Newest Questions
159,065 questions
8
votes
4
answers
520
views
"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 ...
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 ...
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 $P \ll \mu$ ...
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 ...
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 $G$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $...
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 ...
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
$$
a(n\operatorname{XOR}k) = a(n) \operatorname{XOR} a(k)
$$
...
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 ...
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})))=\...
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 ...
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 $\Delta_{g}$....
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 ...
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 ...
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 ...
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: \...
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 ...
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 ...
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 $\omega(G)$ ...
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 ...
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{...
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 ...
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 ...
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....
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 ...
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 ...
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\{...
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(...
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 ...
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 \...
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 $x_n\in\mathbb{R}$ ...
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\...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 } $a_i$ ...
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).
...
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: ...
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{...
-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, ...
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&...
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 ...