# All Questions

**0**

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2 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**

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**0**answers

10 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**

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15 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**

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9 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 ...

**-2**

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26 views

### Montgomery-Odlyzko and Riemann conjecture

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.

**2**

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34 views

### Hodge conjecture for étale cohomology

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

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10 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} \ ...

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7 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**

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13 views

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

In the paper Generalized Modular Functions 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{...

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11 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^{\...

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18 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**

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**1**answer

22 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 ...

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7 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,\...

**5**

votes

**1**answer

105 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 ...

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**1**answer

142 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**

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**1**answer

62 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

**1**answer

74 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**

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**0**answers

49 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**

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**1**answer

149 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**

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**1**answer

54 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**

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**1**answer

36 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**

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**0**answers

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**

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26 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 ...

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18 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**

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**0**answers

40 views

### An $n$-gon is isospectral to a regular $n$-gon

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 ...

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71 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**

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**1**answer

65 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 $\text{PSL}_2(\mathbb C)$?

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49 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**

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**1**answer

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**

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**1**answer

48 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

**2**answers

168 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

**2**answers

112 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

**1**answer

66 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

**1**answer

34 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**

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**0**answers

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

**3**answers

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**

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**0**answers

97 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**

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**0**answers

182 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**

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34 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**

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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**

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**1**answer

40 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**

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31 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**

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59 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**

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**0**answers

64 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**

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**0**answers

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**

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**0**answers

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**

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**0**answers

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

**2**answers

109 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 ...

**0**

votes

**1**answer

115 views

### Do we know an upper bound for the number of possible real parts of the non trivial zeroes of $\zeta$?

Let $n_{\zeta}$ denote the number of possible real parts for the non trivial zeroes of the Riemann Zeta function. RH is equivalent to $n_{\zeta}=1$, and the symmetry arising from the functional ...

**2**

votes

**1**answer

101 views

### Can you reconstruct a simplicial set from an $\infty$-groupoid?

In some categories of things with interesting structure, said structure can be recovered from the category.
For example, in the category of chain complexes of abelian groups, if you're given a chain ...