All Questions
152,891
questions
2
votes
1
answer
148
views
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 ...
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.
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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(...
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 $...
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)...
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 $...
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 [...
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 ...
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 ...
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 ...
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....
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, ...
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\...
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): ...
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 $\Bbb Z[[x]]/(x-p)$ ...
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 ...
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 ...
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 ...
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 ...
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
$$
(x,y) = \sum_{i\in [d]} ...
1
vote
0
answers
74
views
When does a stochastic process have finite second moment?
For a known probability space $(\Omega,\mathcal{F},P)$ where $\Omega$ is the event space, $\mathcal{F}$ a $\sigma$-algebra and $P$ is the (known) probability measure, let $f(x(t,\xi),\xi), \,\, \xi \...
7
votes
1
answer
257
views
Kolmogorov superposition on the Hilbert Cube
A result of Kolmogorov and Arnold says that continuous functions on $\mathbb{R}^n$ can be represented as sums of the form
$$ f(x_1,\dots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^n\phi_{p,q}(x_p)\...
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 ...
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} \...
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/...
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 \...
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 $(\sigma_1, ..., \sigma_n)$:
$$
\chi_{\lambda}(\sigma)...
4
votes
1
answer
163
views
Steenrod algebra: Ádem relations from Milnor product formula
The question is how to deduce the Ádem relations from the Milnor product formula. Straightforward approach leads to certain relation on binomial coefficients mod p.
Could anyone tell me if there is a ...
32
votes
1
answer
2k
views
Does "every" first-order theory have a finitely axiomatizable conservative extension?
I originally asked this question on math.stackexchange.com here.
There's a famous theorem (due to Montague) that states that if $\sf ZFC$ is consistent then it cannot be finitely axiomatized. ...
4
votes
1
answer
490
views
Finite groups with the same character table *including* class types, and square-free order
There are non-isomorphic finite groups with the same (complex) character table, as $D_4$ and $Q_8$.
$$\scriptsize\begin{array}{c|c}
\text{class}&1&2A&2B&2C&4 \newline
\text{...
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}...
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 ...
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 ...
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. ...
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, ...
-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, \...
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 $...
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!=...
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 ...
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$, ...
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)\...
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 ...
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 $\...
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 ...