0
votes
0answers
13 views

Jacobian change of basis matrix for different dimensions

I am considering a real Lie group $G$ acting transitively on an open set $U$ in a real Euclidean space of lower dimension. Given a smooth, compactly supported function $f: G \rightarrow \mathbb{R}$, I ...
3
votes
0answers
32 views

Lifting Abelian Varieties to p-adic fields

Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...
3
votes
1answer
59 views

Subgroups of $SL_3(\mathbb{Z})$ that are finitely generated, Zariski-dense, infinite index, and torsion-free

My question stems from Misha's answer of a MathOverflow question. Misha supplied the following question in his answer: Open question: Does there exist a finitely generated Zariski-dense ...
2
votes
0answers
33 views

What is the best lower bound for 3-sunflowers?

A collection of $t$ sets $A_i$ is called a t-sunflower if $A_i \cap A_j = C $ for all $i \neq j$ for some fixed $C$. A well-known conjecture of Erdos and Rado says that in any $k$-uniform family of ...
3
votes
0answers
23 views

$\Omega$ and $B$ as adjoints between symmetric monoidal $(\infty,n)$- and $(\infty,n-1)$-categories

Given a symmetric monoidal $(\infty,n)$-category $\mathcal{D}$, one obtains a symmetric monoidal $(\infty,n-1)$-category $\Omega \mathcal{D}$ by taking $\Omega \mathcal{D}= ...
2
votes
1answer
64 views

Intersection theory on M_{g,n}

Is there a paper\book that lists the top intersections of Hodge classes and tautological classes on $\overline{\mathcal{M}}_{g,n}$ for small $g$ and $k$, e.g. $g=2,3$ and $k=0,1,2$ ?
1
vote
0answers
49 views

Could one recover the relative K-theory from the quotient derived category?

Let $A\to B$ be a full embedding of exact categories that induces an embedding $D^b(A)\to D^b(B)$. My question is: what can one say about the relation of the homotopy cofibre $K(A)\to K(B)$ (the ...
-3
votes
0answers
57 views

Veronese surface [on hold]

I have a question(Hartshorne ,page 13,exercise 13): If Y be the image of the 2-uple embedding(described in exercise 12) of P^2 in P^5. and if Z(a subset of Y) is a closed curve(variety of dim 1) show ...
2
votes
0answers
33 views

About the rate for one approximation to the heat kernel

Let $G(t,x)$ be the heat kernel $$ G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}. $$ Here is one approximation of $G(t,x)$: $$ G_\epsilon(t,x)=e^{-t/\epsilon} \sum_{k=1}^\infty ...
0
votes
0answers
25 views

Characterizing and counting boolean functions with all influences 1/2

Is there a characterization of boolean functions $f:\{-1,1\}^n \longrightarrow \{-1,1\}$, so that $\mathbf{Inf_i}[f]=\frac{1} {2}$, for all $1\leq i\leq n$? Is it known how many such functions there ...
-2
votes
0answers
7 views

Trigonometric substitution [migrated]

Been out of touch with trigonometry for some time now. Need help proving this expression. Sin2x/2 = 1/2(1-Cosx) Any help will be appreciated. Thanks.
1
vote
1answer
74 views

Bott formula for projective bundles

For a projective space one has Bott formula to compute $h^q(ℙ^n,Ω^p(k))$, where $Ω^p(k)$ is the k-twisted sheaf of sections in the p-th power of the cotangent bundle of $ℙ^n$. I am wondering if there ...
3
votes
1answer
115 views

Existence of a certain subset of natural numbers

I was talking to a friend and the following set $S$ came up. Let $f$ be some real valued function tending to infinity. Let $S$ be a subset of natural numbers such that $|S \cap [1,N]| = N^{\delta}+ ...
3
votes
0answers
38 views

Matching in the Boolean Algebra

We consider the Boolean Algebra of the set $[n]=\{1,\dots,n\}$ and the matching $\psi$ from the $m+1$ subsets of $[n]$ into the $m$ subsets of $[n]$ for $(n+1)/2\leq m+1\leq n$, which is used to show ...
2
votes
0answers
29 views

Where are there defined objects between gerbes and bundle gerbes?

Consider a special kind of/something like a gerbe where there is first given local trivialization data with equivalence over 2-fold overlaps but not isomorpphism. Does this exist in the literature?
1
vote
0answers
33 views

Characterization(?) of coersive(?) elements in the special linear group

Take your favorite matrix norm $\|\bullet\|$ (my favorite is the Frobenius norm $\|A\| = \sqrt{tr A A^t}$). Now consider the set $S_x$ of matrices $A,$ such that $\|A\| < x$ and $\|A^{-1}\| < ...
11
votes
1answer
357 views

Unexpected $\sqrt{3}$

A somewhat lengthy calculation, involving integrals, reveals that the probability an $n\times n$ Hermitian matrix, drawn from the Gaussian unitary ensemble, is positive definite, decays as ...
4
votes
0answers
26 views

Euler number of the complex of basic forms

Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action. I am trying to understand the Hopf bundle ...
2
votes
3answers
336 views

Links between Geometric Group Theory and Number Theory

Do You know any successful applications of the geometric group theory in the number theory? GTG is my main field of interest and I would love to use it to prove new facts in the number theory.
0
votes
0answers
71 views

bound for $|E[\frac{X}{Y}]−\frac{E[X]}{E[Y]}|$

Is there some bound for $|E[X/Y]−E[X]/E[Y]|$ ? where X and Y are both summation of a fixed number of Bernoulli random variables and a constant that is >0, which is to guarantee that the denominator is ...
1
vote
3answers
126 views

Homology of infinite intersection

If $X_1\supseteq X_2\supseteq \ldots$ is a sequence of "nice" compact spaces, I would like to know whether the natural map from $H_*(\cap X_i)$ to the inverse limit $\lim \, H_*(X_i)$ is surjective. ...
2
votes
3answers
197 views

Variations to Cayley's Embedding Theorem for Groups

Early in a course in Algebra the result that every group can be embedded as a subgroup of a symmetric group is introduced. One can further work on it to embed it as a subgroup of a suitable (higher ...
0
votes
0answers
3 views

Abstract Algebra [migrated]

Show that a group that has only a finite number of subgroups must be a finite group.(Fraleigh, A First Course in abstract Algebra-7th Edition,pg.67) I could not show properly so I need help. Thank ...
1
vote
0answers
60 views

Morita equivalence via Kan extension

Are there necessary/sufficient conditions for a functor $f\colon \mathcal C\to \widehat{\mathcal A}$ to induce an equivalence $\text{Lan}_yf=F\colon \widehat{\mathcal C}\leftrightarrows ...
3
votes
3answers
132 views

Existence of nonergodic polygonal billiard

Let $P$ be a polygon in the plane. One can define the billiard flow on the unit tangent bundle of $P$, just following the trajectories of the billiard at speed one. A standard conjecture is that a ...
-1
votes
0answers
28 views

Maximum chi-square distance between norm vectors [on hold]

What is the maximum possible chi-square distance between two normalized vectors? The representation of chi-square distance is below. $d(x,y) = \sum_i \frac{(x_i-y_i)^2}{x_i+y_i}$
1
vote
1answer
75 views

Kaehler form on weighted projective space

The Kaehler potential for the standard Fubini-Study Kaehler form in projective space $\mathbb{C} P^n$ is given by: $$\log(\sum_{i=0}^n |z_i|^2)).$$ What is the analogous formula for a Kaehler ...
2
votes
0answers
34 views

DGBV algebra of symplectic manifold

Let $(M,\omega)$ be a simply connected closed symplectic manifold. Then we have the symplectic codifferential operator $d^{\star}$. Furthermore, $(\Omega^{*}(M),d,d^{\star})$ is a differential ...
-5
votes
0answers
80 views

The problem of Reimann zeta function [on hold]

$\zeta(2)=\sum_0^\infty 1/n^{2}<\pi^2/6=1.644934<2$ From the popular knowledge $\zeta(2)\Gamma(2)=\int_0^\infty x/(e^x-1)dx$ but $\int_0^\infty x/(e^x-1)dx=\int_0^\infty ...
1
vote
0answers
56 views

K-theory of ringed spaces (including henselian and formal schemes); excision and Mayer-Vietoris

Given a ringed topological space $S$ one can easily define its K and K'-theory as the K-theories of the categories of locally free sheaves and of the category of coherent sheaves on it, respectively. ...
-5
votes
0answers
50 views

Prove determinant of nxn matrix is (a+(n-1)b)(a-b)^(n-1)? [on hold]

Prove det(mat) is (a+(n-1)b)(a-b)^n-1 where matrix is nxn matrix with a's on diagonal and all other elements b, off diagonal? For example, suppose matrix with diagonal composed solely of a's. All ...
4
votes
0answers
72 views

$SL(n) \times SL(n)$-invariants of $m$-tuples of matrices

I work over field of complex numbers. Let $G=SL(n) \times SL(n)$, and $(A,B) \in G$ acts on $m$-tuples of matrices $M_{n \times n}(\mathbb{C})^{\oplus m}$ as follows $$ (A,B) \cdot (M_1, \ldots, M_m) ...
3
votes
1answer
169 views

Circle method on things other than the integers

The circle method is often used to estimate the number of solutions to the equation $$x_1 + x_2 + ... x_k = N$$ if for all $i$ $x_i\in A\subseteq\mathbb{N}_0$ and some subset of the nonnegative ...
0
votes
0answers
151 views

Can we construct cohomolgy theory on noetherian separated schemes without Axiom of Choice?

The usual cohomology theory on schemes uses injective or flasque resolutions of quasi-coherent sheaves. Hence it uses Axiom of Choice. However, if the base scheme is a noetherian separated scheme, the ...
-3
votes
0answers
99 views

The problem of Riemann zeta function [on hold]

$\zeta(2)=\sum_0^\infty 1/n^{2}<1.8$ From the popular knowledge $\zeta(2)\Gamma(2)=\int_0^\infty x/(e^x-1)dx$ but $\int_0^\infty x/(e^x-1)dx=\int_0^\infty xe^{-x}/(1-e^{-x})dx$ $=\int_0^\infty ...
3
votes
1answer
160 views

Suslin lines hereditarily Lindelof

I need to prove that every suslin line is hereditarily Lindelof. Any idea will be helpfull.
0
votes
0answers
156 views

Recreating the wheel

I recently finished my Phd in pure maths and I am looking for open problems in my research area, functional analysis. Without going into the details, I stumbled onto an interesting problem and I ...
0
votes
0answers
42 views

Cover one finite subset of integers by another one

Let $A$, $B$ be two finite subsets of integers. We denote by $C(A, B)$ the minimum number of shifts of $A$ to cover $B$. More formally, it can be written as $$ C(A, B)=\min\{|S|: S\subseteq ...
5
votes
1answer
366 views

Is there a sideways-walking rolling convex body?

Let $K$ be a solid, homogenous convex body in $\mathbb{R}^3$. Place $K$ on an inclined plane, and let it roll down the plane, under some reasonable assumptions of friction between $K$ and the plane, ...
0
votes
0answers
31 views

additive support functions of a convex set

Let $K \subset \mathbb{R}^d$ be a compact, convex set. It could be uniquely determined by its support function (for $u$ on the unit-sphere $S^{d-1}$), given by $$h_K(u) = \sup \{ \sum_{i=1}^d x_i ...
-2
votes
0answers
40 views

n-th prime in first order arithmetics [on hold]

Recently I have thought about formalizing Turing machine in first order arithmetics, step by step, starting from the most basic things. But I quickly struck a problem - to continue, I need to find a ...
1
vote
0answers
140 views

A generalized urn-ball matching problem; Complicated combinatoric/probabilistic limit

I'm looking for a generalization to the urn-ball matching problem. As a reminder of what I've got in mind, here's the simple version: Randomly assign (with replacement) $N$ balls to $M$ urns. ...
-1
votes
0answers
82 views

Why integer should have finite many digits? [on hold]

for example, if we take the real part of pi 3.1415926... and write it from right to left like ...6295141, we can get a number, but this number is not a integer, why ? why it is not a integer? Can we ...
1
vote
1answer
22 views

a class of directed hypergraphs

I am interested in a certain class of directed hypergraphs, more precisely in the class of those hypergraphs each of whose hyperedges contain an even number of nodes (not necessarily the same even ...
5
votes
0answers
54 views

When were bordered Heegaard Floer homology's DA bimodules invented?

This is less of a strictly mathematical question and more of a reference request. In their paper "Bordered Heegaard Floer Homology (http://arxiv.org/pdf/0810.0687.pdf)," Lipshitz-Ozsvath-Thurston ...
9
votes
2answers
287 views

What are the invariants of $U\otimes V\otimes W$ under action of $GL(U)\times GL(V) \times GL(W)$

The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case ...
3
votes
1answer
212 views

What year was Hechler forcing created?

Hechler forcing is described on page 278, Jech. Does anyone know when Hechler forcing was first used in a publication?
-2
votes
0answers
34 views

Simplifying Trig Equation with Identifities [on hold]

I have an equation I have been given to solve, I know how to start but I do not know what to do after I use the Trig Identities. Any help? Here is what I was given (cos(A + B) + cos(A - B)) / ...
4
votes
2answers
139 views

Can finitely generated subgroups of limit groups be detected in free group quotients?

In Henry Wilton's excellent paper "Hall's Theorem for limit groups" (Geom. Funct. Anal. 18, pp. 271–303, 2008 ) he proves the following result (see his paper or here and here for the relevant ...
4
votes
0answers
117 views

Universal property of module categories over monads

Let $T$ be a monad on a cocomplete category $\mathcal{C}$. Let's assume that $T$ preserves reflexive coequalizers (or something weaker?). Then the category of $T$-modules $\mathsf{Mod}(T)$ is ...

15 30 50 per page