Newest Questions
159,032 questions
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Does there exist research about equation like $u_{tt}=\det(D_{x}^{2}u)+\dots$?
I have asked this question on Mathematics Stack Exchange yesterday, but there still is no reply.
Does there exist research about equation like $$u_{tt}=\det(D_x^2 u)+\cdots\text{?}$$ That is to say, ...
0
votes
0
answers
130
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Primal optimal attained implies dual optimal attained
Given some optimization problem
$$\min_{x \in S \subset \mathbb{R}^n} f_0(x) \quad \text{s.t.} \quad f_i(x) \leq 0, \quad 1\leq i\leq m$$
we can find the dual problem
$$\max_{\lambda\in\mathbb{R}^m} g(...
- 275
4
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0
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56
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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 ...
- 143
3
votes
0
answers
91
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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$-...
- 515
1
vote
1
answer
142
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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$?
- 149
2
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0
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89
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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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5
votes
1
answer
390
views
Does the simplex map to 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<\...
- 499
3
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0
answers
176
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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 ...
- 20.5k
3
votes
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171
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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 ...
- 656
5
votes
2
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731
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 ...
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2
votes
1
answer
292
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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 ...
- 31
2
votes
1
answer
179
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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 ...
- 101
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vote
0
answers
132
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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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0
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158
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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 ...
- 1,474
4
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116
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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_{10,2}$ . 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
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212
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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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150
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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}$...
- 3,318
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0
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32
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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:\ \...
- 13.2k
7
votes
1
answer
382
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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 ...
- 26.9k
2
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1
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117
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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 ...
- 4,466
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97
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Does any warped product metric with harmonic Weyl curvature admit a structure of zero 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 ...
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1
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260
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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)=...
- 181
2
votes
1
answer
327
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 ...
- 1,433
3
votes
2
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425
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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) $ ...
- 1,219
2
votes
1
answer
264
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 ...
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5
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1
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188
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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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0
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107
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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 ...
- 1,719
4
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0
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176
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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^{\...
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3
answers
2k
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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" ...
- 64k
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1
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169
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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 ...
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2
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0
answers
76
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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 ...
- 64k
1
vote
1
answer
142
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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 ...
- 20.5k
3
votes
1
answer
157
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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 ...
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5
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3
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542
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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 ...
- 20.5k
2
votes
1
answer
93
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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}$$
be integrable random ...
-1
votes
2
answers
240
views
Locus of points for which the sum of the angles subtended there by two different line segments is a constant
Given a line segment AB, the locus of points P such that the angle APB has a constant value is a 'biconvex lens' formed by 2 circular arcs that passes through the points A and B. Special case: if APB ...
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5
votes
1
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235
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Can a solution to this parameterized ODE converge to zero?
Does there exists some $\gamma \ge 0$ such that the solution to the following ODE converges to 0 as $t \to \infty$?
$$y'(t) = \alpha y(t) - \gamma \sigma(t) (1-y^2(t))$$
We are also given y(0) = 2/3, $...
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4
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0
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95
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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 ...
- 3,318
1
vote
1
answer
117
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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 ...
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1
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92
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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.
...
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11
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1
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411
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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 (...
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11
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1
answer
324
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Does the mean ratio of the largest prime factor in prime gaps to the lower bound of the gap converge?
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 $...
- 5,470
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1
answer
84
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Formulas for partial composed product
Let $A(x) = \prod\limits_i (x-\lambda_i)$ and $B(x) = \prod\limits_j (x-\mu_j)$. Then, their composed product is defined as
$$
(A*B)(x) = \prod\limits_{i,j} (x-\lambda_i \mu_j).
$$
Generally, we can ...
- 1,171
3
votes
1
answer
406
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Electricity division and bin packing
In the electricity division problem, there is a powerhouse that supplies $s$ kilowatt of electricity. There are $n$ households. The connection size of household $i$ is $d_i$.
The problem is that $s &...
- 3,559
1
vote
0
answers
137
views
Locus where a family of cycles is rationally trivial is closed?
Let $B$ be a smooth quasi-projective variety over a field of characteristic zero.
Let $\pi\colon \mathcal{X} \rightarrow B$ be a smooth and projective morphism with geometrically integral fibres. Let $...
- 984
4
votes
1
answer
254
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Salvetti complex of dihedral Artin group
The Salvetti complex of a RAAG is well-known and it is fairly simple, since each complete graph gives rise to a tori. The case of Artin groups is wilder, since we do not have tori anymore. The ...
- 911
1
vote
2
answers
260
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Exterior differential systems on compact three-manifolds and Cartan-Kähler theory
Let $M$ be a compact three-manifold. I am interested in the following equation on $M$:
$ d e^i = \sum_{j,k}^3 \Theta^i_{jk} \, e^j\wedge e^k\, , \qquad i =1,2,3$
together with the following condition:...
- 2,818
3
votes
1
answer
247
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What is the minimum and the maximum perimeter of a triangle with area $x$ that can be inscribed in a circle?
This question was posted in MSE but is still open hence posting in MO.
The area of the largest triangle that can be inscribed in a circle of raidus $1$ is $\displaystyle \frac{3 \sqrt{3}}{4}$ for a ...
- 5,470
5
votes
1
answer
192
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Restriction of cuspidal representations of GL(2) to the Borel subgroup
Let $\mathbb{F}_q$ be a finite field and let $B$ be the standard Borel subgroup of $GL(2, \mathbb{F}_q)$ consisting of upper triangular matrices. Is there an explicit description of the restriction ...
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4
votes
1
answer
294
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Why should we expect this odd behavior of negative binomial distributions?
In independent Bernoulli trials with probability $p$ of success on each trial, let $X$ be the number of failures before the $n$th success. Then
$$
\Pr(X=x) = \binom{-n}{\phantom{+}x} (-q)^x p^n \text{ ...
