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martingales and intersection of random walks

Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random walk $X$ as a ...
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13 views

Bousfield Localization of a left proper accessible model category

What is known about the Bousfield localization of a left proper accessible model category by a set of maps ? (I mean not combinatorial which is already known)
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0answers
18 views

Weighted inner product of independent random unit vectors

Let $u=(u_1,...,u_n)$ and $v=(v_1,...,v_n)$ be independent random unit vectors in $\mathbb{R}^n$. Let $\lambda=(\lambda_1,...,\lambda_n)$ be a fixed unit vector in $\mathbb{R}^n$. What is the ...
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0answers
21 views

Extension of probability space problem: Hilbert space valued process V.S. random field

Maybe the question should be "Understanding the measurability: Hilbert space valued process V.S. random field" Consider the SPDE $${\rm d}u+\cdots{\rm d}t=\sigma(t,u){\rm d}W.$$ Consider the ...
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1answer
72 views

Lifting functions between $L^2$

A map $\pi: X \to Y$, $\mu$ is the measure on $X$, and its push forward is defined by $\nu:=\pi_{*} \mu$. If given $f \in L^2(X, \mu)$, can we find $g \in L^2(Y, \nu)$ such that $g \circ \pi= f$, if $...
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0answers
114 views

Categorical Significance of Fibrations

It is well known that the category $\text{Set}$ classifies covering spaces among $1$-categories. That is, for each topological space $X$, there is an equivalence of categories $[ \Pi (X) , \text{Set}]...
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0answers
28 views

Questions Regarding Conditional Entropy

Consider a random variable $G$ distributed in a bounded 2d plane e.g. a square. For any point $g$ sampled from $p(G)$, there are 10 points $s_i$ at the same position as $g$. Assuming all $s$ are from ...
11
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3answers
243 views

A necessary and sufficient condition for $(x_1,…,x_n)$ to be a permutation of $(1,…,n)$

Is there an easy proof of the following statement? $\forall$ $n>0 \in \mathbb N$, $ \exists$ $a\geq0 \in \mathbb N$ such that for any set of integers $(x_1,...,x_n)$ and $1\leq x_i \leq n$: $(...
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0answers
39 views

Provenance of a result on regular simplices with integer vertices

There are several MO questions related to the question of characterizing those integers $n$ for which there exists a regular $n$-simplex in $\mathbb{R}^n$ with integer vertices, e.g., coordinates of ...
3
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1answer
60 views

Ext-vanishing in abelian categories

Given an abelian category $A$ with enough projectives and enough injectives such that projectives do not coincide with injectives. Can we have $Ext^i(I,P)=0$ for any $i>0$ and injective $I$ and ...
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0answers
16 views

Feasibility criteria in Integer Linear Programming

Consider an integer linear programming problem: For $A\in M(m, n, \mathbb{Z})$ and $b\in \mathbb{Z}^m$ find $x=(x_1,\ldots,x_n)^T\in \mathbb{Z}^n_{\geqslant0}$ such that $Ax=b$. Sometimes one ...
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1answer
68 views

Morphisms from projective space to lower dimension spaces [duplicate]

Let $X$ be a variety over a base field $k$ of dimension $n$. Can there be non constant morphisms $P^m \to X$?
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1answer
146 views

What does “can almost be proven in PA” mean regarding Theorem 2 of Timothy Chow's expository article, “The Consistency of Arithmetic”?

In his expository article, "The Consistency of Arithmetic" (MSN), Prof. Chow has the following theorems: Theorem 1. If $a_1, a_2, a_3,\dotsc$ is a sequence of ordinals and $a_i \ge a_j$ whenever $...
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1answer
91 views

Line bundles trivial outside of codimension 3

Let $X$ be a CW complex (possibly a topological/smooth manifold) of dimension $n$, $L\to X$ a complex line bundle and $Y\subset X$ a subcomplex (possibly a submanifold) contained in the codimension 3 ...
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0answers
19 views

General formula for a conditional recursive equation

I've discovered general formulas for other recursive equations but the fact that this one is conditional stumped me. I tried approaches like using Wolfram Mathematica to try and solve it or just ...

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