For $\alpha \in (0,1)$, set $\omega:\mathbb{R^+}\times \mathbb{R^+} \to \mathbb{R}$ defined as following $$\omega(t;\tau):=1-\pi^{-1}\int_0^\infty \frac{e^{-rt-\tau r^\alpha \cos(\alpha \pi)}\sin(\tau r^{\alpha}\sin(\alpha \pi))}{\pi r}dr, \ t, \tau>0$$ Can we prove that exist $a$, $b$, $c$, $\al...

Uncomputable functions: Intro The last month I have been going down the rabbit hole of googology (mathematical study of large numbers) in my free time. I am still trying to wrap my head around the seeming paradox of the existence of natural numbers that are well-defined but uncomputable (in the s...

I have the following situation Let $N_1, N_2 \sim \mathcal{N}(0,1)$ two independent r.v. Let $X = \frac{N_1}{\sqrt{N_{1}^{2} + N_2^2}}$ and $Y = \frac{N_2}{\sqrt{N_{1}^{2} + N_2^2}}$. Now I know how show to that $X$ and $Y$ are not independent, but I don't know how to show $X$ and $Y$ are uncorre...

I am trying to solve the following exercise. What I know: $\tau$ is a vector bundle of dimension $n$ over $\mathbb{R}P^n$. The same is true for the trivial bundle $\mathbb{R}P^n \times \mathbb{R}^n$. Then we find surjective smooth maps $$ \pi: \tau \to \mathbb{R}P^n \\ \tilde{\pi}:\mathbb{R}P^n ...

We consider the group $G = SL(2, 3) $ i.e, the set of $2 \times2$ matrices with determinant 1 and addition and multiplication are performed modulo 3 even in the determinant formula. One can show that $|G| = 24$ a) Let $\alpha = \begin{pmatrix} 2 & 2\\ 2 & 1 \end{pmatrix}$ show that $\alpha \in G$...

According to the Implicit Function Theorem ($2$ dimensional case): if $F:U\subset \mathbb{R}^2\to \mathbb{R}$ is a $C^1$ function defined on the open set $U$ and $(x_0,y_0)\in U$ such that $F(x_0,y_0)=0$ and $F_y(x_0,y_0)\neq 0$, then for some neighbourhoods $I,\,J$ of $x_0,y_0$, respectively t...

Imagine I want to statistically characterise a set of converging points and still get an idea of the converging properties or shape of such set, for example The values of the mean or variance of the $y$ coordinates don't really tell me anything specific about the converging shape of such set and...

During lectures, we got to show that $C_9$, $C_9 + 1$ are subdirectly irreducible, where $C_n$ := $(\{0, 1, . . . , n − 1\}, (0 1 . . . n − 1))$ $C_n + 1$ := $(\{0, 1, . . . , n − 1, n\}, (0 1 . . . n − 1)(n))$ I know the usual definition of subdirect irreducibility - An algebra $A$ is subdir...

I am attempting to prove that the multivariate distribution with maximum entropy for a given covariance is a Gaussian. (PRML, Bishop, problem 2.14). Bishop suggests the use of Lagrange multipliers - concretely, that should maximize $$ \text{H}[x] = -\int p(x)\log(x)dx $$ subject to the constrain...

Description of context Given are independent random variables $n,u$ that are normally and uniformly distributed, $$n\sim\mathcal{N}_{\mu,\sigma}=\frac{1}{\sigma\sqrt{2\pi}}\text{exp}\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right)=f(x)\ \ \ \ \ \text{ for } x\in\mathbb{R}\\ u\sim \ma...

I have a positive integer $n$, and a multiset $S$ of positive integers. $S$ has $n$ elements. For all $s \in S$, $s$ is a divisor of $n$. I believe that there must exist a subset (submultiset) $S' \subset S$ such that the elements of $S'$ sum to $n$. For example, $n$ could be 6, and $S$ could be ...

I have a very simple question but it's confusing me a bit nonetheless. Suppose $(\Omega,\mathcal F,\mu)$ be a measure space and $f\geq0$ is a measurable function. Let $A=\{\omega\in \Omega:f(\omega)>0\}$ and suppose $\mu(A^c)=0$. Suppose $g:\Omega\to\mathbb R_+$ is defined by $g(\omega)=1/f(\o...

There is a big discussion in our class. Teacher says, $f:\mathbb Z^{+}\rightarrow \left\{0\right\}$ and $f(x)=0$ is a constant function. But, student says only $f:\mathbb R\rightarrow \left\{0\right\}$ and $f(x)=0$ is a constant function or $$f(x)=0\,\,\forall x\in\mathbb R$$ Unfortunately, wik...