# All Questions

150,089
questions

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### Primal optimal attained implies dual optimal attained

Given some optimization problem $\operatorname{min}_{x \in S \subset \mathbb{R}^{n}} f_{0}(x)$ $s$.$t$. $f_{i}(x) \leq 0, 1\leq i\leq m$. We can find the dual problem $\operatorname{max}_{\lambda\in\...

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12
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### When does Morita equivalence between two Hopf-von Neumann algebras imply also equivalence of their categories of comodules?

Let $A$ and $B$ be two Hopf-von Neumann (bi)algebras. Furthermore, let us assume that we know that they are Morita equivalent as von Neumann algebras (i.e. their categories of appropriate ...

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0
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24
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### Exponential of Liouville Numbers

By Mahler classification of Transcendental real numbers (into the sets of $S$-, $T$- and $U$-numbers), we know that
Any Liouville number is a $U$-number.
$\log \alpha$ is either an $S$- or a $T$-...

1
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0
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25
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### Complemented C* Algebras

let $A$ and $B$ be unital separable commutative $C^*$ algebras, with $A\subset B$. Is it true that $A$ is complemented in $B$?

1
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0
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26
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### How to define the Sobolev quotient space $H^s(Γ)/{\mathbb R}

Let $\Gamma$ be the boundary of a Lipschitz domain $\Omega\subset \mathbb R^3$. Denote by $H^s(\Gamma)$ the usual scalar Sobolev space for $s\in\mathbb R$. I want to know the definition of the ...

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68
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### Serre's theorem for sheaves

Serre's theorem for (algebraic) vector bundles (VB) says that a VB over a projective variety is the same thing as finitely generated projective module over the ring of algebraic functions $\mathcal{O}(...

2
votes

0
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63
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### Does the oriental inject into the cube?

For every $n \geq 0$ there is an inclusion of the ordered set $\{0<1<\dots<n\}$ into the product
$\{0<1\}^{\times n}$ sending $i$ to the increasing sequence $(0 < \dots<0<1<\...

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0
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24
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### Can the set of parafinite congruences be descriptive-set-theoretically complicated?

Fix an algebra $\mathfrak{A}$ with underlying set $\mathbb{N}$ and finite language $\Sigma$. The set of congruences on $\mathfrak{A}$ is a closed subset $C_\mathfrak{A}$ of $2^\mathbb{N}$ (with the ...

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30
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### Factorization of symmetric polynomials

Let $\Lambda_n$ be the algebra of all symmetric polynomials in $n$ variables, which we also consider as an infinite-dimensional vector $\mathbb{Q}$-space, whose basis is the Schur polynomials.
The ...

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2
answers

85
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### Probabilty measures that are both discrete and continuous

Consider a measure space $\left(S,\Sigma\right)$ where each state $s\in S$ can be expressed as $s=\left(x,c\right)$, where $x\in\mathbb R$ and $c\in\mathbb N$. E.g., suppose $s$ denotes the state of a ...

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60
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### Do balls in expander graphs have small expansion?

Consider a $d$-regular infinite transitive expander graph $G$, and let $B_r$ be a ball of radius $r$ in $G$. Can one place any upper bounds on the expansion of $B_r$?
My intuition is that $B_r$ will ...

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1
answer

58
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### Metropolis-Hastings kernel in measure theory

I'm facing difficulties in formulating the Metropolis-Hastings kernel for a specific problem where I need to sample from a probability distribution involving both discrete and continuous degrees of ...

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0
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39
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### Select random point on elliptic curve

If I have an elliptic curve $E$ over some finite field $F_p$ what is a step by step algorithm to pick a random point that lays on this curve? There is definitely a naive approach to brute force all ...

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43
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### Is the BGQ spectral sequence functorial with respect to morphisms of finite Tor-dimension?

It is well known that the BGQ (Brown-Gersten-Quillen) spectral sequence for the G-theory of a Noetherian scheme of finite Krull-dimension is contravariant with respect to flat morphisms.
My question ...

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69
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### When is a functor of chain complexes triangulated?

Let $\textsf{A}, \textsf{B}$ be abelian categories.
Let $F: \operatorname{Ch}(\textsf{A}) \to \operatorname{Ch}(\textsf{B})$ be an additive functor of chain complexes. If $F$ preserves chain ...

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0
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44
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### Finding inverses of certain elements in the set of normal invariants of a smooth manifold

Let, $V$ denote the Stiefel manifold of 2-frames $V_{n,2}$ . $n$ even. Consider the the map $S_\text{diff} (V) \xrightarrow{\eta} N_\text{diff} (V) $ in the surgery exact sequence of a smooth manifold....

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1
answer

136
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### When is a (co)edge trivial in graph cohomology?

Let $G$ be a connected graph and let $e$ be an edge in this graph. I would like to know if there are necessary and sufficient questions so that $e^{\vee}=0$ in $H^1(G)$? The question must be easy to ...

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0
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75
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### A set inequality problem

There is two different sets called set $a$ and $b$.Let $t$ be a positive integer,and put $t$ objects in another set called set $c$ ,and label the $t$ objects $c^1$,$c^2$...$c^t$.
Next,you put the ...

1
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0
answers

90
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### There exists noncommutative geometric invariant theory?

In this question, I am going to consider noncommutative projective algebraic geometry, as introduced by Artin and Zhang in the seminal paper Noncommutative projective schemes. The $\operatorname{Proj}$...

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0
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19
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### Enumeration of flat integral $K_4$

Question:
What is known about the enumeration of all $(a,b,c,d,e,f)\in\mathbb{N}^6_+: \\ \quad\operatorname{GCD}(a,b,c,d,e,f)=1\ \\ \land\ \exists \lbrace x_1,x_2,x_3,x_4\rbrace\subset\mathbb{E}^2:\ \...

2
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0
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109
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### Large sets of nearly orthogonal integer vectors

This question is motivated by the Question 5 from the 2017 Asia Pacific Mathematical Olympiad. To paraphrase, the question asks what is the largest cardinality of a set $S \subset \mathbb{Z}^n$ such ...

2
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46
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### A problem about the existence of increasing coloring groups

Got stuck on this one for months.
Given a sequence of non decreasing positive integers $a_1, .., a_n$, let there be $a_i$ balls labeled the number i, for each $1 \leq i \leq n$. Suppose there is a k ...

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### How would you find the slope at a specific point on a y=x^2 graph? [closed]

What process should I take to find the slope of a y=x^2 graph at specific coordinates such as (3,9)?

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0
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14
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### Does any warped product metric with harmonic Weyl curvature admit a structure of radial Weyl curvature?

A Riemannian manifold $(M, g)$ has harmonic Weyl curvature iff its Schouten tensor is Codazzi, and if there exists $f: M \to \mathbb{R}$ such that $W(\bullet, \bullet, \bullet, \nabla f) = 0$, one ...

-2
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1
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191
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### On Impossible events

Let's consider a continuous random variable $X$ distributed according to a PDF $p(x):\mathbb{R}\mapsto \mathbb{R}_{\geq 0}$.
Is there a meaningful sense in which one could say that for any $x_0:p(x_0)=...

1
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1
answer

151
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### Groups (not necessarily finite) with a given number of maximal subgroups

It is somewhat easy to see that a group $G$ with exactly one maximal subgroup $M$ must be cyclic: any element in $G\setminus M$ generates $G$.
EDIT: @YCor pointed out in the comments that this ...

2
votes

1
answer

224
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### A question on the manifold $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $

Consider a manifold $ N $ defined as follows
$$
N=\{n\otimes n-m\otimes m:n,m\in S^2,\quad(n,m)=0\}\subset M^{3\times 3},
$$
where $ S^2 $ denotes the two dimensional sphere, $ (\cdot,\cdot) $ ...

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0
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39
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### Tchebychev polynomial and gossip matrix [migrated]

A matrix W is said gossip matrix of a network of edges E if
W is an n × n symmetric matrix,
W is positive semi-definite,
The kernel of W is the set of constant vectors: Ker(W ) = Span(1), where 1 = (...

1
vote

1
answer

129
views

### Continuous path of unitary matrices with prescribed first column?

Consider a continuous curve $u \colon [0,1] \to \mathbb{C}^n$ where $u(t)$ is always a unit vector, $u(t)^* u(t) = 1$.
Question 1: Does there exist a continuous curve $U \colon [0,1] \to \mathbb{C}^{n ...

3
votes

1
answer

138
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### Normalizer of solvable linear group is an algebraic group?

I am trying to read the article "Three-dimensional affine crystallographic groups" of Fried–Goldman (Adv. Math., 1983). At some place, it states that if $G$ is a connected solvable closed ...

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71
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### Evaluating a matrix Pick function via its integral representation

In the proof of Theorem 3.1 of the paper Inequalities for M-matrices, Ando evaluates a matrix function (see equation boxed in orange below) via an integral representation of a Pick function (see ...

3
votes

0
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96
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### Is the test function topology a Mackey topology?

I am a physicist, and I have lately been thinking about distributions as they appear in quantum field theory. In the standard development of the theory of distributions, one considers the space $C^{\...

-1
votes

0
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46
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### The probability to visit a state for the first time after n steps in a markov chain [migrated]

I have the following Markov Chain:
(the probabilities are written above the arrows, and 'a' is a number between 0 and 1)
I want to show that State-1 is a persistent state. To show that, I need to ...

10
votes

3
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565
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### What are some toy models for the stable homotopy groups of spheres?

The graded ring $\pi_ast^s$ of stable homotopy groups of spheres is a horrible ring. It is non-Noetherian, and nilpotent torsion outside of degree zero.
Question: What are some "toy models" ...

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votes

1
answer

47
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### How to integrate an indicator function/constraint into the cost function of a linear program?

I have a mathematical model $P$ for which I optimize two cost functions say $F_1$ and $F_2$ subject to a set of constraints $C1$–$C10$.
In $F_2$, I want it to be included only when its expression ...

2
votes

0
answers

30
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### What are the epis, monos, and extensions in the Freyd Envelope of a triangulated category?

Let $\mathcal T$ be a triangulated category (or homotopy category of a stable $\infty$-category).
Recall that the Freyd envelope of $\mathcal T$ is an abelian category $\mathcal A$ which is ...

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1
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97
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### Congruences that aren't "finite from above," take 2: semigroups

This is a hopefully less trivial version of this question. Briefly, say that a congruence is parafinite if it is the largest congruence contained in some equivalence relation with finitely many ...

3
votes

1
answer

115
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### Creating mazes with colored tiles

Consider the following approach to constructing a maze: Create a rectangular grid of identical square tiles, each colored by one of N colors on a color wheel. For any pair of adjacent tiles, there is ...

3
votes

3
answers

343
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### Congruences that aren't "finite from above"

Let $\mathfrak{A}=(A;...)$ be an algebra in the sense of universal algebra. Say that a congruence $\sim$ on $\mathfrak{A}$ is parafinite iff there is an equivalence relation $E\subseteq A^2$ with ...

1
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1
answer

59
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### Why do distributional isomorphisms preserve joint distribution?

Let $(\Omega,\mathcal{A},\mu)$ and $(\Omega',\mathcal{A}',\mu')$ be probability spaces and
$$f_1,\ldots,f_n:\Omega\to\mathbb R,\; f_1',\cdots, f_n':\Omega'\to\mathbb{R}$$
be integrable random ...

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0
answers

26
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### Understanding Optimality Condition Decomposition (OCD)

I'm studying decomposition techniques for optimization problems and came across this OCD technique in which a problem is decomposed into subproblems, assigning to each a different set of complicating ...

-1
votes

1
answer

61
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### Locus of points for which the sum of the angles subtended there by two different line segments is a constant

Given a line segment AB, the locus of points P such that the angle APB has a constant value is a 'biconvex lens' formed by 2 circular arcs that passes through the points A and B. Special case: if APB ...

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0
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38
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### closed submodule of countable direct product of copies of topologically irreducible modules

Consider the countable direct product $\prod_{i=1}^{\infty}M$ where $M$ is a topologically irreducible module over a ring $A$ with some properties, like noetherian, but not a PID.
Then I'd like to ...

3
votes

1
answer

109
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### Can a solution to this parameterized ODE converge to zero?

Does there exists some $\gamma \ge 0$ such that the solution to the following ODE converges to 0 as $t \to \infty$?
$$y'(t) = \alpha y(t) - \gamma \sigma(t) (1-y^2(t))$$
We are also given y(0) = 2/3, $...

4
votes

0
answers

63
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### List of equivalent conditions for the invariant subalgebra to be polynomial

Let $k$ be a field, $P_n$ the polynomial algebra in $n$ indeterminates, and $G<\operatorname{GL}_n$ a finite group whose order is coprime to the characteristic of $k$, and that acts on $P_n$ by ...

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0
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37
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### $l^2(L^p)$ Decoupling constant of congruent tubes

Demeter's book Fourier Restriction, Decoupling, and Applications give a principle that one cannot decouple in a direction where the manifold is flat. Which is the below proposition:
Proposition 9.5 ...

0
votes

1
answer

43
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### Continuous selectors of a continuous multifunctin on a compact metric space

I am currently working on a continuous selector problem of multifunctions. I am trying to figure out if a continuous multifunction defined on a compact metric space always admit a continuous selector.
...

8
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0
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148
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### Grothendieck purity for Brauer groups of stacks

Let $X$ be a smooth variety over a field $k$ (for the sake of simplicity of characteristic $0$) and $\operatorname{Br}(X) := H^2_{\text{ét}}(X, \mathbb{G}_m)$ its (cohomological) Brauer group (...

0
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0
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63
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### Operator identity

Let $T:\mathcal{D}(A)\to\mathcal{H}$ be a unbounded, self-adjoint, operator with positive spectrum $\sigma(T)\subset [\varepsilon,\infty)$ for $\varepsilon>0$. Hence $T$ is bijective with bounded ...

6
votes

0
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109
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### A limit involving the largest prime factor of a prime gap

Posting in MO since this questions has been unanswered in MSE for 3 months.
Let $p_n$ be the $n$-th prime and $q_n$ be largest among all the prime factors of the composite numbers between $p_n$ and $...