# All Questions

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### If the probability of frame being lost is $P.$ Then, calculate the mean no. of transmission for the frame to make it success

Here the probability of frame being lost is $P.$ So the probability of frame reaching safely would be $(1-P).$ Now lets consider that the frame will reach safely in $k$-th transmission. That means ...
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### Where can I find a proof of the main properties of Weyl Curvature for semi-Riemannian manifolds?

Most of the references I've seen deal with Riemannian geometry, rather than semi-Riemannian geometry. Chens monograph, Pseudo-Riemannian Geometry, $\Delta$-Invariants and Applications is one of the ...
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### Barr's element and the type-A Iwahori–Hecke algebra

Let $\mathfrak{S}_n$ denote the symmetric group on $n$ letters. Barr's element $\mathcal{S}(n) \in \Bbb{R}\bigl[ \mathfrak{S}_n \bigr]$ is defined as \begin{equation} \mathcal{S}(n) := \ \sum_{i=1}^{n-...
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### Characterization of differentiability

For a normed space $(V, \lVert\cdot\rVert_V)$ let us define: \begin{equation} \forall x, y \in V \quad [0,1] \mapsto \gamma_x^y (t) = (1-t)x + ty. \end{equation} I would like to ask whether the ...
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### Inverse of Euler Totient function [closed]

φ(n) = x, where x is an odd integer. I know this has no solution, but how can I formally prove this?
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### Describing the ordinary irreducible characters of a special $p$-group $p^{n+m}$

Let $P$ be a special $p$-group $p^{n+m}$. So $P$ will have $p^m$ linear characters. How does one describe (or determine) the other ordinary irreducible characters of $P$ and will they all be ...
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### What is neutral constructive mathematics

In Mike Shulman's answer to Initiation to constructive mathematics, he discusses how "neutral constructive mathematics" is the fashionable topic in constructive mathematics. When ...
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### Could a nice principle be extracted from this lemma of Gauss

I asked the following question in the math SE, with a bounty of 200 pts, without result. What have the real pros to say about that. question: To prove the quadratic reciprocity law, Gauss needed the ...
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### Corepresentability of involutory objects in monoidal $\infty$-categories

The group $\mathbb{Z}/2$ corepresents the functor $\mathrm{Inv}\colon\mathsf{Mon}\to\mathsf{Sets}$ sending a monoid $A$ to its set of involutory elements (those satisfying $a^2=1_A$). Moreover, this ...
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### Advice on how to deal with an elementary "long-to-prove" statement

I am not entirely sure if this question totally fits here. If it doesn't, I apologise in advance. In a paper I've been working on, we have a very elegant result which, when forgetting about the ...
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### Banach spaces whose second conjugates are separable

It was known that the James space $J$ has separable second conjugate, is non-reflexive and isometric to its second conjugate. I want to know whether there are Banach spaces $X$ with separable second ...
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### Stable $\infty$-categories of derived equivalent varieties

When two varieties have equivalent derived categories of coherent sheaves are the stable $\infty$-categories of coherent sheaves also equivalent? Are the stable $\infty$-categories of varieties "...
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### Alternative Mersenne numbers

Let $\ b\in\mathbb Z,\$ and $\ |b|>1.\$ Call $$M_b(n)\ :=\ \frac{b^n-1}{b-1}$$ to be $n$-th Mersenne number mod $b$. The necessary condition for $\ M_b(n)\$ to be a prime is that $\ n\$ is a ...
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### Involutions in $\infty$-categories

$\newcommand{\id}{\mathrm{id}}$An involution in a category is a functor $\mathbf{B}\mathbb{Z}/2\to\mathcal{C}$, corresponding precisely to an object $X$ of $\mathcal{C}$ together with a $\mathbb{Z}/2$-...
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### Include each point of continuum in a subset so that each subset gets finitely many points

Let $S$ be a set with $\lvert S\rvert=\lvert\mathbb{R}\rvert$. Suppose it has subsets $S_x$ indexed by $x\in \mathbb{R}$ with $\lvert S_x\rvert=\lvert\mathbb{R}\lvert$ for each $x\in \mathbb{R}$. ...
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### Difference between Shannon entropy and min-entropy

I would like to construct a sequence of discrete random variable $X_2, X_3,...,X_n,...$, where $X_n \in\{0,1,2,...,n-1\}$. Given any $\epsilon \in (0,1)$, its Shannon entropy and min-entropy should ...
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### Definition of $\mathcal{O}_{\mathcal{X}}$-modules over a stack $\mathcal{X}$
$\newcommand\Sch{\mathrm{Sch}}$For a stack $\mathcal{X}$ in $(\Sch/S)_{\textrm{ét}}$, there is a site $(\Sch/\mathcal{X})_{\textrm{ét}}$ whose objects are $(T,t)$, where $T$ is an étale $X$-scheme and ...