All Questions

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-...

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 ...

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 \...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

(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 ...

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 ...

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 ...

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&...

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 ...

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 ...

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)

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 ...

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 ...

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 ...

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,...

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 ...

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 ...

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

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 ...

(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.) ...

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 ...

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 ...

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 ...

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: ...

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 $...

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 ...

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

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 ...

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 ...

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 $...

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)...

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}]...

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 ...

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 ...

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 $M$; ...

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 ...

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?

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 $...

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\...

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 ...

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

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 ...

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 ...

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 ...

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 ...