Function with a constant rate of change on the relative interior of a convex...
Let $V$ a convex cone of $\mathbb{R}^n$ having an empty interior. Let $relint(V)$ denote its relative interior.Let $f:V \to \mathbb{R}$. Assume that for all $v \in relint(V)$, $$\lim_{t\to0}\:...
View ArticleThe oscillatory integral $\sup_{b, z > 0} \left| \int_0^b \frac{\cos(z...
I am trying to prove the boundedness of certain oscillatory integrals and I can not deal with the following situation, which I have reduced to a specific example. I claim that$$\sup_{b, z > 0}...
View ArticleBound for the values of a polynomial on [0,1] in term of the partial sum of...
I encountered the following nice little fact in a little problem of mine, and would like to know what proofs you could have in mind !Consider $a_0, \ldots, a_n$$n$ real numbers and form the...
View ArticleHow large can the range of Lebesgue densities of a measurable subset of...
As mentioned in the bounty, I'm actually looking for a set $E$ such that $d(D)=[0,1]$. The original question follows for context.Let $E \subset \mathbb{R}$ be Lebesgue measurable, let $D \subseteq...
View ArticleQuestion about Proof of the Integrability of $f$ and $f_1,f_2,\dots,$ in...
I am self-studying measure theory and got stuck on part of the proof of the Lebesgue's Dominated Convergence Theorem:Theorem$\quad$ 2.4.5$\quad$ (Lebesgue's Dominated Convergence Theorem)Let...
View ArticleProof that $f(x)=\sqrt{x}$ is locally Liptschitz
Let $f:(0, 1] \to \Bbb R$ given by $f(x)=\sqrt{x}$, I want to show that $f$ is locally Liptschitz, this means (for $\lambda>0$, and $x, y \in B(p;r)$)$$|\sqrt x-\sqrt y| \leq \lambda |x-y|$$If we...
View Article$f(x)=f(x+T),T\notin \mathbb Q,\int_0^T f(x)\mathrm dx=-T$. Does ther exist a...
Suppose $f\in C(\mathbb R )$, and we take supremum from all $x\in\mathbb R,k_1,k_2\in\mathbb Z$.I doubt that there won't exist such a function, for $k_1,k_2$ are integers but the function's period is...
View ArticleUnderstanding differentials in an equation in general relativity.
I have not studied physics but I was browsing Carroll's relativity book and randomly stumbled upon the following which I would like to understand mathematically. It says$$"ds^{2} = 0 = - \left( 1 -...
View ArticleCan a non-constant continuous function be constant on these hyperbolas?
Can a non-constant continuous function $f:\mathbb{R}^2\to\mathbb{R}$ be constant on the following...
View ArticleI have made a code for the following exercise, Are the results correct?
I have made a code for the following exercise:Use Romberg integration to calculate the following approximations $\displaystyle\int_{1}^{48}\sqrt[ ]{1+(cosx)^2}dx$1 [Note: The results of this exercise...
View ArticleOn Heine-Borel Theorem for any compact set.
I have recently studied the Heine-Borel theorem i.e. set is compact iff any open cover has a finite subcover. I have proven it only for compact intervals and their cartesian products for higher...
View ArticleMinimum of $\frac{x^3}{x-6}$ for $x>6$ without using derivative?
Find the minimum of $y=f(x)=\dfrac{x^3}{x-6}$ for $x>6$.I can solve the question using derivatives but I have no any idea how to do it without them. Using derivatives, we find $x=9$ and...
View ArticleDoes $A\subseteq \mathbb R^n$ homeomorphic to $\mathbb{R}^n$ mean $A$ is...
Let $A$ be a subset of $\mathbb{R}^n$ (with $n\geq 1$, endowed with its usual topology) homeomorphic to $\mathbb{R}^n$. Is $A$ an open subset of $\mathbb{R}^n$?
View ArticleSolving an "impossible" integral, $\int \frac{dx}{a^{x^2}+b^{x^2}} $
First a saw this problem$$\int \frac{dx}{a^{x^2}+b^{x^2}} $$where a,b are natural numbersThis problem would be easier if $a=b$then error function appears hereI have tried with Wolfram alpha but he show...
View ArticleDominated convergence theorem for integrals depending on a parameter
Is there a theorem mirroring the dominated convergence theorem, but for integrals depending on a parameter instead?Meaning,Let$$\begin{aligned}[t]f:I\times J&\longrightarrow...
View ArticleIs limit when x tend to a derived set point exist( in sequential functions)?...
Suppose that $ f_n(x)$ is uniformly convergent to f on the set E.And for all n ∈ N; $f_n(x)$ is continuous on E.I want to prove that for all derived set point of E (( it means : $N_r(x) ∩ E $ \ { x...
View ArticleProve that the function sequence $f_n(x)=n^2\left(\mathrm{e}^{\frac{1}{n...
Prove that the function sequence $f_n(x)=n^2\left(\mathrm{e}^{\frac{1}{n x}}-1\right) \sin \frac{1}{n x}(n=1,2, \cdots)$ convergent uniformly on $[a,+\infty)(a>0)$ .Proof1. For every $x...
View ArticleUpper bound of $2\vert\cos{\frac{x+y}2}\sin{\frac{x-y}2}\vert$
We have $2\left|\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)\right|$As $\left|\cos\left(\frac{x+y}{2}\right)\right|\leq 1$$\left|\sin\left(\frac{x-y}{2}\right)\right|\leq 1$Is it true?...
View ArticleProve that $\lim_{n \to \infty} \left(\frac{1}{n+1}\right)^2 +...
We want to prove that$$\lim_{n \to \infty} \left(\frac{1}{n+1}\right)^2 + \left(\frac{1}{n+2}\right)^2 + \dots = 0.$$Let $S_n$ be the sum:$$S_n = \left(\frac{1}{n+1}\right)^2 +...
View ArticleA question on how to choose parameters such that a function decays to...
Questionhow to choose $p,\delta$ such that$$n\log(\frac{100}{\delta})-\frac{n^2\delta^2p}{64(1-p)+6\delta}$$ increases when $n$ grows.MotivationI am trying to understand Corollary 5.7 in this paper....
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