1
vote
0answers
22 views

Thales' semicircle theorem in higher dimensions

Thales semicircle theorem says that an angle inscribed in a semicircle is a right angle. Q1. Does a cone with apex on a hemisphere and encompassing the circular base have a solid angle ...
0
votes
0answers
17 views

Derivative of Log_p map

If $\mathscr{M}$ is a $(d-dimensional)$ Riemann Manifold and $p$ is a point therein. What is the derivative of $Log_p$ function?
1
vote
0answers
36 views

The bifunctoriality of co/limits

I recently noticed that there are two senses in which colimits are functorial, and I'm curious about their interplay. Let $C$ be a cocomplete category. Then, on the one hand, for any diagram ...
0
votes
0answers
6 views

regularity of non-local linear elliptic equation

$\alpha\in (0,1)$, $u$ satisfies: \begin{equation*} b\cdot \nabla u(x)+\sum_{i=1}^d \int_{R} \left[u(x+se_i)-u(x)-s\mathbb{I}_{\{|s|\leq ...
5
votes
0answers
56 views

Enriques surfaces over $\mathbb Z$

Does there exist a smooth proper morphism $E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces? By a theorem of, independently, Fontaine and Abrashkin, combined with the ...
0
votes
0answers
10 views

Optimal bound in $L^2$ product on compact Kahler manifold

Let $X$ be a compact Kahler manifold of dimension $n$, equipped with a Kahler metric of volume $1$. There exists a constant $C \geq 1$ such that for any smooth functions $f,g$ on $X$ we have $$ \int_X ...
0
votes
0answers
6 views

How is the structure of spectrum in cap-sets with no strong increments unrealistic if density is too large?

I am reading this excellent paper by Bateman and Katz on improved bounds on the cap set problem. Let A be a set in $\mathbb{F}_3^n$ containing no 3-term arithmetic progression and let $A(x)$ denote ...
0
votes
0answers
12 views

Variance under time-scaling of sample measurements

I am struggling with explain something I read in a Whitepaper. The essence is as follows. Let's begin with a random variable $X$ defined as number of events in an hours. Further, we assume that $X ...
0
votes
0answers
39 views

Does a (non-closed) differential 1-form define a curve?

I am trying to understand under what conditions the following procedure properly defines a curve. Take a manifold $M$ with a (non-closed) 1-form $B$ and an exact 1-form $dA$. Define the functions ...
0
votes
0answers
6 views

When does a symmetric Poisson manifold decompose into homogeneous pieces?

When people study representations, they pay a lot of attention to the irreducible ones. This is justified by the fact that, roughly speaking, every unitary representation decomposes into a direct ...
1
vote
1answer
40 views

Polyhedra containing hexagones only [on hold]

It is well-known that Euler's theorem gives raison d'être to polyhedra containing exactly 12 pentagons if they are connected by 3 in a vertex. The number of hexagons may be arbitrary (in fact >1). In ...
-3
votes
0answers
24 views

why the split extension of a quasicyclic $2$-group $C$ by the cyclic group is not finite by abelien [on hold]

Let $G$ be split extension of a quasicyclic $2$-group $C$ by the cyclic group of ordre 2 generated by the inversion automorphism of $C$ it is clear that $G$ is abelien by (finite cyclic) but why $G$ ...
1
vote
0answers
8 views

Can real-valued Markov processes with continuous surjective sample paths admit a non-trivial “forward-invariant” set?

I have both a more general question (concerning stopping times), and then a more specific application (as described in the title). Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be ...
3
votes
1answer
83 views

Toral rank conjecture

In 1985, S.~Halperin conjectured in the topological context of maximal free torus actions on topological manifolds, that: If $X$ is a topological space, then $$\dim H^*(X;\mathbb Q)\geq 2^{rk(X)}.$$ ...
-1
votes
0answers
42 views

Root in positive Weyl chamber

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. We fix a Cartan subalgebra $\mathfrak{t}\subset \mathfrak{g} $. Let $R\subset \mathfrak{t}^*$ the set of roots. We fix $\Pi\subset R$ the set of ...
2
votes
0answers
39 views

Lower boundedness of the Ricci curvature [on hold]

I know that this is going to be slightly vague, but it's itching me: why is lower boundedness of the Ricci curvature required by some of the very important theorems in Riemannian geometry? The ...
2
votes
0answers
46 views

Geometry of rings and semi-rings

Basically, I wonder whether a theory similar to geometric group theory has been or could be developed for rings and semirings. One direction would be the following. Consider $\mathbb{N}$ (with the ...
10
votes
1answer
224 views

Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?

I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?
1
vote
1answer
68 views

Dense symmetric unitary integer matrix?

Can someone give me a nontrivial example of a symmetric unitary integer matrix? I'm looking for something as dense as possible (i.e., not too many 0's); 5 <= size <= 8 would be ideal.
1
vote
0answers
46 views

Square-free sieve over number fields

I am trying to work on extending various methods to study square-free values of polynomials (or more generally, $k$-free values) over general rings of integers, and a literature review has yielded ...
3
votes
1answer
91 views

$\mathcal S(\mathbb R^n) \hat \otimes_\pi \mathcal S(\mathbb R^m) \simeq \mathcal S(\mathbb R^{n+m})$?

If $S(\mathbb R^n)$ is the Scwartz space of smooth rapidly decaying functions equipped with the topology generated by the family of semi-norms $$\mathcal N_p (\varphi)= \sum_{|\alpha|, |\beta| \leq p} ...
-2
votes
0answers
130 views

Proof theory and the generalized Riemann hypothesis [on hold]

Is there a disproof of the following? CONJECTURE: Let $\chi$ be a Dirichlet character modulo $q$. Let $\varepsilon$ be a positive number with $0 < \varepsilon < \frac{1}{2}$. Let $T$ be a ...
1
vote
0answers
59 views

What is $\int (1-e^{-x})^n dx$? [on hold]

For my purposes, $n$ is a non-negative integer, and $x > 0$. I didn't know how to evaluate this integral, so I plugged it into Mathematica. It told me the solution is $(-1)^n B(e^x; -n, n+1)$ I ...
-1
votes
0answers
52 views

Some questions about the paper, “Hypercontractivity, Sum-of-squares Proofs and Their Applications” [on hold]

I am referring to this famous paper, http://arxiv.org/abs/1205.4484 At the top of page 42, the authors define an equation like $f=Gg$, for two functions $f$ and $g$ defined on a graph $G$. This seems ...
1
vote
0answers
20 views

Is there a generalization of Polya urns to continuous outcome event?

Take for example the simplest model where there are n blue balls and m white balls in an urn. Then, in a first step realization, a white one has been drawn and then c + 1 of this colour had been put ...
3
votes
0answers
90 views

Is surjectivity for morphisms of schemes local on the domain?

It is said so in Knutson's book 'algebraic sapces' in several places for different topologies on schemes, see Chapt. I, 2.19 for Zariski top, 3.13 for flat top., 4.11 for etale topology. But this ...
0
votes
0answers
47 views

Number of “small” subsets to a “large” set [migrated]

For the following we assume the axiom of choice. Let $X$ be a set of cardinality $l$ for some infinite cardinal number $l$, and let $p(X)$ be the number of subsets of $X$ that have cardinality ...
2
votes
0answers
16 views

Conditions for monotone function to take maximal chains to maximal chains surjectively

Suppose that $P$ and $Q$ are graded posets (with rank function $r$) and suppose that all maximal chains of $P$ and $Q$ have length $n$. Let $f:P \to Q$ be a surjective monotone function such that ...
3
votes
0answers
66 views

Scotts Theorem for one ended Fuchsian groups

in Peter Scott's work "Subgroups of Surface groups are almost geometric" is proven that for a closed surface $S$ and any fin. gen. subgroup $U$ of $G:=\pi_1(S,x)$ there exists a finitely sheeted ...
1
vote
0answers
34 views

Subgroups generated by opposite root groups

Suppose $\mathbf{G}$ is a connected reductive (possibly non-split!) group over a field $F$, $\mathbf{S} \leq \mathbf{G}$ a maximal split subtorus and $\mathbf{Z} \leq \mathbf{G}$ its centralizer. For ...
0
votes
0answers
43 views

orbit space of $\mathbb{Z}_p$ action over complex projective space by permuting the homogeneous coordinates

$Z_p$:=cyclic group of order $p$. I want to understnd $H_\ast(\mathbb{C} P^{n}/Z_{n+1})$ with $(n+1)$ being a prime number,and the action is given by permuting the homogeneous coordinates. For ...
0
votes
0answers
117 views

In set theory, is there a name for a function which maps the empty set to zero and all the others to one? [on hold]

I would like to avoid inventing something which might be standard. Thus, I'am asking if there is a name for a function which is defined as $f$: Let $S$ be any set, then $f(S)=0$ if $S$ is empty and ...
5
votes
2answers
284 views

Texts about Dwork's work

I want to ask about references to papers, that probably exist, which explain the articles of Bernard Dwork starting from "The rationality of the zeta function of an algebraic variety" to "On the ...
0
votes
1answer
57 views

A problem with pointwise stabilizer subgroups of fixed-point subspaces II

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$. Let ...
7
votes
1answer
240 views

Applications of set theory in physics

In the introduction of the paper "Links between physics and set theory", the following quote of Eris Chric is stated: ...
3
votes
0answers
71 views

Simply connected Lie groups homeomorphic to R^n are solvable

I have found the following claim in many proofs "Simply connected Lie groups homeomorphic to $\mathbb{R}^n$ are solvable". But the universal covering of $SL(2,\mathbb{R})$ satisfies the hypothesis of ...
2
votes
0answers
54 views

Homotopical categories, the 2-out-of-6 property, and saturation

A homotopical category is a cateogry with a class $\mathcal W$ of arrows called weak equivalences which satisfies the 2-out-of-6 property. The nlab article shows a deep connection between $\mathcal ...
3
votes
0answers
68 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

I asked this question on math.stackexchange a week ago, but did not get an answer. First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general ...
4
votes
2answers
441 views

Why not develop a Hamiltonian-based Morse theory?

I have begun to learn the basics of Morse theory and Floer homology. I understand that Floer homology is the natural theory for symplectic manifolds, but from my preliminary knowledge of Morse theory ...
4
votes
0answers
49 views

Combinatorics of palindromic decompositions

This is sort of a companion to my question Number of trivializations of a trivial word in the free group (which in turn is motivated by my earlier question here). It turns out that that question may ...
5
votes
1answer
102 views

A “quadratic” triangular inequality

In a Euclidian space (Hermitian as well), say $\ell^2_n$, the following inequality holds true $$(QI)\qquad |b|\cdot|c-a|\le|c|\cdot|a-b|+|a|\cdot|b-c|,\qquad\forall a,b,c\in\ell^2_n.$$ In other words, ...
1
vote
0answers
26 views

Concise disambiguation of Voronoi boundaries

Say that $x_1,\dots,x_n$ are points in the plane, with a Voronoi diagram $V_1,\dots,V_n$. The Voronoi diagram is typically defined by $$V_i = \{x:\|x-x_i\|\leq \|x-x_j\|~\forall j\}~.$$ Is there any ...
0
votes
1answer
61 views

A problem with pointwise stabilizer subgroups of fixed-point subspaces I

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$. Let ...
0
votes
1answer
48 views

Double coset separability and the existence of vanishing sequences for surface group

Definition: Let $G$ be a group. $G$ is said to be double coset separable if given any finitely generated subgroups $H$ and $K$ in $G$, given any $g\in G$ and $h\not\in HgK$, there exists a finite ...
-4
votes
2answers
165 views

Brocard's problem [on hold]

According to Brocard's problem $$x^{2}=n!+1 \Rightarrow (x+1)(x-1)=n!$$, where $(x+1)/2$ , $(x-1)/2$ are consecutive integers, assume one of them is z, another is (z-1). z and (z-1) have all primes ...
1
vote
1answer
72 views

Loop defects in Walker-Wang model

My question is about the description of general defects (specially loop defects) in the Walker-Wang (WW) model. Elementary excitations in the WW model can be point particles, loop defects and more ...
1
vote
1answer
136 views

Smoothing transverse intersections

Let $S$ be a complex surface with ample canonical class. Let $C_1$ and $C_2$ be smooth complex curves in $S$ that intersect transversally at $n $ points. Furthermore, assume that the self-intersection ...
3
votes
1answer
110 views

Is the derived category of perfect complexes idempotent complete?

Let $\mathcal{C}$ be a category. We call a morphism $\alpha: X\rightarrow X$ an idempotent if $\alpha^2=\alpha$ in $\mathcal{C}$. We call $\mathcal{C}$ is $\textit{idempotent complete}$ if any ...
-2
votes
0answers
83 views

Defining Reinhardt Cardinals in Choiceless Models of $ZF$ [on hold]

Can Reinhardt cardinals be defined in choiceless models of $ZF$? This question is motivated by the following remark by Prof. Hamkins in his comment to me regarding Andrea Nespola's question ...
2
votes
1answer
100 views

Two equivalent descriptions of a physical system yielding a non-trivial mathematical formula

First I would like to admit that this question may not be entirely appropriate for this site, but I will give it a go none the less. One often hears stories about how string dualities lead to highly ...

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