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Existence of $\sqrt{2}$ in a finite group algebra over $\mathbb{Q}$

I cannot find a finite group $G$ such that $\exists x\in \mathbb{Q}[G]$ with $x^2=2e$, where $\mathbb{Q}[G]$ is the group algebra of $G$ over $\mathbb{Q}$. I also could not prove it does not exist. ...
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2 votes
0 answers
14 views

Fourier transform of the hyperboloid

Equip $\mathbb{R}^{d+1}$ with the Lorentzian form $\langle x, y\rangle=-x^0y^0+{\bf x}\cdot{\bf y}$ where $x=(x^0,{\bf x})$ and $\cdot$ is the usual Euclidean dot product. We define the hyperboloid $\...
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  • 181
1 vote
0 answers
24 views

Cardinality of set of powers and roots of an element in a group $G$

For a group element $a$, let $b = a^{\frac{1}{k}}$ be any solution of $x^k = a$ if it exists. Using this, for a finite group $G$ and $a \in G$, we define $${\rm Pow}(a) := \left\{b \in G : b = a^n \...
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  • 1,149
0 votes
0 answers
36 views

Surjective sheaf homomorphisms induced by morphisms of schemes

Let $S$ be a scheme and $X\to Y$ be a morphism over $S$. Then we have an induced homomorphism of sheaves $h_X=\mathrm{Hom}_S(-, X)\to h_Y=\mathrm{Hom}_S(-, Y)$ over the small étale site $S_{étale}$. ...
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  • 387
-1 votes
0 answers
22 views

is there a rule that sayas if fof(x)= the inverse of fof(x) |||| is f(x)=the inverse of f(x) [closed]

at the end fof(x)= inverse of fof(x) is there a rule that says when you have fof(x)=inverse fof(x) ===>>>> f(x)=the inverse of f(x)
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-1 votes
0 answers
39 views

Strongly polynomial theorem for QP Basis Identification Algorithm seem to imply $NP=P$

Based on Berkelaar, Jansen, Roos, and Terlaky - Basis- and tripartition identification for quadratic programming and linear complementarity problems: Theorem 4.2 If there exists a strongly polynomial ...
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-1 votes
0 answers
16 views

How to find a family of probability density functions satisfying the following properties

Let $\mathcal{F} = \{f_{w}\}_{w \in \mathbb{R}_{\geq 0}}$ be a family of probability density functions satisfying the following properties. The support domain of $f$ is $[0, 2w]$, i.e., $\int_{\...
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  • 193
0 votes
0 answers
63 views

No way to seventh heaven--the cursed couple [5][2]

The refined Eulerian partition polynomials $[E]$ of OEIS A145271 are intimately related to Graves-Lie infinitesimal generators / exponentiated vector fields and provide the coefficients of the ...
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  • 8,292
3 votes
1 answer
52 views

How to get by with only functorial cylindrical objects?

In the excellent "A Handbook of Model Categories" (2021), cofibrant and fibrant homotopies are defined exactly as it seemed natural to me: immediately through functorial cylindrical objects /...
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0 votes
0 answers
17 views

Simple-path decomposition of random tours

let $V$ denote a set of $n$ vertices, $\lbrace\lbrace v_i,v_j\rbrace\subset V\rbrace$ the set $E$ of edges and $\lbrace\omega_{ij}=\omega_{ji}\in\mathbb{R}:\lbrace i,j\rbrace\mapsto\omega_{ij}\rbrace$ ...
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3 votes
0 answers
54 views

Equation $wxyz(w+x+y+z)=1$ in $\mathbb{Q}_+^4$

In this thread on Math.SE, Noam D. Elkies give the following parametric family of solutions in $\mathbb{Q_+}^3$ of the equation $xyz(x+y+z)=1$ (found by Euler) : $$ x = \frac{6 t^3 (t^4-2)^2} {(4 t^4 +...
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  • 463
-3 votes
0 answers
31 views

Probability que. Of bacteria [closed]

There is a bacteria. It can die with a prob 1/4. Splitting into 2 prob 1/2 & probability of splitting into 3 is 1/4. All the bacteria are identical. What is the probability that chosen bacteria ...
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1 vote
0 answers
21 views

Banach algebras satisfying $pq=qp=q \Rightarrow \|q\|\leq\|p\|$ for all idempotents $p$ and $q$

This question could be way below the level of MO, so apologies in advance. I posted the same question in MS about 10 days ago without a definitive answer so far. Let $A$ be a Banach algebra with the ...
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  • 1,239
0 votes
1 answer
52 views

Connectivity of a matroid is at least its rank?

The connectivity $\eta(X)$ of a simplicial complex $X$ is defined as the $$1+\min_j\{j \mid \tilde{H}_j(X)\neq 0\}.$$ If no such $j$ exists, then $\eta(X):=\infty$. (See here for this definition, ...
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  • 201
0 votes
0 answers
26 views

Integral involving Bessel and Laguerre function

Is there a formulas for the following integral $$\int^\infty_0 e^{-ar^2}L^1_k(b r^2)J_1(cr)r^d dr $$ where $L^1_k$ is the Laguerre polynomials of type 1 and $J_1$ is the Bessel function with $a,...
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2 votes
2 answers
60 views

Maximal uncountable chains in ${\cal P}(\omega)$

Let ${\cal P}(\omega)$ denote the power-set of $\omega$. We order it by set inclusion $\subseteq$ and say that ${\cal C}\subseteq {\cal P}(\omega)$ is a chain if for all $A, B\in {\cal C}$ we have $A\...
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2 votes
1 answer
34 views

Minimal digraph covering with no 2-path edge sets is of size $\left( 1 + o \left( 1 \right) \right) \log_2 \chi(G)$

The last problem in 2022 IMC Day 1 strongly correlates with graph theory. In its official solution, the fundamental approach can be rephrased as follows. Give a digraph $G=(V,E)$. We call a subset of ...
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1 vote
1 answer
60 views

Space filling curves

The classic Hahn-Mazurkiewicz theorem has the following consequence: Let $X$ be a compact, connected topological manifold. Then there is a continuous surjective map $f: [0,1] \rightarrow X$. It is ...
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2 votes
0 answers
49 views

Entropy of a sequence

I am reading the paper Sign Changes in Hecke Eigenvalues by Matomaki and Radziwill, and in one place they mention the following, It would be interesting to rule out the possibility of $\lambda_f(n)$ ...
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-1 votes
1 answer
136 views

In Galois theory, why solvable groups must have their quotient groups be Abelian? [closed]

The definition of solvable groups can be regarded as two constraints, one is that there must be a sequence of normal subgroups, and the other is that the quotient groups between these sequences are ...
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  • 1
3 votes
0 answers
49 views

Candidates of nonsmoothable homology 4-spheres

I'm not trying to duplicate a question Are there non-smoothable homotopy/homology spheres? But in Igor Belegradek's answer, he pointed out that "Some of them (homology spheres) have large ...
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  • 908
0 votes
1 answer
51 views

Characterization of extendible distributions

I asked this question on Mathematics Stackexchange, but got no answer. I found the following question which characterize the extension of a distribution in $\mathbb{R}$: Let $f \in L_{\text{loc}}^{1}(...
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0 votes
0 answers
54 views

What to do when the second moment method does not provide a sufficient bound for $P(X=0)$

We have that for a real valued random variable $X$, $$ P(X=0) \leq \frac{\text{Var}(X)}{\left(\mathbb{E}(X)\right)^2} $$ known as Chebyshev's inequality. Consider a random variable $X \in \{0,1,2,\...
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  • 486
6 votes
0 answers
165 views

A coincidence or a fact: determinants of two matrices

While playing around with the MO question Determinant with factorials is not 0? about a determinant of the Hankel matrix of entries $(i+j-2)!$, having the value $\prod_{k=0}^{n-1}k!^2$, I stumbled on ...
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0 votes
0 answers
39 views

Diameter of a Cartesian product of two graphs

If I am looking at a Cartesian product of two graphs $G_1$ and $G_2$ (defined here https://en.wikipedia.org/wiki/Cartesian_product_of_graphs). I am trying to bound the diameter of the graph $G_1 \...
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2 votes
0 answers
82 views

How to simplify this homotopy totalization coming from an arc-cover into a pullback?

My question concerns the proof of Proposition 4.2 in Bhatt-Mathew’s paper on the arc-topology, but my confusion is completely general and anyone familiar with limits in $\infty$-categories would know ...
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  • 1,035
-2 votes
1 answer
159 views

Digit sum of a prime number

Let 𝑝 be a positive integer (𝑝 < 997) and 𝑞 = 𝑆(𝑝) be the digit sum of 𝑝 such that 𝑞 + 1 ≡ 0 (mod 2). Is it that if 𝑝 is prime then 𝑞 is also prime? e.g. 𝑝=47(prime)-> 𝑞=4+7=11 (prime)...
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0 votes
0 answers
32 views

Inclusion-exclusion in a set of multivariables

I want to determine the risk of a multi-asset portfolio, in a way different from previous attempts. It is because current methods focus on just the correlation coefficient of two variables, while in a ...
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  • 221
2 votes
0 answers
42 views

Cluster expansion, Mayer expansion and perturbative renormalization group

This is a second part of my previous question, which I decided to split into two parts not to mix up different topics at one giant question. Again, according to V. Rivasseau (section 1.5 of ...
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  • 1,077
3 votes
0 answers
76 views

What is a large field problem?

I was reading Constructive Renormalization Group by V. Rivasseau and I got some points which I would like to clarify. On page 2, Rivasseau talks about the large field problem and, if I understood it ...
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  • 1,077
-2 votes
0 answers
95 views

Let $V$ be a variety. A point $P \in V$ is nonsingular iff $\dim_k(M_P/M_{P}^{2})=\dim(V)$ [closed]

First of all, we consider $k$ to be an algebraically closed field, and by $M_P$ I denote the maximal ideal of the coordinate ring $k[V]$ at $P$. As for the statement, I have managed to understand how ...
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  • 121
1 vote
0 answers
34 views

Control of solutions to nonlinear elliptic equations away from boundary

Let $\Omega$ be a bounded domain in $\mathbb R^3$ with a smooth boundary. Consider a smooth real valued function $F:\overline\Omega \times \mathbb R \to \mathbb R$ with the property that $\partial_s F(...
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  • 3,007
4 votes
0 answers
89 views

Dense triangle-free graphs and their independent sets

Recall that a graph is triangle-free if it does not contain a copy of $K_3$. Also, for a graph $G$, $\alpha(G)$ shall denote its independence number. Lastly, we will write $o(1)$ to denote quantities ...
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0 votes
0 answers
46 views

A normal form for an operator within its Algebra

Let $H$ be a Hilbert space. For a compact bounded linear operator $T$ on $H$ of norm less than or equal to one and any continuous function $f$ on the unit interval $[0,1]$ with $f(0)=0$, we define the ...
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  • 3,007
0 votes
1 answer
87 views

3-dimensional hyperbolic space

In the 3-dimensional hyperbolic space there are given a plane $\mathcal{P}$ and four distinct lines $a_1, a_2, r_1, r_2$ in such positions that $a_1$ and $a_2$ are perpendicular to $\mathcal{P}$, $r_1$...
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  • 1
0 votes
0 answers
11 views

Benefit of a reformulation of $k$-factor calculation

In the article "An algorithm for computing simple k-factors.”, Meijer, Henk, Yurai Núñez-Rodríguez, and David Rappaport the authors essentially address the problem of finding $k$-factors of ...
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  • 11.2k
1 vote
0 answers
86 views

Abelian subvarieties corresponding to vector subspaces

Let $S$ be a connected smooth projective surface. Let $C$ a smooth curve on $S$ In page 9 of the paper "https://arxiv.org/abs/1704.04187v1" a read the following: Let \begin{equation*} r: ...
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  • 431
1 vote
0 answers
28 views

The solid hull and weak*-closure in Banach lattices

Let $E$ be a Banach lattice. A subset $A$ of $E$ is called solid if $|x|\leq |y|$ for some $y\in A$ implies that $x\in A$. For a subset $A$ of $E$, the solid hull of $A$ is the smallest solid set ...
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3 votes
0 answers
50 views

Visiting zero-sum triples in a vector space

Is it true that for any set $A\subset\mathbb F_5^n$ satisfying $A\cap(-A)=\varnothing$, there is a subset $A'\subseteq A$ such any triple $(a,b,c)\in A\times A\times A$ with $a+b+c=0$ has exactly one ...
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  • 21.7k
5 votes
1 answer
90 views

Intermediate value property for Sobolev functions

Let $d \geq 2$, and let $f \in W^{1, 1} (\mathbb R^d)$ be a Sobolev function. Question: For any $a, b \in \mathbb R$ such that $\text{essinf } f \leq a < b \leq \text{esssup } f$, is it true that $\...
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  • 1,197
2 votes
0 answers
84 views

Possible characterisation of compactly generated weakly Hausdorff spaces

Is it true that, in the category $\mathbf{Top}$ of topological spaces and continuous maps, the compactly generated weakly Hausdorff spaces are precisely the spaces arising as filtered colimits of ...
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3 votes
0 answers
48 views

When does the null-cone consist entirely of eigenvectors?

Let $V$ be a finite-dimensional representation of a complex reductive Lie algebra $\mathfrak g$. For our purposes, we may define the null-cone like this: $v\in V$ belongs to the null-cone if and only ...
7 votes
2 answers
181 views

Examples of homology sphere that bound a nonsmoothable contractible 4-manifold

Freedman’s theorem shows that all 3-dimensional homology spheres bound topologically a contractible 4-manifold. It is well known that the Poincaré homology sphere does not bound a smooth contractible ...
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  • 2,553
9 votes
0 answers
180 views

If $A+A+A$ contains the extremes, does it contain the middle?

Let $b \ge 1$ and $A\subseteq [0,b]$ be a set of integers (all intervals will be of integers). Write $hA := \underbrace{A + \ldots + A}_{h\text{ summands}} = \{ \sum_{i=1}^h a_i ~|~a_i \in A,\, \...
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  • 815
8 votes
2 answers
408 views

Homotopy coherent generalization of classifying space theory

Classically, given a compact Lie group $G$, there is a topological space $BG$ which classifies principal $G$-bundles. This means that there is an equality of sets {principal $G$-bundles up to ...
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2 votes
0 answers
78 views

Ash–Stevens for Hilbert modular forms

In the theory of mod-$p$ modular forms, I learned a while ago about an interesting result that I think is technically due to Serre and Tate, though the proof was first published by Jochnowitz in ...
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  • 107
7 votes
0 answers
82 views

How nice can sets of reals be under $\mathsf{ZF} + \mathsf{BPI}$?

It's well known that the full axiom of choice is not needed to prove the existence of non-measurable subsets of $\mathbb{R}$. In particular, the Boolean prime ideal theorem ($\mathsf{BPI}$) is ...
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  • 6,534
3 votes
0 answers
31 views

how to efficiently find level sets (using a modification of a root-finding algorithm)?

I'm trying to find a set of points $\{ a_i | f(a_i) = c_i \}_{i=1}^k$ where $f$ and $\{ c_i \}_{i=1}^k$ are given in sorted order. All $c_i > 0$, $f$ is continuous and monotonically increasing, $f(...
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5 votes
0 answers
151 views

A $p$-adic homotopy theory for non-simply connected spaces?

I'm looking to understand the state of the art for $p$-adic (unstable) homotopy theory of non-simply connected (non-nilpotent!) spaces. Ideally, I'd also like integral versions, e.g. things like ...
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6 votes
0 answers
106 views

Surprising symmetry in the Ramanujan bound

The condition for a connected $(q+1)$-regular graph to be Ramanujan is that every nonzero eigenvalue $\lambda$ of the graph Laplacian satisfy $$q+1-2\sqrt{q}\le \lambda\le q+1+2\sqrt{q}.$$ With a ...
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