# All Questions

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### 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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### Advice on Family Index theorem

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 ...
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### 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 ...
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### An inequality in product space $V$ conjecture

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 ...
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### Fourier transform of complex functions

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-...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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 $O(2,1)$ ...
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### 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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### 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 ...
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### 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, $m$ ...
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### 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 ...
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### 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 ...
[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 $\... 1answer 61 views ### Positive semidefinite ordering for covariance matrices Suppose that X and Z are matrices with the same number of rows. Let $$D = \left[\begin{array}{cc} X' X & X'Z \\ Z'X & Z'Z \end{array} \right]^{-1} - \left[\begin{array}{cc} (X' X)^{-1} & ... 1answer 33 views ### Describing the action of ^2E_6(q) One of the constructions of the group ^2E_6(q) was presented by Tits in his paper "Les «formes réelles» des groupes de type E_6". It is being constructed by looking at the action of ^2E_6(q) on ... 0answers 44 views ### Proof that a function is injection [on hold] I want to show that the function f(x) := \frac{x}{\sqrt(x^2+1)}, x\in \mathbb{R}, is the bijection of \mathbb{R} onto B:={y:0<y<1} Firstly, I use the horizontal line test. Taking {x_1}... 3answers 64 views ### Regular tournaments Let T=(V,E) be a tournament. We call it regular if all vertices have the same out-degree. It is not hard to see that there are no regular tournaments on an even number of points. Let n>0 be an ... 0answers 95 views ### p-adic representations of the fundamental group of a smooth proper curve over a finite field This question is very general. Let C be a smooth and proper curve over a finite field {\bf F}_p. Are there any general results or conjectures on continuous non abelian representations$$ \pi_1(C)\... 0answers 178 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 ... 0answers 31 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$n-1$... 0answers 64 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 ... 1answer 38 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-... 0answers 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$F\in\mathbb{C}^{m\times n}$... 0answers 56 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,... 0answers 59 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 $$C\|a\| \... 0answers 57 views ### An innocuous second order linear ODE [on hold] Is there much work done on equations of the form$$ 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 ... 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. ... 0answers 45 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 ... 2answers 96 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 ... 0answers 89 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 ... 1answer 100 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 ... 0answers 99 views ### Variants of the Angel problem The original Angel problem, first proposed by Conway, Berlekamp, and Guy has two players: the devil and the angel. The angel is placed at the origin of an infinite 2-D chessboard. The angel's ... 1answer 77 views ### Can this equality hold for a nonzero b? Please may you kindly assist me on this integration exercise: For real a, b with a \neq 0, consider the equality$$\int_1^\infty f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^\infty f(x)\sin(a\... 3answers 364 views ### Homotopy type of some lattices with top and bottom removed The only reaction to this question on math.SE was 21 views in 4 days, so I decided to repost it here. I am not changing anything. There was an interesting question on MO which OP removed by some ... 1answer 111 views ### GCD for two Cullen numbers The$n$'th Cullen number is$C_n = n\cdot2^n+1$. If$m$and$n$are natural numbers, what can one say about$\gcd(C_n,C_m)$, where$m$and$n$are different positive integers? 1answer 41 views ### Cluster algebra structure compatible with Poisson brackets Let$X$be a Poisson variety. There is a concept "cluster algebra structure compatible with Poisson structure" introduced in the paper. Suppose that we construct a maximal independent set of ... 0answers 30 views ### Mersenne number with small Carmichael function Let$\lambda(\cdot)$be the Carmichael function. I'm trying to understand the magnitude of the smallest values of$\lambda(2^n - 1)$, when$n$runs over the positive integers. Precisely my question is:... 0answers 36 views ### I can't derive the integrating factor of this first order linear Equation [on hold] I can't derive the integrating factor of this first order linear Equation (x2 - y2 - y) dx - (x2 - y2 - x) dy = O. the answer is: integrating factor = 1/(x2 - y2) 0answers 26 views ### Jordan curve in$C^2$[migrated] Can we find a Jordan curve$\gamma$in$\mathbf{C}^2$of class$C^1$such that the projection to the first coordinate plane divides the plane into infinite components of connectivity. 2answers 86 views ### Asymptotic Growth of Markov Chain I asked the following question one week ago at math.stackexchange but didn't receive a response, so I want to give it here another try: I'm interested in the following problem: We have got a time-... 2answers 62 views ### Is there a full-rank map with connected graph and simply connected image that is not injective? I want to find a continuously differentiable function$F:X\to Y$, where$X\subseteq\mathbb{R}^n$,$Y\subseteq\mathbb{R}^m$are open ($n\le m$) with${\rm rk}\, \frac{\partial F}{\partial x}(x) = ...
The polar topology originates from the $S$-topology and is used in duality pairs. Due to the connection between the original topology and the weak topology, we can rephrase the original topology in ...