All Questions
46,955
questions with no upvoted or accepted answers
4
votes
0
answers
61
views
weak factorization systems (co)generated by an arbitrary class of morphisms
Under what assumptions can one prove existence of weak factorization systems (co)generated by a class of morphisms ?
Are there counterexamples ? I am interested both in assumptions on the class of ...
- 493
4
votes
1
answer
241
views
Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?
This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$
are ...
- 1,417
4
votes
0
answers
182
views
Cohomological methods in intersection theory and derived categories
Are there any enumerative questions akin to: “What is the number of planes containing a given line tangent to a given cubic surface in $\mathbb{P}^3$” that we can answer using derived categories? I've ...
- 373
4
votes
0
answers
94
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Duality for finite quotient groups of finitely generated free abelian groups
$\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Hom}{{\rm Hom}}
$ The following lemma is certainly known.
Lemma (well-known).
Let $B$ be a lattice (that is, a finitely generated ...
- 13.6k
4
votes
0
answers
130
views
Reverse Sobolev inequality for family of holomorphic functions
Denote by $P_m$ the space of polynomials of degree $m$ in a single complex variable $x$. This is a "Reverse Sobolev inequality":
Theorem. Let $U \subset \mathbb{C}^2$ be an open domain and $...
- 847
4
votes
0
answers
398
views
Is there anything special about the Honda formal group?
The "standard" Morava E-theory $E_n$ (at a prime $p$) is typically defined using the so-called "Honda formal group law", the unique FGL $\Gamma_n$ over $\mathbb{F}_{p^n}$ ...
- 1,768
4
votes
0
answers
259
views
Does this "join-like complex" of $K_5$ and $K_3$ embed in $\Bbb R^4$?
Consider the following 2-dimensional CW-complex: its 1-skeleton is $K_8$, which we write as an edge-disjoint union $K_5\cup K_{5,3}\cup K_3$.
Then for any two edges $ab\in E(K_5)$ and $cd\in E(K_3)$ ...
- 12.6k
4
votes
0
answers
131
views
Can a polytopal graph be "centrally symmetric" in more than one way?
Let $P,Q$ be two centrally symmetric convex polytopes, potentially of different dimensions and combinatorial type, but with the same edge-graph $G$.
The central symmetry of $P$ induces an involutory ...
- 12.6k
4
votes
0
answers
105
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Does integration induce a Kan fibration between the mapping spaces of CDGAs and cochain complexes over $\mathbb{Q}$?
For two CDGAs $A$ and $B$ over $\mathbb{Q}$, the mapping space $\text{Map}_{CDGA}(A,B)$ is the simplicial set with $n$-simplices
$$
\Phi : A \to B \otimes \Omega^*(\Delta^n)
$$
and simplices maps ...
- 285
4
votes
0
answers
131
views
Derived subgroup of rational points vs. rational points of derived subgroups
Let $G$ be a connected split reductive group over a field $k$. In general, we have an inclusion
$$
f: [G(k), G(k)] \rightarrow [G,G](k).
$$
If $k$ is not algebraically closed, $f$ is not necessarily ...
- 2,681
4
votes
0
answers
133
views
Finite subgroups of $\mathrm{SL}_2(\mathbb{O})$
What are the finite subgroups (subloops that are groups) of $\mathrm{SL}_2(\mathbb{O})$? In particular, are there any not contained in a $\mathrm{SL}_2(\mathbb{H})$?
- 2,689
4
votes
0
answers
78
views
On Carmichael function and aliquot parts of odd perfect numbers
I've asked nine months ago this question on Mathematics Stack Exchange with identifier 4430381 and same title. There is not answer for this question on Mathematics Stack Exchange, I wondered if this ...
4
votes
0
answers
80
views
Bounding the Betti numbers of Čech complexes in Euclidean space
Let $S$ be a set of $n$ points in $\mathbb{R}^d$, where $d \ge 2$. Then let $C=C(S)$ denote the union of closed balls of radius $1$ centered at points of $S$.
For $0 \le j \le d-1$, how large can the ...
- 7,755
4
votes
0
answers
204
views
Cartan–Remmert reduction of an algebraic variety
Let $V$ be a normal connected algebraic (say, quasi-projective) variety over complex numbers. Assume that underlying complex analytic space $V^\text{an}$ is holomorphically convex, and thus admits the ...
- 1,115
4
votes
0
answers
140
views
Formal and informal proofs: Is there any "bilingual corpus"?
There are extensive libraries of formalized mathematics like those of Lean, Coq or Isabelle/HOL. What I am interested in is documents of formalized mathematics that closely follow certain informal ...
- 1,218
4
votes
0
answers
156
views
Effective way for studying PDEs
I am new to this stack, and thought my question belongs here.
I am a first-year graduate student currently taking my second course on PDEs (basically covering Evans ch. 5 and onwards). I am planning ...
- 141
4
votes
0
answers
83
views
What are known properties of the boundary curves of J-holomorphic curve with boundary
Suppose $\Sigma$ be a punctured Riemann surface with punctured boundary, and $(M, J)$ be a $2n$-manifold with almost complex structure $J$. Let $L$ be a totally real submanifolds of $M$ in smooth ...
- 265
4
votes
1
answer
157
views
Probability of a number being a bound for roots
Consider the polynomial $p(z)=\sum_0^na_iz^k$ where $a_n=1$ and $a_k \sim N(0,1)$, $k=0,1,2,\dotsc,n-1.$
What is the probability that 2 will be a bound of the roots of the polynomial? How can we find ...
- 808
4
votes
0
answers
248
views
Can we have full choice prior to Reinhardt cardinals?
Working in $\sf ZF + Reinhardt \ cardinal$, can we have full choice over all stages $V_{\alpha < \kappa}$ where $\kappa$ is the Reinhardt cardinal, i.e., the critical point of the elementary ...
- 9,276
4
votes
0
answers
125
views
Local integrability of $\log|f(x)|$ in several variables
If $f(z)$ is an analytic function in a complex neighborhood (in $\mathbb C^n$) of a real point $x^0 \in \mathbb R^n \subset \mathbb C^n$, then $\log|f(x)|$ is integrable over some neighborhood $U \...
- 585
4
votes
0
answers
474
views
What is the computational complexity of Arnoldi algorithm for diagonalization?
What is the space and time computational complexity of finding $k$ eigenvalues of an $N\times N$ matrix using the iterative Arnoldi algorithm?
I know that exact diagonalization scales like $O(N^3)$, ...
- 141
4
votes
0
answers
128
views
Can a convex frame hold all circles of radius $1/n$ immobile?
Here is a frame that holds circles of radius $1, \frac{1}{2}, \frac13, ..., \frac17$ immobile.
By "immobile", I mean no circle can move without overlapping other circles or the frame, ...
- 2,673
4
votes
0
answers
175
views
The closest ellipse to a given triangle
Definition: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set.
Question: Given a general triangle T, to ...
- 5,493
4
votes
0
answers
122
views
Is there a projective bundle formula for Deligne cohomology?
Given a projective bundle $\mathbb{P}(E) \to X$ on a complex manifold $X$, is there a projective bundle formula for Deligne cohomology? That is, can Deligne cohomology $H_D^n(\mathbb{P}(E),\mathbb{Z}(...
- 91
4
votes
0
answers
158
views
Definitions of torch ring
Well, I find myself in a tangled web of references, definitions and doubt. How do I get myself into these things? Get ready for a rollercoaster of definitions.
An FGC ring is a commutative ring whose ...
- 1,603
4
votes
0
answers
180
views
Physical intuition for curvature on higher order frame bundles?
$\DeclareMathOperator\SO{SO}$A priori: I apologize if this isn't up to Mathoverflow standards, I've had very little luck getting questions on this subject answered elsewhere.
I'm looking for a physics ...
- 230
4
votes
0
answers
124
views
Weak version of (elliptic analog) Artin's primitive roots conjecture
Let $E/\mathbb{Q}$ be an elliptic curve, and $P\in E(\mathbb{Q})$ be any non-torsion point. Given any $\varepsilon>0,$ how often it is true that $\mathrm{ord}(P \pmod p)>p^{1-\varepsilon},~p~\...
- 521
4
votes
0
answers
161
views
The homotopy type of the space of symplectic structures
While reading the book Introduction to the $h$-Principle by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures ...
- 506
4
votes
0
answers
196
views
Are $\pi$ and $e^{\pi i \sqrt{d}}$ algebraically independent?
Let $-d\in\mathbb{N}$ be square-free. Nesterenko (https://doi.org/10.1070/SM1996v187n09ABEH000158) proved that $e^{\pi\sqrt{-d}}$ and $\pi$ are algebraically independent. Is it known whether $e^{\pi\...
- 141
4
votes
0
answers
142
views
Finitistic dimension conjecture — why artin algebras?
As I understand it, the finitistic dimension conjecture says that if a ring $A$ is finitely generated over a commutative artinian ring $K$, then the finitistic dimension of $A$ is finite.
My question ...
4
votes
0
answers
66
views
harmonic envelope of holomorphy
Let $D$ be a (real) domain in $\mathbb R^n=\mathbb R^n+i\lbrace 0 \rbrace\subset \mathbb C^n$. Then, due to P. Lelong, there exists a maximal (complex) domain $\tilde D\subset\mathbb C^n$, $D=\tilde D\...
- 246
4
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0
answers
178
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Strongly constant divergence vector fields
Inspired by this question on homothety vector field we ask the following question
Let $M$ be a manifold equiped by a volum form $\Omega$. A strongly constant divergence vector field is a vector ...
- 312
4
votes
0
answers
154
views
Upper bound for the first eigenvalue of the Laplacian on surfaces with boundary
Let $\Sigma$ be a compact smooth surface with boundary. Define
$$\Lambda(\Sigma) := \sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ is a smooth Riemannian metric on $\Sigma$} \}$$
...
- 2,085
4
votes
0
answers
298
views
Loop-suspension of degree d map of sphere
Let $f\colon S^n \rightarrow S^n$ be a basepoint preserving map of degree $d \geq 1$. We then get an induced map $\Omega \Sigma(f)\colon \Omega \Sigma S^n \rightarrow \Omega \Sigma S^n$. There is ...
- 41
4
votes
1
answer
413
views
Exponential sum vs. exponential integral via Poisson summation
When we want to estimate an exponential sum
$$
\sum_{M<m\le M'}e(f(m))
\quad\text{with}\quad
1\le M\le M'\le 2M
\quad\text{and}\quad
e(x):=\exp(2\pi ix)
$$
where $e(x):=\exp(2\pi ix)$
and the phase ...
- 343
4
votes
0
answers
147
views
On decomposition of the space of automorphic forms (via central characters)
Let $F$ be a number field and $\mathbb{A}$ be its adele. For simplicity, we assume $G$ is a connected reductive group.
Given a unitary central character $\chi: Z_{G}(F)\backslash Z_{G}(\mathbb{A}) \to ...
- 141
4
votes
0
answers
158
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base change property of Topological Hochschild homology
What is the "base change property" of topological Hochschild homology?
In Proposition 11.10 of Bhatt-Morrow-Scholze's paper "Topological Hochschild homology and integral p-adic Hodge ...
- 91
4
votes
0
answers
91
views
Lifting theorem for modules over a DGA
In the survey paper "Cartan's Constructions, the homology of $K(\pi,n)$'s, and some later developments" by J. C. Moore ( http://www.numdam.org/item/AST_1976__32-33__173_0/ ) he states a ...
4
votes
0
answers
184
views
What are the modularity conjectures for Artin motives?
Classically, singular cohomology is an important tool for studying topological spaces, in particular, complex varieties. In the mid-twentieth century it was realized that there are many analogues of ...
- 509
4
votes
0
answers
289
views
Pohozaev identity for linear equations
For $-\Delta u =0$, the Pohozaev identity on say $B_1$ says
$$ \int_{S_1} |u_T|^2 \,d\sigma = \int_{S_1} |u_N|^2 \,d\sigma + (n-2) \int_{B_1} |\nabla u|^2 \ dx$$
Here $u_T$ are the tangential ...
- 483
4
votes
0
answers
101
views
Interplay beween simplicial and Weyl algebra identities
Recall that the (first) Weyl algebra is the algebra generated by $x,y$ with the relation $xy-yx=1$. It can be realized as the algebra of differential operators on $k[x]$, where one generator acts as ...
- 6,484
4
votes
0
answers
189
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History of tropical mathematics
This is a follow-up to this question about the origin of tropical mathematics.
Are there any articles, websites or books which deal with the history of tropical mathematics?
I have been trying to find ...
- 211
4
votes
0
answers
69
views
Conjugacy of cocharacters from conjugacy of labelled diagrams
Everything to follow is over some fixed algebraically closed field $k$. Although all the definitions make sense regardless of characteristic, the meat of the question is about small positive ...
- 11.4k
4
votes
0
answers
124
views
An uncountable Baire γ-space without an isolated point exists?
An open cover $U$ of a space $X$ is:
• an $\omega$-cover if $X$ does not belong to $U$ and every finite subset of $X$ is contained in a member of $U$.
• a $\gamma$-cover if it is infinite and each $x\...
- 1,071
4
votes
0
answers
362
views
Are smooth irreducible affine varieties set theoretical complete intersection?
I'm trying to prove a lemma, so I wanted to use the next claim (which I do not know if it is true or if there is a counter example): every smooth irreducible affine variety is set theoretical complete ...
4
votes
0
answers
93
views
Is there a 5-cell-600-cell honeycomb?
Is there a convex uniform tiling of hyperbolic 4-space with 5-cells and 600-cells as its facets and a snub 24-cell as its vertex figure?
- 2,689
4
votes
0
answers
105
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Greatest common divisors of some binomial coefficients
This is cross-posted from math.stackexchange.
While making some computation, I stumbled upon a curious relation among some binomial coefficients.
Consider the sequence of binomial coefficients $a(k,n)$...
- 690
4
votes
0
answers
191
views
Almost conjugate subgroups of compact simple Lie groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$Let $G$ be a compact connected Lie group.
Definition:
Two finite subgroups $H_1,H_2$ of $G$ are said to be almost ...
- 2,321
4
votes
0
answers
232
views
Road map for learning cluster algebras
I'm a PhD student and I would like learn about cluster algebras. I'm wondering what is a good reference (i.e., has detailed explanations, examples, etc) to learn from the basic and what are some of ...
- 767
4
votes
0
answers
170
views
infinite families in stable homotopy groups
The question is about infinite families in stable homotopy groups. Yes, there are some Q&A about the topic.
But I wonder if the order of Mahowald's elements is known?
in Green Book it mentioned ...
- 171