Simply connected connected compact Lie groups are well-known to be in bijective correspondence with complex semisimple Lie algebras. Now every connected simply connected complex Lie group has a unique compact real form. So it would seem that this gives a bijection between connected simply connect...

I apologize if this is put inelegantly as I'm new to this type of mathematics, but I thought of this experiment and can't stop turning it over in my mind. Take any set of points on a 2d plane and connect every point to every other point using line segments. Now take the midpoints of each of those...

$\newcommand{\Mod}{\mathbf{Mod}}\newcommand{\Ab}{\mathbf{Ab}}\newcommand{\from}{\colon}\newcommand{\Id}{\mathrm{Id}}\newcommand{\tensor}{\otimes}$ Let $R$ be a ring and $\Mod(R)$ the category of (right) $R$-modules. When you learn abstract algebra you meet a number of constructions following roug...

The theory of groups and the theory of abelian groups differ by only one axiom, commutativity. "Forgetting" this axiom induces the inclusion functor from Ab to Grp, which has a left adjoint, the abelianization functor. What is the correct generalization of this fact? In general, the class of stru...

I just began reading Ordinary Differential Equations by Arnol'd, and in the first few paragraphs of the first chapter, Arnol'd says the following: The theory of ordinary differential equations is one of the basic tools of mathematical science. This theory makes it possible to study all evolution...

Suppose $\Omega$ is a bounded and simply connected domain in $\mathbb{R}^3$ with a smooth boundary $\partial \Omega$. Assume that there is a strong solution $u \in H^2(\Omega)$ to the following equation \begin{equation} \left\{ \begin{aligned} \Delta u&= u & \hspace{15pt} &\text{in $\Omega...

I have an integral which involves hyperbolic functions. The integral is $I =\displaystyle \int_{0}^{L} \frac{\cosh(K(x-L))}{\sqrt{\cosh(KL) - \cosh(K(L - x))}} \mathrm dx$ I tried solving it as follows but was unable to proceed further. Using the hyperbolic cosine identity: $\cosh(a - b) = \cos...

Let us work with $A$-modules for simplicity. Suppose that I have a functor $F$ from $A$ modules to $A$ modules, and I can prove that for every sequence : $$N_1\rightarrow N_2\rightarrow N_3$$ which is exact at $2$, then: $$F(N_1)\rightarrow F(N_2)\rightarrow F(N_3)$$ Does this imply that $F$ is a...

This is the link for geo gebra applet https://www.geogebra.org/classic/g5bb7ngf (sorry for some different labellings) Given two circles internally say $X_1$ and $X_2$, $X_1 $ is the larger, centers of $X_1, X_2 $ are $A, D $ respectively, internally tangent to each other at some point $ B $ let...

Let $G$ be a Lie group. Suppose any irreducible unitary representation is one-dimensional. (Irreducible meaning having no proper closed invariant subspaces.) Is $G$ abelian? This great answer seems to imply that this is the case, but I would like a more complete answer or a good reference. Citing...

The problem Consider the inscribable quadrilateral $ABCD$ with $AC=a, CD=c, DA=d$. Let the points $X$ and $Y$ be on the lines $AB$ and $BC$, respectively, such that $A\in (BX), C\in (BY) $ and $BX=BY=BD$. If the angle bisector $DAC$ intersects the line $XY$ at the point $Z$, prove that: $\vec{DZ}...

There is a technique that allows you to sample points uniformly ins a sphere by combining three Gaussian distributions as described in wolfram alpha. Additionally, there is a concept called spherical guassian, commonly defined as: $$ Ce^{\lambda(\mu \cdot v - 1)^2} $$ There is also a generalizati...

Assume $S_0 = 0$ and let $N_k$ be the number of times of visiting level k before returning to origin. My textbook first claims that $P(N_k > 0) = \frac{1}{2k}$ and $P(N_k > j+1|N_k > j) = \frac{1}{2} + \frac{1}{2} \frac{k-1}{k}$ and hints that this can be used to show that $E(N_k) = 1$. I am lost...

It is really well-known that the binomial distribution with parameters (n,p) can be approximated when n is large, p is small, and np is a constant (some textbooks say moderate). In my case, $n=x\ln x$, and $p=1/x$. $x$ grows large so the first two conditions are satisfied. however, $np=\ln x$ so ...

In order to find the derivative of the expression $$\frac{\mathrm{d}}{\mathrm{d}x} [(x-a)^2-(x-b)^2],$$ is the chain rule needed at all or is it possible to just use the sum rule and the power rule? I took this from a video that explains optimization for machine learning. The professor mentions t...