1
vote
0answers
97 views

Is there a non-integer in the dimension spectrum for the Heisenberg group?

Let $\Gamma = \langle a,b,c \ | \ c=aba^{-1}b^{-1}, \ ac=ca, \ bc = cb \rangle$ be the discrete Heisenberg group. Let $\ell: \Gamma \to \mathbb{N} $ be the word length on $\Gamma$. This group has a ...
3
votes
0answers
97 views

Kunneth decomposition of the relative diagonal of a projective bundle

Let $\mathcal{E}$ be a projective bundle of rank $r$ over a smooth complex quasi projective variety $B$, and form its associated projective bundle $\chi :=\mathbb{P}(\mathcal{E})$. Let $\pi : \chi ...
-3
votes
0answers
47 views

Find function $h$ so that $h(U,V)$ equals density of $f(a)da$ for $f(a)=\frac{1}{2}e^{-\small|a|} ,a \in \mathbb R$ [on hold]

Let $f(a)=\frac{1}{2}e^{-\small|a|}$, $a \in \mathbb R$ and let $U,V$ be two independently uniformly distributed random variables on $[0,1]$. Now I want to find a function $h$ so that $h(U,V)$ is ...
2
votes
0answers
103 views

Semistability of a sheaf on nodal curve

Suppose $X$ is a projective, connected, nodal curve (can be reducible) over an algebraically closed field $k$ of arbitrary characteristic. Let $F$ be a pure sheaf on $X$ and denote by $\pi^{*}(F)$ its ...
-5
votes
0answers
40 views

Difficult derivative of the log of a function [on hold]

Can someone help me figure out how the derivative of log p(x) wrt. theta becomes to solution below ? $ -\frac{\delta logp(x)}{\delta \theta} = \frac{\delta F(x)}{\delta \theta } - \sum p(x) ...
3
votes
0answers
58 views

How many unimodular lattices does it take to fill a cube with high probability?

Consider $C_a$ in $\Bbb Z^n$ a cube of height $a$ at origin in positive coordinates with one corner at origin. Consider the set $M_c$ of all unimodular matrices in $\Bbb Z^{n\times n}$ with each ...
4
votes
0answers
77 views

Singular symplectic reduction in infinite dimension

In 1991, Sjamaar and Lerman [1] introduced the notion of stratified symplectic spaces. Namely, if $M$ is a symplectic manifold and $G$ a Lie group acting properly (but not necessarily freely) on $M$ ...
3
votes
0answers
44 views

Are Bipartite Matching and General Matching Really Different Problems?

Questions: Have there been attempts to either prove or disprove, that every general matching problem can be transformed into a bipartite matching problem, from whose solution the solution ...
0
votes
0answers
56 views

Reference request: Uniformly totally bounded classes of compact metric spaces are Gromov-Hausdorff precompact

The following Theorem can be found for instance here (Theorem 7.4.15): Theorem. (author ?) Any uniformly totally bounded class $\mathfrak X$ of compact metric spaces is pre-compact in the ...
7
votes
0answers
97 views

Determinants of octonionic hermitian matrices

For quaternionic hermitian matrices (i.e. quaternionic square matrices $(a_{ij})$ satisfying $a_{ji}=\bar a_{ij}$) there is a nice notion of (Moore) determinant which can be defined as follows. ...
5
votes
0answers
51 views

Lower bound for smallest eigenvalue of symmetric doubly-stochastic Metropolis-Hasting transition matrix

For my master's thesis research, I stumbled upon a question concerning the Metropolis-Hasting transition matrix $W$. Context $\quad$ Let me start with some context. I consider connected undirected ...
0
votes
0answers
80 views

Mathematical consulting or bioinformatics related careers for mathematicians with good statistics and coding experience in West Europe [on hold]

Before I start, apologies if the question is very specific, but these are exactly what I want to be. I should mention that I already studied: "Industry"/Government jobs for mathematicians ...
2
votes
1answer
66 views

IFS maps on circle

A systems $<f_0,f_1>$ is minimal if the set $\{h(x): h=f_{i_n}\circ f_{i_{n-1}}\circ...\circ f_{i_1}, i_k \in \{0,1\},n>0\}$ is dense in $S^1$, for every $x\in S^1$. Consider $f:S^1 \to S^1, ...
1
vote
1answer
35 views

Maximizing joint entropy?

I'm stuck trying to find the maximum entropy probability distribution taking into account a joint distribution. Basically, I want to find the maximum entropy expression for $p(x,y)$ when the marginal ...
0
votes
0answers
61 views

Derandomizing AP existence in $A\subseteq \{1,\ldots,N\}$ for $\delta(A) \geq 1/k$

In the answer to the mathoverflow question here, it was established that if we let $p$ be the probability of including point $v$ in $A\subseteq \{1,\ldots,N\}$ and this is done independently for all ...
12
votes
1answer
261 views

Measuring the failure of pushforward to commute with Steenrod squares

Let $f \colon X \rightarrow Y$ be a map of topological spaces. Let's say that they are (closed) manifolds (not necessarily orientable), though to be honest I'm really interested in the more general ...
3
votes
0answers
48 views

Real interpolation of weighted Sobolev spaces with different weights

Let $\Omega \subseteq \mathbb{R}^n$ be open and let $w_0$ and $w_1$ be measurable and almost everywhere positive and finite functions defined on $\Omega$. Let $L^2_{w_0}(\Omega)$ be the weighted ...
1
vote
0answers
77 views

Decompositon of the Euler class in the ideal generated by Weyl-invariant polynomials

Let $G$ be a complex reductive Lie group, $B$ be a Borel subgroup, $T\subset B$ be a maximal torus, $W$ be the Weyl group. Then the space $X:=G/B$ is a complex manifold of dimension $n$, denote by ...
0
votes
0answers
55 views

Splitting of totally geodesic Riemannian foliations

Let $\mathcal F$ be a non-singular Riemannian foliation on $(M,g)$ whose leaves are totally geodesic. Suppose further that the leaves are Riemannian products of irreducible manifolds $L=L_0\times ...
5
votes
1answer
166 views

Bounding the degree of an algebraic extension containing solutions to polynomials

Also posted on math.stackexchange... Let $F$ be a field, and let $f_{1},\ldots, f_{s}$ be polynomials in $F[x_{1},\ldots, x_{t}]$. Assume that the degree of the polynomials is bounded by $d$, by ...
5
votes
1answer
114 views

Density of non-algebraic leaves in the characteristic foliation

Let $X$ be a compact complex manifold equipped with a holomorphic symplectic form $\omega$. Let $D$ be a smooth divisor on $X$. At each point of $D$, the restriction of $\omega$ to $D$ has ...
6
votes
0answers
175 views

Universal Property of Fontaine's Period Ring $B_{dR}^+$

In the introduction to his Asterisque Expose "Le Corps des Periodes p-Adiques", Fontaine announces a characterization of $B_{dR}^+$ by some universal property. Unfortunatly, at least for $B_{dR}^+$ ...
0
votes
0answers
39 views

On a count of certain number of primes in an interval

Fix a prime $p$, $\alpha\in(0,1)$ and $\beta\in(1,2)$ and let $\mathcal U$ be primes in $[p^\alpha,\beta p^\alpha]$ such that if $b\in\mathcal U$ and if $d$ is multiplicative inverse of $b$ in $\Bbb ...
1
vote
3answers
115 views

Hadwiger number and minimal degree

Suppose $G$ is a finite simple graph and $\eta(G)$ is the maximal $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$. If $\delta(G)$ is the minimal degree, do we have $\delta(G)\leq\eta(G)$?
11
votes
0answers
154 views

Concept associated to the Eudoxus reals

I am aware of three different constructions of the field of real numbers : The Cauchy sequence construction : in this case, we see the field $\mathbb{Q}$ as a metric space and $\mathbb{R}$ is the ...
0
votes
0answers
283 views

Becoming a Mature Mathematician [on hold]

I am currently a sophomore in my undergraduate mathematics program. It has taken me a while to take school seriously; I was one of "those" students who just skated by without studying until Linear ...
2
votes
0answers
122 views

Are these moduli problems of curves “well-behaved”?

Let X be a smooth projective surface over $\mathbb C$, and let $d\geq 3$ be an integer. Suppose that all smooth hypersurfaces of degree $d$ are of genus $g\geq 2$. Let $H_{X,d}$ be the Hilbert scheme ...
-6
votes
0answers
27 views

applications that involves the Legendre Polynomials [on hold]

i am requested to make a basic research about the applications that involve the Legendre Polynomials.. thanks in advance
1
vote
1answer
165 views

harmonic analysis on $p$-adic $x^2 + y^2 + z^2 = 1$?

Are there p-adic analogues to spherical harmonics? In the case of $K = \mathbb{R}$, the spherical harmonics form a basis to $L^2 [SO(3)]$ where What happens in the $p$-adic case? Is there sphere ...
1
vote
0answers
17 views

History of the Vertex Disjoint Cycle Cover with Minimal Edgeweight Sum

Questions: who first posed the problem of determining a collection of (directed) cycles, whose edgeweight sum is minimal and, for which each vertex belongs to exactly one of the cycles? who came up ...
2
votes
0answers
43 views

If $f$ is dynamically coherent, is there a unique invariant foliation tangent to $E^{c}$?

Let $f$ be a diffeomorphism of a closed manifold $M$ such that $f$ is partially hyperbolic if the tanget bundle of $M$, $TM$ splits into three invariant sub-bunbles $$ TM = E^{s} \oplus E^{c} \oplus ...
-4
votes
0answers
38 views

finding eigenvector [on hold]

I have where λ1 = λ2 = 6 and λ2 = λ3 = 0. I wish to find the eigenvectors for these eigenvalues above. I've tried to turn it into equations and trying to solve them (this is for λ1 & λ2): ...
0
votes
0answers
11 views

Stucture of inverse (MP) of totally positive rectangular matrix

The special structure of inverse of non-singular totally positive square matrix (whose all entries are positive) discussed in MO(see here). The inverse has a special structure (M-matrix). With some ...
0
votes
0answers
41 views

SVD alternatives for symmetric matrices

Given any symmetric real valued matrix $A \in \mathbb{R}^{n\times n}$, I can decompose $A$ as the product of two complex matrices $$ A = E'E $$ Practically this can be done easily using SVD ...
0
votes
0answers
60 views

How to find a basis of weight vectors? [on hold]

I have to following Lie Algebra $L=\{x\in End(\mathbb{C}^6)\colon x^tS+Sx=0\}$, where $S=[\begin{smallmatrix} 0&I_3 \\ I_3&0 \end{smallmatrix}]$, and the subalgebra $H$ given by the diagonal ...
0
votes
0answers
27 views

Mixed PDE/finite difference equation

I have the following mixed pde/finite-difference equation for $f(t,x,y)$: $a x^2 f_{xx} + bxf_x + f_t - bxy + c\sinh(d\delta) = 0$ subject to $f(T,x,y)=0$, $x>0,\ t\geq 0,\ y\in\mathbb Z$, ...
15
votes
0answers
203 views

What are the indecomposable $U_q\mathfrak{sl}(2)$-modules?

Let $\mathfrak g=\mathfrak{sl}(2)$. Let $\zeta$ be a primitive root of unity of even order. Say $\zeta=e^{2\pi i/6}$, for concreteness. Let $U_q\mathfrak g$ be Lusztig's integral form of the ...
2
votes
1answer
140 views

projectivity with assumption of big and semi-amplness

Let $X$ be a compact Kaehler manifold with $D$ be an effective divisor on $X$ such that $K_X+D$ is semi-ample and big then $X$ is projective?
3
votes
0answers
58 views

Correspondence between dual center and linear characters of finite reductive group

Let $(G,F)$ be a connected reductive group defined over $\mathbb{F}_q$ via the Frobenius $F$ and let $(G^*,F^*)$ be a group in duality with $(G,F)$ with respect to rational maximal tori $T \subseteq ...
2
votes
1answer
124 views

(quasi)metric on Riemannian manifolds via Brownian Motion?

Given points $a$ and $b$ on a Riemannian manifold $M$, I would like a (quasi)metric that corresponds to some property of Brownian Motion from $a$ to $b$ (or rather, to $\epsilon$-ball $B = \{ x : |x - ...
-2
votes
0answers
24 views

Cumulative distribution function and sum of random variables [on hold]

For two continuous (iid) random variables $X$ and $Y$, we have (ref): $$ \mathbb{P}(X+Y \le a) =\int_{-\infty}^\infty \int_{-\infty}^{c-x} \big ( f(x,y) dy \big ) dx$$ with $f$ being the joint density ...
1
vote
0answers
89 views

Presentation of hyperbolic groups [on hold]

Is it true that all hyperbolic groups are finitely presented? If yes, what is the right reference for that?
3
votes
1answer
78 views

geodesics on $G/K$ which are not the orbits of a 1-parameter subgroup of $G$

Let $G$ be Lie group and $K \subset G$ a closed subgroup, such that there exists a $v \in T(G/K)$ whose isotropy-group $G_v$ is discrete (so iff $\dim G_v =0$). Lets assume $g$ acts properly on ...
-1
votes
1answer
51 views

Hadwiger number of total graph

Let $G=(V,E)$ be a finite, simple, undirected graph, and let $T(G)$ be its total graph. The Hadwiger number $\eta(G)$ is the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$. Is there an ...
9
votes
0answers
269 views

What's wrong with Advanced Studies in Contemporary Mathematics (Kyungshang)?

By some reason the Journal mentioned in the title is no longer covered by the AMS Math. Reviews. On the MathSciNet web page it says: Last Issue: 24, no. 1  2014 Indexed cover-to-cover Status: No ...
1
vote
1answer
149 views

s(n) = kn or s(n) = n/k? [on hold]

This is not an important question, just for fun. Definition: $\sigma (n)$ = sum of the positive divisors of $n$. $s(n)$ = sum of the proper positive divisors of $n$. For $s(n) = kn$ , where $k$ ...
-5
votes
0answers
41 views

Homomorphism, Group Theory [on hold]

Let G=Z4, the group of integers modulo 4, and let H be the Klein four group, let f: G->H be a homomorphism. Why does the kernel of f must contain the element of 2 of G?
3
votes
0answers
111 views

Critical case of Sobolev Embedding

I got stuck in the following lemma: Lemma 1: Let $B$ be the unit ball in the 4 dimensional Euclidean space. Suppose that $u\in W^{2,2}(B)$, then $e^{u}\in L^{q}$ for any $q\geq1$. Actually, I ...
4
votes
1answer
51 views

$G$-orbits in Springer resolution (or, stabilizer actions on Springer fibers)

This may be an elementary question, but I'm having trouble coming up with an answer: Let $\tilde{N} = T^*(G/B)$ be the Springer resolution of the nilpotent cone. Does it have finitely many ...
30
votes
2answers
3k views

Why is this new result such a big deal?

This popular article reports a recent result in reverse mathematics, showing that a certain theorem in Ramsey theory is provable from RCA$_0$, the base theory in SOSOA. Then there are a bunch of ...

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