Does every triangle satisfy $\frac{1}{a-b+\pi R} + \frac{1}{b-c+\pi R} -...
Experimental data show that in any triangle with sides $(a,b,c)$ and circumradius $R$, if $x > 2$ then,$$\frac{1}{x} - \frac{4}{x^2 - 4} < \frac{R}{a-b+Rx} + \frac{R}{b-c+Rx} - \frac{R}{c-a+Rx}...
View ArticleIntegration of the partial derivative of a homogeneous function of degree zero
Suppose $f\in C^{\infty}(\mathbb{R}^n \backslash \{0\})$ satisfies $f(\lambda x)= f(x)$ for $\lambda >0$, and it has zero average on $\mathbb{S}^{n-1}$.My question is: how can we show that its n-th...
View ArticleHow to prove the dual to the dual norm of a norm it's itself in Euclidean...
RBecause Eucildean space $\mathbb{R}^n$ has many good properties such as complete and reflexive or etc. And since it's finite dimensional, so all kinds of norms in $\mathbb{R}^n$ are equivalent, and in...
View ArticleBaby Rudin Chapter 2 Problem 6 Advice/Solution Verification [closed]
My friend and I are learning reading Baby Rudin currently and we both have reached the problem set for chapter 2, and my friend asks me to solve this problem and here is my solution:Original Problem:...
View ArticleBeginner Real Analysis Question on Convergence
Hi I'm just starting to learn some analysis and I am wondering if when proving convergences of sequences, where a represents terms of the sequence, and L is the limit, is it sufficient to say:∀ε> 0...
View ArticleDouble integrals with kernels
Let $f, g$ be functions in the Schwartz space $\mathcal S(\mathbb R^n, \mathbb R)$. Let $a\in (0, 1)$ and consider the kernel $K(u, v)=|x-y|^{-N-a-1}$, being both $u, v, z\in\mathbb R^n$ vectors such...
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Problem with dense set
On ' Set theory with an introduction to real point sets'(Dasgupta, Abhijit ,2014) i found this exercise:This is interesting because compare the topological (left,1) and order (right,2) definition of...
View Articleconvolution of the fundamental solution with the homogeneous solution
I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero?Let $U$ and $E$ be...
View ArticleIntuition behind the exponential convergence(e-convergence)
I'm studying a concept called e-convergence for sequences of probability densities. The definition states:A sequence $(g_n)_{n \in \mathbb{N}}$ in $M_{\mu}$ is e-convergent to $g$ if:$(g_n)_{n \in...
View ArticleConfusion on defining uniform distribution on hypersphere and its sampling...
Fix a dimension $d$. Write $S^{d-1}$ for the surface of a hypersphere in $\mathbb{R}^d$, namely set of all $x = (x_1, \ldots, x_d) \in \mathbb{R}^d$ such that $|x|^2 = x_1^2 + \cdots + x_d^2 = 1$. I...
View ArticleA Hölder norm of square root of a $C^2$ function
BackgroundI am reading a proof of the Calabi-Yau theorem from these notes. In page 15 he claims the following statement without proof (calling it elementary): Let $M$ be a compact manifold. There...
View ArticleEvaluate the limit $\lim_{n\to\infty}\prod_{r=1}^n\frac{4r}{4r+3}$
Evaluate the limit $$\lim_{n\to\infty}\prod_{r=1}^n\frac{4r}{4r+3}$$My Attempt:I am trying to do by Squeeze theorem and then idea of telescopic product but not able to get anywhere.Using $A.M\geq G.M$...
View ArticleWhy does $\lim(\inf (a_n +b_n))=\lim( \inf(a_n) )+ \lim (\inf(b_n))$ fail?
I was working on the proof that for bounded sequences $a_n$ and $b_n$, $\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty} (b_n) \leq \liminf_{n\to\infty} (a_n+b_n)$. I got to a point where I concluded...
View ArticleProve that If $A \subset R^n$ a function $f : A \to R^m$ is continuous iff...
($\implies$) direction was easy enough, but I am confused on $(⟸)$. So we have that If $A \subset R^n$, for every open set $U \in R^m$ there is some open set $V \in R^m$ such that $f^{-1}(U) = V \cap...
View ArticleI can't prove a wiki statement about convolutins
Here is the link:https://en.wikipedia.org/wiki/Fundamental_solution#Application_to_the_exampleIt states that :$$∫_{-∞}^∞\frac{1}{2} |x-y| \sin(y) dy=-sin(x)$$ as distributionsThe best I can come up...
View ArticleExample of proof of an infimum?
I'm doing an excercise about infimums and supremums and I've seen different examples of proving a =inf(S) $\iff a\le s :\forall s \in S, \forall \epsilon >0\exists s\in S\colon a+\epsilon>s$ and...
View ArticleQuasi-concavity and contour sets
Hi I'm trying to prove a function f is quasiconcave if the upper contour set of f with cutoff a is convex. Could someone please give me some idea on how to start or think about the direction of this...
View ArticleEvaluate $\lim_{n \rightarrow \infty} \int_{-n}^n f(1+\frac{x}{n^2}) g(x) dx$
I want to evaluate $\lim_{n \rightarrow \infty} \int_{-n}^n f(1+\frac{x}{n^2}) g(x) dx$, where $g: \mathbb{R} \rightarrow \mathbb{R}$ is (Lebesgue)-integrable, and $f:\mathbb{R} \rightarrow \mathbb{R}$...
View ArticleDf $F$ with regularly varying Tail $\overline{F}$ with index $-\alpha$ for...
I am reading the proof of Theorem 3.3.7 in the book "Modelling Extremal Events" by Embrechts, Klüppelberg and Mikosch. I don't understand the first couple of lines of the proof: If $F$ is a df with...
View ArticleHow to find a formula for the terms of this sequence?
I saw a problem on this forum concerning the number$$T = 1 + \frac{2 +\frac{3+ \frac {4+...}{5+...}}{4+\frac{5+...}{6+...}} }{3 + \frac{4+\frac{5+...}{6+...}}{5+\frac{6+...}{7+...}}}$$whose rule is...
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