Evaluating $\lim_{n\to\infty}\sum_{i=1}^n...
Let $X_1,...,X_n\stackrel{iid}{\sim} \mu$ where has a density with respect to the Lebesgue measure on $\mathbb{R}$: $\mu(dx)=\rho(x)dx$. For every $x$ show$$\lim_{n\to\infty}\sum_{i=1}^n...
View ArticleAbout density in $\mathbb{R}^{3}$
Suppose that $\frac{\lambda_2}{\lambda_1}, \frac{\lambda_3}{\lambda_1}\in\mathbb{R}^{+}\setminus\mathbb{Q}$.Does anyone have any suggestions to prove the...
View ArticleA complete metric space with midpoints is a length space
Let $X$ be a complete metric space and suppose that for all $x,y\in X$ there exists a midpoint $z\in X$ such that $$\frac{1}{2}d(x,y)=d(x,z)=d(y,z)$$To show that this complete metric space is a length...
View Articleprove that a closed ball in Euclidean space is perfect
I've just been working with just the complex numbers z less than or equal to 1, but am curious as to how one would prove this for all closed balls in Rn as well. I only have up to Rudin page $33$ in my...
View ArticleConvergence in Total Variation vs implies Uniform Convergence
Let $(f_n)_{n\in \mathbb{N}}$ and $f$ real function defined on an interval $[a,b]$ s.t. $f, f_n$ are bounded variation function on $I$, and suppose that $TV(f_n) \to TV(f)$. I should prove that it...
View ArticlePrincipal argument of $z^n$
I understand the process of finding the principal argument of a complex number if it has been raised to the power of n, i.e., $z^n$.If we simply had z = x + iy then our principal value is defined to be...
View ArticleContinuous dependence of inverse functions on parameters
If we have a bijective, continuous function $f\colon \mathbb{R} \to \mathbb{R}$, then it is proven in every introductory course to analysis that the inverse function $f^{-1}\colon \mathbb{R} \to...
View ArticleHow can I compute the limit with an integral？
I came across a problem that I couldn't solve in a mathematical analysis textbook：Let $\alpha, L \in \mathbb{R}$ and $L \neq 0$:$$\mbox{If}\quad\lim_{n \to \infty}\dfrac{\displaystyle...
View ArticleSuppose $1\leq p_1
I need to prove the following result:Suppose $1\leq p_1<p_2<+\infty$ and $\mu$ a finite measure, then $\mathscr{L}^{p_2}(X,\mathscr{A},\mu)\subseteq\mathscr{L}^{p_1}(X,\mathscr{A},\mu)$.Here is...
View ArticleProof for the following nascent delta function
Let $f(x)$ be a (nonzero) rapidly decreasing function in $\mathbb{R}$ (i.e. $f(x)$ and all its derivatives go to zero as $x\to\pm\infty$ faster than any negative power of $x$). The sequence of...
View ArticleInequality related to function $f(x)=x\ln x-\frac{x^2}e+ax$
I'm trying to solve a hard question about function and here is the question below:For the given function $f(x)=x\ln x-\frac{x^2}e+ax$ (where $a$ is a parameter), we have such...
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Explanation of Shakarchi's proof of 1.3.4 in Lang's Undergraduate Analysis
I'm currently working through Lang's Undergraduate Analysis, and trying to understand Rami Shakarchi's proof of the following:Let $a$ be a positive integer such that $\sqrt a$ is irrational. Let...
View ArticleFirst decomposition theorem of fuzzy set
I read if standard union in fuzzy set have definition:Union of two fuzzy sets $\tilde{A}$ and $\tilde{B}$ in universe $X$ denoted $\tilde{A}\cup\tilde{B}$ is fuzzy set in universe $X$ with membership...
View ArticleProve that this sequence is eventually one
Let $ \ \mathbb{N} = \{ 0,1,2,3,4,...\} \, $, $ \ O = \{ n \in \mathbb{N} : n \text{ is odd} \} \ $ and $ \ T: O \to O \ $ be such that, for all $ \ n \in \mathbb{N} \, $,\begin{align*}T(8n+1) & =...
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What is the feasible region for convex combinations of powers of cosines?
Let $x = \sum_{i=1}^N a_i \cos^2(\theta_i)$ and $y = \sum_{i=1}^N a_i \cos^3(\theta_i)$ be the two convex combinations of powers of cosines, where the variables satisfy the following constraints(1)....
View ArticleCompletion of a vector-valued function to form a diffeomorphism
Let $\varphi_r \colon \mathbb{R}^n \to \mathbb{R}^r$ be a $\mathcal{C}^1(\mathbb{R}^n, \mathbb{R}^r)$ vector-valued function.Do we know sufficient conditions on $\varphi_r$ for the existence of a...
View ArticleExtracting a subsequence common to infinitely many sets from an uncountable...
Let $\{a_n\},\{b_n\}$ be strictly increasing sequence of positive integers satisfying $a_1<b_1<a_2<b_2<a_3<b_3<\ldots$ and $(b_n-a_n) \to \infty$. Define $I_n:= [a_n,b_n]$, meaning...
View Articleasymptotic approximation of the integral $\int\limits_{0}^{\infty}e^{\alpha...
Consider the integral $F(\alpha):=\int\limits_{0}^{\infty}e^{\alpha x}f(x)dx$, where $\alpha>0$ is a real parameter and $f$ is a kernel (or a pdf). I want to describe a class of kernels $f$ so that...
View ArticleDoubt about a proof in complex analysis
In proving that a continuous function$$f:S(0,1)\to \mathbb{R}$$Has a continuous extension to$$h:\overline{B(0,1)}\to \mathbb{R}$$Such that $h|_{B(0,1)}$ is harmonic, one needs to show that the...
View Article$e^{-\lVert x \rVert ^2}$ is integrable over $\mathbb{R}^n$ and...
$e^{-\lVert x \rVert ^2}$ is lebesgue integrable over $\mathbb{R}^n$ and $\int_{\mathbb{R}^n} e^{-\lVert x \rVert ^2} = \pi^{n/2}$. I'm a bit lost on this, I've tried to use a particular result about...
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