# All Questions

104,371
questions

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### 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

**1**answer

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 ...

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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 ...

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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 ...

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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 ...

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**1**answer

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 ...

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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 ...

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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 ...

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**1**answer

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 ...

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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,...

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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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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)

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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 ...

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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 ...

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votes

**1**answer

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 $...

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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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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**

votes

**3**answers

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**

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**0**answers

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

**1**answer

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 ...

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**0**answers

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

**1**answer

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 $P^m \to X$?

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**1**answer

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

**1**answer

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 ...

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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 ...

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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 ...

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**1**answer

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 ...

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**1**answer

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 ...

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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**

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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**

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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**

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**1**answer

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

**1**answer

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

**3**answers

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 ...

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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 ...

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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
$$R\vee R' := \langle R\cup R'\rangle,$$...

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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 ...

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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 ...

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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 ...

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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} ...

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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 ...

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**2**answers

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**

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**1**answer

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**

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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 ...

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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 $r\geq \epsilon n$. ...

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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 ...

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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?

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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)=\...

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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 \...

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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 ...