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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\...
wsz_fantasy's user avatar
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0 answers
12 views

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 ...
szantag's user avatar
0 votes
0 answers
24 views

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 $T$-...
Jean's user avatar
  • 485
1 vote
0 answers
25 views

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 $B$?
user44155's user avatar
  • 131
1 vote
0 answers
26 views

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 ...
SAKLY's user avatar
  • 53
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0 answers
68 views

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}(...
Yilmaz Caddesi's user avatar
2 votes
0 answers
63 views

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<\...
willie's user avatar
  • 381
1 vote
0 answers
24 views

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 ...
Noah Schweber's user avatar
1 vote
0 answers
30 views

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 ...
Leox's user avatar
  • 526
-1 votes
2 answers
85 views

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 ...
Iris Allevi's user avatar
1 vote
0 answers
60 views

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 ...
user3521569's user avatar
-1 votes
1 answer
58 views

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 ...
Iris Allevi's user avatar
0 votes
0 answers
39 views

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 ...
R Artur's user avatar
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0 answers
43 views

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 ...
Boris's user avatar
  • 411
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0 answers
69 views

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 ...
Jannik Pitt's user avatar
  • 1,073
1 vote
0 answers
44 views

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....
Sagnik Biswas's user avatar
1 vote
1 answer
136 views

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 ...
divergent's user avatar
0 votes
0 answers
75 views

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 ...
A Math guy's user avatar
1 vote
0 answers
90 views

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 $\operatorname{Proj}$...
jg1896's user avatar
  • 2,435
0 votes
0 answers
19 views

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:\ \...
Manfred Weis's user avatar
  • 12.4k
2 votes
0 answers
109 views

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 ...
Stanley Yao Xiao's user avatar
2 votes
0 answers
46 views

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 ...
John Jiang's user avatar
  • 4,352
-4 votes
0 answers
27 views

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)?
user517545's user avatar
0 votes
0 answers
14 views

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 ...
Matheus Andrade's user avatar
-2 votes
1 answer
191 views

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)=...
matteogost's user avatar
1 vote
1 answer
151 views

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 ...
semisimpleton's user avatar
2 votes
1 answer
224 views

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) $ ...
Luis Yanka Annalisc's user avatar
0 votes
0 answers
39 views

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 = (...
over dose's user avatar
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 ...
ccriscitiello's user avatar
3 votes
1 answer
138 views

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 ...
LeeM's user avatar
  • 31
0 votes
0 answers
71 views

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 ...
Pietro Paparella's user avatar
3 votes
0 answers
96 views

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^{\...
Jon's user avatar
  • 31
-1 votes
0 answers
46 views

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 ...
JoeHills's user avatar
10 votes
3 answers
565 views

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" ...
Tim Campion's user avatar
  • 58.5k
0 votes
1 answer
47 views

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 ...
LyLa's user avatar
  • 3
2 votes
0 answers
30 views

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 ...
Tim Campion's user avatar
  • 58.5k
0 votes
1 answer
97 views

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 ...
Noah Schweber's user avatar
3 votes
1 answer
115 views

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 ...
Travis's user avatar
  • 75
3 votes
3 answers
343 views

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 ...
Noah Schweber's user avatar
1 vote
1 answer
59 views

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 ...
Pavlos Motakis's user avatar
-1 votes
0 answers
26 views

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 ...
Iakl's user avatar
  • 1
-1 votes
1 answer
61 views

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 ...
Nandakumar R's user avatar
  • 5,109
-3 votes
0 answers
38 views

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 ...
Ang's user avatar
  • 95
3 votes
1 answer
109 views

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, $...
icecuber's user avatar
4 votes
0 answers
63 views

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 ...
jg1896's user avatar
  • 2,435
0 votes
0 answers
37 views

$l^2(L^p)$ 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 ...
Vstal's user avatar
  • 101
0 votes
1 answer
43 views

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. ...
Saito's user avatar
  • 49
8 votes
0 answers
148 views

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 (...
Tim Santens's user avatar
0 votes
0 answers
63 views

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 ...
B.Hueber's user avatar
  • 811
6 votes
0 answers
109 views

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 $...
Nilotpal Kanti Sinha's user avatar

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