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$k$-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 ...
jonathan's user avatar
  • 486
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}...
T. Amdeberhan's user avatar
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 "...
fuser0909's user avatar
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 ...
Exit path's user avatar
  • 2,969
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 $$...
Harry Gindi's user avatar
  • 19.4k
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 $...
user521337's user avatar
  • 1,189
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 $...
user521337's user avatar
  • 1,189
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 (...
Ben Sprott's user avatar
  • 1,281
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 ...
Cogicero's user avatar
  • 167
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. ...
BrianT's user avatar
  • 1,197
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?
Ali Taghavi's user avatar
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 ...
algebrachallenged's user avatar
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^{\...
Jhoan Perez's user avatar
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{...
Anne F.'s user avatar
  • 131
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 ...
Meisam Soleimani Malekan's user avatar
-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{...
tobias's user avatar
  • 739
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^\...
Aurel's user avatar
  • 4,878
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 ...
Ben Sprott's user avatar
  • 1,281
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\|,\...
Manfred Weis's user avatar
  • 12.6k
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 ...
ness1's user avatar
  • 121
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 ...
Taras Banakh's user avatar
  • 40.8k
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 ...
ness1's user avatar
  • 121
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 $P$ ...
H A Helfgott's user avatar
  • 19.3k
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\}}...
Christoph Mark's user avatar
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 $\...
M. Reza. K's user avatar
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 ...
Dominic van der Zypen's user avatar
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 ...
TheStudent's user avatar
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 ...
Dominic van der Zypen's user avatar
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 "...
მამუკა ჯიბლაძე's user avatar
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)-...
soodehMehboodi's user avatar
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? ...
Ken's user avatar
  • 397
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(...
Alexander's user avatar
  • 861
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 ...
natalie Kop's user avatar
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 $\mathfrak f$...
Shimrod's user avatar
  • 2,335
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,...
Yi-Hsuan Lin's user avatar
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 ...
Taras Banakh's user avatar
  • 40.8k
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 ...
Joseph O'Rourke's user avatar
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?
Joakim Færgeman's user avatar
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,...
David Halitsky's user avatar
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 ...
darij grinberg's user avatar
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}...
hao chen's user avatar
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 $\...
Xing Wang's user avatar
  • 119
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\...
Manfred Weis's user avatar
  • 12.6k
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: $0 \to \mathbb Z \to \mathcal O_X \to \mathcal O_X^* \to 0$ ...
Asvin's user avatar
  • 7,648
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 ...
wonderich's user avatar
  • 10.3k
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 ...
Ehud Meir's user avatar
  • 4,969
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?
user09127's user avatar
  • 765
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 ...
Marie Anderlecht's user avatar
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 $$...
math110's user avatar
  • 4,230
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 ...
galois1989's user avatar

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