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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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 $U$ $(\Omega,\mathcal A,\... 1answer 38 views Algebraic structure on homotopy groups of spheres It is about a "conjecture" I heard (when I was student). There would exist an algebraic structure on the homotopy groups of spheres such that this algebraic structure would be the free algebraic ... 1answer 118 views Number of fixed points in Zagier's involution (Fermat's Theorem) Zagier's has found a famous one sentence proof for Fermat's theorem on sums of two squares. It centers on the following involution of the set$S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace $having ... 1answer 52 views Why vanish the integer m of an ample line bundle in the Kodaira embedding theorem? I try to understand the following version of the Kodaira embedding theorem: Let$X$be a compact Kähler manifold. A line bundle$L$is positiv if and only if it is ample. I have a problem with the '... 1answer 63 views Finitely generated subrings of$\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,\...
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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 ...