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8 votes
4 answers
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"Upside-down unimodal" sequences in combinatorics

Recall a sequence $a_0,\ldots,a_n$ of positive integers is unimodal if $a_0 \leq \cdots \leq a_m \geq \cdots \geq a_n$ for some $0 \leq m \leq n$. Unimodal integer sequences are abundant in ...
Sam Hopkins's user avatar
  • 24.2k
1 vote
1 answer
98 views

How large can a subset of computable reals, whose comparison function is computable, grow?

How large can a subset of computable reals, whose comparison function is computable, grow? For example, rational numbers are computable reals, and its comparison function is computable. As another ...
Hexirp's user avatar
  • 325
3 votes
1 answer
118 views

Can you metrize "convergence in probability with respect to every probability measure absolutely continuous"

Let $X_n$ be a sequence of real valued (or more general) random variables and let $\mu$ be some Borel measure on $\mathbb R$ - not necessarily finite. Suppose for every probability measure $P \ll \mu$ ...
user479223's user avatar
  • 1,924
2 votes
1 answer
377 views

Maximal subgroups of projective general linear group

$\newcommand{\sc}{\mathrm{sc}}$All the groups below are algebraic groups over an algebraically closed field, From Page $163$ of Malle and Testerman's book "Linear algebraic groups and finite ...
user488802's user avatar
6 votes
1 answer
286 views

Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$

This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763. It got upvotes, but no answers or comments, and so I ask it here. Let $G$ ...
Mikhail Borovoi's user avatar
9 votes
1 answer
651 views

What is the value of $j(2\sqrt{-163})$?

My question is how to calculate the value of $j(2\sqrt{-163})$ and its minimal polynomial, where the $j$ is elliptic modular function (see https://mathworld.wolfram.com/j-Function.html). The class ...
GuoJi's user avatar
  • 245
16 votes
2 answers
797 views

Operations on the set of large cardinal axioms

Here's a question from a non-set-theorist, but a sometime-user of large cardinals. The name Cantor's attic is pretty evocative for the collection of large cardinal axioms: looking through the pages ...
Tim Campion's user avatar
1 vote
0 answers
124 views

Abelian varieties with endomorphism structure

Let me stick to principally polarised abelian varieties $X$ over $\mathbb C$. I have seen several definitions of what it means for $X$ to have real multiplication by a totally real field $F$: There ...
Aitor Iribar Lopez's user avatar
3 votes
0 answers
73 views

Is the discrete logarithm equivalent to solving polynomial discrete logarithms?

Suppose we can quickly solve the discrete logarithm modulo $p$. Let's say $2$ is a generator so we can quickly find $l$ for which $2^l =h$ for any given target $h$. An interesting observation is that ...
mtheorylord's user avatar
2 votes
1 answer
132 views

The projective structure of a non-distributive modular lattice

Thrall states ("On the Projective Structure of a Modular Lattice", 1951, https://scholar.google.com/scholar?cluster=4998496641867146321): One measure of the complexity of a modular lattice ...
Dale's user avatar
  • 429
1 vote
0 answers
161 views

Higher dimensional Seifert surfaces and link numbers of higher knots

In 3-manifold topology, the notion of Seifert surface is well known. It is then used to define link numbers of knots. Question: Consider embedding $N^n \rightarrow M^{2n+1}$ of n-dimensional manifold $...
0x11111's user avatar
  • 593
11 votes
1 answer
4k views

Understanding the application of two inequalities?

I am reading the paper "The long-time behaviour of a stochastic SIR epidemic model with distributed delay and multidimensional Levy jumps" by Driss Kiouach and Yassine Sabbar. I have two ...
Math's user avatar
  • 185
2 votes
0 answers
90 views

Unexpected recursion for the A193231 (blue code of $n$)

Let $a(n)$ be A193231, blue code of $n$ i.e. self-inverse permutation of non-negative integers such that $a(n)<2^k$ iff $n<2^k$ and $$ a(n\operatorname{XOR}k) = a(n) \operatorname{XOR} a(k) $$ ...
Notamathematician's user avatar
11 votes
2 answers
858 views

Spectral sequences and short exact sequences

Suppose I take a short exact sequence of filtered chain complexes: $$0\to A\xrightarrow{p} B\xrightarrow{q} C\to 0$$ We assume that $p$ and $q$ are filtration-preserving, so that $p(F_rA)\subseteq ...
Richard Hepworth's user avatar
5 votes
1 answer
555 views

Betti numbers of non-orientable $3$-manifolds

Let $M^3$ be a compact $3$-manifold with boundary $\partial M$. If $M$ is orientable, then it is known (see Lemma 3.5 here) that $2\dim(\ker(H_1(\partial M,\mathbb{Q})\rightarrow H_1(M,\mathbb{Q})))=\...
Alessio Di Prisa's user avatar
1 vote
1 answer
138 views

Recognizability/unique composition property for substitution tiling

This may be a very basic question, but I have not found an answer to it so far in my search. The question is whether there is an "algorithmic" way to check unique-composition/recognizability ...
Keen-ameteur's user avatar
2 votes
0 answers
188 views

Self-adjointness of fractional laplacian

Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
B.Hueber's user avatar
  • 1,171
2 votes
0 answers
55 views

Tangential normal invariant isomorphism

Recently, I was reading the paper "Finite Group Actions on Kervaire Manifold" by Crowley, Hambolton. But I am having problem understanding a definition. Here it is, In page 15-16 they are ...
Sagnik Biswas ma20d013's user avatar
7 votes
1 answer
332 views

Mapping spaces in complete Segal spaces and quasi-categories

Complete Segal spaces and quasi-categories are two common models for the theory of $(\infty,1)$-categories, and both are equipped with a natural notion of hom spaces. For complete Segal spaces, which ...
ChrisLazda's user avatar
  • 1,838
7 votes
1 answer
560 views

Proof of global Peano existence theorem in ZF?

By global Peano existence theorem I mean the existence of a maximal interval of solution of a first order ODE $x'=f(x,t)$ with continuous $f$. The proofs of the global Peano Theorem found in the ...
Mikhail Katz's user avatar
  • 16.6k
5 votes
0 answers
137 views

Parametrizing polynomials with given Galois group

Consider a transitive group $G \subset S_n$, and the set $E$ of polynomials in $\mathbb{K}[x]$ of degree $n$ with Galois group $\subset G$. I am looking for a rational surjective mapping $\varphi: \...
T. Combot's user avatar
  • 231
0 votes
1 answer
113 views

Polynomial equations with linear inequalities

Given a set of polynomials equations in variables $x_1\dots x_n$ and a set of linear inequalities $L_k(x_1\dots x_n)\ne0$, is the set of solutions an algebraic set? If it is, what is the corresponding ...
Alm's user avatar
  • 1,207
3 votes
0 answers
296 views

Explicit family of polynomials describing embedded torus in complex projective space

This question is cross-posted (with modifications) from MSE. The original question is probably unfit for MathOverflow (although a professor I asked said that this is very nontrivial), but I'm hoping ...
Paul Cusson's user avatar
  • 1,763
2 votes
0 answers
104 views

Clique-coclique and uncertainty

The clique-coclique inequality states that for a graph $G$ on $n$ vertices that is either distance-regular or vertex-transitive, the independence number $\alpha(G)$ and the clique number $\omega(G)$ ...
Seva's user avatar
  • 23k
3 votes
2 answers
293 views

On convergence of convex-concave functions

Let $(f_n)$ be a sequence of twice differentiable functions on $\mathbb R$ such that for each $n$ there exists some $x_n\in\mathbb{R}$ such that: $f_n$ is strictly convex on $(-\infty,x_n)$, $f_n$ is ...
Iosif Pinelis's user avatar
3 votes
1 answer
243 views

Existence and uniqueness of solutions for continuous and directionally differentiable ODE

Given $f:\mathbb{R}^n \to \mathbb{R}^n$ continuous and directionally differentiable (i.e., such that the directional derivative of $f$ exists for any direction) at a neighborhood $N$ of $x_0\in\mathbb{...
Todd Chavez's user avatar
1 vote
0 answers
27 views

Spectrum of the convolution of the Maxwell collision kernel with a distribution

Given the Maxwell collision kernel $A(z) = |z|^2I_d - z \otimes z$, where $I$ denotes the $d\times d$ identity matrix and $z\otimes z = zz^T$ is the outer product, it is easy to see that $A(z)$ has ...
Vasily Ilin's user avatar
5 votes
1 answer
157 views

Simplicial objects in quasicategory which come from homotopy coherent nerve

Let $\mathcal{C}$ be a simplicially enriched category whose Hom-objects are all Kan complexes. Denote by $N\mathcal{C}$ the homotopy-coherent nerve of $\mathcal{C}$, which is a quasicategory. Suppose ...
K. Strong's user avatar
  • 423
12 votes
1 answer
706 views

How many integrals can give multiples of $\pi$?

This question notes a few families of rational functions whose integrals (from $0$ to $1$) give rational multiples of $\pi$. A fairly straightforward explanation is given there and in the related Math....
Steven Stadnicki's user avatar
13 votes
1 answer
700 views

When is $\mathrm{gcd}(k,p^k-1)=1$ true?

Let $p$ be a prime. Is there a classification of the numbers $k \geq 1$ such that $\gcd(k,p^k-1)=1$? If not, can we at least produce an explicit infinite subset? What is known about these $k$? For the ...
Martin Brandenburg's user avatar
3 votes
1 answer
511 views

Girth 5 graphs with diameter 2

Is there an infinite class of graphs with diameter 2, girth 5, and minimum degree at least 2? Girth 5 is necessary, since otherwise complete bipartite graphs are an answer. Minimum degree at least 2 ...
David Wood's user avatar
  • 1,319
0 votes
0 answers
103 views

On MSB and LSB of Diffie Hellman

Given generator $g$ of multiplicative cyclic group modulo $p$ a prime and two elements $h_1$ and $h_2$ such that there are $x_1$ and $x_2$ respectively satisfying $g^{x_i}=h_i\bmod p$ at every $i\in\{...
Turbo's user avatar
  • 13.9k
6 votes
1 answer
181 views

Expected value of the length of the shortest non-zero vector in a lattice?

$\DeclareMathOperator\SL{SL}$What is the expected value of the length of the shortest non-zero vector in a (unimodular) lattice? I.e., let $G=\SL_n(\mathbb{R})$ with Haar measure $\mu$, $\Gamma=\SL_n(...
yoyo's user avatar
  • 609
1 vote
0 answers
163 views

Can the Constructible Universe be built in absence of Unions and Power?

Can $L$ be built in $\sf ZF$ $\sf-Regularity-Union-Power+ Boolean \ Union$? We know that $L$ can be built in $\sf KP$, but here we don't have Set Union. If the answer is to the negative, then would ...
Zuhair Al-Johar's user avatar
2 votes
0 answers
162 views

Root system terminology

Let $\Phi$ be a root system. In a paper I'm writing, I need to work with subsets $\Phi' \subset \Phi$ satisfying the following two conditions: For all $\lambda_1,\lambda_2 \in \Phi'$ and $c_1,c_2 \...
Eric's user avatar
  • 21
1 vote
1 answer
300 views

Convergence of concave/convex function

Let assume that you have a sequence of twice differentiable functions $(f_n)_{n\in\mathbb{N}}\in\mathscr{C}^2(\mathbb{R})^{\mathbb{N}}$. Let suppose that for each $f_n$, it exists a $x_n\in\mathbb{R}$ ...
NancyBoy's user avatar
  • 393
0 votes
1 answer
178 views

Primes above the distant prime neighbors

Let $\ \mathbb P\ $ be the set of all natural primes. Pair $\ (p\ q)\ $ are prime neighbors $\ \Leftarrow:\Rightarrow$ $$ \{x\in\mathbb Z: p\le x\le q\}\cap\mathbb P\,\ =\,\ \{p\,\ q\} $$ Prime $\ x\...
Wlod AA's user avatar
  • 4,786
1 vote
0 answers
137 views

A comparison between packing and covering as classes of problems

We continue from Bounds for the Dispersal Problem in convex regions and Bounds for minimax facility location in a convex region Let us consider the classes of problems: Given a convex region $R$ and ...
Nandakumar R's user avatar
  • 5,979
10 votes
1 answer
256 views

How much of second-order arithmetic do you need for $\mathbf{\Sigma}^1_1$-determinacy to give you countable transitive models of $\mathsf{ZFC}$?

This is in some sense a follow-up to this question. The answer there says that over $\mathsf{Z}_2$ (second-order arithmetic), (boldface) $\mathbf{\Sigma}^1_1$-determinacy is enough to entail the ...
James E Hanson's user avatar
0 votes
0 answers
66 views

Boundedness of dimension of representations that restrict to a fixed representation of a normal subgroup

Let $G$ be a compact Lie group, $H$ a closed subgroup and $W$ an irreducible real representation of $G$. Then it follows from Frobenius reciprocity and Bott’s definition of induced representation that ...
rick's user avatar
  • 201
0 votes
1 answer
664 views

What is this three dimensional curve that looks like an infinity sign called?

What is this three dimensional curve that looks like an infinity sign called? (Is there a known parametric equation for it?) It was generated with this Sagemath - script, where you can zoom in 3d in ...
mathoverflowUser's user avatar
1 vote
0 answers
158 views

What does $\nabla^i f$ mean?

I am reading the article Some Geometric Calculations on Wasserstein Spaces of John Lott and there is this covariant index in the covariant derivative: $\nabla^i$. And I don't quite understand it. In ...
Gomes93's user avatar
  • 169
5 votes
0 answers
145 views

Tensor product - Vertex / Chiral algebras

Two questions regarding tensor product of modules over vertex / chiral algebras: First question: For (rational?) vertex operator algebras there is a notion of fusion product of modules inducing a ...
E. KOW's user avatar
  • 834
0 votes
1 answer
103 views

Probabilistic bounds of random polynomials

This is follow-up question to my previous question about the expected number of roots . I am considering a random polynomial given by $$p(z) = \sum_{i=0}^{n} a_i z^i$$, where each coefficient } $a_i$ ...
AgnostMystic's user avatar
5 votes
0 answers
204 views

Preservation of (co)limits under taking derived categories

Let $R$ be a commutative ring. Let $\{A_i\}_{i \in I}$ be a diagram of $R$-linear $1$-categories, indexed by a finite poset $I$. (If this matters, assume that the $A_i$ have finitely many objects). ...
Laurent Cote's user avatar
4 votes
0 answers
156 views

The smallest sequence without differences among Fibonacci numbers

Given a subset $\mathcal S\subset \mathbb N\setminus\{0\}$ of (strictly) positive integers, we can consider subsets $A$ of $\mathbb N$ (or $\mathbb Z$) with no differences in $\mathcal S$. Examples: ...
Roland Bacher's user avatar
2 votes
1 answer
293 views

Is there a *relative* moduli stack of objects functor?

Toen and Vaquie have constructed for any dg category $\mathcal{C}$ a stack $\mathcal{M}_\mathcal{C}$ parametrising objects in $\mathcal{C}$. Its definition is $$\mathcal{M}_\mathcal{C}(R)\ =\ \text{...
Pulcinella's user avatar
  • 5,711
-1 votes
1 answer
159 views

Classification of real Clifford algebras

$\DeclareMathOperator\Cl{Cl}$Let $V$ be a real vector space of dimension $p+q$. Let $Q$ be a non-degenerate quadratic form on $V$ of signature $(p,q)$ where $p$ is the number of positive eigenvalues, ...
asv's user avatar
  • 21.8k
3 votes
1 answer
274 views

Exact decay for solutions of fractional Laplacian equation

Let $s\in (0,1), N\ge 2$ and $U$ be the unique radially decreasing solution of \begin{equation} \ \ \left\{\begin{aligned} (-\Delta)^s U+ U &=U^p &&\text{ in } \mathbb{R}^N\\ U&...
sadiaz's user avatar
  • 402
2 votes
0 answers
60 views

upper bound for the exponential conjugacy growth rate for non-virtually nilpotent polycyclic groups

Given $n ≥ 0$, the conjugacy growth function $c(n)$ of a finitely generated group $G$, with respect to some finite generating set $S$, counts the number of conjugacy classes intersecting the ball of ...
ghc1997's user avatar
  • 823

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