6
votes
1answer
101 views

Stochastic Analogue of Stokes Theorem

Dynkin's formula can be thought of as the stochastic version of the Fundamental Theorem of calculus, $$E^x[f(X_{\tau})]=f(x)+E^x\left[\int_0^{\tau}Af(X_u)du\right],$$ where $\tau$ is a first exit time ...
0
votes
1answer
35 views

Length of longest directed circuit in random tournament

Build a random tournament $T=(V,E)$ on $V=\{1,\ldots, n\}$ in the following fashion: for $i < j\in \{1,\ldots, n\}$ let the probability be $0.5$ whether $(i,j)\in E$ or $(j,i)\in E$ (in a ...
5
votes
1answer
144 views

Random Cantor sets on the unit interval

Denote $A=\{0\}, B=\{0,1\}$. Then any subset of $\Omega:=\{A,B\}^{\mathbb N}$ is a continuum provided the number of $B$'s is infinite. We treat these as binary expansions of numbers in $[0,1]$. For ...
1
vote
1answer
109 views

Yoneda extension of a faithful functor is faithful

Let $F: \mathcal C \to \mathcal D$ be a functor with $\mathcal D$ cocomplete, and let $\mathscr P \mathcal C$ be the free cocompletion of $\mathcal C$ (i.e., the category of small presheaves on $\...
0
votes
1answer
106 views

Necessary and sufficient conditions for Kolmogorov's Extension Theorem

Let $(X_n,\mathcal{X}_n)$, $n=1,2,\ldots$ be measurable spaces. Define $Y_n = \prod_{k=1}^n X_k$ and let $\mathcal{Y}_n$ be the corresponding product $\sigma$-algebra. Similarly let $Y=\prod_{k=1}^\...
-1
votes
0answers
60 views

Estimating an exponential sum of a particular type

I was trying to estimate the following exponential sum: For given irrationals $\alpha$ and $\beta$, and given integer $x$ , let $$S(x,\alpha,\beta)=\sum_{n\leq N}\sum_{m\le M}A(m)B(m,n)e(x(m\alpha-n\...
0
votes
0answers
61 views

2D sequence of integers [closed]

I found the following sequence of integers $a_n^k$, where $n$ and $k$ to extend to infinity. Each row are coefficients of a polynomial, similar to the coefficients of the Legendre polynomials. The ...
4
votes
0answers
102 views

Automorphisms of an infinite graph built from a finite motif

Suppose we have a lattice $L$ in $\mathbb{R}^n$ for which we choose some fundamental domain $D\subset \mathbb{R}^n$ homeomorphic to a closed ball. Translates of $D$ by distinct elements of $L$ ...
4
votes
1answer
193 views

Mapping a group to a finite group s.t. the image of each generator is nontrivial

Recall that a group $G$ is called residually finite if for any nontrivial element $g\in G$ there exists a finite group $H$ and a homomorphism $f$ from $G$ to $H$ such that $f(g)\neq1$. My question is ...
0
votes
1answer
34 views

Convergence of an inhomogeneous markov chain

A markov chain is defined as $X_t=F(X_{t-1})X_{t-1}$, where $X_t$ and $X_{t-1}$ are both vector. So the transition matrix depends on the current states. I want to show that for any given initial ...
3
votes
1answer
120 views

When does every $\infty$-localization correspond to a Bousfield localization?

Let $\mathcal{M}$ be a model category presenting an $\infty$-category $\mathcal{C}$. I believe that every left Bousfield localization $\widetilde{\mathcal{M}}$ of $\mathcal{M}$ corresponds to a ...
4
votes
0answers
88 views

A color interpolation lemma

I need the following "color interpolation lemma". Actually I know a way to prove it, but I'm not very satisfied with that proof. Lemma. Let $G=(V,E)$ be a (properly) colored graph with colors $1, \...
1
vote
0answers
17 views

Examples of partially permutative left-distributive algebras

An algebra $(X,*)$ that satisfies the identity $x*(y*z)=(x*y)*(x*z)$ is said to be a left-distributive algebra. Let $L:X^{2}\rightarrow X^{2}$ be the mapping defined by $L(x,y)=(x*y,x)$ and let $T:X^{...
5
votes
0answers
93 views

For which Ramsey type results density versions are wrong?

I look for examples of Ramsey-type statements, for which the density counterparts do not hold. Example: usual Ramsey theorem. If all edges of a complete graph $K_n$ are colored in $c$ colors, there ...
-3
votes
0answers
62 views

Determinant of a tensor product [closed]

Let V and W be two vector spaces over a field of characteristic zero. Give a formula for the top exterior power of V tensor W.
5
votes
2answers
103 views

Stable unions without stable images

A regular category is one with finite limits and pullback-stable images (i.e. (regular epi, mono) factorizations). A coherent category is a regular category that also has pullback-stable finite ...
0
votes
1answer
98 views

Doubling metrics, doubling measures, Lebesgue density

As stated in this question, Lebesgue differentiation theorem holds on locally doubling space? and proved here, http://www.math.uiuc.edu/~tyson/595f15lecture2.pdf the Lebesgue differentiation theorem (...
4
votes
1answer
141 views

Asymptotics on the number of ways to pair off $\{1, 2, \dots, 2n\}$ into primes

Given $S = \{1, 2, \dots, 2n\}$, we can always pair off elements into $n$ pairs such that each sum to a prime. The proof of this fact is easy and follows from Bertrand's postulate. Now, let $\gamma(n)...
0
votes
0answers
47 views

When are these sums consecutive integers? [closed]

It is possible to construct $\frac{n(n-1)}{2}$ sums which each contain two distinct summands chosen from a set $n$ numbers. For which $n\geq 3$ do there exist a set of $n$ (distinct) integers such ...
3
votes
0answers
143 views

Fields generated by torsion points of CM elliptic curves

I'm using the same setup as Corollary 1.7 on p. 44 of de Shalit manuscript (Iwasawa theory of elliptic curves with complex multiplication). I think there is a mistake in his Corollary 1.7 and I'm ...
2
votes
0answers
83 views

Threshold for prophet inequality

The prophet inequality is related to the following scenario: Suppose there are $n$ independent positive random variables $X_1,\dots,X_n$. They might not be identically distributed. We reveal them ...
1
vote
2answers
155 views

Maps between symmetric powers of the natural module for $SL_2 (k)$ in prime characteristic

Let $G=SL_2(k)$ considered as a linear algebraic group over an algebraically closed field of prime characteristic. Let $E$ be the natural module for $G$ and denote by $S^r (E)$ its $r-$th symmetric ...
2
votes
0answers
69 views

An question about Cauchy Problem in General Relativity [closed]

Yesterday, in Brazilian School on Differential Geometry, a friend asked me the question: Given an (non-trivial) initial data set $(M,g,k)$ for the Cauchy problem in General Relativity. Is there ...
-5
votes
0answers
47 views

Is a continuous two variables function also continuous with respect to each variable? [closed]

I have a simple question, let $f:X\times Y\rightarrow Z$ be a map with two variables, and $X,Y,Z$ are topological spaces, I want to know if $f$ is continuous, then how about $f_{x_{0}}:Y\rightarrow Z$ ...
3
votes
1answer
273 views

Geometric intuition for the condition of Galois descent

Continuing in my attempts to understand bits and pieces of Borceux and Janelidze's Galois Theories, I've just realized that I don't have any geometric intuition for the most convenient ...
3
votes
0answers
47 views

Which blow ups in the base of a conic bundle preserve the “standard” condition?

Assume we are given a nontrivial standard conic bundle $\pi: X\rightarrow S$, that is $X$ and $S$ are smooth projective varieties (say over $\mathbb{C}$), $\pi$ is flat and furthermore we have $Pic(X)=...
-4
votes
0answers
43 views

How can i integral of this function? [closed]

I want to know how can i solve this function. $\int (1-y^d)^n \, dy$ Is it possible to solve it? If you know the method, please teach me.
1
vote
0answers
53 views

Is the category of prederivators cartesian closed?

The question is in the title. ${\bf PDer} = Fun({\bf Cat}^\text{op}, {\bf CAT})$ is obviously cartesian since $\bf CAT$ is. The usual argument for presheaf categories does not apply directly since 1-...
1
vote
0answers
29 views

Strict/strong functors are co/reflective inside lax functors, the coendy way

Bozapalides' remarks on lax presheaves show that the category $[{\cal A}^\text{op}, {\bf Cat}]$ is reflective and coreflective inside the category of lax functors, lax natural transformations and ...
2
votes
1answer
53 views

How to compute bounding coefficients for McDiarmid's inequality?

I am trying to understand the proof in Sec. A2 of Gretton et al.. To make the question self-contained, I summarize below the key ingredients. At the end of the post, I state my question. Given a ...
3
votes
1answer
143 views

Kähler classes for surfaces of general type with $c_1^2=3c_2$

Given a smooth, compact complex surface with ample canonical bundle satisfying $c_1^2=3c_2$, is it true that every Kahler class is a multiple of $c_1$? This seems to be the case for fake projective ...
0
votes
0answers
10 views

Numerical methods for variational inequality involving the Dirichlet-Neumann operator

I am currently writing my master thesis about the numerical computation of a solution to the following variational inequality by means of the time-domain boundary element method. Let $Q\subset \...
3
votes
0answers
100 views

A question about smooth convex lattice polygons

Let $P$ be a smooth convex lattice polygon in $\mathbb{R}^2$ (the lattice being $\mathbb{Z}^2$). Here smooth means that at any vertex of $P$, the two primitive integer vectors (i.e. vectors whose ...
1
vote
0answers
78 views

Monomial algebras and depth

Let $R:=k[x_1,\ldots, x_n]$ be the standard polynomial ring. Let $I\subseteq R$ be a monomial ideal of height $\ge 2,$ and $\{\ell_1, \ldots, \ell_{t}\}\subseteq R_1$ an $R$-regular sequence. Assume $...
0
votes
0answers
13 views

Does a vector belongs to a simplicial subcone when it belong to cone with more than n generators?

Assume $x_{0}\in \text{cone}(a_{1},\dots,a_{N})$, where $a_{i}\in \mathbb{R}^{n}_{+}$ ($a_{i}\in \mathbb{R}^{n}$, and $a_{i}\geq 0$) for $i=1,\dots,N$ (i.e., $x_0$ lies in the cone generated by $a_{i}$...
0
votes
0answers
78 views

a modular character problem [closed]

Let $B\in$Bl$(G|D)$ and suppose that $\sigma\in$Aut$(G)$ fixes every $\chi\in$Irr$(B)$. If $d\in D$, show that $d$ and $d^\sigma$ are $G$-conjugate. It is a problem from Navarro's book "characters and ...
3
votes
1answer
60 views

Gap-opening perturbations of the periodic Schrödinger operator

I am trying to understand this short paper and I am getting stuck right at the end. Let $V(x)$ be $C^\infty$ and 1-periodic (that is, $V(x)=V(x+1)$). We are considering the operator $$A=-\dfrac{d^2}...
0
votes
0answers
24 views

Super classical r-matrices and Poisson Lie supergroups

In classical case, given an r-matrix $r$ for $sl_n$, we can compute the corresponding Poisson bracket on $SL_n$ by using the formula $\{L \otimes L\} = [r, L \otimes L]$. For example, let $g=sl_2$ and ...
3
votes
0answers
168 views

Artin conjecture on L-functions

Artin conjecture on Artin $L$-functions asserts that the Artin $L$-function $L(\rho,s)$ of a non-trivial irreducible representation $\rho$ of the Galois group $\Gamma$ of a number field admits ...
2
votes
0answers
33 views

Partially permutative matrices

Let $V$ be a finite dimensional vector space over a field $K$. Then a map $L:V\otimes V\rightarrow V\otimes V$ is said to satisfy the Yang-Baxter equation if $(L\otimes I)(I\otimes L)(L\otimes I)=(I\...
-1
votes
1answer
47 views

About the critical points of quasi-convex functions

What do we know about the structure of critical points of quasi-convex functions? I am looking for statements like "the critical points of a quasi-convex function are always either a global minima ...
6
votes
0answers
64 views

k-flats in homogeneous spaces

In a symmetric space of rank $k$ (and I'll take $k > 1$) every geodesic is contained in a $k$-flat: a totally geodesic, flat, connected, and closed submanifold of dimension $k$. Question. Are ...
2
votes
1answer
132 views

Size of automorphism group of random regular graph

If I pick a random regular graph on $n$-vertices and degree $d$ from uniform distribution what is the probability that its automorphism group is of size at least $m$? -- I want to know what is the ...
18
votes
0answers
286 views

Follow-up to Steinberg's problem (12) in his 1966 ICM talk?

Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (with ...
3
votes
0answers
88 views

The number of monotone Boolean functions

In the paper "The number of monotone Boolean functions" A. D. Korshunov calculates an asymptotic number of the number of monotone Boolean functions (see https://en.wikipedia.org/wiki/Dedekind_number)...
3
votes
0answers
64 views

Minimal algebraic degree of symmetric unit distance embedding of Heawood graph

I'm looking at embeddings of the Heawood graph in the plane as unit distance graph. Apparently the first such embedding was given by Gerbracht, 2009 and has algebraic (over the rationals) coordinates ...
0
votes
0answers
79 views

What is the sharpest bound of this sum?

Fix $y\geq 1$ and let $\delta$ be a small enough positive real number. Put $$\mathcal{D}^{+}=\left\{d=p_{1}...p_{l}: p_{l}<...<p_{1},\ p_{m} \leq y_{m} \ \textrm{for all odd} \ m \right\},$$ ...
0
votes
1answer
80 views

A uniform Lebesgue density theorem

The Lebesgue density theorem in $\mathbb{R}^n$ may be stated as follows. For a Lebesgue-measurable $A\subseteq\mathbb{R}$ and $r>0, x\in\mathbb{R}^n$, define $$ \chi_{A,r}(x)=\frac{\mu(A\cap B_r(x))...
-4
votes
0answers
72 views

How do I go about solving the following problem? [closed]

Given $(m_1,m_2, ...,m_r)\in Z^r_{\geq 0}$, and $a_1, a_2, · · · , a_r \in \mathbb{N}$ such that: $\sum_{i=1}^r a_im_i=qL$ where $L$ denotes the least common multiple of $a_1, a_2, · · · , a_r$ and $...
3
votes
1answer
130 views

Number of distinct variables used in axiomatizating (Classical) Propositional Logic

The first part of the present question is concerned with Classical Propositional Logic (CPL). The second part involves its fragments or alternative logical systems. There are in the literature many ...

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