# 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-...

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votes

**1**answer

81 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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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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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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votes

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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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vote

**1**answer

57 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 ...

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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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51 views

+50

### Reference: Hajlasz-Sobolev Spaces with Values in a Metric Space

Let $(X,d,\mu)$ be a separable metric measure space on which every ball has positive but finite measure.
I've come across the definition of a homogeneous Fractional Hajlasz-Sobolev spaces which are ...

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54 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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votes

**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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votes

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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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votes

**4**answers

3k views

### Complex torus, C^n/Λ versus (C*)^n

I'm having trouble distinguishing the various sorts of tori.
One definition of torus is the algebraic torus. Groups like $SU(2,\mathbb{C})$ and $SU(3,\mathbb{C})$ have important subgroups that are ...

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votes

**1**answer

111 views

### Integral zeros of the Newton polynomial

I'm trying to understand the following result;
Statement: A newton polynomial of the form
$$a_1 {x\choose c_1}+a_2{x\choose c_2}+a_3{x\choose c_3}+⋯+a_s{x\choose c_s},$$
where $0 ≤c_1<c_2<c_3&...

**4**

votes

**1**answer

327 views

### A conjecture about the submatrix of orthogonal matrix

Let $U$ be an $n\times n$ orthogonal matrix, i.e. $U\in\mathbb{R}^{n \times n}$. For any non-empty ordered sets $S_1,S_2\subset\{1,2,...,n\}$, define $U_{S_1S_2}$ to be an $|S_1|\times|S_2|$ submatrix ...

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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 ...

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**0**answers

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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**0**answers

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 ...

**0**

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**0**answers

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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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,...

**5**

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**1**answer

147 views

### Studying for primes $q_k \neq 2$ the sets $\{q_1!+q_k,…,q_{k-1}!+q_k\}$

For a prime $q_k \neq 2$ we can study the corresponding set $\{q_1!+q_k,...,q_{k-1}!+q_k\}$, where $q_1,...q_{k-1}$ are all primes strictly less than the prime $q_k$.
Peter and Mathphile computed ...

**122**

votes

**45**answers

15k views

### Old books you would like to have reprinted with high-quality typesetting

There are some questions on mathoverflow such as
What out-of-print books would you like to see re-printed?
Old books still used
with answers that tell us things such as:
Mathematicians prefer to ...

**14**

votes

**3**answers

308 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 $1\leq x_i \leq n$:
$(...

**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 ...

**5**

votes

**0**answers

318 views

### Eilenberg-Mac Lane spaces for surface group extensions.

(The question has been edited. It was pointed out in the comments that $\Gamma_G$ could be a surface group, thought of as a finite extension of another surface group, in which case $G$ is finite.)
...

**55**

votes

**6**answers

11k views

### What is an intuitive view of adjoints? (version 1: category theory)

In trying to think of an intuitive answer to a question on adjoints, I realised that I didn't have a nice conceptual understanding of what an adjoint pair actually is.
I know the definition (several ...

**3**

votes

**1**answer

101 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 ...

**6**

votes

**1**answer

194 views

### Idempotent functions on Sp(1)

The quaternion group $Sp(1)\simeq S^3$ can be understood as $(z,w)\in\mathbb {C}^2$ with $|z|^2+|w|^2=1$ where multiplication is defined by $(z,w)(t,s)=(zt-\bar{s}w,zs+\bar{t}w)$.
Question: How do ...

**7**

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**0**answers

105 views

+100

### Variously pointed closed sets

A tree $A\subseteq \omega^{<\omega}$ - possibly with dead ends - is pointed iff every path $p\in[A]$ has $p\ge_TA$. This lifts to two distinct notions of pointedness for closed sets in Baire space: ...

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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 $...

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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 ...

**1**

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**1**answer

108 views

### $0$-“norm” minimization with least-squares regularization

I have the following optimization problem in $\mathbf{x} \in \mathbb{R}^{K \times 1}$
$$\min_{\mathbf{x}>0} \quad \|\mathbf{A}\mathbf{x}\|_0 + \alpha \|\mathbf{B}\mathbf{x}-\mathbf{c}\|_2^2$$
...

**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 ...

**1**

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**0**answers

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 $...

**3**

votes

**1**answer

124 views

### An application of Itô's formula to an SDE on a Lie group

I'm trying to understand a calculation in this paper (equation (3.8)). With some details removed, the setup is as follows.
Let $G$ be a Lie group, and $g(t)$ a curve in $G$ satisfying the SDE
$$dg(t)...

**4**

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**0**answers

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 ...

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**0**answers

67 views

### Convergence of non-collapsing sequences of Riemannian manifolds with biliterally bounded sectional curvature

EDIT: I heard that there is the following result:
Given a sequence $\{(M_i^n,g_i)\}$ of compact smooth $n$-dimensional Riemannian manifolds with uniformly bounded absolute value of sectional ...

**6**

votes

**1**answer

172 views

### Asymptotic bound on minimum epsilon cover of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a compact smooth $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\varepsilon)$ denote the minimal cardinal of an $\varepsilon$-cover $P$ of $M$; ...

**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 ...

**9**

votes

**2**answers

1k views

### Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?

Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split?
If not, is there an example?

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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 $...

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101 views

### Annihilator of an element and Jacobson radical

Let $R$ be a commutative ring with 1. Is there any characterization for an element $a$ of $R$ such that $\operatorname{ann}(1-a)\subseteq J(R)$ and $a\in J(R)$, where $\operatorname{ann}(x):=\{r\in R\...

**7**

votes

**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 ...

**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 $P^m \to X$?

**43**

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**0**answers

2k views

### Uniformization over finite fields?

The following is a question I've been asking people on and off for a few years, mostly out of idle curiosity, though I think it's pretty interesting. Since I've made more or less no progress, I ...

**1**

vote

**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 ...

**1**

vote

**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 ...

**0**

votes

**0**answers

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 ...