# All Questions

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+100

### Orthogonality relation for integral with Bessel functions

I want to calculate the following integral $$\int_0^1 dx \,x \,\left(J_n'(x \,x_{nm})J'_n(x\,x_{nm'})x_{nm'} x_{nm}+\frac{n^2}{x^2}J_n(x \,x_{nm})J_n(x\, x_{nm'})\right)$$ where $x_{nm}$ is the $m$ th ...
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### Probability of error for two independent process

Let assume that we have two possible event $a_1$ and $a_2$ that may happen with probability $p(a_1)$ and $p(a_2)$ with $p(a_1)+p(a_2)=1$. Now let us consider a detector which is affected by some noise ...
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### Recursive funtions (R.Soare)

I started learning Computability theory by myself and got stuck on this one problem from R.Soare's book "Recursively enumerable sets and degrees", problem 3.11. It doesn't seem to be too ...
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### Two ways of explaining the notation $\mathbb{R}^2$

I found two ways to explain the notation $\mathbb{R}^2$. First is by Cartesian product: $\mathbb{R} \times \mathbb{R}$. Secondly, regarding $2$ as any set with two elements. suppose it's $\{0,1\}$. ...
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### Poincaré-Hopf Theorem for manifolds with boundary in Milnor's book

I've just read Milnor's topology from the differentiable viewpoint, though I haven't been able to figure out the case of manifolds with boundary. Milnor says that the problem is that in this case the ...
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### Need help with gradient descent convergence calculation

We are given $\alpha=1-8\eta$, where $\eta$ is the step size. We are also given the following ($z$ is irrelevant, but basically it's the output of the algorithm at step $k$): We are asked what values ...
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### Proving a prior distribution defined by a trignometric basis on $L^2(\mathbb{T})$ is Hölder continous

Consider $L_2(\mathbb{T})$ with the basis $$\phi_{2k}(x)=\sqrt{2}\cos(2\pi k x)\\ \phi_{2k-1}(x)=\sqrt{2}\sin(2\pi k x)$$ for $k\in\mathbb{N}$. The functions $\phi_k$ belong to the domain $H^{2p}$ of ...
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### Compute exact integrals with quaternions

It's common knowledge that complex analysis is helpful in computing a bunch of exact real integrals. Is there any occurence of quaternions/quaternion formalism helping in the same way? If not, what ...
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### If $\rho(z)<1$ then exists some $N$ such that $\lVert R(z)^N \rVert <3/4$ for all $z\in U$

Let $\rho(z)$ denote the spectral radius of $R(z)$, where $R:U \to \text{Hom}(\mathcal{L},\mathcal{L})$ and $U$ is a closed subset of $\mathbb{C}$, here $\mathcal{L}$ can be understood as a Banach ...
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### A naive question about the Godement resolution

So, I have read several accounts on Godement resolution, including an exposition by Godement himself in "Topologie algébrique et théorie des faisceaux". And there is still one thing that ...
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### Forward characterization of measurable functions?

In topology, the standard definition of continuity works in the "backward" direction, since it puts a condition on the pre-images of a function rather than images: $f$ is continuous if the ...
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### How to most compactly define addition, a norm, and a metric in this space?

I have a vector space with some unusual equivalences between vectors, and I'm struggling to define a norm and metric which will preserve those equivalences in a simple way. The way I need to represent ...
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### Prove or disprove: $1+\frac{\ln^c(ax)-\ln^ca}{\ln^c(2a)-\ln^ca}-x\leq 0$, where $1<x$, $a>2$, and $c$ is chosen so that $f'(2)=0$

Define: $$f(x)=1+\frac{\ln^{c}\left(ax\right)-\ln^{c}\left(a\right)}{\ln^{c}\left(2a\right)-\ln^{c}\left(a\right)}-x$$ where $1< x$ and $a>2$. Assume further that the parameter $c$ is chosen so ...
Suppose $R$ is a principal ideal domain. Let $S$ be a multiplicatively closed subset of $R$ not containing $0$. Show that $S^{-1}R$, the localization of $R$ by $S$, is a principal ideal domain. I ...