Let $f$ be a bounded analytic/holomorphic function on $\Bbb{D}$ such that $|f(z)| = 1$ for almost every $z \in \partial \Bbb{D} = S^1$, where the unique circle is given the standard Lebesgue measure (this is the definition of an inner function). Is it possible to show that $|f'| \le 1$ on $\Bbb{D...

There was some recent media reporting about a purported Google breakthrough on applying machine learning techniques to tackle the protein folding problem, as told for example in this news article, DeepMind AI handles protein folding, which humbled previous software. Unfortunately there is not muc...

I was trying to get a closed form of the sum $$\sum_{k=1}^{n}k!$$ Mathematica gives the answer $$(-1)^n\Gamma(n+2)(!(-n-2))-!(-1)-1$$ Here the ! sign before a number is the subfactorial. Here is my try: \begin{align} \sum_{k=1}^{n}k!&=\sum_{k=1}^{n}\Gamma(k+1)\\ &=\sum_{k=1}^{n}\int_{0}^{\infty}x...

We have \begin{align} \lim_{n\to\infty}\left(\frac{1}{a}+\frac{2}{a^2}+...+\frac{n}{a^n}\right)&=\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k}{a^k}\\ &=\sum_{k=1}^{\infty}\frac{k}{a^k} \end{align} Let $1/a=x,$ so we have \begin{align}\sum_{k=1}^{\infty}\frac{k}{a^k}&=\sum_{k=1}^{\infty}kx^k\\ &=x\sum_...

A fractal is usually defined to be a self similar shape (this is the informal definition). But, the formal definition is: A fractal is a set for which the Hausdorff dimension strictly exceeds the topological dimension (formal definition). Before proceeding to the question, I will tell what this...

Let $H$ be the Hawaiian earring, i.e. it is the union of circles $S_n$ with centre $(1/n, 0)$ and radius $1/n$. By a universal cover I mean a covering $p\colon X\to H$ (with $X$ connected) such that for any other covering $p'\colon X'\to H$, there is a map $f\colon X\to X'$ such that $p'\circ f =...

Consider two matrices acting on points in $(0,1)^2$ in the real plane, $$h_s=\begin{pmatrix} e^{-e^{s}} & 0 \\ 0 & e^{-e^{-s}} \end{pmatrix}.$$ and $$g_s=\begin{pmatrix} 1-e^{-e^{s}} & 0 \\ 0 & e^{-e^{-s}} \end{pmatrix}.$$ How do you compose these transformations? I tried to do $h_s...

Let $f$ be a bounded analytic/holomorphic function on $\Bbb{D}$ such that $|f(z)| = 1$ for almost every $z \in \partial \Bbb{D} = S^1$, where the unique circle is given the standard Lebesgue measure (this is the definition of an inner function). Is it possible to show that $|f'| \le 1$ on $\Bbb{D...

I want to prove that there does not exists a complete surface of revolution with constant Gaussian curvature $K=-1$. To do this, I classified the surface of revolution with $K=-1$: If $\gamma(t)=(x(t),0,z(t))$, then the surface revolution from $\gamma$ has $K=-1$ if and only if $x(t)= c_1e^t+c_2e...

So the function is $$u_n(x)=\begin{cases} a&\mbox{ if } x\in (\frac{i}{n},\frac{i+1/2}{n}], i=0,\dots,n-1, \\ b&\mbox{ if } x\in(\frac{i+1/2}{n},\frac{i+1}{n}],i=0,\dots,n-1.\end{cases}$$ in $\textit{L}^1((0,1))$ and i want to show $u_n \rightharpoonup (a+b)/2$. I think i have successfully proven...