# All Questions

156,886
questions

-1
votes

0
answers

6
views

### Did Lebesgue like non-measurable set or not?

I was surprised by the following paragraph in Bressoud's A radical approach to Lebesgue's theory of integration, quoted by Caicedo's in his comment to this question:
Vitali's nonmeasurable set, ...

0
votes

0
answers

8
views

### Integral involving exp(i*cos(x)) and exp(i*sin(x))

Does somebody know if there is a formula for the integral
$\int \limits_0^{2\pi}e^{i a cos x+i b sin x }e^{imx }dx$
if m is integer.
The formulas 3.937(1,2) in [Gradshteyn, I. S., & Ryzhik, I. M. ...

0
votes

0
answers

12
views

### Functional equations with coupled arguments and additive sructure

Let $G$ be a locally compact abelian group and let $f: G \to \mathbb{R}^+$ be a continuous function satisfying the functional equation
$$f(x + \phi(y)) + f(y + \phi(x)) = 1 + f(x+y)$$
for all $x, y \...

-2
votes

0
answers

21
views

### Rigorous definition for the finite sum

Normally, one defines for $n,m \in \mathbb{N}$ and given $a_k \in \mathbb{R}$ for each $m\leq k \leq n$ the finite sum as $$\sum_{k=m}^n a_k:= \begin{cases} a_m+\dots +a_n &, n\geq m\\ 0 &, n&...

0
votes

0
answers

18
views

### A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$

Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...

2
votes

0
answers

31
views

### Canonical basis in equivariant K-theory of the Springer resolution

In Definition 15.0.2 of the notes from a course by Bezrukavnikov there is a characterization of canonical basis in K-theory of a Springer fiber which is due to Lusztig. This characterization is in ...

1
vote

0
answers

13
views

### Diameter bounds by mean curvature and area

I'm wondering about a generalization of Simon/Topping/Wu-Zheng's results on bounding diameter by the mean curvature, which roughly says: given a closed $\Sigma^{n-1} \subseteq M^n$,
$$\text{diam}(\...

-5
votes

0
answers

78
views

### Arxiv rejected my paper: What should do next [closed]

I got this below email and stating my work has lack of originality ( it is a survey paper under review): the email. It was on hold for 2 weeks and then rejected with this. Should I give another try on ...

-5
votes

0
answers

43
views

### about unprovability of the Riemann Hypothesis [closed]

I want to task my thesis is true or false.
About Riemann Hypothesis unprovability.
I have asked in quora, too. There are little supporter.
YouTube
https://www.youtube.com/watch?v=a5_eIqq_YQA
Original ...

5
votes

0
answers

61
views

### Infinite cardinals and learnability of probability distributions

Two players play as follows. Player one chooses a secret finitely supported probability distribution $P$ on $ω_k$ (or another known set with $\aleph_k$ elements), and randomly takes $n+1$ samples ...

5
votes

1
answer

34
views

### Example of two dinatural transformations between finite categories that do not compose

It is often stated that dinatural transformations do not compose. It is clear from their definition that there is no reason to expect them to compose. However, I have found it surprisingly difficult ...

0
votes

0
answers

28
views

### Lower bound for the size of a family of sets

Consider a family $\mathcal{G} = \{ A_1,B_1,\ldots,B_m \}$ of $m+1$ non-empty finite distinct sets with the following property:
$$A_1 \cap B_k = \emptyset, 1 \le k \le m$$
Let $\mathcal{F} = \{A_1 \...

1
vote

0
answers

70
views

### Serre functors and global dimensions

Let $k$ be a field.
Let $\mathcal{C}$ be an abelian category (over $k$).
$\mathcal{C}$ has a finite global dimension if there exists integer $n > 0$ such that
$$
\operatorname{Ext}^i(M,N) =0
$$
...

2
votes

0
answers

42
views

### Ind-completion commutes with category product

$\def\A{\mathcal{A}}
\def\C{\mathcal{C}}
\def\D{\mathcal{D}}
\def\ind{\operatorname{Ind}}
\def\op{\mathrm{op}}
\def\Hom{\operatorname{Hom}}$I am trying to understand the following result from ...

-5
votes

0
answers

38
views

### Do you think e equal to the sum of all 1 cubic meter existence in the high dimension universe? [closed]

Do you think e equal to the sum of all 1 cubic meter existence in the high dimension universe ? (What is the ultimate meaning of e ? )

0
votes

2
answers

93
views

### What is the smallest area of a central section of the unit hypercube?

Let $\mathcal{U} \subseteq \mathbb{R}^n$ denote the unit hypercube i.e. $\mathcal{U} = [0,1]^n$, and assume that for some $d \in \mathbb{R}^n$ one denotes by
$$
\mathcal{H} = \left\{x \in \mathbb{R}^n ...

1
vote

1
answer

90
views

### Why is $2A_0(X)=0$ for a cubic threefold $X$ containing a line, over an arbitrary field $k$

I can't quite follow Proposition $2.1$ of "UNIVERSAL UNRAMIFIED COHOMOLOGY OF CUBIC FOURFOLDS CONTAINING A PLANE". I posted this on Math stackexchange but got no answer.
Let $X$ be a smooth ...

0
votes

0
answers

20
views

### What is weak convergence of random permutons?

In various papers on permutons you can find statements similar to this (see Maazoun's thesis)
For any $n$ let $\sigma_n$ be a random permutation of size $n$. TFAE:
$(\mu_{\sigma_n})_n$ converges in ...

1
vote

0
answers

24
views

### Genus of binary quadratic forms: $f(x,y), g(x,y)$ in same genus if and only if represent same values in $(\mathbb Z/m\mathbb Z)^\ast$ for all $m$

In David Cox's book: Primes of the form $x^2+ny^2$, second edition, there is a theorem(Theorem 3.21, page 52) characterize whether two binary quadratic forms in the same genus. The contents of the ...

0
votes

0
answers

110
views

### Algebraic varieties over finite fields

Let $\mathbb{F}_q$ be a finite field of order $q$, and let $V \subset \mathbb{F}_q^d$ be an algebraic variety of dimension $m$. Let $\mathcal{C}$ be a finite set of structural constraints imposed on $...

0
votes

0
answers

45
views

### Polynomial iterations and cyclic unit groups

Let $n$ be a positive integer, and denote by $(\mathbb{Z}/n\mathbb{Z})^*$ the multiplicative group of units modulo $n$. Let $f(x) \in \mathbb{Z}[x]$ be an arbitrary polynomial with integer ...

0
votes

0
answers

94
views

### Quantum labyrinth

Let $M$ be a compact, connected, orientable 3-manifold with boundary $\partial M$. We interpret $M$ as a "quantum labyrinth." Let $x_0, x_1 \in \partial M$ be two distinct points ...

2
votes

1
answer

116
views

### Some questions on a paper of Rellich

I was trying to read the paper "Über das asymptotische Verhalten der Lösungen von $\Delta u+\lambda u =0$ in unendlichen Gebieten" by Franz Rellich (MR17816, Zbl 0028.16401).
Since it is in ...

13
votes

1
answer

311
views

### We have $\binom{62}{26}^2+\binom{62}{27}^2=\binom{62}{28}^2$. How many other Pythagorean triples are contained in a single row of Pascal's triangle?

At MSE I asked, "Does any row of Pascal's triangle contain a Pythagorean triple?" The answer is yes; the example $\binom{62}{26}^2+\binom{62}{27}^2=\binom{62}{28}^2$ was given.
Now my ...

0
votes

0
answers

21
views

### Optimal rectification and dissociated set decompositions in finite abelian groups

Let $G$ be a finite abelian group of order $n$. For a subset $A \subseteq G$, we define its rectification threshold $R(A)$ as the smallest integer $R$ such that there exists some $\lambda \in G^*$ for ...

1
vote

0
answers

51
views

### Counting conjugacy classes of completely reducible subgroups in general linear groups

Let $G = GL(n, q)$ be the general linear group over the finite field $\mathbb{F}_q$ with $q$ elements, where $q$ is a power of a prime $p$. Let $m$ be a positive integer dividing $q-1$. Suppose $\...

3
votes

3
answers

653
views

### Why is resonance such a widespread phenomenon?

It is easy to mathematically describe the motion of a mass which is attached to a spring and also pushed around by a sinusoidal force. We get a differential equation of the form:
$$\frac{\mathrm{d}^2x}...

3
votes

0
answers

26
views

### Small deviation asymptotics for sub-gaussian diffusions in dirichlet spaces

Let $(X,d,\mu)$ be a metric measure space equipped with a strongly local, regular Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ on $L^2(X,\mu)$. Assume that the associated heat kernel $p_t(...

2
votes

1
answer

78
views

### Finiteness and bounds for elliptic curves realizing a given galois representation

Let $\rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}_p)$ be a continuous, irreducible Galois representation. Consider the set $\mathcal{L}_\rho$ of all elliptic curves $...

0
votes

0
answers

28
views

### Compact HKT structures on moduli spaces of connections with torsion

Let $M$ be a smooth, compact, oriented 4-manifold equipped with a Riemannian metric $g$. Let $H$ be a non-vanishing, closed 3-form on $M$. A hypercomplex structure on $M$ is a triple of endomorphisms $...

1
vote

0
answers

65
views

### Identifying $d_1$ in the Atiyah-Hirzebruch-Serre spectral sequence

In A Primer on Spectral Sequences (also later published in More Concise Algebraic Topology), J. Peter May describes the Serre Spectral Sequence for any homology theory. To recap, suppose $p\colon E\...

3
votes

1
answer

164
views

### Rigorous statistical mechanics: difficulty of realistic models

Soft question: I am a mathematician self-learning statistical mechanics. The (mathematical) literature is concentrated on lattice models like the Ising model and the lattice-gas model. I understand ...

-1
votes

0
answers

29
views

### Optimal Stratification of Time-Evolving Relational Structures with Constrained Update Mechanisms

Let $(S, T, \preceq)$ be a totally ordered set of timestamps, where $\preceq$ is the natural ordering on timestamps.
Define a relational structure $R$ as a tuple $(I, A, V, \tau)$ where:
$I$ is a ...

2
votes

0
answers

71
views

### Galois cohomology for rational torsion of elliptic curves

Let $E/K$ be an elliptic curve over a number field.
Let $M=K(E[p])$.
I want to know $H^1(M/K,E[p])$: for $p=2$, it is $0$, but what about the case $p>2$?
Is it always zero?
In fact, I want to know ...

0
votes

1
answer

39
views

### Is a continuum in the plane regular for the Dirichlet problem at all points?

As the title asks. Let me elaborate; suppose $\mathcal K$ is a continuum (compact, connected) set of $\mathbb C$ (with at least two points!). Let's say that $g(z;a)$ is the green's function of the ...

6
votes

0
answers

86
views

### Projections of closed geodesics under the modular function

In the answers to this question it was shown that for closed geodesics on $\mathbb{H}^2/\Gamma(2)$, the projection under the modular function $\lambda$ is an immersed topological component of a real ...

3
votes

0
answers

79
views

### Why the hyperbolic Laplacian?

In the theory of automorphic forms there is the weight $k\in\mathbb{Z}$ Laplacian
\begin{align*}
\Delta_k:=-y^2 \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+iky\frac{\...

-2
votes

0
answers

18
views

### Excesive couples in simplicial homology [migrated]

I am reading "Elements of topological algebra" by Munkres and here is an exercise:
Prove, that if K1 and K2 are ...

3
votes

1
answer

63
views

### Direct characterization of finite-dimensional $1$-injective Banach spaces

It follows from Kelley's Theorem that the only finite-dimensional $1$-injective Banach spaces are $\ell^\infty_n$, $n\in\mathbb N$. Is there a simple direct proof of this fact, without having to talk ...

1
vote

0
answers

35
views

### Bounding the Bloch-Kato Selmer group of a twisted symmetric power of a Tate module

Let $E$ be an elliptic curve over $\mathbb{Q}$, with good reduction at $p$, and let $V = H^1_{et}(\overline{E}, \mathbb{Q}_p)$ be (the dual of) its (rationalized) Tate module.
Let $S^nV$ denote its $n$...

-1
votes

0
answers

28
views

### What does the decision boundary of a hypothesis in a logistic regression actually signifies and how to visualise it?

Suppose I am solving a linear regression of the form $h_\theta(x) = \theta^T x$ which is shown on the left of the image. Now if I solve it as a logistic regression problem by using a sigmoid function ...

2
votes

1
answer

132
views

### Field extensions and completions at possibly infinite places

In Serre's Corps Locaux, Chapter 2 §3, is presented a classical proof. We are in an "ABKL" setup, where $K/L$ is finite, $A$ is Dedekind, $B$ is the integral closure of $A$ and $B$ is $A$-...

2
votes

1
answer

123
views

### Exponential sums over a linear subspace

I'm looking into certain type of exponential sums, which are summed over a linear subspace, and I couldn't find a good reference for that.
The (simplified) setting is the following. Let $p$ be a prime,...

7
votes

0
answers

214
views

### Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$

I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by:
$$
Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy
$$
This operator ...

4
votes

1
answer

203
views

### Is Morava K-theory of a classifying space of a compact Lie group a Noetherian ring?

Let $p$ be a prime and $n > 1$ a height. My conventions for Morava K-theory are that $K_p(n)^*(pt)=\mathbb{F}_p[v_n,v_n^{-1}]$, $|v_n|$ (the degree of $v_n$) is $2(p^n-1)$.
Question: If $G$ be a ...

4
votes

0
answers

135
views

### Is $\overline{\mathcal{M}}_{g,n}$ a Koszul space?

In https://arxiv.org/abs/1902.06318 Dotsenko proved that $\overline{\mathcal{M}}_{0,n+1}$ is a Koszul space, i.e. it is both formal and coformal. Equivalently, a space is Koszul if it is formal and ...

4
votes

0
answers

64
views

### Reinforced Maximum Principle

Let $U\subset{\mathbb R}^n$ be a bounded open domain with smooth boundary. I assume that $U$ is diffeomorphic to a ball. You may think of $L=\Delta$ and $U$ is the unit ball.
Let $L=\operatorname{div}(...

1
vote

0
answers

78
views

### Euler-Lagrange equation of fractional Laplacian

The following result is in "An extension problem related to the fractional Laplacian" Section 3.2 by Caffarelli-Silvestre. I’m confused how to show it and wish to have some help.
Suppose $u:\...

-3
votes

0
answers

81
views

### representation of finite $p$-group [closed]

Suppose $G$ is a finite non-abelian $p$-group, $F_{p}$ is finite field of $p$ elements. If there is a injective homomorphism of $G$ to $GL(n,F_{p})$,then
$$n\geq \frac{|G|}{p} ?$$

5
votes

1
answer

129
views

### Can a scattered profinite set continuously surject onto a non-scattered profinite set?

A topological space is scattered if every nonempty subset has an isolated point. Are there any continuous surjections from a scattered profinite set to non-scattered profinite set?