All Questions
152,889
questions
5
votes
2
answers
193
views
$k$-invariants as extension classes
Let $X$ be a connected cell complex with fundamental group $G$
and $(n-1)$-connected universal covering space. Let $\Pi=\pi_n(X)$.
We may construct a $K(G,1)$ complex $K$ by adjoining cells of ...
5
votes
2
answers
861
views
Searching for a proof for a series identity
The below identity I have found experimentally.
Question. Is this true? If so, may you provide a "slick" (or any) proof.
$$6\sum_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum_{k=1}^{\infty}...
1
vote
0
answers
50
views
Vertical and horizontal percolation on heterogeneous honeycomb lattice
I have a regular honeycomb lattice where a bond in the unit cell aligns with $(1,0)$; call this the horizontal direction. Each horizontal bond in the lattice is open with probability $p$ and each "...
9
votes
4
answers
1k
views
Localization of $\infty$-categories
In ordinary category theory, the localization $C[S^{-1}]$ at a class of morphisms $S$ (with possibly some assumptions on $S$) is a category $C[S^{-1}]$ together with a map $L:C \to C[S^{-1}]$ such ...
5
votes
1
answer
177
views
Factorization of colimits through slices?
I could swear I remember a result of the following form:
Suppose we have a pair of functors $$C\xrightarrow{F}D\xrightarrow{G} X,$$ with $X$ cocomplete.
then we obtain a functor $$D\to X$$ sending $$...
1
vote
0
answers
48
views
On different norms of the interpolating operator
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
4
votes
1
answer
148
views
Find $p$ s.t. there is a sequence of nodes in $[0,1]$ s.t. sequence of interpolating polynomials of every continuous function converges in $p$-norm
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
0
votes
0
answers
88
views
Define a directed-complete partial order of lists
The List monad takes a set and produces the set of lists on that set. The elements of the set become the symbols or objects of the list. I would like to define a directed-complete partial order (...
1
vote
1
answer
2k
views
Any inequalities / estimates for a lower bound of the $L^2$ inner product of a quantity and its derivative?
For some numerical analysis of a fluid, I am wondering if there is any inequality that provides a lower bound (in the $L^2$ norm), for the $L^2$ inner product of a quantity with its derivative. In ...
1
vote
0
answers
74
views
On some finiteness properties of cohomological algebras of complex tori
Denote by $A := \mathbb{C}[u,u^{-1}]$, $u = (u_1,...,u_n)$, the algebra of polynomial functions on the complex torus $(\mathbb{C} \setminus \{ 0 \})^n$, which we consider as a $\mathbb{C}[u]$-module.
...
4
votes
1
answer
599
views
Updated background on Hilbert 16th problem?
What is the current situation of the second part of the Hilbert 16th problem? What are the most updated news on this problem?
5
votes
2
answers
289
views
Seifert fiberings of zero euler number which are semi-bundles
Let M be a closed oriented manifold which has the structure of a "semi-bundle" (See Section 1.2. of Hatcher's notes on three-manifolds) over an interval I. Assume that M is Seifert fibered over a base ...
0
votes
1
answer
155
views
Nonlocal integral
I have a little problem with the next integral,
$$ \int{d^3{\bf r^{\prime}}\left[\frac{exp(-ar^{\prime})}{r^\prime}\right]u({\bf r}-{\bf r^\prime}})=\int{4\pi r^\prime dr^{\prime}exp(-ar^{\...
13
votes
1
answer
528
views
On non-representability of certain hom schemes
Let $k$ be an algebraically closed field. It is well-known that the isom-sheaf Isom$(\mathbb{A}^1_k,\mathbb{A}^1_k)$ is not representable by an algebraic space. (To be clear, the functor Isom$(\mathbb{...
3
votes
1
answer
191
views
An analytical zero divisor
Let $G$, $\mathbb C[G]$ and $\text{vN}(G)$ be a torsion free group, it's group ring and group von Neumann algebra, resp.. Let $0\neq\alpha\in\mathbb C[G]$ and $0\neq p\neq1$ is a projection in the ...
-2
votes
1
answer
193
views
Strong estimates for the zeta function on natural numbers
Let $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}$$
be the Riemann zeta function (here we just consider real $s$).
We do have a description given by
$$\zeta(s) = \frac{s}{s-1}-s\int_{1}^\infty \frac{...
8
votes
1
answer
300
views
Are there partially algebraic Hecke characters?
$\newcommand{\C}{\mathbb{C}} \newcommand{\U}{\mathbb{U}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}}$
Let $F$ be a number field.
Let $\chi\colon \mathbb{A}_F^\...
6
votes
1
answer
376
views
Map from the Multiset Monad to the Giry Monad: From Data to Probabilities
The Mulitiset monad, aka the free commutative monoid monad or "Bag" monad, takes a set to the set of all Multisets for that set. A Multiset is like a set, but can have duplicates. It is used in ...
2
votes
1
answer
252
views
Checking planar convexity of 4 points with Stewart's formula
Is the following conjecture correct?
Conjecture:
If $A,B,C,D$ are four points in general position in the euclidean plane, with
$a:=\|C-B\|,\ \ b:=\|C-A\|,\ \ c:=\|B-A\|$
$a':=\|D-A\|,\...
3
votes
2
answers
421
views
Asymptotically invariant maps and strongly ergodic actions
Let $\Gamma$ be a countable group which acts strongly ergodically on a probability measure space $(X,\mu)$. Let $\sigma_k:X \rightarrow Y$ be a sequence of measurable functions into a complete metric ...
5
votes
1
answer
311
views
Is the compact-open topology on the dual of a separable Frechet space sequential?
Let $X$ be a separable Frechet space (= Polish locally convex linear metric space) and $X'_c$ be the space of linear continuous functionals on $X$, endowed with the compact-open topology (= the ...
1
vote
0
answers
37
views
Strong ergodicity of a countable subgroup of $PO(3,1)$
If we identify the boundary at infinity of the hyperbolic $3$-space $\mathbb{H}^3$ with the complex projective line $\mathbb{P}^1(\mathbb{C})=\mathbb{C} \cup \{ \infty\}$, we know that the ideal ...
18
votes
3
answers
3k
views
Entropy and total variation distance
Let $X$, $Y$ be discrete random variables taking values within the same set of $N$ elements. Let the total variation distance $|P-Q|$ (which is half the $L_1$ distance between the distributions of $P$ ...
3
votes
1
answer
152
views
Automorphisms of homogeneous space $F_4/P_{\{\beta_2\}}$ over the exceptional group $F_4$
Let $F_4$ be the connected, simply connected, simple, complex, linear algebraic group of type $\mathsf{F}_4$, with Dynkin diagram
$$
\beta_1-\beta_2\Rightarrow\beta_3-\beta_4\,.
$$
Let $P_{\{\beta_2\}}...
1
vote
1
answer
79
views
A property on some unbounded metric spaces
Suppose that $(X,d)$ is a metric spaces. Which condition(s) can guaranties the following property:
$\forall x, \forall y \in X, \exists \{z_n\}$ such that $\lim_{n\to +\infty } d(x,z_n)=+\infty$ and $\...
3
votes
1
answer
98
views
Maximal matchings in connected graphs
Let $n\in\mathbb{N}$ be a positive integer. Is there a connected graph $G$ such that $G$ cannot be coloured with less than $n$ colours, and every two maximal matchings have non-empty intersection?
(I ...
19
votes
1
answer
671
views
Proof of Minkowski theorem using harmonic analysis
I am trying to properly write a proof of Minkowski's theorem in a self-contained way and understandable by (good) undergraduates.
Theorem (Minkowski)
Let $L$ be a lattice of $\mathbb{R}^n$ and ...
2
votes
2
answers
147
views
Graphs in which all maximal matchings intersect
Let $G=(V,E)$ be a simple, undirected graph. A matching is a set $M\subseteq E$ consisting of pairwise disjoint edges. We say $M$ is maximal if it is maximal amongst all matchings in $G$ with respect ...
5
votes
1
answer
292
views
Conceptual and practical reasons and consequences of inverting weak equivalences
Although dealing with this in one or other form for many years, to my shame this question only struck me now.
One of the most radical differences between categories of "algebraic" and "...
0
votes
1
answer
266
views
Background on the functional equation $F(x+1)+F(x)=f(x)$ [closed]
In the theory of indefinite sums, anti-differences and finite calculus, the following difference functional equation and its solutions are very important:
$$\bigtriangleup F(x):=F(x+1)-...
2
votes
1
answer
109
views
$3$-uniform hypergraph with $n$ vertices and $O(n^{3/2})$ hyperedges
Suppose $H=(V,E)$ is a $3$-uniform hypergraph with $n$ vertices $V$ and $O(n^{3/2})$ hyperedges, $E.$
My question is: How many vertices do I need to delete to make sure all hyperedges are destroyed? ...
8
votes
2
answers
824
views
Anisotropic algebraic groups have no unipotent elements
I have found the following fact stated in a number of places:
If $k$ is any field, a connected reductive group $G$ is anisotropic if and only if its only unipotent element is $e$ and $\mathrm{Hom}_k(...
7
votes
0
answers
146
views
$p$ -adic periods of modular curves X_0(71)
I have seen in some papers computation of $p$-adic periods of modular curves $X_0(N)$. Can somebody please explain to me what are the possible applications of such computations?
as a concrete ...
3
votes
1
answer
400
views
Roots of modular functions
Let $\mathfrak f(\tau)=e^{-\pi i/24}\frac{\eta\left(\frac{\tau+1}{2}\right)}{\eta(\tau)}=q^{-1/48}\prod_{n=1}^{\infty}\left(1+q^{n+1/2}\right)$ be the Weber modular function. The function $\mathfrak f$...
2
votes
1
answer
64
views
Compute the hull of nonnegative linear combinations of a finite set, and the extreme points of the intersection of two polyhedra
Let $\mathbb{R}^d$ be $d$-dimensional Euclidean space
Let $\Delta=\{x\in\mathbb{R}^d_+:\sum_{i=1}^dx^i\leq1\}$ ($x^i$ is the i-th coordinate of $x$)
(Equivalently, $\Delta$ is the convex hull of $\{(0,...
2
votes
1
answer
331
views
The completeness of spaces of continuous functions with the compact-open topology
For a Tychonoff space $X$ let $C_k(X)$ denote the space of continuous real-valued functions on $X$, endowed with the compact-open topology.
Problem. Is the space $C_k(X)$ Polish if it is Polishable ...
2
votes
0
answers
97
views
8-partition of a planar convex body by 4 concurrent lines
It is known1 that any convex body $K$ in the plane can be
partitioned into $6$ equal-area pieces by $3$ concurrent lines
which meet at a point in $K$.
Call this a $6$-partition.
This result cannot be ...
1
vote
0
answers
205
views
If the kernel of an irreducible character contains the derived subgroup - is it then linear? [closed]
It is of course true that any linear character of a group contains the commutator subgroup. But is the converse also true? If not - do you know of a counterexample?
0
votes
1
answer
1k
views
The generalized Kronecker delta and three sets of 16 tetrahedra defined by 192 of the 240 roots of E8 (vertices of Gosset's 8-polytope 4_21)
Original question (without additional information from Wendy):
Using 192 of the 240 roots of E8 (vertices of 4_21), Wendy Krieger has defined 48 disjoint tetrahedra this way:
Taking the E8 as {128,...
2
votes
1
answer
246
views
Gale order on multisets of elements of a lattice
The question
Let $L$ be a lattice (in the sense of combinatorics, not number theory).
An $L$-bag will mean a finite multiset of elements of $L$.
Given an $L$-bag $A$, we consider three possible ...
2
votes
1
answer
435
views
Minimax Approximation to Sine Function on interval [-K, K]
Let $p_{n,K}(x)$ be the polynomial of degree $n$ that minimizes $\epsilon_{n,K} = ||p_{n,K}(x) - sin(x)||_\infty$ on the interval $[-K, K]$. Question: what is the asymptotic behavior of $\epsilon_{n,K}...
2
votes
1
answer
94
views
Quotient with positive second derivative in the limit?
I am studying the quotient of
$$f(\varepsilon) = \sum_{i=1}^{\infty} \frac{i^2}{2^{\varepsilon i^2}}$$
and $$g(\varepsilon) = \sum_{i=1}^{\infty} \frac{1}{2^{\varepsilon i^2}}$$
for some $\...
4
votes
1
answer
378
views
Mathematical Structure and Objects Induced by Pairs of Disjoint Subsets
Let $\mathcal{S}$ be a finite, discrete and non-empty set, i.e.,
$$\begin{align} \operatorname{card}\left(\mathcal{S}\right) & =:n\in\mathbb{N}^+\\ V& :=\{v\subset\mathcal{S}\ |\ v\ne\...
3
votes
0
answers
313
views
Finding the torsion of the Neron Severi group in the first homology group
Let X be a variety over $\mathbb C$. I will implicitly identify this with us complex analytification. Consider the exponential sequence:
$0 \to \mathbb Z \to \mathcal O_X \to \mathcal O_X^* \to 0$
...
3
votes
0
answers
208
views
Cell structure on $B\mathbb{G}$ and the bar resolution of $\mathbb{G}$
Consider $\mathbb{G}$, which can be viewed as a group, as well as a 2-group.
(For example, given a short exact sequence
$$
1 \to BG_2 \to \mathbb{G} \to G_1 \to 1
$$
and the fiber sequence:
$$
B^2G_2 ...
5
votes
0
answers
154
views
lifting of idempotents in group ring
Let $G$ be a finite group, and let $\pi:G\to Q$ be a surjective group homomophism. The map $\pi:G\to Q$ does not necessarily split, but we can always find a set theoretical splitting $s:Q\to G$. In ...
1
vote
1
answer
71
views
Why is a certain space of linear isometries paracompact
Let $V$ be a finite dimensional real inner product space and $U$ a real inner product space of countable dimension. Why is the space of linear isometries from $V$ to $U$ paracompact?
9
votes
1
answer
202
views
Separability of compact quantum groups
In the theory of compact quantum groups due Woronowicz, we assume usually that the C*-algebra of the compact quantum group is separable. Is the assumption essential in the theory? Will it eventually ...
5
votes
2
answers
916
views
On the upper bound of $\sum_{i=1}^{n}x^m_{i}$ subject to the conditions $\sum_{i=1}^{n}x_{i}=0$ and $\sum_{i=1}^{n}x^2_{i}=n$
The following question has been posted on mathematics stackexchange: inequalities problem, perhaps arising from a question on expectations.
Let $x_{1},x_{2},\cdots,x_{n}$ are real numbers, and such
$$...
4
votes
0
answers
251
views
About Hopf invariant
I am trying to understand properly the Hopf invariant. So, one way to compute the Hopf invariant from a smooth map $f:S^3\to S^2$ consists on taking two regular values $p,q\in S^2$ and computing the ...