Is $a_n < \epsilon s_n$ for a strictly increasing sequence $(a_n)$, its...
Let $a_n$ be a strictly increasing sequence so that its partial sums $s_n$ tend to infinit.I want to prove that for every positive real $\epsilon$ there is a natural number $N$ so that if $n>N$,...
View ArticleProve $\|u\|_{C^{2,\alpha}(\mathbb{R}^3)}\leq...
Assume $ 0 < \alpha < 1 $, and the function $ f \in C^{0, \alpha}(\mathbb{R}^3) $ has compact support. Define the function$$u(x) = \frac{1}{4\pi} \int_{\mathbb{R}^3} \frac{f(y)}{|x - y|} \,...
View ArticleEquivalence of two definitions of differentiability on non-open sets
I am looking for a reference, or simple argument, that would shed light on the following two definitions of the derivative of a multivariate function.Let $E\subseteq\mathbb R^n$, not necessarily open,...
View ArticleHow to adjust fraction parts to arrive to accurate rate
Let say the target rate is:$$0.2312076819339227309219662514224185$$How to adjust this fraction $984398920/4257639330$ that gives this $\mbox{rate}\...
View ArticleOn the continuity a function given by evaluating compact subsets of...
Let $B$ be a closed ball in $\mathbb{R}^n$ and write $C(B)$ for the Banach space (with respect to the supremum norm) of the continuous real-valued functions on $B$.Now given a compact subset $K$ of...
View Articleprove that $f(||x+y||) = f(||x||) + f(||y||)$ assuming that is a convex...
I'm trying to solve the following question:assume $f : [0, \infty) \rightarrow \mathbb{R}$ is convex and continuously differentiable where $f(0)=0$ and $f'(0)=0$. $X$ is a normed vector space on...
View ArticleDirect proof of $\sqrt{2}$ is irrational [duplicate]
Is there a way to directly prove that $\sqrt{2}$ is irrational without using contradiction - the most popular method? I was thinking that maybe it could be done using a corollary of the completeness...
View ArticlePolynomials with only even terms are dense in the set of polynomials in...
Let ${\cal P}$ set of all polynomials on one variable in $[0,1]$. It can be equipped with the supremum norm to obtain a normed space. Consider ${\cal Q}= \{a_0 + a_2x^2+a_4x^4+\dots a_{2k}x^{2k}:...
View ArticleAlternative definition of derivative in real analysis
I am working on the following definition of derivative.Let $I$ be an open interval and $f \colon I \rightarrow \mathbb{R}$ be a function, and let $x \in I.$ The function $f$ is differentiable at $x$ if...
View ArticleIntegrating a smooth 1-form over shrinking loops in the complex plane is...
Let $U \subseteq \mathbb{C}$ be an open subset with $U \ni 0$, let $\varphi \in C^\infty(U,\mathbb{C})$, and consider the smooth $1$-form $\omega := \varphi(z) \frac{dz}{z}$ on $U \setminus \{0\}$. If...
View ArticleA closed set with empty interior
How do I prove that the set$L_n = \{f \in C([0, 1]), \exists \, x_0 \text{ such that } \forall \, x \in [0, 1], |f(x) - f(x_0)| \leq n|x - x_0|\}$is closed with empty interior? Furthermore, how do I...
View ArticleDefintion of distributions why not define with complex conjugate
For a complex valued locally integrable function $f$ on an open set $U \subset \mathbb{R}^{n}$, I saw many sources defined distribution induced by $f$ as $\phi \mapsto \int f\phi \, dx$.If $\phi$ is...
View ArticleWhat does $(\frac{2}{3}+C_{n-1})$ mean in the representation of the *n*th...
In this Wikipedia's page on Cantor set, the recursive formula for the nth member is as follows:$C_{n} := \frac{C_{n-1}}{3} + (\frac{2}{3} + C_{n-1}).$But I don't quite understand what $+$ does in this...
View ArticleOn a formula for $\pi e$.
I'm looking for ideas on how to represent $\pi e$ as an infinite series of rational numbers.It is easy to build meaningless formulas using double summations by distributing a series for $\pi$ over a...
View ArticleConfusion with calculating a limit $F(x):=\int_{x}^{2x}e^{-2t}t^{-1}dt$
The question is given in the following way:Define $F:(0,\infty)\rightarrow \mathbf{R}$ by $$F(x):=\int_{x}^{2x}e^{-2t}t^{-1}dt$$ Determine whether or not $\lim_{x\rightarrow 0^+}F(x)$ exists.In the...
View ArticleOn Theorem 2.7 of Montgomery and Vaughan's MNT
In Theorem 2.7 of Montgomery and Vaughan's Multiplicative Number Theory, they show that a constant arising in an earlier calculation is precisely the Euler-Mascheroni constant. While I understand the...
View ArticleLoomis-Whitney inequality from generalized Hölder inequality
I've proven the following "generalized Hölder inequality",Suppose $q^{-1}=\sum_{i=1}^np_i^{-1}$ for $1\leq q,p_1,...,p_n\leq\infty$, then $$||\prod_{i=1}^nf_i||_q\leq\prod_{i=1}^n||f_i||_{p_i}$$and am...
View ArticleWhy is the inverse of a differentiable function $f: \mathbb{R}^n \to...
I came across a problem in my multivariable calculus studies whose proof I don't fully understand. The problem states:Let $f: \mathbb{R}^n \to \mathbb{R}^m$ be a differentiable and invertible function....
View ArticleRecovering irrational number $\alpha$ from behavior of the sequence $ (\{...
Given an irrational number $\alpha > 0$ (I specifically care about the case where $\alpha$ is a quadratic integer which is also a Pisot number, but the question can be asked in general), it is well...
View ArticleProve that $f(\| x+y \| ) = f(\| x \| ) + f(\| y \| )$ assuming that is a...
I'm trying to solve the following question:Assume that $f : [0, \infty) \rightarrow \mathbb{R}$ is convex and continuously differentiable where $f(0)=0$ and $f'(0)=0$. $X$ is a normed vector space on...
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