0
votes
0answers
10 views

Clarification of the proof of the main theorem of the paper of Huluse

I am trying to understand some open steps in the following article The Sign of Fourier coefficients of Half-integral Weight Cusp Form by Huluse, Mehmer and Chan , I found the following : Let $f\in ...
0
votes
0answers
16 views

Topological spaces having many isolated points

Consider a topological Hausdorff space $X$ with the following property: Any infinite closed subset contains a non-empty open set. I want to show that $X$ is $\alpha$-scattered, i.e., the set of ...
0
votes
0answers
7 views

Comparison inequalities for Hamiltonian mechanics with convex potential - analogue to Rauch's theorem?

I'm asking about an area (Hamiltonian mechanics) that I don't know at all well; thus, I keep the question somewhat vague. In differential geometry, there are a number of results saying that geodesics ...
1
vote
0answers
17 views

The composition of a dissipative operator and a positive operator is dissipative?

Consider the following bilinear system on a open and bounded domain $\Omega$ \begin{equation} \left\{\begin{array}{r c l} \displaystyle\frac{dy(t)}{dt} &=& Ay(t)+u(t)By(t)\\ y(0) &...
-2
votes
0answers
30 views

Old question : What would be the group $G$ (with $Γ_1$, $Γ_2$) with the bijection $v$?

The question What would be the group $G$ (with $\Gamma_1$, $\Gamma_2$) with the bijection $v$? doesn't have been answered. It should be interesting to get an answer to this answer. Could anyone be ...
1
vote
1answer
15 views

Non-lattice Veech groups

I was thinking of Veech surfaces, which are translation surfaces whose stabilizer under the $\mathrm{Sl}_2(\mathbb{R})$ action is a lattice in $\mathrm{Sl}_2(\mathbb{R})$. They seem to have been ...
-3
votes
0answers
43 views

Montgomery-Odlyzko and Riemann conjecture [on hold]

If the Montgomery-Odlyzko law is solved, does it follow the Riemann conjecture? I think the Montgomery-Odlyzko law know well about the distribution of primes.
3
votes
1answer
83 views

Hodge standard conjecture for étale cohomology

It is known that Hodge standard conjecture is true for étale cohomology for field $k$ in characteristic zero. It means that the following pairing $$(x,y)\mapsto (-1)^{i}\langle L^{r-2i}(x),y\rangle$$ ...
1
vote
0answers
16 views

$L^\infty(\Omega)$-regularity for strongly damped wave equation

I am interested in the following IBVP for the strongly damped wave equation: \begin{equation} u_{tt}-c^2\Delta u-b\Delta u_t+eu_t=f(x,t) \quad \text{in} \ \Omega \times (0,T), \\ u=0 \quad \text{on} \ ...
0
votes
0answers
10 views

Matrix minimization

Is there an explicit solution to the problem of minimizing $||X-X_0||_F^2+||X^{-1}-Y_0||^2_F$, with respect to matrix X, where $X_0$ and $Y_0$ are given, and all matrices are real $n\times n$ and ...
1
vote
0answers
24 views

Parabolic characters of subgroups $\Gamma \subset \operatorname{SL}_2(\textbf{Z})$ generated by parabolic and elliptic elements

In the paper Generalized Modular Forms from Knopp and Mason, one can read in page $6$: Remark. It is not too hard to prove that a subgroup $\Gamma$ of finite index in $\operatorname{SL}_2(\textbf{Z})$...
0
votes
0answers
14 views

Is the inverse of the criterion of fully-faithfulness of derived tensor product also true?

Let $A$ and $B$ be differential graded algebras over a field $k$ and $D(A)$ and $D(B)$ be the derived categories of differential graded modules. Let $N$ be an $A$-$B$ bimodule. Then $(-)\otimes^{\...
2
votes
1answer
38 views

Projectively equivalent toric varieties

Say I have two normal lattice polytopes $P$ and $Q$ in $\mathbb{R}^n$ (lattice $\mathbb{Z}^n$) with the same number of lattice points $N+1$. Then they define two toric varieties $X_P$ and $X_Q$ which ...
1
vote
1answer
26 views

Limit of stochastic subsequence of stationary ergodic sequence

Let $\{X_k\}_{k\in\mathbb{N}}$ be a stationary ergodic sequence on a probability space $(\Omega,\mathcal{F},P)$ with shift $T$. Also, let $\{v_k\}_{k\in\mathbb{N}}$ be a sequence of random variables ...
0
votes
0answers
8 views

Derive a SPDE of evolutionary type for $u$ from ${\rm d}X(t)=u(t,X(t)){\rm d}t+\xi(t,X(t)){\rm d}W(t)$

Let $U$ and $V$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ $(\Omega,\mathcal A,\...
6
votes
2answers
168 views

Algebraic structure on homotopy groups of spheres

It is about a "conjecture" I heard (when I was student). There would exist an algebraic structure on the homotopy groups of spheres such that this algebraic structure would be the free algebraic ...
0
votes
1answer
153 views

Number of fixed points in Zagier's involution (Fermat's Theorem)

Zagier's has found a famous one sentence proof for Fermat's theorem on sums of two squares. It centers on the following involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace $ having ...
0
votes
1answer
67 views

Why vanish the integer m of an ample line bundle in the Kodaira embedding theorem?

I try to understand the following version of the Kodaira embedding theorem: Let $X$ be a compact Kähler manifold. A line bundle $L$ is positiv if and only if it is ample. I have a problem with the '...
3
votes
1answer
82 views

Finitely generated subrings of $\mathbb{R}$ are finitely approximable

In Ivanov's Finite Approximability of Modular Teichmüller Groups, for the proof of Lemma 2, the following is stated: Let $G$ be a finitely generated group and $\tau: G \to \operatorname{PSL}(2,\...
-3
votes
0answers
50 views

Advice on Family Index theorem [on hold]

I am reading Bismut's paper Family Index and Heat equation, but I have knowledge on the probability or stochastic. Could anyone give some advice or introduce some ref. on probability to understand ...
4
votes
1answer
157 views

Fermionic Wick Theorem

Assume we are given are fermionic many-particle system with creation operators $c_1^\dagger,c_2^\dagger,...$ acting on some Hilbert space $\mathcal{H}$. That is, the $c_i^\dagger$ and their ...
0
votes
1answer
61 views

An inequality in product space $V$ conjecture [on hold]

I found an inequality as following: Let $x, y, z$ be three complex numbers then: \begin{equation*} \frac{1}{2}(|y+z-x|+|x+z-y| + |y+x-z|) \le |x| + |y|+|z|+\frac{1}{2}|x+y+z| \end{equation*} (1) The ...
1
vote
1answer
38 views

Fourier transform of complex functions [on hold]

I want to know what is the meaning of Fourier transform for a complex function, say function $F:\mathbb C \to \mathbb C$? Up to my knowledge, Fourier transform is defined for real-valued or complex-...
-4
votes
0answers
43 views

On the symmetricity of the Riemann zeta function [on hold]

Yesterday, i stumbled upon something quite interesting about the Riemann zeta function, and i posted it on MSE. However, i could not get a conclusive response, hence i decided to post it here, though ...
0
votes
0answers
27 views

Reference Request: “Resolutions” of $K_n$ for $n$ odd

A resolution (in the combinatorial design sense) of $K_{n}$ is a collection of sets of edges of $K_{n}$ so that within each set of edges, each vertex appears once, and over the entire collection, each ...
0
votes
0answers
19 views

Counting different residue classes of certain form modulo $4n-1$?

I like to calculate number of different residue classes of the form $b^2c$ modulo $4abc-1$, where $a,b,c$ are positive integers and $\gcd(a,b)=1$. I could prove that if $abc$ is prime there are only 3 ...
3
votes
0answers
51 views

A $n$-gon is isospectral to a regular $n$-gon (Isospectral $\implies$ isometry ?)

If an $n$-gon $P$ is isospectral to a regular $n$-gon $Q$, what could we say about the shape of the $P$. Otherwise, what could we say about $Q$? In fact, some hints or simply some ideas should be ...
0
votes
0answers
73 views

construction of the group $O(2,1)$ [on hold]

I am reading a paper "Proper actions and Pseudo-Riemannian space forms" by R. Kulkarni, and I am interested are there any references/papers which decribe the construction of the group $O(2,1)$ ...
1
vote
1answer
72 views

Polish Group Topologies on PSL(2,C)

Does anyone know how many Polish group topologies (or where to begin to look for this information) can be put on $\text{PSL}_2(\mathbb C)$?
1
vote
0answers
52 views

Embedding of Gorenstein orbifold as a hypersurface

I am trying to understand if three complex dimensional orbifold singularity $\mathbb{C}^3 / \Gamma$ can be embedded as hypersurface in $\mathbb{C}^4$. The condition of being Gorenstein and having ...
3
votes
1answer
54 views

Is every regular Borel outer measure topologically additive?

If $m$ is a regular Borel outer measure is it true that $m$ is topologically additive? If so what is a proof or a counterexample? Definitions: Topologically Additive: $X$ is a topological space, $m$ ...
3
votes
1answer
52 views

Simplicial Model Categories

Let $D$ be a combinatorial simplicial model category (e.g $SSet$ with the standard model structure) and let $C$ be a small simplicial category. Of course, we can consider the projective model ...
2
votes
2answers
169 views

Need help with paper written in Russian… Yorgov's paper on self-dual codes with automorphisms of odd order

I came across this paper by V.I.Yorgov named "Binary self-dual codes with automorphisms of odd order". (Actually I was first reading another paper by Borello that cited this paper. It later become ...
4
votes
2answers
119 views

rho invariant of manifolds

[I thought that I had already posted this question, but I couldn't find it in a search, so I apologize if I'm posting twice.] Let $G$ be a finite group. Then the rational oriented bordism ring $\...
5
votes
1answer
67 views

Positive semidefinite ordering for covariance matrices

Suppose that X and Z are matrices with the same number of rows. Let $$ D = \left[\begin{array}{cc} X' X & X'Z \\ Z'X & Z'Z \end{array} \right]^{-1} - \left[\begin{array}{cc} (X' X)^{-1} & ...
6
votes
1answer
35 views

Describing the action of $^2E_6(q)$

One of the constructions of the group $^2E_6(q)$ was presented by Tits in his paper "Les «formes réelles» des groupes de type $E_6$". It is being constructed by looking at the action of $^2E_6(q)$ on ...
-4
votes
0answers
44 views

Proof that a function is injection [on hold]

I want to show that the function $f(x) := \frac{x}{\sqrt(x^2+1)}$, $x\in \mathbb{R}$, is the bijection of $\mathbb{R}$ onto $B:={y:0<y<1}$ Firstly, I use the horizontal line test. Taking ${x_1}$...
0
votes
3answers
65 views

Regular tournaments

Let $T=(V,E)$ be a tournament. We call it regular if all vertices have the same out-degree. It is not hard to see that there are no regular tournaments on an even number of points. Let $n>0$ be an ...
5
votes
0answers
99 views

$p$-adic representations of the fundamental group of a smooth proper curve over a finite field

This question is very general. Let $C$ be a smooth and proper curve over a finite field ${\bf F}_p$. Are there any general results or conjectures on continuous non abelian representations $$ \pi_1(C)\...
6
votes
0answers
185 views

Properties of Grothendieck ring for field of characterictic $p$

In this article there is a proof that for field $k$ of characteristic zero Grothendieck ring $K(\mathbf{Var}_k)$ is not an integral domain. In many articles I found statement that similar theorem for ...
4
votes
0answers
35 views

Number of classes $\pmod p$ represented by $b_1s^{n-1} + \dots + b_n$ where $ord_p(s) = n$

Let $n \in \mathbb Z$ with $n \ge 3$ and let $p$ be a prime number such that $n|p-1$. Let $a_1,a_2,\dots,a_{2n-1} \in \mathbb Z/p\mathbb Z$. Suppose that the same class is represented by at most $n-1$ ...
3
votes
0answers
66 views

Elementary symmetric functions of reciprocals of monic polynomials in function fields

Let $q$ be a prime power and $\mathbb{F}_q$ the field of cardinality $q$. Let $A = \mathbb{F}_q[T]$ and let $A_+ \subset A$ be the monic polynomials. Choose any ordering $<$ of $A_+$ and let $k$ be ...
2
votes
1answer
44 views

Moerdijk Model Structure on Bisimplicial sets

Let $s^2Set$ denote the category of bisimplicial sets, i.e. simplicial objects in the category of simplicial sets. Recall that in the Moerdijk model structure on $s^2Set$, weak equivalences are "point-...
4
votes
0answers
33 views

Fast matrix-vector product for structured matrices

Let $X\in\mathbb{C}^{m\times n}$ be a matrix that satisfies the Sylvester equation $$AX-XB = F,\qquad A\in\mathbb{C}^{m\times m}, \quad B\in\mathbb{C}^{n\times n},$$ where $F\in\mathbb{C}^{m\times n}$...
5
votes
0answers
61 views

Examples of nonstable ∞-categories in which sifted colimits commute with finite limits

What are some natural examples (if any) of nonstable ∞-categories in which finite limits commute with sifted colimits (or rather just colimits over Δ^op)? Stable ∞-categories do satisfy this property,...
4
votes
0answers
65 views

C$^*$-algebras in which the spectral radius is comparable to the norm

For every commutative C$^*$-algebra the spectral radius is equal to the norm. My question is: For which C$^*$-algebras $\mathcal A$ does there exist a constant $C>0$ such that $$C\|a\| \...
0
votes
0answers
58 views

An innocuous second order linear ODE [on hold]

Is there much work done on equations of the form $$ y'' + \alpha(t)y = 0,$$ where $\alpha(t) \in C^\infty([0,\infty))$ and $\alpha(t) > 0$? In particular, I am looking for some blow-up results. I ...
1
vote
0answers
12 views

Non-negative polynomials $f(p), p\in P$ from Polynomial ideal where $P$ compact polytope?

Partially related to Non-negative Polynomials from Polynomial Ideal? where focus on graph theory but now I want to understand more generally how to determine non-negativity in more general case. A. ...
2
votes
0answers
50 views

A specific mollified functions in the Sobolev space H^1(R)

Let $u>0$ be in $H^{1}(\mathbb{R})=W^{1,2}(\mathbb{R})$, we know that the set of $C^{\infty}$ functions with compact support are dense in the Sobolev space $H^{1}(\mathbb{R})$. Hence, we have a ...
5
votes
2answers
114 views

Do character tables determine association schemes up to isomorphism?

I am interested in commutative, but not-necessarily-symmetric association schemes. I noticed that the conjugacy class scheme of the dihedral group of order 8 has the same character table as the one ...

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