All Questions
152,891
questions
39
votes
4
answers
3k
views
The symmetric group theory of natural numbers
Sometimes it is not easy to formulate a correct question. Here is a better version of this question (I still do not know if it is optimal, but it is better than the previous one).
We say that a set $...
4
votes
0
answers
1k
views
Generalizing Autonne-Takagi factorization
Autonne-Takagi factorization (Léon Autonne (1915) and Teiji Takagi (1925)) says that:
A complex symmetric matrix can be 'diagonalized' using a unitary matrix: If $A$ is a rank-$n$ complex symmetric ...
5
votes
1
answer
339
views
Diagonalizable pro-algebraic group in Kottwitz's 1985 Compositio paper
In Kottwitz's 1985 Compositio paper,
Isocrystals with additional structure, first page, paragraph 4:
Let $\mathbb{D}$ be the diagonalizable pro-algebraic group over $\mathbb{Q}_p$ with character ...
0
votes
0
answers
119
views
Maximizing the sum of hook lengths
Given positive integers $a\geq b$, and $n\in\{1,2,\dots,ab\}$ I am looking for a partition of $n$ into at most $b$ parts of size at most $a$ which maximizes the sum of the hook lengths in the ...
5
votes
0
answers
181
views
Which rings are the endomorphisms ring of some abelian groups?
Which rings are (isomorphic to) the endomorphisms ring of some abelian group? Is there any necessary and sufficient condition?
3
votes
1
answer
154
views
Is the level set of a product of affine linear functions comprised of convex curves?
Internet searches haven't helped. Can you?
Let $\, f = \prod_{i=1}^n (a_i x + b_i y + c_i).$
Is each component of $\, f^{-1}(1)$ a convex curve?
I expect so, and can prove it for $n=2,$ but I'm ...
9
votes
2
answers
497
views
Does Easton forcing preserve measurable cardinals?
The question is in the title. For Easton's theorem see Wikipedia. Loosely speaking we can use forcing to manipulate the powerset function on regular cardinals as much as we like given we satisfy the ...
6
votes
1
answer
414
views
Transitive homeomorphisms of Erdős spaces
A surjective homeomorphism $h:X\to X$ is minimal if $$\overline{\{h^n(x):n\in \mathbb N\}}=X$$ for every $x\in X$. In other words, the orbit of each point is dense.
Does either of the Erdös spaces $\...
7
votes
0
answers
270
views
What are $(m,n)$-pseudoplanes?
An incidence geometry is a set $P$ (the "points"), a set $L$ (the "lines"), and a relation $I\subseteq P\times L$ ("incidence"). Equivalently, a bipartite graph with the halves of the partition ...
9
votes
3
answers
448
views
Minimal combinatorial data needed to define a polytope [duplicate]
Suppose I give a list of vertices $(v_1, v_2, ..., v_n)$, and a list of "adjacencies", i.e. pairs of vertices $(v_i,v_j)$. Does it exists a unique polytope that has this vertices and realises the ...
11
votes
0
answers
719
views
Fast computation of matrix product $AXA^T$ with fixed $A$?
Suppose we have two $n$-by-$n$ matrices $X$ and $A$, where $A$ is known and $X$ may change in different invocations, and we want to compute $AXA^T$. Is there an algorithm that beats the naive one of ...
8
votes
1
answer
385
views
projective plane cubics with exactly 9 real points
It is not hard to construct such curves explicitly, e.g. my favourite example is a curve $U$ singular at $(3:4:5i)$ and also passing through $(1:0:0)$, $(0:1:0)$, $(0:0:1)$, $(1:1:0)$, $(1:0:1)$, $(0:...
3
votes
1
answer
251
views
Strong polynomial algorithm for linear programming
What is the current state of finding a strong polynomial algorithm for linear programming? Is there any reference?
1
vote
2
answers
232
views
Is there a standard name for this type of multidigraph?
A digraph (direct graph) consists of a set $V$ of vertices and a set $E$ of directed edges $v\to v'$. A multidigraph is a digraph in which $E$ is a multiset, so edges may appear multiple times in $E$, ...
2
votes
0
answers
288
views
Local coefficients system
Let $G$ be a compact group. Then there exists a universal principal $G$-bundle $G\rightarrow E_{G}\rightarrow B_{G}$. Let $X$ be a paracompact $G$-space and suppose that $X\rightarrow X_{G}\rightarrow ...
3
votes
3
answers
1k
views
A few general questions about pre-sheaves and sheaves
I am no specialist in sheaf theory, so I would be glad to get some help regarding the following:
I have a pre-sheaf $F$ of abelian groups above a topological space $X$, and I have found an open ...
5
votes
0
answers
195
views
When a coherent sheaf on DM stack is locally free?
A coherent sheaf $\mathcal{F}$ on a variety $X$ is locally free if every fiber $\mathcal{F}|_x$ is of the same dimension. My question is if such theorem is also true on a Deligne-Mumford stack, or ...
1
vote
1
answer
675
views
Restriction of vector bundles
I am trying to compute the Chern classes of the restriction of a rank two vector bundle on $\mathbb{P}^3$, denoted by $E$, with fixed Chern classes, $c_1(E) = c_1$ and $c_2(E) = c_2$, to a hyperplane $...
6
votes
1
answer
598
views
The group of isometries of Shahshahani metric
Edit: 28 January 2023 I just realized that this metric is frequently used in this paper
https://hal.science/hal-01382281/document
Let $$M=\{(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n\mid x_i>0,\;i=1,2,\...
3
votes
0
answers
151
views
exact sequence of fundamental groups associated to "almost" smooth families of curves
Suppose I have a proper, flat family of curves $X \to S$ that has a section. Fix a basepoint $s \in S$ and let $X_s$ denote the corresponding fiber. Let $\mathbb{L}$ be a set of primes which does not ...
3
votes
0
answers
116
views
Obstruction to the existence of a complex-valued determinant function
Let $\mathcal A$ be a Type $II_1$ von Neumann algebra, equipped with a finite trace $\tau: \mathcal A \to \mathbb C$. Further, denote by $\mathcal A^\times \subset \mathcal A$ the group of invertible ...
3
votes
3
answers
335
views
Intersection of a line and a conic in a surface
Does there exists a smooth projective surface $X$ which contains a projective line $L$ and a smooth conic $C$ such that $L\cap C=$empty?
4
votes
2
answers
277
views
Quasisimple group with cyclic Sylow p-subgroup and weakly real p-elements?
Does there exist a quasisimple group $G$ and an odd prime $p$ such that $G$ has cyclic Sylow $p$-subgroups and a weakly real element of $p$-power order?
From Strongly real elements of odd order in ...
8
votes
1
answer
209
views
Categorification of monotone maps via tilting modules?
It is well known that for the algebra of $n \times n$-upper triangular matrices over a field the number of tilting modules is equal to the Catalan number $C_n$. This is just the (hereditary) Nakayama ...
1
vote
1
answer
890
views
Hironaka's theorem and smooth completion
Hironaka's theorem states that for any algebraic variety (analytic space) $X$ there exists a smooth variety (complex manifold) $X'$ and a morphism $f : X' \rightarrow X$ such that $f$ restricted to $X ...
4
votes
0
answers
133
views
Ambidextrous Yoneda structures, admissibles and adjunctions
Let $\cal K$ be a 2-category, supporting two Yoneda structures, induced by a pseudoadjunction $P^\sharp\dashv P : {\cal K}^\text{coop}\to \cal K$; this means that there is such an adjunction, or in ...
7
votes
2
answers
261
views
Functors in Isbell duality exchange $f^*a$ and $f_*a$
As you maybe remember, Isbell duality is an adjunction
$$\mathcal O : [A°,Set] \leftrightarrows [A,Set]° : {\cal S}pec$$
as defined here; since every functor $f : A\to B$ defines both
a functor $f^...
1
vote
1
answer
68
views
Example of a certain partitioned manifold
I'm looking for an example of a non-compact spin manifold $M$ and a compact subset $K\subseteq M$ such that $\partial K$ is a compact hypersurface in $M$ with $\hat{A}(\partial K)\neq 0$.
(At first I ...
1
vote
0
answers
74
views
Has this type of pathwise (S)DE been studied before?
I thought of a possible type of pathwise-defined nonautonomous/stochastic differential equation, and I was wondering if it has been studied before.
Let $(G,\ast)$ be an abelian $C^1$ Lie group....
0
votes
2
answers
197
views
Prime factors of a sequence of integers which differ by consecutive prime differences
[Edited following Gerhard's answer. I had forgotten to state a crucial assumption; my apologies for the confusion. I have reworded slightly to try to make things clearer.]
Let $p$ be the $k$-th odd ...
4
votes
0
answers
379
views
Can we efficiently factor $n$ given that $n=pq$ where $p,q$ are primes satisfying $p=a^2+b^2, q=2ab+1$ for some $a,b$
Suppose we're given a particular number $n \in \mathbb{N}$.
We're also given that $n=pq$ where $p,q$ are unknown primes satisfying
$$
p=a^2+b^2
$$ and
$$
q=2ab+1
$$
for some $a,b$.
Is there an ...
2
votes
0
answers
40
views
Asymptotic flag in terms of geometry of the stratum of abelian differentials?
Let $C$ be a closed Riemann surface of genus $g\geq 1$. Fix a holomorphic 1-form on $C$; it endows $C$ with a flat structure (i.e. a metric of trivial holonomy which has conical singularities at a ...
5
votes
0
answers
193
views
Semisimplicity of the p-adic étale Tate module over $F_p(t)$
Let $k$ be a finitely generated field of positive characteristic p.
Let $A$ be an abelian variety over $k$ and write $T_p(A)$ for the $p$-adic étale Tate module of $A$. Is it known if the natural ...
10
votes
0
answers
402
views
Using the universal property of K-theory
A paper of Blumberg, Gepner and Tabuada gives a universal property of K-theory: from their abstract "connective algebraic K-theory is the universal
additive invariant, i.e., the universal functor with ...
1
vote
0
answers
47
views
What separates a cyclic polytope from a projective polytope?
I am having trouble understanding the difference between a cyclic polytope and a convex projective polytope as positive geometries.
The link https://arxiv.org/pdf/1703.04541.pdf is the source of ...
4
votes
0
answers
102
views
Compact subspace of sober space
We know from lemma 1.2.5 in part C of Sketches of an Elephant (by Johnstone) that both open and closed subspaces of a sober space are again sober. This raises the following question.
Question: Is a ...
5
votes
1
answer
310
views
Analytic families of compact self-adjoint operators: eigenvalue extension
Suppose that $A(t), t \in \mathbb{R}$, is an analytic family of compact self-adjoint operators on a Hilbert space. The Kato-Rellich theorem says that every non-zero eigenvalue of $A(t)$ splits into ...
6
votes
1
answer
731
views
Are there non-projective, but algebraic, hyperkahler varieties?
Let $k$ be an algebraically closed field of characteristic zero. I am not sure what the right definition of a hyperkahler variety over $k$ is, but I think the following might be close enough.
...
2
votes
1
answer
461
views
On the paradox that $n$-coskeletal simplicial sets model all homotopy types
Please help me resolve the following paradox:
False claim: Let $X$ be an $n$-coskeletal, $n$-connected simplicial set. Then $X$ is weakly contractible.
Actually, I suppose the claim is ...
7
votes
1
answer
217
views
Five-dimensional manifolds fibering over a fixed hyperbolic surface
I am aware of the classical work by Smale and Barden computing the diffeomorphism type of smooth simply connected 5-manifolds in D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965),...
3
votes
1
answer
223
views
Isogeny of Drinfeld module
Let $\phi$ be a Drinfeld module over a function field $K$. It is known that if the rank of $\phi$ is 1, then it is isomorphic over $K$ to one defined over $O_K$. Is it true that if the rank of $\phi$ ...
0
votes
0
answers
392
views
What can be said about the level set of the real part of an analytic function?
I am cross-listing this question from math.stackexchange since I did not find a satisfactory answer there. This is my first time posting a question on MO, so if this is not the appropriate community I ...
0
votes
1
answer
123
views
Sets of integers represented by products of $q(q^n-1)$
Consider $X=\{q(q^n-1)|q \ \text{is some power of a prime number}, n\in \Bbb N^*\}$, $S=\{ s \in \Bbb N| s=\prod_i s_i, s_i \in X\} $, I am interested in which integers are in $S$. For example, $2, 6 \...
1
vote
1
answer
674
views
finite dimensional C*-algebras
Let $A$ be a C*-algebra. Suppose that every cyclic representation of $A$ is finite dimensional.
Q. Is $A$ finite dimensional?
3
votes
1
answer
261
views
Reconstructing the Green's function of an initial-value problem of partial differential equation
Consider a partial differential equation that is of the following form:
\begin{equation}
(-\partial_x^2+g(x))f(x, t)=i\partial_tf(x, t)
\end{equation}
where $g(x)$ is a real function. Suppose that $f(...
8
votes
2
answers
866
views
What is the polynomial functor for the Bag monad
I may be wrong, but we should be able to write the Bag monad in a polynomial form. The bag monad, is exectly the multiset monad whose category of algebras are the commutative monoids. Another name ...
3
votes
0
answers
86
views
Archimedean components of base changed automorphic representations
Let $\pi$ be an automorphic (cuspidal) representation of ${\rm GL}(n)_{/\Bbb Q}$.
Let $K\supset\Bbb Q$ a numberfield. The existence of the base change lift $\pi_K$ is known in a number of cases.
...
6
votes
1
answer
285
views
Why is the number of Hamiltonian Cycles of n-octahedron equivalent to the number of Perfect Matching in specific family of Graphs?
In OEIS A003436, it is written that the number of inequivalent labeled Hamilton Cycles of an n-dimesnional Octahedron is the same as the number of Perfect Matchings in a the complement of the Cycle ...
4
votes
1
answer
127
views
Fréchet–Urysohn subspaces in $[0,1]^{[0,1]}$
A topological space $X$ is called a Fréchet–Urysohn space if for each subset $A\subseteq X$ and for each point in its closure, $x\in\overline{A}$, there is a sequence (not just a net, but a sequence) $...
8
votes
2
answers
1k
views
The Ricci Form and the First Chern Class
Let $(M, \omega)$ denote a compact Kähler manifold. Since $d\omega =0$, $\omega$ represents a cohomology class in $H^2(M, \mathbb{R})$. Let $\rho$ denote the Ricci form of $M$, in local coordinates, ...