All Questions
15,614 questions
0
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119
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Reference request: hyperfinite cross product
Given a countable essentially free ergodic non-singular group action $G \curvearrowright (X, \mu)$ on a standard measure space, suppose $\mu$ is a non-atomic probability measure and $\alpha: G \...
2
votes
0
answers
30
views
An algorithm to decompose a directly indecomposable permutation group into a wreath product
I am considering the following two binary operations on permutation groups:
the direct product, and
the wreath product.
It turns out that there is an efficient algorithm to factor a given ...
- 5,822
12
votes
1
answer
816
views
Do linear groups over a commutative ring satisfy the Tits alternative?
A group $G$ is said to satisfy the Tits alternative if any finitely generated subgroup of $G$ is either virtually solvable or contains a nonabelian free subgroup. Tits proved this for linear groups ...
- 863
0
votes
1
answer
70
views
Asymptotic expansion inverse discrete Fourier transform
Let $\ell^1(\mathbb{Z})$ be the space of biinfinite sequences $f = (f(n))_{n \in \mathbb{Z}} \subset \mathbb{C}$ such that it is absolutely summable. The discrete Fourier transform or Fourier series ...
4
votes
0
answers
123
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Minimal model for $A_\infty$-categories
Is there a reference for existence and construction of the minimal model of an $A_\infty$-category? Most references I found ultimately refer to Lefèvre-Hasegawa's thesis but there doesn't seem to be a ...
- 489
4
votes
0
answers
90
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Reference for fact about flags of vexillary permutations
Vexillary permutations are an important family of permutations in Schubert calculus. There are several definitions, for example that they avoid the pattern 2143.
Recall the Lehmer code of a ...
- 1,989
1
vote
0
answers
99
views
Shortest loop through vertices of a convex polytope
Let $P$ be a convex polytope in Euclidean space $\mathbf{R}^3$ and $\Gamma$ be a closed curve which passes through all vertices of $P$. How small can the length $L$ of $\Gamma$ be? More specifically, ...
- 7,149
8
votes
4
answers
1k
views
Counting with trees
Let $\mathcal{U}_n$ denote the set of unrooted unlabelled trees with $n$ edges. For $T\in\mathcal{U}_n$, let $1^{u_1}2^{u_2}\cdots n^{u_n}$ be its degree distribution, that is, $u_i=\#$ of vertices ...
- 43.2k
1
vote
0
answers
60
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Reference for Gröbner-Shirshov algorithm in free restricted Lie algebras
I am searching for a reference on the Gröbner-Shirshov algorithm specifically for free restricted Lie algebras. I have already consulted the textbook by Bokut et al (Gröbner–Shirshov Bases Normal ...
- 1,013
5
votes
0
answers
78
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Is there a generalization of the Diameter Sphere Theorem to orbifolds?
The Diameter Sphere Theorem of Grove and Shiohama asserts that if $M$ is a compact Riemannian manifold with sectional curvature bounded from bellow by 1 and diameter greater than $\pi/2$, then $M$ is ...
- 51
8
votes
1
answer
321
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Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:
Q. Does every cancellative duo semigroup embed into a group?
A (multiplicatively ...
- 10.5k
4
votes
0
answers
70
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Permutation matrix in terms of an $\mathfrak{su}(r)$-basis (generalised Gell-Mann matrices)
Let $V \cong \mathbb{C}^r$ be the defining representation of $\mathfrak{su}(r)$. Then the permutation on $V \otimes V$ can be expressed as
$$ P = \frac{1}{r} \, 1 \otimes 1 + \frac{1}{2} \sum_{a=1}^{r^...
- 1,996
1
vote
1
answer
106
views
Iterated optimal transport
Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
- 13
1
vote
0
answers
89
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The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results.
Let $A$ be a finitely generated abelian group,...
- 823
9
votes
1
answer
223
views
$p$-adic analytic pro-$p$ group satisfies a pro-$p$ identity?
Let $p$ be a prime. Let $w$ be an element of a free pro-$p$ group $F_r$ of finite rank $r\geq 2$. Then we say that a pro-$p$ group $G$ satisfies the pro-$p$ identity
$w$ if for every homomorphism $ f:...
- 667
8
votes
2
answers
596
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If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 10.5k
3
votes
0
answers
152
views
What is known about the word problem on free algebraic models?
Consider the fragment of first-order logic with equality and universal quantification as the only logical symbols; we call this logic the logic of universal algebra. I am interested in languages $\...
16
votes
2
answers
1k
views
Identifying short introductory book on non-commutative geometry I read c.2008
I’m trying to identify a book I remember reading and enjoying in grad school around 2008–9; I’ve forgotten its author and title, and haven’t been able to find a book matching my memories in half an ...
- 21.3k
7
votes
2
answers
487
views
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
- 10.5k
3
votes
1
answer
326
views
Holomorphic homotopy conjecture
Let $X$ be a smooth projective variety over the complex numbers, and let $\text{Coh}(X)$ be the category of coherent sheaves on $X$. Consider the dg-category $\text{Perf}(X)$ of perfect complexes on $...
- 31
0
votes
1
answer
70
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N-soliton, The Lax operator and the transmission coefficient
I'm interested in the soliton stability result given in HERBERT KOCH and DANIEL TATARU's paper
"MULTISOLITONS FOR THE CUBIC NLS IN 1-D AND THEIR
STABILITY", published in IHES. However, I ...
- 695
7
votes
0
answers
265
views
Herbrand's consistency proof
Jacques Herbrand's thesis "Investigations in proof theory: The properties of true propositions" (or in the original French "Recherches sur la théorie de la démonstration", with the ...
- 161
1
vote
0
answers
95
views
Reference for structure of subhomogeneous C*-algebras?
Recall that a C*-algebra is subhomogeneous if there is some $n$ such that the dimension of $H$ is at most $n$ for every irreducible representation $\pi \colon A \to \mathcal{B}(H)$ on some Hilbert ...
- 733
0
votes
0
answers
78
views
Definition of Moore-Penrose inverse for unbounded self-adjoint operators?
I know there is a concept of Moore-Penrose or pseudoinverse of a matrix. I would like to know if one can define it for densely defined unbounded self-adjoint operators on Hilbert spaces as well. ...
1
vote
1
answer
84
views
$H^2$-elliptic regularity (up to the boundary) for operators with lower order terms for Lipschitz/convex domains
Let $\Omega$ be a bounded domain which is Lipschitz or convex. Given an elliptic operator of the form
$$\langle Au, v \rangle = a_{ij}u_{x_i}v_{x_j} + b_i u_{x_i}v + cuv$$
are there any elliptic ...
- 93
3
votes
0
answers
389
views
Status of motives in higher category theory: motives and algebraic cycles through a higher categorical perspective
A while ago this interesting question was asked Derived Algebraic Geometry and Chow Rings/Chow Motives.
Primary question:
Have there been any recent developments/advances on the above question? If not,...
1
vote
1
answer
222
views
Sum over three squares
Let $x$ be a sufficiently large number. Is there an explicit or asymptotic formula for the following sum
$$\sum_{\substack{n\leq x\\ n=a^2+b^2+c^2}} 1.$$ Any reference would be helpful.
- 1,247
2
votes
0
answers
82
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Beyond the Bring Radical: What is known about "generating radicals" for roots of polynomials of a given degree?
Famously, there is no general solution by radicals to find roots of polynomials (real, say) with degree $d\geq 5$. Somewhat less famously, there is a general solution[?] in degree $5$ using the so-...
5
votes
0
answers
156
views
Reference Request: Completeness of the space of all Whittaker models(a lemma in JPSS1981)
$\DeclareMathOperator\GL{GL}$There is a lemma in the proofs of local converse theorem stated as
Suppose $F$ is a non-archimedean local field, $\psi$ is a non-trivial addtive character on $F$. $N_n$ ...
- 53
0
votes
0
answers
101
views
Identities for Prime Coefficients of Certain Cusp Forms
While working with Fourier expansions of cusp forms of congruence subgroups of the modular group, I observed the following patterns in their prime coefficients.
Let $a(n)$ be the Fourier coefficients ...
3
votes
0
answers
147
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Solutions of a quadratic Diophantine equation over algebraic integers
The problem is not very exactly Diophantine in the classical way. I am trying to find some algebraic integer solutions in a number ring to a quadratic equation over the same ring.
Precisely, let $\...
- 287
23
votes
4
answers
2k
views
Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 43.2k
2
votes
2
answers
362
views
Size of $\zeta'(s)$ at its zeros
How large can the derivative of the Riemann zeta function be at its zeros?
More specifically, let $\rho$ be a zero of the zeta function with $\Im(\rho)\in (0,T]$. What can we say about $|\zeta'(\rho)|...
user536681
3
votes
0
answers
61
views
Is this bipartite equivalent of 1-walk-regular graphs known?
A graph $G$ is 1-walk-regular if
for each vertex $v$ the number of closed walks of length $\ell$ starting (and ending) at $v$ depends only on $\ell$ but not on $v$.
for each edge $vw$ the number of ...
- 13.6k
1
vote
0
answers
53
views
Moduli space of stable bundles with fixed Chern class is bounded
This is an excerpt on a set of lecture notes I'm going over:
Classically, a lot of interest in stable vector bundles is due to the fact that stability allows the study of moduli of vector bundles via ...
- 73
3
votes
1
answer
109
views
Literature request: Covariance operators for Gaussian measures
I am looking to answer the question:
If $\mathcal{B}$ is a separable Banach space and $R: \mathcal{B}^*\to\mathcal{B}$ is a symmetric and positive operator, then $\phi: \mathcal{B}^*\to\mathbb{R}, \...
- 171
1
vote
1
answer
277
views
Moduli space of complex and anti-complex tori?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SL{SL}$By Will Sawin's answer to Moduli Spaces of Higher Dimensional Complex Tori the moduli space of complex $d$-tori is $X ...
- 261
1
vote
0
answers
52
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How large can the normalizer of $\mathrm{Ad}(G)$ in $\mathrm{GL}(\mathfrak{g})$ be?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ad{Ad}$Let $G$ be a real Lie group with Lie algebra $\mathfrak g$ (say reductive/semisimple if it makes the question easier).
I am interested in ...
- 1,942
11
votes
2
answers
515
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Knots having the same Alexander module which are not S-equivalent
As is discussed in this answer, S-equivalence class can be regarded as the Alexander module plus Blachfield pairing, an isomorphism of some duals of Alexander modules.
There are examples of knots ...
- 548
9
votes
2
answers
491
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Reference Request for global Hölder continuity of solutions to elliptic PDEs
This is question that was also posted on MathStackExchange Link, where it was suggested I post this question on MathOverflow. Please note that the answer given in Link does not help as I do not have ...
- 133
-1
votes
1
answer
61
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Asking for some references on correlations of joint optimization problems
Here are two problems that I am trying to understand, and it would be nice if someone could provide references on whether there is some structure theorem for these problems that have been studied in ...
1
vote
0
answers
52
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High order parabolic PDEs on manifolds: Reference request
I recently became interested in parabolic PDEs of order 4 on surfaces. However, I have a very little background in parabolic PDEs. I discovered Lunardi's book (Analytic semigroups and optimal ...
- 363
0
votes
0
answers
67
views
Projection of a gaussian random vector onto a convex body
Let $K \subset \mathbb{R}^n$ denote a convex body. Let $\Pi_K$ denote the projection onto $K$,
$$
\Pi_K(y) = \mathrm{arg\,min}_{x \in K} \|y - x\|,
$$
where $\|\cdot\|$ denotes the usual Euclidean ...
- 460
1
vote
0
answers
86
views
Gamma convergence via density argument: Looking for references
I am looking for a reference or result dealing with Gamma via density argument.
Let me elaborate more my wish. I am actually trying to establish the Gamma convergence (precisely only the liminf) of a ...
- 1,101
16
votes
2
answers
1k
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CH in non-set theoretic foundations
I asked this question on stack exchange and got little attention, barring a nice example I intend to look into. The original post can be found here: https://math.stackexchange.com/q/4941233/1053681
I ...
- 261
3
votes
1
answer
159
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Reference request: generalized Jacobian variety for higher dimensional variety
Let $X\subset \mathbf{P}^n$ be a hypersurface such that the singular locus of $X$ consists of a single ordinary double point. I'm trying to find a reference to the "generalized" intermediate ...
- 389
2
votes
2
answers
242
views
A necessary and sufficient condition for three diagonals of a hexagon to be concurrent
When talking about the condition for the three diagonals of a hexagon to be concurrent, we will think of Brianchon's theorem. Using software, I discovered a necessary and sufficient trigonometric ...
- 1,776
7
votes
2
answers
283
views
Bounded geometric morphisms, origin and motivation for the terminology
Bounded geometric morphisms serve as a generalization of Grothendieck topoi; $T$ being a Grothendieck topos iff global section $T \rightarrow Set$ is bounded. I managed to only track this down to the ...
- 1,347
0
votes
0
answers
54
views
Number of indecomposable modules over representation-finite hereditary algebras
Let $A$ be a finite dimensional $K$-algebra over a field $K$ that is hereditary and of finite representation type.
It is well known that they are classified by Dynkin diagrams.
For algebraically ...
- 26.5k
3
votes
1
answer
135
views
Concentration of sample median for iid Gaussians
Let $X_1, \dots, X_n$ be iid according to $\mathcal{N}(0, 1)$, and let $M_n$ be the median of the $X_1, \dots, X_n$. I recall reading a concentration inequality for $M_n$ that was (roughly) as follows:...
- 75