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14 views

Specialization of an irreducible polynomial and monogeneous ring of integers

Let $P\in\mathbb{Q}[X_1,\ldots,X_n][T]$ be an irreducible polynomial, which is monic in $T$ of degree $d\geq 1$. For $x=(x_1,\ldots,x_n)\in \mathbb{Q}^n$, let $P_x=P(x_1,\ldots,x_n,T)$. It is well-...
6
votes
1answer
84 views

Was Jacobi the first to notice the ambiguity in the partial derivatives notation? And did anyone object to his fix?

In his 1841 article De determinantibus, Jacobi remarked that the notation $\frac{\partial z}{\partial x}$ for partial derivatives is ambiguous. He observed that when $z$ is a function of $x,y$ as well ...
0
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0answers
22 views

List of all known Riesz representation theorems

Due to the history and development of measure and integration theory and different mathematical schools, there is a huge variety and inconsistency of definitions for concepts like tightness of a ...
0
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0answers
9 views

$L^p $ Space with Values in Metric Space Homeomorphic

Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete $\lambda$-doubling metric measure spaces and $p \in [1,\infty)$. Moreover, suppose that there exists a homeomorphism $\Phi$ from $(Y,d_Y)$ to some ...
0
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0answers
20 views

Calculation of the sphere surface using only intrinsic variables

How would we calculate the surface of a 2 sphere if we had no understanding the third dimension and wanted to find the surface which should be circumference squared over π. By intrinsic it's also ...
1
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1answer
23 views

Mixing time of random walks on graphs

Suppose that we start a lazy random walk on a connected graph. However, the starting node is picked from a distribution of $\mu$ and $||\mu-\pi||_{TV}<1/8$, where $\pi$ is the stationary ...
0
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0answers
45 views

Crystals and nilpotence

Fix a prime $p$ and a height $n \geq 1$, then there is a closed substack $\mathcal{M}^{\geq n}$ of the stack of $\mathbb{F}_p$-formal groups consisting of formal groups having height $\geq n$. A ...
1
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0answers
37 views

Exact sequence in example in Grothendieck's Tohoku paper resulting from the Cech-to-derived-functor spectral sequence

Grothendieck gives in his Tohoku paper in example 3.8.3 an example for that $\check{\mathrm{H}}^{2}(X,\mathcal{F}) \neq \mathrm{H}^{2}(X,\mathcal{F})$. In the beginning he states that there exisits ...
0
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1answer
19 views

Different definitions of a relatively compact operator

(Cross-post from Math Stackexchange, where some work has been done in the comments) Let $T,K$ be unbounded operators on a Hilbert space $H$. I've seen the following definition of a relatively compact ...
1
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0answers
28 views

Image of the Lang-Steinerg on disconnected centralizers of semisimple elements

Let $\newcommand{\dbF}{\mathbb F}\dbF_q$ be a finite field and let $G\subseteq\mathrm{GL}_N(\bar{\dbF}_q)$ be a connected reductive group defined over $\dbF_q$. Let $F$ be the associated Frobenius map,...
1
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0answers
27 views

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 ...
3
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0answers
34 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)
0
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0answers
22 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 ...
1
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0answers
27 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 ...
0
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1answer
76 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 $...
4
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0answers
143 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
35 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 ...
14
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3answers
309 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 $1\leq x_i \leq n$: $(...
2
votes
0answers
47 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
votes
1answer
80 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 ...
0
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0answers
18 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 ...
2
votes
1answer
73 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 $P^m \to X$?
0
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1answer
157 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 $...
3
votes
1answer
102 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 ...
0
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0answers
20 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 ...
0
votes
0answers
14 views

Approximate method to extract behavior of a Laplace transform in an intermediate region

In the theory of random walks, Tauberian type theorems are often applied to extract the small or large-time behavior from a difficult equation. For example, the Montroll-Weiss formula describing a ...
1
vote
1answer
33 views

Reference request for (weak*) metrizability of a bounded space of signed Radon measures on a compact set

I know the following is true and I know how to prove it (cf. exercise 50 on page 171 in Folland, Theorem 7.18 in Folland), but per my adviser's instructions, it would be better to find a source to ...
7
votes
1answer
210 views

Is there any conditions on a finite abelian group so that it cannot be class group of any number field?

The Cohen-Lenstra paper says the probability that the odd part of a class group being cyclic is close to 0.98. So I was thinking: can we find any conditions on a finite abelian group so that it cannot ...
1
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0answers
36 views

Curvature operator $[\Lambda,i \nabla^2]$ for a Kähler manifold

Let $(M,g)$ be a compact Kähler manifold, and let $L$ and $\Lambda$ be the associated Lefschetz and dual Lefschetz operator. Consider $E$ a vector bundle over $M$ equipped with an Hermitian metric ...
5
votes
0answers
32 views

Quantitive and computational improvement of the Oseledets multiplicative ergodic theorem for irrational rotation

Consider irrational rotation $T:S^1\to S^1, T(x) = x + \alpha$ where $\alpha\notin \mathbb{Q}$ (you may assume additional number theoretic properties of $\alpha$, say $\alpha = \sqrt{2}$ is already ...
3
votes
0answers
68 views

Geodesics (Local vs Global)

Let $M$ be a Riemannian manifold, and $p,q\in M$ two points. Now, if $M$ is a complete metric space the Hopf–Rinow theorem ensures that there is a geodesic $C\subset M$ joining $p$ and $q$ that ...
1
vote
1answer
58 views

Lie algebra elements commuting with a principal nilpotent element

Let $\mathfrak{g}$ be a semisimple complex Lie algebra, $A \in \mathfrak{g}$ a principal nilpotent element (i.e. its centralizer is of dimension equal to the rank of $\mathfrak{g}$). I wish to ...
2
votes
1answer
45 views

Estimating the probability density of a component of a mixture distribution

Let $X \in \mathbb{R}^d$ be a random variable with probability distribution $P$. Let $f:\mathbb{R}^d \to \mathbb{R}^d$ be an invertible function and let $P_{f}$ be the distribution of random variable $...
5
votes
3answers
417 views

Link of a singularity

I would like to understand the topological type of a link of a singularity in a simple example. Consider for instance the cone ${xy-z^2=0}\subset\mathbb{C}^3$. If we set $x = x_1+ix_2, y = y_1+iy_2, z ...
0
votes
0answers
89 views

Two equations and a question related to a well-known conjecture from number theory

On the Wikipedia page, the Beal´s conjecture is stated as: If $A^x+B^y=C^z$, where $A,B,C,x,y,z$ are positive integers with $x,y,z>2$, then $A$,$B$, and $C$ have a common prime factor. I think ...
1
vote
0answers
21 views

Equivalence relations: Cosimplicial semilattice?

For $n\ge 0$, let $E_n$ be the set of all equivalence relations on $[n]:=\{0,\dotsc,n\}$. Now given two equivalence relations $R,R'\in E_n$, we build their join $$R\vee R' := \langle R\cup R'\rangle,$$...
0
votes
0answers
29 views

Girsanov Theorem - Dependence of the probability space

I'm trying to understand the proof Novikov Condition and other works in this field (Kazamaki etc) and for this, i have to understand the Girsanov Theorem, which is also a part of the proof. In ...
1
vote
0answers
29 views

Do we have $C_L^G/\mathrm{Nm}_G(C_L)\simeq \mathrm{Cl_K}$?

According to wiki about Artin reciprocity law, in the proof we have $H^0(\mathrm{Gal}(L/K), C_L)\simeq H^{-2}(\mathrm{Gal} (L/K),\mathbb Z)$ for Galois extension of global fields where $C_L$ is ...
5
votes
0answers
80 views

Permutations, skew-symmetric forms and degeneracy

Define a skew-symmetric form $(\cdot,\cdot)$ on $\mathbb{R}^{2k}$ by $$(e_i,e_j) = \begin{cases} 1 &\text{if $i<j$},\\ -1 &\text{if $i>j$},\\ 0 & \text{if $i=j$.}\end{cases}$$ Given ...
1
vote
0answers
227 views

A (surprising?) expression for $e$

I apologise if this is off topic. Consider the quantity $$ F(m,n,k)=\frac{(m)_k}{k!n^{k-1} } $$ where $m,n \in \mathbb{N}.$ For moderately large $n$, it seems that the approximation $$ \sum_{k=1}^{K} ...
1
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0answers
55 views

Gaussian-weighted area of triangle

I'm trying to find how to generalise the calculation of the Gaussian-weighted area to triangles for convolution purposes. Let's start with how that works when there's only one line. If there's a line ...
1
vote
2answers
82 views

on exponential distributions and dot products

Let $a,b$ be two variables drawn from an exponential distribution with parameter $λ_1$. Let $c,d$ be two variables drawn from an exponential distribution with parameter $λ_2$. I am interested in the ...
2
votes
1answer
77 views

Uniqueness of presentation for semi-abelian varieties

Let $k$ be any field and $G$ a semi-abelian variety over $k$, i.e., an algebraic group that fits into an exact sequence $$ 1 \to T \to G \to A \to 1$$ of algebraic groups, where $T$ is an algebraic ...
2
votes
0answers
80 views

Volume doubling, uniform Poincaré, counterexample

The Poincaré inequality and the volume doubling property are important notions related to heat kernel estimates. Pavel Gyrya and Laurent Saloff-Coste obtain the two sided heat kernel estimate of ...
3
votes
0answers
51 views

Does an antisymmetric matrix with high rank have a minor with disjoint rows and columns and high rank?

This is a generalized version of Does a non-singular matrix have a large minor with disjoint rows and columns and full rank? Let $A$ be an $n$-by-$n$ antisymmetric matrix of rank $r\geq \epsilon n$. ...
1
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0answers
70 views

Tensor schemes “with relations”

In "The geometry of tensor calculus I," Joyal and Street introduce the notion of a tensor scheme, a set of abstract objects, together with formal morphisms between words in this set. They then prove ...
-1
votes
0answers
79 views

The equation $x^n=y$ in $p$-groups [on hold]

Is there an algorithm or a method to find the set $G^n=\{x^n:x\in G\}$ where $G$ is a $p$-group and $n$ is integer?
0
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0answers
71 views

If the coefficient of the polynomial positive

I want to know what is following sum coefficient looks like. We sum over all integers $p$, $q$ also we put the condition that $q$ is even. Also, it should depend on the parity of $k$ $$\bar{S}(k)=\...
1
vote
0answers
131 views

Trace of Frobenius on $p$-adic Tate module

Let $k$ be a finite field of characteristic $p>0$ with $q$ elements. Denote $K=W(k)[1/p]$. Let $E$ be an elliptic curve over $W(k)$ with good reduction. Choose a lifting $\mathrm{Frob} \in \...
-3
votes
0answers
83 views

is $p$ is prime, then the nonzero elements of $ℤ_p$ form a group of order $p-1$ multiplication [on hold]

first of all I'm sorry to post this, maybe it's not as professional as the other post. I defined $ℤ^*_p = \big\{\overline{a}∈ℤ _p\vert \overline{a}≠0\big\}$.The order of $ℤ_p$ is $p$ so $ℤ^*_p $ has ...

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