Articles of reference request

Does anybody know of a site that has a set of all theorems?

I mean, if there exists a site that his function is to show and save theorems with their proofs?

The comultiplication on $\mathbb{C} S_3$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=S_3$, the algebra structure of $\mathcal{A}$ is $\mathbb{C} \oplus \mathbb{C} \oplus M_2(\mathbb{C})$. Let $\{ e_1,e_2,a_{11}, a_{12}, a_{21}, a_{22} \}$ be a matrix […]

efficient and accurate approximation of error function

I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$

Proving that $X/R$ is Hausdorff $\implies$ $R$ closed.

As the title says, I’m trying to prove that if $X/R$ is a Hausdorff space then $R\subset X\times X$ is closed. I have several questions about this: $(1)$ What exactly is $R$? I thought of $R$ as an equivalence relation, but I never thought of that relation (or any) as a set. What are the […]

Is there any surprising elementary probability problem that result in surprising solution like the Monty Hall problem?

For recreational purpose, i haven’t seen a interesting elemetary probability question quite a while. Is there any surprising elementary probability problem that result in surprising solution like the Monte Hall problem? Please give a few examples.

On integral of a function over a simplex

Help w/the following general calculation and references would be appreciated. Let $ABC$ be a triangle in the plane. Then for any linear function of two variables $u$. $$ \int_{\triangle}|\nabla u|^2=\gamma_{AB}(u(A)-u(B))^2+ \gamma_{AC}(u(A)-u(C))^2+\gamma_{BC}(u(B)-u(C))^2, $$ where $$ \gamma_{AB}=\frac{1}{2}\cot(\angle C), \gamma_{AC}=\frac{1}{2}\cot(\angle B), \gamma_{BC}=\frac{1}{2}\cot(\angle A). $$ What is a good reference for the formula? Is it due to R. Duffin? […]

References for coordinate-free linear algebra books

I’m looking for some (or one) good book(s) that teach linear algebra either purely coordinate-free or ones that present the standard bag-of-tools alongside coordinate-free alternatives or discussions. Thanks!

Suggested textbook for Multivariable Calculus

This question already has an answer here: Multivariable Calculus Book Reference 4 answers

Error in Billingsley?

Problem 8.25 in the third edition of Probability and Measure by Billingsley (1995, p. 142) is as follows: Suppose that an irreducible [Markov] chain of period $t>1$ has a stationary distribution $\{\pi_j\}$. Show that, if $i\in S_{\nu}$ and $j\in S_{\nu+\alpha}$ ($\nu+\alpha$ reduced modulo $t$), then $$\lim_np_{ij}^{(nt+\alpha)}=\pi_j.\tag{$\diamondsuit$}$$ Show that $$\lim_n n^{-1}\sum_{m=1}^np_{ij}^{(m)}=\pi_j/t\quad\text{for all $i$ and $j$.}\tag{$\clubsuit$}$$ Here, the sets […]

Trying to understand the limit of regular polygons: circle vs apeirogon (vs infinigon?)

In the definition of regular polygon at the Wikipedia, there is this statement about the limit of a n-gon: “In the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed.” My question is: I […]