvergence of $\sum_{n\geq1}\frac{\sin^n(n)}{n}$
Determine whether the sum converges $\sum\limits_{n=1}^{\infty} \frac{\sin^n(n)}{n} $I tried to use all the standard signs, but there's nothing to understand about them in general. It seems to me that...
View Articlehow to make a bijection map of a set of sequences
i dont know to approach this problem , how can i make bijection map for such a setConsider the set S of all sequences of natural numbers, such that each sequencein S has values 0 from some point...
View ArticleOrder $\leq$ on the real numbers defined via Cauchy sequences
I like to define the real numbers $\mathbb{R}$ as equivalence classes $[(a_n)_{n\geq 0}]$ of Cauchy sequences $(a_n)_{n\geq 0}$ of rational numbers by the equivalence relation: $(a_n)_{n\geq 0}$ is in...
View ArticleEquivalences for $G$-regularity and $F$-regularity in measure theory
I am doing some exercises from my real analysis class and got stuck trying to prove the following equivalences:Let $ A \subseteq \mathbb{R} $. Prove that the following statements are equivalent:a) $ A...
View ArticleDoes the sequence $\{\sin^n(n)\}$ converge?
Does the sequence $\{\sin^n(n)\}$ converge? Does the series $\sum\limits_{n=1}^\infty \sin^n(n)$ converge?
View ArticleProve that $a^x$ is continuous
I'm having trouble with proving the following:Let $a > 0$ be a positive real number. Show that the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) := a^x$ is continuous.I'm a first...
View ArticleDensely defined operator adjoint domain is $\{0\}$ [duplicate]
What would be an example of an operator $T$ defined on a densely defined domain (subspace of banach space) such that domain of $T^*$ is $\{0\}$? In the text I am reading, I am presented of unbounded...
View ArticleMinimizing half concave half convex function
I am trying to prove the following theorem:Theorem:Let $ x_1, x_2, \ldots, x_n $ be real numbers satisfying the following conditions:$ x_1 \leq x_2 \leq \cdots \leq x_n $,$ x_1, x_2, \ldots, x_n \in...
View Article$A = \left\{ \frac{x}{x+1} : x > 0 \right\}$. Prove inf $A$ = $0$.
$A = \left\{ \frac{x}{x+1} : x > 0 \right\}$The set A is bounded, because $0 < 1 - \frac{1}{x+1} < 1$ for all $ x > 0 $.My attempt:inf $A$ = $0$, because:$ 0 < 1 - \frac{1}{x+1}$ for all...
View ArticleQuestions about S. Solecki, Analytic ideals and their applications.
I read from "S. Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic, 99 (1999),51–71."In the first; I can't understand what is the analytic ideal? I know what the ideal is but I...
View Article$ G_f(x)=\int dy \frac{f(y)}{4\pi|x-y|}$ implies $\frac{\partial}{\partial...
exercise prove that if $f\in L^1(\mathbb R^3)\cap L^{\infty}(\mathbb R^3)$ and said $ G_f(x)=\int dy \frac{f(y)}{4\pi|x-y|}$ we have$\frac{\partial}{\partial x_i} G_f(x)=\int dy...
View ArticleQuestion on distributions and topology
Let $X$ and $Y$ be two topological spaces and $f:X\to Y$ a function. It is well-known that continuity of $f$ implies sequentially continuity, while the reverse is in general only true if $X$ is first...
View Article$f(x) = x e^x$. Let $L : (0, \infty) \to (0, \infty)$ be its inverse...
Consider the function $f : (0, \infty) \to (0, \infty)$ given by $f(x) = x e^x$. Let $L : (0, \infty) \to (0, \infty)$ be its inverse function. Which of the following statements is correct?$\lim_{x \to...
View ArticleO_p definition using a divergent sequence.
A random variable $X_n$ satisfies $X_n=O_p(n^{-1/2})$. In my study, I defined an event $A_n$ such that $|X_n| \leq C_0 n^{-1/2}$ occurs for some positive constant $C_0$. I want to use the evnet $A_n$,...
View ArticleNotes claim that a lower semi-continuous function attains its minimum on...
I am currently reading Touchettes notes: https://arxiv.org/abs/0804.0327 on large deviation results and in Appendix B the author states that (in page 81, below eq. B3) that:The lower semi-continuity of...
View ArticleQuestion on the symmetry of the joint density of two identically distributed...
Given $f(x,y)$ a positive smooth function on $[0,1] \times [0,1]$ that satisfies the condition$\int f(x,y) dx = \int f(y,x) dx$ for all $y \in [0,1]$.Is it true that $f(x,y) = f(y,x)$?If not, could you...
View ArticleWhy does $\mathbb{E}(X)=0$ hold for any r.v. with a law $\mu$ on $\mathbb{R}$...
Let $X$ be a r.v. on $(\mathbb{R},\mathcal{B}(\mathbb{R}), m)$ with $m$ the Lebesgue measure. Assume that $X$ has a law $\mu$ on $\mathbb{R}$ and assume that $X$ is both symmetric (also known as an...
View ArticleAsymptotic behavior of the Hermite functions [closed]
I would like to understand the asymptotic behavior of the Hermite function :$$\psi_k(x) = \frac{1}{\sqrt{2^k k!}}H_k(x) e^{-\frac{x^2}{2}},$$where $H_k(x)$ is the $k-$th Hermite polynomial. For...
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