0
votes
0answers
10 views

Lower boundedness of the Ricci curvature

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 ...
0
votes
0answers
20 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 ...
7
votes
1answer
106 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
30 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.
0
votes
0answers
25 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 ...
2
votes
1answer
57 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} ...
-1
votes
0answers
91 views

Proof theory and the generalized Riemann hypothesis

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 ...
0
votes
0answers
42 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
37 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
12 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 ...
1
vote
0answers
58 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
13 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 ...
2
votes
0answers
37 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 ...
0
votes
0answers
21 views

Subgroups generated by opposite root groups

Suppose $\mathbf{G}$ is a connected reductive (possibly non-split!) group over a local field $F$, $\mathbf{S} \leq \mathbf{G}$ a maximal split subtorus and $\mathbf{Z} \leq \mathbf{G}$ its ...
0
votes
0answers
37 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
105 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
1answer
173 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
0answers
39 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
0answers
160 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
65 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
42 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
56 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
414 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
40 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
0answers
85 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
21 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
56 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
37 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
138 views

An equation of prime counting function [on hold]

I have encountered the below problem- Given, $z(z-1)$ is divisible by all prime < $\sqrt{z} <n$ and all prime divisor of $z(z-1)$ are consecutive primes. Prove(or disprove)- $$ ...
0
votes
0answers
40 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
126 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
104 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
75 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
92 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 ...
6
votes
0answers
38 views

Angle subtended by the shortest segment that bisects the area of a convex polygon

Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could ...
0
votes
1answer
54 views

Mathematical statistical qm book-recommendation

I feel that there are quite a few good and rigorous books on the mathematical foundations of quantum mechanics, but I am currently looking for a book that covers mathematical statistical quantum ...
4
votes
0answers
35 views

Anything known about the Grundy Ordinal of Sylver's Coinage

Sylver's coinage is an example of an unbounded finite (if slightly modified) combinatorial impartial game. Quoth wikipedia: The two players take turns naming positive integers that are not the ...
0
votes
0answers
21 views

Pascal and Brianchon's theorems generalized for hyperbolic paraboloid

I know that giving a general version of these two theorems for quadrics can be quite tricky, but if we restrict ourselves to a verssion that holds for the hyperbolic paraboloid only it should be ...
2
votes
0answers
104 views

One question about iteration on groups

Let $G$ be a finitely generated group, $H$ a subgroup of $G$ of index $n$, with $a_i$ a set of coset representatives and $$G=\displaystyle\bigcup_{i=1}^nH{a_i}.$$ Let $\phi:H\rightarrow G$ be a ...
1
vote
1answer
90 views

Beilinson-Bernstein localization: $\mathfrak{g}$ action on $G$-equivariant sheaf

I have a few elementary questions related to Beilinson-Bernstein localization. Let $G$ be a semisimple algebraic group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$. Consider the setup of ...
3
votes
0answers
86 views

A property of the Frechet filter and every ultrafilter

(Joint question with Piotr Szewczak.) Definitions and notation. By filter we mean a filter on $\omega$ containing the cofinite sets at least. For a filter $\mathcal{F}$, let ...
10
votes
1answer
139 views

A classic cardinal characteristic of the continuum in disguise?

We believe the answer to the following question, that is relevant to a joint research project with Piotr Szewczak, should be known. We would appreciate any help or pointer. Needed definitions may be ...
10
votes
4answers
770 views

Are there any Algebraic Geometry Theorems that were proved using Combinatorics?

I'm collaborating with some algebraic geometers in a paper, and when writing the introduction I mentioned the interaction of Combinatorics and Algebraic Geometry, and gave some examples like the ...
-2
votes
1answer
70 views

About structure of the set of perfect matchings of $K_{n,n}$

Are there any special properties known about the set of perfect matchings of $K_{n,n}$? Like any global structure of this set? Some natural way to partition it? Like is there some algebraic structure ...
0
votes
0answers
52 views

absolutely continuous of two probability measures

Suppose $X_t$ satisfies $$X_t=\int_0^t b(X_s)ds+ L_t,\quad t\in[0,1]$$ where $L_t, t\in[0,1]$ is a $\alpha-$stable process. Let $P_L$ be the law of $L$, $P_X$ be the law of $X$. ($P_L, P_X$ are ...
-4
votes
0answers
54 views

Probability- My homework is confusing me [on hold]

The question is that there is a game, it has 38 congruent pieces, 18 are orange, 18 are blue, and 2 are white. To win you have to get either orange or blue and you get 2$, to play you pay 1 dollar, ...
1
vote
0answers
71 views

Poincare-Lefschetz duality, universal coefficients, and middle cohomology [migrated]

Sorry if the question is too simple, algebraic topology is not my strong suit. Let $(M,\partial M)$ be a $2n$-dimensional manifold with boundary, with one-dimensional middle cohomology. By ...
6
votes
0answers
61 views

Is there a universal straightedge and compass construction of a segment incommensurable to a given one in the hyperbolic plane?

"Universal" means that the construction steps are independent of the length of the given segment. In the Euclidean plane one can take the diagonal of a square built on it. Without the "universal" the ...
1
vote
1answer
46 views

Injecting premises into two implicational premises connected by a tensor (multiplicative conjunction) in linear logic

I have another question regarding linear logic: I want to get to the proof E, using the premises in (1-4). Is this at all possible? 1: $A$ 2: $C$ 3: $(A\multimap B)\otimes(C\multimap D)$ 4: ...

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