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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 $a$, $b$, $c$, $\al... 3 hours later… 3:41 AM 7 Uncomputable functions: Intro The last month I have been going down the rabbit hole of googology (mathematical study of large numbers) in my free time. I am still trying to wrap my head around the seeming paradox of the existence of natural numbers that are well-defined but uncomputable (in the s... 1 hour later… 4:57 AM 3 I have the following situation Let$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$are uncorre... 2 hours later… 6:45 AM @Feeds Answers to this question are eligible for a +250 reputation bounty. Knocker379 wants to draw more attention to this question: > Find the area of A with some explanation (I'll comment any questions I have about your answer. Bounty is 250) @Feeds Answers to this question are eligible for a +250 reputation bounty. Bernhard Boehmler wants to draw more attention to this question. @Feeds Answers to this question are eligible for a +100 reputation bounty. SAWblade wants to draw more attention to this question. @Feeds Answers to this question are eligible for a +100 reputation bounty. Jarbas Dantas Silva is looking for an answer from a reputable source: > if the constants requested in the conditions of the question,$a$,$b$,$c$and$\alpha$exist, the reason for their existence must be explained mathematically in detail. In the case of non-existence, it must also be explained with mathematical rigor and in detail. I do not accept gaps to be filled in and every result used must be cited. I thank any help. @HNQmath.se Removed from HNQ by adding MathJax. 3 hours later… 9:31 AM 4 I am trying to solve the following exercise. What I know:$\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 $$\pi: \tau \to \mathbb{R}P^n \\ \tilde{\pi}:\mathbb{R}P^n ... 9:56 AM 5 We consider the group G = SL(2, 3) i.e, the set of 2 \times2 matrices with determinant 1 and addition and multiplication are performed modulo 3 even in the determinant formula. One can show that |G| = 24 a) Let \alpha = \begin{pmatrix} 2 & 2\\ 2 & 1 \end{pmatrix} show that \alpha \in G... 2 hours later… 11:47 AM 0 According to the Implicit Function Theorem (2 dimensional case): if F:U\subset \mathbb{R}^2\to \mathbb{R} is a C^1 function defined on the open set U and (x_0,y_0)\in U such that F(x_0,y_0)=0 and F_y(x_0,y_0)\neq 0, then for some neighbourhoods I,\,J of x_0,y_0, respectively t... 0 Imagine I want to statistically characterise a set of converging points and still get an idea of the converging properties or shape of such set, for example The values of the mean or variance of the y coordinates don't really tell me anything specific about the converging shape of such set and... 2 hours later… 1:29 PM @Feeds Answers to this question are eligible for a +50 reputation bounty. Nikolaos Skout is looking for an answer from a reputable source. @Feeds Answers to this question are eligible for a +50 reputation bounty. sam wolfe wants to draw more attention to this question. 2:18 PM 0 During lectures, we got to show that C_9, C_9 + 1 are subdirectly irreducible, where C_n := (\{0, 1, . . . , n − 1\}, (0 1 . . . n − 1)) C_n + 1 := (\{0, 1, . . . , n − 1, n\}, (0 1 . . . n − 1)(n)) I know the usual definition of subdirect irreducibility - An algebra A is subdir... 3:01 PM @Feeds Answers to this question are eligible for a +100 reputation bounty. Tereza Tizkova is looking for an answer from a reputable source. 3:34 PM 2 I am attempting to prove that the multivariate distribution with maximum entropy for a given covariance is a Gaussian. (PRML, Bishop, problem 2.14). Bishop suggests the use of Lagrange multipliers - concretely, that should maximize$$ \text{H}[x] = -\int p(x)\log(x)dx $$subject to the constrain... 4:26 PM 2 Description of context Given are independent random variables n,u that are normally and uniformly distributed,$$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... 5:09 PM @Feeds Answers to this question are eligible for a +50 reputation bounty. BadProgrammer is looking for an answer from a reputable source. @Feeds Answers to this question are eligible for a +50 reputation bounty. granular bastard wants to draw more attention to this question. 3 hours later… 8:09 PM 4 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 ... @Feeds Answers to this question are eligible for a +50 reputation bounty. isaacg wants to draw more attention to this question: > I'd like to see some progress on this problem. I think it's subtly difficult. 8:58 PM 1 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...

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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...

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