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Concavity, martingales and stopping time

Suppose $(x_t)_t$ is a bounded $\mathbb F_t$ martingale and $f(t,x)$ is continuous, bounded, and concave in $x$. So, for any $s \ge t$, $$\mathbb E_t f(s,x_s) \le f(s,\mathbb E(x_s)) = f(s,x).$$ Does ...
avk255's user avatar
  • 543
2 votes
1 answer
67 views

Analogue of spectral values of automorphisms in vN algebra

Is there any analog of studying spectral properties of automorphisms of von Neumann algebra? Does it make sense, if anybody knows please give a reference.
user136400's user avatar
4 votes
1 answer
332 views

Dihedral extension of $\mathbb Q$ with small discriminant

Let $K$ be a fixed quadratic number field, say $K=\mathbb Q(\sqrt 5)$. For any integer $n \geq 3,$ I would like to build a number field $D_n$ such that $D_n/\mathbb Q$ is Galois, with Galois group ...
A. Bailleul's user avatar
  • 1,164
6 votes
1 answer
392 views

Isomorphisms in enriched categories

Let $(M,\otimes,1)$ be closed monoidal category and $C$ an $M$-enriched category. Assume we have $C$-objects $X$ and $X'$ and a morphism $f:1\to C(X,X')$ in $M$. We call $f$ an isomorphism if there is ...
FKranhold's user avatar
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0 votes
0 answers
89 views

Formal definition of episodic Markov Decision process?

David Silver, in his lecture 4 from his Youtube lectures, speaks about episodic Markov Decision Processes (MDPs) and Monte-Carlo Policy Evaluation. I could not find a formal definition of episodic ...
Vasileios Anagnostopoulos's user avatar
17 votes
6 answers
1k views

Is the concept of a "numerable" fiber bundle really useful or an empty generalization?

Numerable fiber bundles are defined by Dold (DOLD 1962 - Partitions of Unity in theory of Fibrations) as a generalization of fiber bundles over a paracompact space : the trivialization cover of the ...
ychemama's user avatar
  • 1,326
2 votes
0 answers
43 views

Standard name for rational Levi subgroups of rational parabolic groups

I am looking for a standard name for the groups above. They appear in Harish-Chandra theory. It would be convenient to have a shorter name for them. I tried looking online but I did not find useful ...
Jesua Israel Epequin Chavez's user avatar
2 votes
1 answer
148 views

Boundary regularity type results of fractional laplacian

Let $u$ be a solution of $-\Delta u= f \text{ in } \Omega$ with $u=0 \text{ on } \partial \Omega.$ If $f\in L^2(\Omega)$, then $u\in H^2(\Omega)\cap H^{1}_{0}(\Omega)$provided $\Omega$ is a smooth ...
GabS's user avatar
  • 407
1 vote
0 answers
75 views

Probabilistic method with multiple objective functions

Let $\mathcal X$ be a finite set and $f$ a function from $\mathcal X$ to $\mathbb R^+$. Basic probabilistic method says that if I can find a probability distribution on $\mathcal X$ and show that $E[f(...
EEStudent's user avatar
5 votes
0 answers
341 views

Serre functors for non-proper categories

One usually defines a Serre functor to be a functor on a $k$-linear category $\mathcal{C}$ which has finite dimensional $Hom$s over $k$. In that case, the standard definition is that a Serre functor $...
skr's user avatar
  • 512
1 vote
0 answers
119 views

Generalization of a result of Frey

In Proposition (2) in the paper [1], in below, it is proved that: Assume that $K$ is a number field and that $C$ has infinitely many points of degree $\leq d$ over $K$. (Here $d\geq 2$ is an integer)...
user131222's user avatar
5 votes
2 answers
412 views

Ordered union of Borel sets

Let $\mathfrak{A}$ be an uncountable collection of Borel sets in $\mathbb{R}^d$ such that for any $A,B\in\mathfrak{A}$, either $A\subset B$ or $A\supset B$. Then is it necessarily true that the union $...
user137602's user avatar
6 votes
2 answers
366 views

Hyperelliptic Jacobians with (or without) CM

Let $C$ be a hyperelliptic curve $y^2 = f(x) $ defined over $\mathbb{Q}$, where $f(x) \in \mathbb{Q} [x]$ is a polynomial of degree $n=5$ or $6$, and $J = Jac(C)$ its Jacobian. I know Zarhin's result [...
Kazuki  Sato's user avatar
18 votes
2 answers
729 views

Properties of categories that can not be proven by abstract nonsense

What are examples of properties of particular categories that can be formulated in categorical language and "feel" like they ought to be provable formally but they actually are not? I think that this ...
user avatar
6 votes
1 answer
248 views

$p$-adic equivalence of spectra with $G$-action

In Krause and Nikolaus' "Lectures on topological Hochschild homology and cyclotomic spectra" (which can be found here), there is a lemma I'm trying to understand, but one line in the proof ...
Stahl's user avatar
  • 1,049
2 votes
0 answers
67 views

Computing several Fourier terms to machine precision

For periodic not-necessarily smooth $f$ and a range of $m$, say $0\ldots 31$, I want to compute $\int_{-\pi}^\pi f(t) \cos (mt) dt$ (and maybe the same integral with $\sin$ instead of $\cos$) to ...
JCK's user avatar
  • 71
5 votes
0 answers
194 views

K(1)-localization and homotopy fixed points in motivic homotopy theory

It is known in classical stable homotopy theory that there is an equivalence $$ L_{K(1)}S\simeq KU^{h\mathbb{Z}_p^\times} $$ which gives an especially convenient way to compute the K(1)-local sphere....
CWcx's user avatar
  • 628
1 vote
0 answers
368 views

Riemann Surfaces of Infinite Genus and Transcendental Curves

I'm a graduate student struggling to find a topic for his doctoral dissertation which hasn't already been explored. At present, my hope was to see if classical results about Jacobian Varieties, ...
MCS's user avatar
  • 1,256
6 votes
1 answer
194 views

Two monoidal structures and copowering

Let $(\mathbf{M},\otimes,1)$ be a closed monoidal category and $(\mathbf{C},\oplus,0)$ an $\mathbf{M}$-enriched monoidal category. Furthermore, assume that we have a copowering $\odot:\mathbf{M}\times\...
FKranhold's user avatar
  • 1,623
2 votes
0 answers
36 views

Exclusion processes from point of view of a tagged particle

I'm interested in the simple exclusion processes on $Z^d$ and the ergodic theorems that can be proved from the point of view of the particle. Ellen Saada proved the following in 1987 (Annals of Prob): ...
arjun's user avatar
  • 921
3 votes
0 answers
166 views

Simple transfinite generalization of $p$-adic integers

One way to define the ring of $p$-adic integers is as a quotient of the formal power series semiring $\Bbb N[[x]]/(x-p)$. One can likewise start with the formal power series ring $\Bbb Z[[x]]/(x-p)$ ...
Mike Battaglia's user avatar
0 votes
1 answer
67 views

Nonsmooth dynamical system (DAE) - systematic way to calculate period numerically?

What I have in mind is a mechanical system that is described by an implicit system of ODEs or a system of DAEs (differential algebraic equations). The system is asymptotically stable, meaning that ...
SimpleProgrammer 's user avatar
2 votes
1 answer
88 views

Can a Multilayer Perceptron fit any binary function?

Consider a perceptron $F(x) = \phi(x * w - b), \ x \in \mathbb{R}^n,$ (with Heaviside activation function $\phi$) and a dataset consisting of a finite subset $\Omega \subseteq \mathbb{R}^n$ with ...
GM1's user avatar
  • 23
7 votes
2 answers
263 views

The union of all coreflective Cartesian closed subcategories of $\mathbf{Top}$

Very often, in topology, one restricts to a coreflective Cartesian closed subcategory of $\mathbf{Top}$ in order to freely use exponential laws for mapping spaces, which imply things like "the ...
Jeremy Brazas's user avatar
4 votes
1 answer
378 views

Idempotent completion of linear categories and Yoneda

Let $ \text{Vect} $ be the category of finite dimensional vector spaces over an algebraically closed field. The idempotent completion of a $\text{Vect}$-category $ \mathcal{C} $ may be though of in ...
Arthur's user avatar
  • 1,379
4 votes
0 answers
152 views

What are the zonal spherical functions for a finite unitary group acting on a unit sphere?

Given a prime power $q$ and a dimension $d$, consider the Hermitian form $(\cdot,\cdot) \colon \mathbb{F}_{q^2}^d \times \mathbb{F}_{q^2}^d \to \mathbb{F}_{q^2}$ given by $$ (x,y) = \sum_{i\in [d]} ...
Dustin G. Mixon's user avatar
1 vote
0 answers
74 views

When does a stochastic process have finite second moment?

For a known probability space $(\Omega,\mathcal{F},P)$ where $\Omega$ is the event space, $\mathcal{F}$ a $\sigma$-algebra and $P$ is the (known) probability measure, let $f(x(t,\xi),\xi), \,\, \xi \...
laurila's user avatar
  • 11
7 votes
1 answer
257 views

Kolmogorov superposition on the Hilbert Cube

A result of Kolmogorov and Arnold says that continuous functions on $\mathbb{R}^n$ can be represented as sums of the form $$ f(x_1,\dots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^n\phi_{p,q}(x_p)\...
James Hanson's user avatar
  • 10.3k
4 votes
0 answers
106 views

Hilbert space compression of lamplighter over lamplighter groups

$C_2 \wr \mathbb{Z}$ is the lamplighter group but I'm currently looking at the lamplighter group with this group as a base space. Question: Consider the group $C_2 \wr (C_2 \wr \mathbb{Z})$, what is ...
ARG's user avatar
  • 4,342
1 vote
1 answer
105 views

Algorithm for (binary) integer programming

I am looking for an algorithm that can solve the following (binary) integer programming problem. The problem description is given below: \begin{align*} &\max\sum_{i\in I}\sum_{j\in J}g_{ij}w_{ij} \...
sankha's user avatar
  • 85
7 votes
0 answers
124 views

Contractible affine surfaces of log Kodaira dimension 2

The first examples of contractible smooth affine algebraic surfaces (over the complex numbers) of log Kodaira dimension 2 were constructed in a famous paper of Ramanujam https://www.jstor.org/stable/...
Daniel Pomerleano's user avatar
0 votes
1 answer
647 views

Equivariant Sheaf: Explanation on Stalks

I have a question about the explanation of the data defining a so called equivariant sheaf $F$ on a scheme X from wiki: https://en.wikipedia.org/wiki/Equivariant_sheaf. Let denote by $\sigma: G \...
user267839's user avatar
  • 5,948
1 vote
0 answers
96 views

Two versions of the Murnaghan-Nakayama rule

I have always see the following Murnaghan-Nakayama rule for a partition $\lambda$ and a permutation $\sigma \in \mathfrak{S}_n$ of cycle structure $(\sigma_1, ..., \sigma_n)$: $$ \chi_{\lambda}(\sigma)...
eti902's user avatar
  • 835
4 votes
1 answer
163 views

Steenrod algebra: Ádem relations from Milnor product formula

The question is how to deduce the Ádem relations from the Milnor product formula. Straightforward approach leads to certain relation on binomial coefficients mod p. Could anyone tell me if there is a ...
Dr.Martens's user avatar
32 votes
1 answer
2k views

Does "every" first-order theory have a finitely axiomatizable conservative extension?

I originally asked this question on math.stackexchange.com here. There's a famous theorem (due to Montague) that states that if $\sf ZFC$ is consistent then it cannot be finitely axiomatized. ...
Oscar Cunningham's user avatar
4 votes
1 answer
490 views

Finite groups with the same character table *including* class types, and square-free order

There are non-isomorphic finite groups with the same (complex) character table, as $D_4$ and $Q_8$. $$\scriptsize\begin{array}{c|c} \text{class}&1&2A&2B&2C&4 \newline \text{...
Sebastien Palcoux's user avatar
3 votes
1 answer
242 views

Is this unipotent group, over characteristic 2, connected?

Let $E_{ij}(x)\in \mathrm{Mat}_{7\times7}(\overline{\mathbb{F}}_2)$ be the matrix with zeros everywhere, except for the value $x$ at $ij$. Set $$a(x)=1+E_{12}(x)+E_{34}(x)+E_{56}(x),\quad b(y)=1+E_{23}...
Dror Speiser's user avatar
  • 4,563
2 votes
0 answers
108 views

Equivalence De Rham and Dolbeault groupoids

I believe there is an error or incompleteness in Goldman's and Xia's proof of the equivalence of the De Rham and Dolbeault groupoids, contained in Rank One Higgs Bundles and Representations of ...
Anderson Gama's user avatar
1 vote
0 answers
39 views

Reference on graphs such that contracting 2 non-adjacent vertices increases the Hadwiger number

Suppose $G=(V,E)$ is a finite simple graph. Let $h(G)$ denote the Hadwiger number, that is, the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$. We say that a finite, simple, undirected ...
Dominic van der Zypen's user avatar
1 vote
2 answers
322 views

Regarding Haagerup $L^{P}$ spaces

There is a definition in Haagerup's paper on $L^{P}$ spaces for weights, my question is after putting the norm is it become semifinite $L^{P}$ space on the crossed product? I am not clear please help. ...
user136400's user avatar
2 votes
0 answers
90 views

A character sum $\sum_{0<n\leq Y}\chi_4(n)\chi_0^{(k)}(n)$ estimate

I'm reading the paper 'Jutila, Matti. "On the Mean Value of $L (1/2, χ)$ FW Real Characters." Analysis 1.2 (1981): 149-161.' Let $\chi_4(n)$ be the real primitive nonprincipal character of modulo 4, ...
LWW's user avatar
  • 653
-2 votes
1 answer
188 views

Flat cohomology and finite direct sum

Let $X$ be a scheme (we can assume $X$ is smooth over a field $k$). Let $\mathcal F_1$ and $\mathcal F_2$ be two sheaves of abelian groups on $X$. Is it always true that $H^i_{\text{flat}}(X, \...
grontim's user avatar
  • 167
4 votes
2 answers
578 views

A proper flat family with geometrically reduced fibers

Let $R$ be a Noetherian commutative unital ring. Let $f:X\rightarrow Y=\mathrm{Spec}\,R$ be a proper flat morphism of schemes with geometrically reduced fibers. We want to prove that the induced map $...
vanya's user avatar
  • 41
5 votes
1 answer
300 views

"Oddity" of Fibonacci-Catalan numbers

As a follow up to my previous two MO questions, here and here, let's consider the below inquiry. Define the Fibonacci-Catalan numbers by $FC_n=\frac1{F_{n+1}}\binom{2n}n_F$ where $F_0=0, F_1=1, F_0!=...
T. Amdeberhan's user avatar
15 votes
2 answers
826 views

Turning simplicial complexes into simplicial sets without ordering the vertices

Given an abstract simplicial complex $K$, one can make a simplicial set $X(K)$ with $n$-simplices given by sequences $(x_0, \dotsc, x_n)$ such that $\{x_0, x_1, \dotsc, x_n\}$ is a simplex of $K$. The ...
Omar Antolín-Camarena's user avatar
4 votes
0 answers
73 views

List of Small Non-Abelian Symmetrically Presented Groups

Let $F_n$ be the free group generated by $x_1,\ldots,x_n$ and let $S_n$ be the symmetric group on $\{1,\cdots,n\}$. Let $w=x_{i_1}^{\pm1}\cdots x_{i_s}^{\pm1}$ be a word and for each $\sigma \in S_n$, ...
Zuriel's user avatar
  • 1,098
1 vote
1 answer
369 views

Hyperbolic embedding of a directed acyclic graph defined over strings

For integer $n$ and alphabet $\Sigma$ we construct a DAG (directed acyclic graph) $G=(V,E)$ over strings $s\in\Sigma^\star$ as follows: $$V = \{s\in\Sigma^\star\colon |s|\le n\}$$ $$E = \{(s_1,s_2)\...
kvphxga's user avatar
  • 175
3 votes
0 answers
103 views

Numbers with large algebraic independency

Define the measure of algebraic independence of the numbers $a_1, \ldots, a_n \in \mathbb{C}$ as $\Phi(a_1, \ldots, a_n; m, H) = \min |P(a_1, \ldots, a_n)|$, where the minimum is taken over all ...
Alexey Milovanov's user avatar
1 vote
1 answer
236 views

Infinitesimal deformation of strict transform

Let $X$ be an affine, complex surface with isolated singularities and $\pi:\widetilde{X} \to X$ be a resolution of singularities (not necessarily minimal) i.e., $\widetilde{X}$ is non-singular and $\...
Ron's user avatar
  • 2,116
6 votes
2 answers
1k views

Products and sum of cubes in Fibonacci

Consider the familiar sequence of Fibonacci numbers: $F_0=0, F_1=1, F_n=F_{n-1}+F_{n-2}$. Although it is rather easy to furnish an algebraic verification of the below identity, I wish to see a ...
T. Amdeberhan's user avatar

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