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lower bounding...
Could we expect that $$\inf_{\theta\neq \text{regular polygon}}S(\theta)\sim f(n)$$where $S(\theta):=\inf_{\theta\neq \text{regular...
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 ArticleConvolution preserve the boundary condition
Here, I want to know if convolution will preserve the Neumann condition or not. Suppose $K$ is a continuous function on some interval $[-L,L]$, and $u$ is some 'good enouth' function on $[0,L]$ that...
View ArticleSeeking Algorithm to Solve a Convolution Integral or Directly Convolve Two CDFs
Hi everyone,I'm working on a project where I need to find a way to directly convolve two cumulative distribution functions (CDFs) given in polynomial form, and solve the following convolution integral:...
View ArticleThe pile of sand principle in Pugh’s Real Mathematical Analysis
Exercise 1.25.d in Pugh’s Real Mathematical Analysis asks for a solution of the following:given $\epsilon\gt 0$ prove that finitely many disjoint closed discs can be drawn inside the unit square so...
View ArticleQuestion about proof of Lebesgue Decomposition Theorem for the case of...
I am asked to proof the following Lebesgue Decomposition Theorem for the case of $\sigma$-finite positive measure:Lebesgue Decomposition Theorem$\quad$ Let $(X,\mathscr{A})$ be a measurable space, let...
View Articlefinding a sequence that diverges
let $a_n$ be a sequence where for all $n\ge3$$a_n \in (a_{n-1},a_{n-2})$ or $a_n \in (a_{n-2},a_{n-1})$i need to find an example to such sequence that diverges.i figured that such sequence is of the...
View Articlehow to generalize a version of IVT theorem
We know, with the IVT theorem: if $f(0)=0,f'(0)>0$ and $f(a)<0$ for a continuous function, one can deduce that there exists at least $c \in ]0, a[$ such that $f(c)=0$.could we generalize this...
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Determine the continuity of a function
The following question appeared in a graduate-level entrance exam. In this question more than 1 option can be correct.I determined that option A is incorrect since $p(x)$ and $q(x)$ are unbounded on...
View ArticleA limit without invoking L'Hopital: $\lim_{x \to 0} \frac{x \cos x - \sin...
The following limit$$\ell=\lim_{x \rightarrow 0} \frac{x \cos x - \sin x}{x^2}$$is a nice candidate for L'Hopital's Rule. This was given at a school before L'Hopital's Rule was covered. I wonder how we...
View ArticleConsequence Vitali covering lemma
Let $f\colon[a,b]\to\mathbb{R}$ be a real valued function and put$$A=\{x\in(a,b)\colon f\text{ is differentiable at }x,\; f'(x)=0\}.$$Let $\lambda$ denote the Lebesgue measure on $[a,b]$, then the...
View ArticleSupremum of a derivative of a family of holomorphic functions at a given point
I have been working on the following exercise: Let $\Omega \subsetneq \mathbb{C}$ simply connected, open set and $a,b \in \Omega$ fixed points. Let $C(\Omega, a ,b) = \sup \{|f'(a)| ; \quad f: \Omega...
View ArticleWhy do Dedekind cuts use sets?
As far as I understand Dedekind cuts, they are really about ranges, not sets. In other words, they are about all of the values greater than or less than some number. But by using sets, I'm led to...
View ArticleEvaluate $\lim_{x \to...
Evaluate $$\lim_{x \to \infty}x^4\left(\arctan\frac{2x^2+5}{x^2+1}-\arctan\frac{2x^2+7}{x^2+2}\right)$$My SolutionDenote$$f(t):=\arctan t.$$By Lagrange's Mean Value Theorem，we...
View ArticleIntegral to series: $\int_0^1e^{\sin(\log x)}\...
I was looking for an explicit evaluation of the integral, when I stumbled across this series, that seems to converge to the value of the integral.$$\mathcal I=\int_0^1e^{\sin(\log x)}\...
View ArticleDoes this polynomial admit more than a single real root?
Consider the polynomial $p(x)$ such that:$$ p(x) = 1 + 2x + 3x^2 + 4x^3 + \cdots + (2n+2)x^{2n+1} $$where $ n $ is a natural number. Because $p(x)$ has odd degree, then from the Complex conjugate root...
View ArticleAbout countable family of sets in R satisfying a hitting condition
I was working on a problem when I hit a little snag about a geometric problem. To properly describe it, let me introduce some notions.We say that a point $p$ hits a set $X$ if $p\in X$.We say that a...
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Differential Equation Involving Minimum of Two Functions
This one has completely stumped me. It's from an ODE general exam, I'm given the IVP$$ y' = \text{min}(y^2, M),$$$$ y(0) = 1.$$With $M>1$ and I'm asked to give an explicit solution and discuss...
View ArticleLet $f$ be twice differentiable function on $(0,1)$.
Let $f$ be twice differentiable function on $(0,1)$. Given that for every $x\in(0,1)$, $|f''(x)|\leq M$ where $M$ is a non-negative real number. Prove that $f$ is uniformly continuous on $(0,1)$.My...
View ArticleIf the slope of a secant is always irrational, is the function linear?
If $f : \mathbb{R} \rightarrow \mathbb{R}$ is a function, such that all distinct $x,y \in \mathbb{R}$ we have that:$$\frac{f(x)-f(y)}{x-y} \not\in \mathbb{Q}$$Can we conclude that $f$ is a linear...
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