All Questions
15,616 questions
2
votes
0
answers
167
views
Centralizer of PSL in PGL and of SL in GL: reference request
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\PSL{PSL}$Consider the general linear group $\GL(n,q)$ over a finite field with $q$ elements and ...
- 44.4k
1
vote
0
answers
85
views
Unitary representations of the symmetric group over finite fields
I am interested in understanding the unitary representations of the symmetric group over $\mathbb{F}_{q^2}$. In general, some comments here are relevant
Unitary representations of finite groups over ...
- 571
9
votes
1
answer
1k
views
Is the number of varieties of groups still unknown?
A variety of groups is a class of groups satisfying a specified set of equations. Equivalently, it is a class of groups that is closed under homomorphic images, subgroups, and direct products. A ...
- 21.3k
2
votes
1
answer
431
views
Shadows of partitions of lcm
$\DeclareMathOperator\lcm{lcm}$Fix an integer $n\geq1$. Denote the least common multiple $L_n=\lcm(1,2,\dots,n)$.
QUESTION. Is the following true? For each integer partition $\lambda=(\lambda_1,\...
- 23.7k
4
votes
0
answers
97
views
Characterization of Vilenkin group
It is shown in [1, Section 1] by C.W. Onneweer that every infinite compact, metrizable, zero-dimensional commutative group is a Vilenkin group. My question is does this implication also hold if we ...
- 63.9k
2
votes
1
answer
276
views
Estimating a sum over set partitions
Let $[n]:=\{1,\dots,n\}$. Fix a set partition $\rho$ of $[n]$, with an abuse of notation we shall use $\rho\vdash [n]$.
I would like to estimate the following alternating sum.
QUESTION. Is this true?
...
1
vote
0
answers
69
views
Unique continuation of Laplace eigenforms
Let $M$ be a compact Riemannian manifold and $\Delta = d\delta + \delta d$ denote the (positive definite) Hodge Laplacian acting on differential forms. Call a smooth differential form $\omega$ a ...
- 1,407
7
votes
2
answers
1k
views
Selberg class definition and Riemann hypothesis
Looking at the Selberg class definition on Wikipedia, under "Comment on definition", there is this paragraph:
"The condition that the real part of $\mu_i$ be non-negative is because ...
1
vote
0
answers
101
views
Locating volume 2 of certain conference proceedings in analytic number theory
Does anyone know where one might locate "Analytic Number Theory: Proceedings of a Conference in Honor of Heini Halberstam, Volume 2"? There exists Volume 1 here: https://link.springer.com/...
- 1,890
1
vote
0
answers
74
views
Asymptotically small submatrices of random matrices
Consider an ensemble of $N \times N$ random Hermitian matrices distributed according to some unitarily invariant measure
$$P(M) \mathrm{d}M = \frac{1}{Z_{N}} e^{-\mathrm{tr}[ Q(M)]}\mathrm{d}M,$$
for ...
- 131
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?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 105k
49
votes
3
answers
3k
views
What happens if you strip everything but the “between” relation in metric spaces
Given a metric space $(X,d)$ and three points $x,y,z$ in $X$, say that $y$ is between $x$ and $z$ if $d(x,z) = d(x,y) + d(y,z)$, and write $[x,z]$ for the set of points between $x$ and $z$.
Obviously,...
- 32.1k
9
votes
1
answer
223
views
$p$-adic analytic pro-$p$ group satisfies a pro-$p$ identity?
Let $p$ be a prime. Let $w$ be an element of a free pro-$p$ group $F_r$ of finite rank $r\geq 2$. Then we say that a pro-$p$ group $G$ satisfies the pro-$p$ identity
$w$ if for every homomorphism $ f:...
2
votes
4
answers
212
views
Efficient algorithm for graph problem
Let $D=(V,E)$ be a directed graph, $S,T\subset V$ and $f:V\rightarrow \{1,\ldots, k\}$ a positive, bounded weight-function and $l\in \mathbb{N}$, find a path $v_1,\ldots, v_l\in V$ with $v_1\in S$ and ...
- 63.9k
14
votes
1
answer
678
views
History of well founded relations
I have changed the title in the hope of attracting the attention of someone who knows about the history of set theory as well as intuitionistic logic:
Who was the first to state the definition of well-...
- 8,481
2
votes
0
answers
37
views
Theta series of well-rounded lattices
I've started looking into well-rounded Euclidean lattices and I was interested in learning whether their theta series have any interesting properties, but haven't found much in terms of bibliography ...
- 223
55
votes
2
answers
3k
views
Is it known? A sum over lattice parallelograms of area one is equal to $\pi$
I recently discovered a formula, my proof is really a high school proof in three lines.
$$4\sum_{x, \, y \, \in \, \mathbb Z_{\geq 0}^2, \, \det(x \ \ y) = 1} \frac{1}{\lVert x\rVert^2\cdot\lVert y\...
- 5,053
3
votes
1
answer
286
views
Derived Koszul complex
Let $X$ be a projective variety over $\mathbb{C}$ and $V$ be a vector bundle over $X$. Let $\pi: V\to X$ be the natural projection.
Let $i: X\to V$ be the zero section map. Let $V^\vee$ be the dual ...
- 2,806
3
votes
1
answer
231
views
Are principal parabolic group scheme bundles Zariski locally trivial?
Let $P$ denote a parabolic subgroup scheme of $\operatorname{Sp}(2n;F)$, where $F$ is a field (I am interested in $K=\mathbb{Q}_p$ so possibly okay to assume local with characteristic $0$ if it makes ...
- 777
16
votes
2
answers
590
views
Can you perturb an inscribed polytope so all its edges grow?
Consider the family of convex simplicial polytopes with vertices in the unit sphere of $\mathbb{R}^n$ which have the origin as an interior point.
My question is the following:
Let $P, P'$ be two non-...
- 13.6k
3
votes
0
answers
287
views
What did Mirimanoff say about Intuitionism?
Dmitry Mirimanoff, "L'intuitionisme", Alma Mater n° 6, Geneva, 1945.
Most of Mirimanoff's work was in number theory, but he wrote three papers about set theory that were way ahead of their ...
- 8,481
7
votes
0
answers
160
views
What happens if we add an initial object to a Lawvere theory?
Motivation
There is a theory of smooth spaces (for example, diffeological spaces) as certain sheaves on the site $\mathrm{Cart}$, which is the "category of cartesian spaces" whose objects ...
- 355
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 ...
- 151
4
votes
1
answer
203
views
Stationary phase formula for a complex valued phase
I'd be interested in computing an asymptotic expansion when $h \rightarrow 0$, of an integral of the form
$$
I_h = \int_{\mathbb{R}}{e^{\frac{i}{h}\varphi(x)}dx}
$$
where $\varphi : \mathbb{R} \...
- 6,367
6
votes
1
answer
168
views
Laplacian is surjective from $\mathcal{C}^{\infty}(B)$ to $\mathcal{C}^{\infty}(B)$
Let $B$ denote the open unit ball in $\mathbb{R}^n$. Let $\mathcal{C}^{\infty}(B)$ represent the space of smooth functions on $B$.
Is the Laplacian operator $\Delta$ surjective as a map from $\mathcal{...
- 16.4k
7
votes
2
answers
284
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 ...
- 35.5k
3
votes
0
answers
148
views
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
5
votes
1
answer
250
views
About generalized Springer theory for spin groups
I am interested in the detailed computation of the generalized Springer theory for spin groups (type B or D). In the last sentence in Section 14 of Lusztig's Intersection cohomology complex on a ...
- 12.9k
0
votes
1
answer
119
views
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 \...
- 17k
1
vote
1
answer
118
views
Reference request for the isomorphism $H^1(G_{K_v},E)[n]\cong (E(K_v)/nE(K_v))^*$ in the context of Tate-duality
Let $E/K$ be an elliptic curve over a number field $K$. Let $M_K$ be the set of all places of $K$. Let $K_v$ be a completion of $K$ at $v$.
I'm searching for a reference for the statement of the ...
- 8,945
8
votes
2
answers
896
views
Can you do geometry with persistent homology?
Setup
In practice, persistent homology of data $X$ is often used to infer the homology of the underlying (Riemannian) manifold $M$ that the data is sampled from.
However most filtrations (Vietoris, ...
- 363
7
votes
1
answer
394
views
Inverse limit in the category of $C^{\ast}$-algebras or operator spaces
Does the inverse limits (projective limits) exist in the category of $C^{\ast}$-algebras or operator spaces?
I tried to search but could not find a proper reference. Any reference or comments about ...
CommunityBot
- 1
1
vote
1
answer
72
views
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 ...
- 1,826
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
4
votes
0
answers
70
views
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
175
views
Original text about Riesel numbers?
Riesel numbers appeared in the answer to my question
Solving $2^{x+1} m -1=p^y$ for prime $p$ and natural $x,y$.
The original reference
Riesel, Hans (1956). "Några stora primtal". Elementa. ...
- 105k
8
votes
4
answers
1k
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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
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 ...
- 63.9k
9
votes
4
answers
3k
views
Textbook suggestions for rigorous fluid dynamics
I am interested in studying fluid dynamics and am searching for a good introductory textbook. I know just the very basics of fluids on the physics side. For mathematical prerequisites, I have ...
- 35.5k
16
votes
2
answers
887
views
Mumford–Tate 1962 "Algebraic geometry seminar" citation
In FGA 3.V, there is a citation for
Mumford D. and Tate J., Séminaire de géométrie algébrique, Harvard University, Spring term 1962 (à paraître).
This seems to be the same seminar mentioned by ...
- 101
14
votes
2
answers
851
views
Where is Grothendieck's “Résidus et Dualité: Prénotes pour un Séminaire Hartshorne”?
About the book Residues and Duality, from R. Hartshorne, Giacomo Aiello wrote in a footnote to Grothendieck's bibliography by topic:
This is the famous seminar held by Robin Hartshorne at Harvard in ...
- 101
2
votes
1
answer
215
views
Number of binary matroids of rank $r$ on a ground set with $n$ elements
How many simple binary matroids are there, up to isomorphism, of rank $r$ on an $n$-element ground set, where $r \le n < 2^r$? Write this number as $a_r(n)$. Is there somewhere where I can get this ...
- 331
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
4
votes
0
answers
124
views
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
91
views
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
60
views
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
4
votes
1
answer
194
views
When is one dynamical system an approximation of another?
I've been thinking about the question of when a discrete time dynamical system $f : X \to X$ (or possibly other objects) can be said to approximately model another dynamical system. So far I've mostly ...
- 4,459
8
votes
1
answer
321
views
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
5
votes
0
answers
78
views
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
7
votes
2
answers
488
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