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This is a follow-up on the previous question. Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb …

For a real-valued function $f$ on $[0,1]$, define its quadratic variation by the formula $$[f]:=\limsup\sum_{j=1}^n(f(t_j)-f(t_{j-1}))^2,$$ where the $\limsup$ is taken over all "partitions" $0=t_0<\c …

$\newcommand\P{\mathcal P}$A "partition" $P$ (of the interval $[0,1]$) is a finite sequence $(t_0,\dots,t_n)$ such that $0=t_0<\cdots<t_n=1$; then the mesh of $P$ is $\|P\|:=\max_{1\le j\le n}(t_j-t_{ …

For real $s>0$, let $$S(s):=\sum_{n=-\infty}^\infty e^{-n^2/(2s^2)} =\vartheta _3\left(0,e^{-1/(2 s^2)}\right),$$ where $\vartheta$ is the elliptic theta function. Plotting suggests that the identity …

Suppose we have a sequence of entire functions $f_n$ such that $$\text{$f_n(z)\to0$ for each natural $z$}\tag{1}$$ (as $n\to\infty$). Is it possible to give general additional conditions on the sequen …

Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb R)$. Does it then necessarily follow that the …

$\newcommand\R{\mathbb R}$Let $f\colon\R^p\to\R$ be a continuous function. For $u=(u_1,\dots,u_p)$ and $v=(v_1,\dots,v_p)$ in $\R^p$, let $[u,v]:=\prod_{r=1}^p[u_r,v_r]$; $u\wedge v:=\big(\min(u_1,v_1 …

$\newcommand{\R}{\mathbb R}$Consider the following construction. For real $u$, let \begin{equation} f(u):=\frac{2u^2}{1+u^4}, \end{equation} so that the function $f\colon\R\to\R$ is continuous, $0 …

Let $a_n$ and $b_n$ be two different expressions in natural $n$ with values in the set of all nonnegative integers such that we have the identity $a_n=b_n$ for all $n$. As a simplest example, we may c …

Are there sources that treat questions like the following ones? Suppose that $f\colon\mathbb{C}\to\mathbb{C}$ is an entire function such that $f(x)$ is real for all real $x$ and $f(x)\sim1/x$ as $x …

$\newcommand\B{\mathscr B}\newcommand\A{\mathscr A}\newcommand\si{\sigma}$Let $I:=[0,1]$, and let $\B$ and $\B^2$ denote the Borel $\si$-algebras over $I$ and $I^2$, respectively. Let $\A$ stand for t …

$\newcommand\Z{\Bbb Z}\newcommand\De{\Delta}$For a natural $n$ and a function $g\colon\Z\to\Bbb R$, let $B_n g$ be the corresponding Bernstein polynomial, so that $$(B_n g)(x)=\sum_{k\in\Z} g(k)\binom …

$\newcommand\R{\Bbb R}$Let $F$ be the set of all functions of the form $\max(a,b,c)$, where $a,b,c$ are affine functions from $\R^2$ to $\R$ and the maximum is taken pointwise. Let $G$ be the set of a …

This question was raised in the comment by Todd Trimble at how to proof there is a natural number n, the first four digits of n! Is 2018?. I thought the question may be posted separately, as even part …

Identity \begin{equation} \sum_{k=0}^n\binom nk^2 \binom k{n-m}=\binom nm \binom{n+m}m \tag{1} \end{equation} was used in an answer here. As shown in that answer, (1) easily reduces to \begin{alig …