# All Questions

128,669
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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13 views

### 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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29 views

### 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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75 views

### 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 ...

**-3**

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60 views

### 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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42 views

### 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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**1**answer

205 views

### 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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**2**answers

197 views

### 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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73 views

### 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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117 views

### 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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**1**answer

49 views

### On a quotient of a finite extension group $G=p^{n+m}.Q$

Let $G=p^{n+m}.Q$ be an extension group of the special $p$-group $p^{n+m}$ by a group $Q$. Now $p^{n+m}=p^n{{}^\cdot}p^m$. How does one show that $\frac{G}{p^n}\cong p^m.Q$? Or equivalently that $G \...

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37 views

### Derivation of taylor approximation

We have a random variable given as follows:
$\tilde{x} = k*\mu +\tilde{y}$
$\mu$ is the mean, k is positive and close to zero, and $\tilde{y}$ is a zero-mean component of $\tilde{x}$.
Now assume the ...

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34 views

### How to measure perceived note similarity in music / simplicity of ratios?

I have discovered a method to measure the similarity of two successive musical notes which I wanted to share with a question:
It is known in music theory that two successive pitches $a,b$ which sound “...

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31 views

### $L_p$ estimate in mixed boundary problem for elliptic equation

Let $Q$ be convex polygon, $\Gamma$ be a portion of boundary
$\partial Q$ and $H^1_\Gamma=\lbrace u\in H^1(Q):
u|_\Gamma=0\rbrace$. For $f\in (L_2(Q))^2$ consider the problem
$$
\int_Q A(x)\nabla u \...

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27 views

### Expressing the singular values of a 2-by-2 real-valued matrix by the norm of the two columns and the angle between them

I'm looking for an elegant way to show the following claim.
Claim: Let $m_1, m_2 \in \mathbb{R}^2$ be the two columns of matrix $M \in \mathbb{R}^{(2 \times 2)}$. The singular values of the matrix are ...

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97 views

### Actions of rings (and other algebraic structures) on abelian categories

On the project I am currently working on, there are abelian, Krull-Schmidt categories $\mathcal{C}$ where it seems natural to equip $\mathcal{C}$ with the action of a ring $R$ (in some cases a ...

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23 views

### Local well posedness for a stochastic wave equation

Suppose we have a stochastic wave equation, with Itô's derivative in the place of the usual Newtonian ones.
Does it make sense to talk about local well posedness?

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106 views

### In Deligne-Lusztig theory which degrees do irreps show up in?

In Deligne-Lusztig theory we take an alternating sum over cohomology in all degrees. Given an irrep of a finite group of Lie type can we trace back which degree it shows up in?

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74 views

### Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?

It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G.
More generally here (MO275769) Qiaochu Yuan ...

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87 views

### Realize the discrete series of $\mathrm{SL}_2(\mathbb{F}_q)$ in an abelian variety

Is there an abelian variety $A/\mathbb{F}_q$ and an embedding $\mathrm{SL}_2(\mathbb{F}_q)\to \mathrm{Aut}_{\mathbb{F}_q}(A)$ such that $H^1(A\otimes \overline{\mathbb{F}_{q}}, \mathbb{Q}_l)$ contains ...

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86 views

### A question about mod $p$ local Langlands for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$

In the mod $p$ local Langlands correspondence for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$, the irreducible supercuspidal representation $\left(\mathrm{ind}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}_{\mathrm{GL}_{2}(...

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65 views

### Tangent bundle of Milnor manifold

As I have been studying about Milnor manifold defined above, I want to understand its tangent bundle structure. I could not find anything related to that anywhere.
I am aware of the fact that $H(m,n)$ ...

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**1**answer

61 views

### Bounded variation of the partial derivatives of a convex function

Let $f:\mathbb R^2\to\mathbb R$ be a convex function. For simplicity, assume that $f\in C^1$. A general theorem which can be found in the book of Evans and Gariepy says that the gradient $\nabla f$ is ...

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357 views

### Categories disguised as other structures

It is common to hear that category theory unifies many apparently disparate areas of mathematics. One way it does so is by allowing us to take other mathematical structures and organize them into ...

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43 views

### Minimum level principal congruence subgroup coming from neat open compact subgroup

For a connected reductive group $G/\mathbb{Q}$ what is known about the minimum level such that the respective principal congruence subgroup is the intersection of a neat open compact subgroup of $G(\...

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**1**answer

51 views

### Quadratic cusp shape

Which hyperbolic $3$-manifolds are known to have quadratic cusp shape?
Explanations: Cusps of hyperbolic $3$-manifolds are products torus x interval. They lift to horoballs in hyperbolic $3$-space, ...

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**1**answer

129 views

### Fundamental groups of degree 2 covers of projective spaces

Does being a degree 2 cover of a projective space impose restrictions on the fundamental groups of non-singular complex projective varieties? For curves it does not.

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67 views

### Can the Boolean group $C_2^\omega$ be covered by less than $\mathfrak b$ nowhere dense subgroups?

Let $\mathrm{cov}_H(C_2^\omega)$ be the smallest cardinality of a cover of the Boolean group $C_2^\omega=(\mathbb Z/2\mathbb Z)^\omega$ by closed subgroups of infinite index. It can be shown that
$$\...

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87 views

### Bounds for a covering number of the circle group $\mathbb T$ by some its small subgroups

$\newcommand{\w}{\omega}\newcommand{\A}{\mathcal A}\newcommand{\F}{\mathcal F}\newcommand{\I}{\mathcal I}\newcommand{\J}{\mathcal J}\newcommand{\M}{\mathcal M}\newcommand{\N}{\mathcal N}\newcommand{\x}...

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137 views

### 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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134 views

### 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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89 views

### 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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101 views

### 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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109 views

### 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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69 views

### 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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64 views

### Outer and inner automorphism of $\mathrm{Pin}$ groups

$\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSO{...

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77 views

### Has anyone seen a red-themed website about theorems and proofs of probability statistics?

I've seen a website before that has very detailed theorems on various probability and statistics topics, and the proofs are attached below the theorems. The site was red-themed. Now I can't find it, ...

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**3**answers

427 views

### Does there exist some $p(x) \in \mathbb{Q}[x]$, deg$(p) > 1$, which maps $\mathbb{Q}$ onto itself surjectively?

Clearly this is impossible for $p$ of even degree, and I imagine that Cardano’s formula quickly reveals it to be impossible in the cubic case, although I have not checked in detail. My guess is that ...

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77 views

### Delooping monoidal $(\infty,1)$-categories into $(\infty,2)$-categories

This is the one categorical level higher version of the question Delooping monoidal $\infty$-groupoids into $\infty$-categories.
The classical, bicategorical, setting.
Given a monoidal category $(\...

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**1**answer

98 views

### Delooping monoidal $\infty$-groupoids into $\infty$-categories

The classical setting.
Given a monoid $A$, there's a category $\mathbf{B}A$, called the delooping of $A$, having a single object $\star$ and satisfying $\mathrm{Hom}_{\mathbf{B}A}(\star,\star)\overset{...

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275 views

### Is there an even number $a$ such that $a^{2^{n}}+1$ is prime for infinitely many $n$?

Is there an even number $a$ such that $\{n: a^{2^{n}}+1 \text{ is prime} \}$ is an infinite set?
Let $a$ be even. Is there infinitely many $n$ such that $a^{2^{n}}+1$ is composite?

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110 views

### A consequence of the Min-Max Principle for self-adjoint operators

Let $H=(H, (\cdot, \cdot))$ be a Hilbert space. Let $T_1,T_2:D \subset H \longrightarrow H$ be a self-adjoint operators (not necessarily bounded). It's well-know that the spectrum $\sigma(T_i)$ of $...

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110 views

### 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 ...

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70 views

### Minimal $b_2$ in Sarkisov's construction

In the paper On the structure of conic bundles. Math. USSR, Izv.,
120:355–390, 1982, Theorem 5.10, Sarkisov constructed the first example of non-rational, rationally connected $3$-fold $X$ with $H^{3}...

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57 views

### Weights in a mixture distribution

I'm trying to come up with a reasonable way to determine the weights in a mixture distribution. Let us consider the following example:
There are two districts ($i=1,2$) in a city, both of which share ...

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**3**answers

121 views

### Determining polynomial approximations of piecewise constant functions

Let $t_1 < t_2 < \cdots <t_m$ be real, and $X = \cup_{i=1}^{m-1} (t_i, t_{i+1})$ be a union of real open intervals. Let $f:X \rightarrow \{-1, 1\}$ be any piecewise constant function of form
...

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181 views

### Different Bialgebra/Hopf algebra structures on coalgebras

Given a coalgebra $C$, can there exist more than one algebra structure on $C$ giving it the structure of a bialgebra? I will also ask the same question for Hopf algebras.

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**1**answer

85 views

### Is the Poisson formula valid when the boundary condition is $ L^2 $?

Dirichlet problem for Laplace equation as follows
\begin{eqnarray}
\Delta{u}&=&0\text{ in }B_r(0)\\
u&=&g\text{ on }\partial B_{r}(0),
\end{eqnarray}
where $ g $ is continuous.
It is ...

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122 views

### Non-singular variety covered by pairwise disjoint singular subvarieties

Can you cover a non-singular algebraic variety by pairwise disjoint singular closed subvarieties? Varieties are over an algebraically closed field of characteristic other than $2$ and $3$.
In ...

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83 views

### Least positive value of a random polynomial

Fix a positive even integer $d$ and consider the polynomial $f(x)=c_d x^d+\ldots+c_1x+c_0$, where the $c_i$ are independent random variables that follow the uniform distribution in the interval $[-1,1]...