# All Questions

137,605
questions

2
votes

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27
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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. ...

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 $\...

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 \...

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 $S_{étale}$.
...

-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)

-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 ...

-1
votes

0
answers

16
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### 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_{\...

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 ...

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 /...

0
votes

0
answers

17
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### 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 $\lbrace\omega_{ij}=\omega_{ji}\in\mathbb{R}:\lbrace i,j\rbrace\mapsto\omega_{ij}\rbrace$ ...

3
votes

0
answers

54
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### 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 +...

-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 ...

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 ...

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, ...

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,...

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\...

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 ...

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 ...

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 $\lambda_f(n)$ ...

-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 ...

3
votes

0
answers

49
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### 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 ...

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}(...

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,\...

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 ...

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 \...

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 ...

-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)...

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 ...

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 ...

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 ...

-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 ...

1
vote

0
answers

34
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### 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(...

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 ...

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 ...

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 $\mathcal{P}$, $r_1$...

0
votes

0
answers

11
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### 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 ...

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: ...

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 ...

3
votes

0
answers

50
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### 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 ...

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 $\...

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 ...

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
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### 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 ...

9
votes

0
answers

180
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### 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,\, \...

8
votes

2
answers

408
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### 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 ...

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 ...

7
votes

0
answers

82
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### 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 ...

3
votes

0
answers

31
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### 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(...

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 ...

6
votes

0
answers

106
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### 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 ...