# All Questions

**1**

vote

**1**answer

109 views

### Determining conjugacy class of a subgroup from the sizes of its intersections with the conjugacy classes

Let $C_1,C_2,\ldots,C_n$ be the conjugacy classes of a finite group $G$. It is possible that there are two non-conjugate subgroups $K \leq G$ and $H \leq G$ such that $|H \cap C_i|=|K \cap C_i|$ for ...

**2**

votes

**0**answers

61 views

### Number of multipartite partitions with odd components

For some positive integer $r$, by an $r$-vector I will mean an $r$-tuple $(a_1,a_2,\dots,a_r)$ with $a_1,\dots,a_r$ nonnegative integers not all zero, and I will call it odd if $a_1,\dots,a_r$ are all ...

**3**

votes

**1**answer

157 views

### G-Correlation of Vectors

Let $\vec{a},\vec{b} \in \mathbb{R}^{n}$. Consider the function $f: S_n \to \mathbb{R}$ given by $f(\sigma):= \sum_{i=1}^{n} a_i b_{\sigma(i)}$. Let $G$ be a subgroup of $S_n$, given by generators.
...

**0**

votes

**0**answers

49 views

### Square root of a measure preserving action

Let $G \curvearrowright (X,\nu)$ be probability measure preserving action of a countable discrete group. When does there exist a probability measure preserving action $G \curvearrowright (Y,\mu)$ such ...

**7**

votes

**1**answer

128 views

### Is the word problem decidable for free finitely generated self-square groups?

A self-square group is a group with extra structure, which encodes the fact that the group is isomorphic to its own direct square.
To be exact, the group $G$ has a special element $1$, a unary ...

**0**

votes

**0**answers

45 views

### A PDE problem on the Heat-Like differential equations

I came across the following questions in part of my work:
Consider the Heat-Like equation of the form $\frac{\partial u}{\partial t}=\hat{H}u + f(x,t)u + g(x,t)$ where $\hat{H}$ is a Sturm-Liouville ...

**10**

votes

**2**answers

349 views

### BSD and congruent numbers

Let $n$ be a positive integer, and let $E_n$ denote the elliptic curve $y^2=x^3-n^2x$. By work of Tunnell, it's known that if $E_n$ satisfies the BSD conjecture, then there is an algorithm to decide ...

**3**

votes

**0**answers

63 views

### Topos Theory, internal Heyting Algebra

Given a topos $\mathcal{E}$ with subobject classifier $\Omega$.
If we denote by $N\Omega$ the former of all local operators on $\Omega$, that is, Lawvere–Tierney topologies of $\mathcal{E}$, it is ...

**9**

votes

**1**answer

405 views

### Categorical or simplicial introduction to modern homotopy theory

I've heard some people saying that it would be easier to study homotopy theory first through simplicial sets. First, i thought that these people were not being serious. But it caught me wondering...
...

**0**

votes

**1**answer

78 views

### For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$

Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...

**30**

votes

**3**answers

2k views

### A curious determinantal inequality

In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...

**2**

votes

**0**answers

115 views

### Maximal abelian extension and tamely ramify extension

Let $K$ be a number field and $v$ a finite place of $K$ lying above a prime number $p$. Assume further that $v$ is unramified over $p$. If $K^{ab}$ is the maximal abelian extension let $w$ be a ...

**1**

vote

**0**answers

57 views

### The normalizer problem for group rings

I recently studied about The Normalizer problem (NP) which states that given an integral group ring $\Bbb{Z}G$, $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}$ denotes centre of $\cal{U}$ = ...

**-3**

votes

**0**answers

15 views

### Test correlation between 3 variables by hand [on hold]

Do the three areas (courtside, lower deck, upper deck) differ in soda sales per hour?
Courtside Lower Deck Upper Deck
38 35 11
42 37 25
40 ...

**25**

votes

**3**answers

704 views

### A point set of power series with coefficients in {-1, 1}. Connected or not?

Let $z$ be a fixed complex number with $|z|<1$ and consider the set
$$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$
What can be said about the set $M$ ...

**4**

votes

**0**answers

236 views

### Euler characteristic, character of group representation and Riemann Roch theorem

I am considering the following set up:Let $G$ be a finite group,let $Rep(G)$ denote the category of finite dimensional representations over $\mathbb{C}$. Let $V,W$ be representations of $G$ in ...

**-1**

votes

**2**answers

88 views

### Do graphs with $\omega(G) = \chi(G)$ grow “common” as $|V|$ grows large?

On the set $[n]:= \{1,\ldots,n\}$ we consider the set $${\cal P}_2([n]) = \big\{\{a,b\}: a,b \in [n], a\neq b\big\}.$$
Since $$|{\cal P}_2([n])| =2^{n \choose 2}$$ there are exactly $2^{n\choose 2}$ ...

**4**

votes

**3**answers

190 views

### Ergodic theory: from Dynamics to Gibbs measure

I'm trying to understand the ergodic theory approach to statistical mechanics, namely how ergodic measure preserving dynamics lead to the Gibbs measure.
I have a compact space $X$, a probability ...

**0**

votes

**0**answers

64 views

### Are there unconditional results for boundedness of finitely many rational points on $f(x,y)=n$ for all $n$?

As the title says, over the rationals, are there unconditional results for boundedness by the degree of $f(x,y)$ of the number of finitely many rational points on
$f(x,y)=n$ for all rational $n$?
...

**49**

votes

**4**answers

4k views

### Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?

The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first ...

**11**

votes

**3**answers

696 views

### Why does the bitxor function appear in Nim?

I am conducting research in Combinatorial Game Theory (CGT). Although I have done a considerable amount of reading, I do not completely understand why the bit-xor function also known as the nim-sum ...

**2**

votes

**0**answers

82 views

### Representations of Hamilton's real/complex quaternions algebra

A lot of works and questions deal with classifying representations of a simple central algebra of given dimension over a non-archimedean field, for instance here.
But do we know precisely such a ...

**11**

votes

**5**answers

972 views

### Is $x^p-x+1$ always irreducible in $F_p[x]$?

It seems that for any prime number $p$ and for any non-zero element $a$ in the finite field $\mathbb F_p$, the polynomial $x^p-x+a$ is irreducible over $\mathbb F_p$. (It is of course obvious that ...

**3**

votes

**2**answers

361 views

### What does it mean for a differential equation “to be integrable”? [duplicate]

What does it mean for a differential equation "to be integrable"?
Are "integrable" and "solvable" synonyms?
The first thing that comes to my mind is to say: it's integrable if we can find the ...

**5**

votes

**0**answers

49 views

### Upper bound for a sum including Andre polynomial coefficients

Let $ c_{n,k} $ be a sequence defined by the following relations: $\displaystyle c_{n,0} = 1, \hspace{0.1cm} (n \ge 1)$;
$$ c_{n,k} = (k+1) c_{n-1,k} +(n-2k+1) c_{n-1,k-1}, \hspace{0.5cm} (1 \leq k ...

**4**

votes

**0**answers

43 views

### General approaches to extension theorems as Caratheodory

I would like to know if there are some general studies about extension-like theorem, in the sense which i'm going to describe. This paragraph is not rigorous; I just would like the idea to be clear.
I ...

**4**

votes

**2**answers

115 views

### Extraction of Coefficients in the Exponential Function of a Series

Question: Let $f(x) \in x\mathbb{C}[[x]]$. What is the (asymptotically) fastest algorithm for calculating the coefficient of $x^n$ in $e^{f(x)}$?
Naive Solution 1: Using fast polynomial ...

**4**

votes

**2**answers

115 views

### “Natural” action of a semidirect product of groups

Suppose we have a group $N$ acting on a set $X$, and a group $Q$ acting on $Y$. Moreover, we are given $\phi: Q \to \mathrm{Aut}(N)$.
My question is, is there a "natural" action on a set that can be ...

**7**

votes

**0**answers

117 views

### $k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set ...

**3**

votes

**1**answer

82 views

### Is there a unique line bundle in the Kummer surface which pulls back to a totally symmetric line bundle?

Let $X=Jac(C)$ be an abelian surface over $\mathbb{C}$, the Jacobian of a genus 2 curve. Let $L$ be a symmetric line bundle. Let $Y$ be the Kummer surface, quotient of $X$ by the action of involution. ...

**0**

votes

**0**answers

17 views

### Bayesian model to estimate the parameter of a Bernoulli law

Suppose we have iid boolean variables $X_1,...,X_T = X_{1:T}$ and the associated deterministic parameters $k_1,...,k_T=k_{1:T}$ and $c_1,...,c_T=c_{1:T}$, where for each $t \in \mathbb{N}$, $k_{t} \in ...

**1**

vote

**1**answer

97 views

### Sum of Eigenvectors Entries of an Adjacency Matrix

I have a question regarding the sums $\sum_{i=1}^{n}v_{j}\left(i\right)$ where $v_j$ are eigenvectors of adjacency matrix $A$ which have been normalized to unit length.
Ordering the eigenvectors by ...

**4**

votes

**0**answers

137 views

### Wild automorphisms of a free group

Let $F_X$ be a free group on a countably infinite set $X$. Let $\alpha$ be an automorphism of $F_X$ and $H$ a closed subgroup of $F_X$ in the profinite topology.
Is it possible that $\alpha(H) ...

**3**

votes

**2**answers

237 views

### A property of uncountable almost disjoint families

Let $\mathcal{A}$ be an uncountable almost disjoint family (not necessarily maximal) of infinite subsets of $\mathbb{N}$. Denote by $\mathcal{A}_{\subseteq}=\{ B\subseteq\mathbb{N}:|B|=\omega \wedge ...

**4**

votes

**1**answer

192 views

### Unit in cyclotomic field

Let $n \in \mathbb{N}$ and $\zeta$ be a primitive $n$-th root of unity. I want to know for which $n$ the element $1+2(\zeta+\zeta^{-1})$ is a unit in the ring of integers of $\mathbb{Q}[\zeta]$. Can ...

**-3**

votes

**0**answers

52 views

### Closed form for Stirling numbers of second kind [on hold]

so yeah, a friend and I have been trying to find a closed form for the Stirling numbers of second kind with little success.
https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
The ...

**2**

votes

**0**answers

52 views

### what is the link between plethysm in regular representation of the symmetric group and plethysm in Schur functions.

I am trying to understand first how one can define the plethysm say $s_\lambda \circ s_\mu$ as a module in the regular representation of the symmetric group.
1)How is it connected to the plethysms ...

**1**

vote

**0**answers

108 views

### Local cohomology commuting with fiber

Let $A$ be a nice commutative ring (say, $A=k[t_1,\ldots , t_n]$, ring of polynomials over an algebraically closed field $k$).
Let $M$ be an $A[x]$-module, which is finitely generated as an ...

**5**

votes

**0**answers

88 views

### Eigenvalue inequality for regular graphs

I recently proved an inequality relating some of the eigenvalues of a regular graph with each other, and I was wondering if it is already known. I was unable to find it online, and a quick skim ...

**11**

votes

**2**answers

578 views

### Number theory and physics

I was following some lectures by Edward Frenkel about Langlands correspondence. He was describing some analogies between number theory and theoretical physics (Mirror symmetry). At some point ( my ...

**2**

votes

**1**answer

200 views

### Subgroups of hyperbolic groups

Let $G$ be a finitely generated hyperbolic group, and let $H \leq G$ be a subgroup whose profinite completion is finitely generated. Must $H$ be finitely generated?
In view of Ian Agol's answer, I ...

**2**

votes

**1**answer

75 views

### Prescribing an induced metric

We know that, if we have a surface $z=f(x,y)$ with Euclidean space being ambient manifold, the induced metric is as follows (in matrix form):
$$g=\begin{bmatrix}
1+\left ( \frac{\partial ...

**0**

votes

**0**answers

80 views

### $C = V(x^3 - xz^2 - y^2z)$, linear equivalence [on hold]

Let $C = V(x^3 - xz^2 - y^2z) \subset \mathbb{P}^2(\mathbb{C})$. Let $p_0 = [0, 1, 0]$, $p_1 = [0, 0, 1]$, $p_2 = [1, 0, 1]$, $p_3 = [-1, 0, 1]$. I have two questions.
Is $2p_0$ linearly equivalent ...

**9**

votes

**1**answer

364 views

### Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan

The following is a re-post from MSE because I did not get any answer even after offering a bounty.
Towards the end of G. N. Watson's (one of the joint authors of famous book "A Course of Modern ...

**2**

votes

**1**answer

191 views

### In a fibration, where does the generic object live?

In Bart Jacobs. Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics 141. North Holland, 1999. isbn: 9780444508539, the author writes, p. 326:
Sometimes, for ...

**105**

votes

**27**answers

14k views

### Extremely messy proofs

Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery. What I am not being taught, largely, is what ...

**1**

vote

**0**answers

91 views

### Centerless finite groups with exactly three prime divisors

Let $G$ be a centerless finitee group and $\pi(G)=\{p,q,r\}$ be the set of prime divisors of $|G|$. I have some questions about these groups.
Is there any classification of such groups?
In what ...

**5**

votes

**0**answers

172 views

+150

### A characterization of quadratics similar to an inverse sieve problem

Suppose $\mathscr{A} \subset \mathbb{N}$ enjoys for all large enough cutoffs $X$ the following properties:
$|\mathscr{A} \cap [1,X]| > \sqrt{X}/10$; and
the discriminant $\prod_{\alpha \neq ...

**5**

votes

**0**answers

200 views

### Recurrence Formula for Zernike polynomials

I'm not sure if this is research level, so if this result is known, please excuse the intrusion. I am trying to find a relation between solutions of the Laplacian equation in $4$ dimensions and those ...

**17**

votes

**1**answer

490 views

### Which philosophy for reductive groups?

I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the groupe $G$ we work on is to be ...