All Questions
150,089
questions
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4
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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\...
- 123
0
votes
0
answers
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 ...
- 1
0
votes
0
answers
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 $T$-...
- 485
1
vote
0
answers
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 $B$?
- 131
1
vote
0
answers
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 ...
- 53
0
votes
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}(...
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<\...
- 381
1
vote
0
answers
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 ...
- 18.9k
1
vote
0
answers
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 ...
- 526
-1
votes
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 ...
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 ...
- 21
-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 ...
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 ...
- 1
0
votes
0
answers
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 ...
- 411
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votes
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 ...
- 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....
- 111
1
vote
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 ...
- 23
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 ...
- 129
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 $\operatorname{Proj}$...
- 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:\ \...
- 12.4k
2
votes
0
answers
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 ...
- 24.8k
2
votes
0
answers
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 ...
- 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)?
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 ...
- 619
-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)=...
- 141
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 ...
- 837
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) $ ...
- 1,079
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 = (...
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 ...
- 509
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 ...
- 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 ...
- 1,711
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^{\...
- 31
-1
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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 ...
- 9
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" ...
- 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 ...
- 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 ...
- 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 ...
- 18.9k
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 ...
- 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 ...
- 18.9k
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 ...
-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 ...
- 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 ...
- 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 ...
- 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, $...
- 41
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 ...
- 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 ...
- 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.
...
- 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 (...
- 188
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 ...
- 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 $...
- 4,576