-1
votes
1answer
124 views

number of partitions from 0 to n^2 [on hold]

You are given the numbers 0 to n^2. You must use n numbers with no number greater than n to form all the partitions of the numbers 0 to n^2. For example with n=4 you want to find the partitions of 7:...
-3
votes
1answer
146 views

Does one need an external, peer-reviewed grant to become tenured faculty in this field? [on hold]

As a secondary question, how important is it to be awarded grants to remain employed as a PhD mathematician at an academic research institution? Is it common or uncommon for PhD mathematicians to have ...
2
votes
0answers
62 views

Kahler Einstein metric with minimal singularities

Let $X$ be a Kahler variety with snc divisor $D$ such that $K_X+D$ is ample. then there is a Kahler metric $\omega_E$ such that $Ric(\omega_E)=-\omega_E$ on $E=X\setminus D$, then $h=\frac{1}{\...
3
votes
1answer
113 views

Elements in the group von Neumann algebra which are not summable

Let $G$ be a discrete group. I am wondering if there is a recipe which can be applied to find elements in the group von Neumann algebra which are not absolutely summable, i.e. $T \in VN(G)$ while $T \...
0
votes
1answer
47 views

Count Functional digraph [on hold]

Given a set of nodes, how can I count the number of different functional digraph containing a specific number of connected components? With a restriction that no node can have an edge point to itself. ...
1
vote
0answers
34 views

Finding the analytic Zariski decomposition singular hermitian metric on a relative line bundle

Let $f:X\to S$ be a proper surjective projective morphism between complex manifolds with connected fibers and let $D$ be an effective $\mathbb Q$-divisor on $X$ such that $$S^°=\{s\in S| f\text{ is ...
6
votes
1answer
186 views

How to compute $[CP^2, G/PL]$?

Let $E^4$ be the two stage Postnikov space appearing in the homotopy type of the classifying space $G/PL$. One of its properties is that it only has two nontrivial homotopy groups $\pi_2(E)=Z/2Z$ and $...
3
votes
0answers
68 views

Gauss Bonnett on a flat surface with border

I'm reading the article "Euler Characteristics of Teichmuller Curves in genus two" by Matt Bainbridge and there's a point I don't understand in the proof of theorem 5.5. Maybe you can help me clarify ...
2
votes
0answers
70 views

Some queries on Moduli Space of Parabolic Vector Bundles on curve

In "Moduli of Vector Bundles on curves with Parabolic Structures"-Bulletin of the American Mathematical Society Volume 83, Number 1, January 1977 the author announces the following result on moduli ...
2
votes
0answers
90 views

Are all zeros of $\dfrac{\Gamma'}{\Gamma^2}(s) \pm \dfrac{\Gamma'}{\Gamma^2}(1-s)$ either real or on the line with $\Re(s)=\frac12$?

In this post it was proven that, except for a finite few, all zeros of $\Gamma(s) \pm \Gamma(1-s)$ are either real or reside on the line $\Re(s)=\frac12$. I have been searching for similar reflexive $...
1
vote
0answers
36 views

Singular canonical hermitian metric

Let $M$ be a complex manifold , take $$K_{M,m}:=\sup \{|\sigma|^{\frac{2}{m}}; \sigma\in H^0(M,\mathcal O_M(mK_M)), |\int_M(\sigma\wedge\bar\sigma)^{\frac{1}{m}}|\leq 1\}$$ Let $$K_{M,\infty}:=\lim\...
0
votes
0answers
48 views

What is combinatorially possible, for a singular minimal surface in $\mathbf{R}^3$?

A cubic planar graph gives a cell decomposition of a two-sphere, whose dual complex is a triangulation. If I understand Plateau's laws here, a soap film gives a cell decomposition whose dual complex ...
-2
votes
0answers
23 views
4
votes
0answers
182 views

Stopping times for Brownian motion

Let $B_t, t\geq 0$ be standard Brownian motion. Let $\big(\mathcal{G}_t, t\geq 0\big)$ be the natural filtration, defined by $\mathcal{G}_t=\sigma(B_s, 0\leq s\leq t)$. Define also a filtration $\...
0
votes
0answers
37 views

How to build C-Corn for Coq from source? [on hold]

Trying to install C-Corn via opam install. The problem is that since two weeks I do not see any progress, I see only Processing: make . I have i7 , 8 GB RAM. Is that normal or has something gone ...
6
votes
2answers
279 views

Simple proof for $\sum_{i=1}^{n} a^{gcd(i,n)} $ is divisible by n

Burnside's Lemma Deduce That: $$\sum_{i=1}^{n} a^{gcd(i,n)} $$ is divisible by n it's a beautiful result. but i want to prove it without any abstract algebraic tools such as Burnside's Lemma... is ...
1
vote
1answer
109 views

Affine open subsets for algebraic group actions

Let $G$ be a reductive algebraic group over an algebraically closed field $K$ of characteristic zero. Assume that $G$ acts on an affine variety $X$. Assume that $X$ contains an open orbit $U$ (so $\...
2
votes
0answers
48 views

A specific spanning property of a family of vectors

Let $v_1, v_2, v_3, \dots$ be a family of vectors in $\mathbb R^n$ that span $\mathbb R^n$. Given this family, define the family of vectors \begin{align*} \begin{pmatrix} v_1 \\ \alpha_1 v_1 \end{...
0
votes
1answer
66 views

Heat kernel upper bounds on a complete Riemannian manifold

Let $M$ be a complete Riemannian manifold, and $p(t, x, y)$ denotes its heat kernel. I am trying to find sufficient conditions for when the following holds: $$ p(t, x, y) \leq Ct^{-n/2}, \forall x, y, ...
0
votes
0answers
28 views

Evaluation modules of $U_q(L(sl_2))$

Let $a \in \mathbb{C}^{\times}$, $r \in N$. Let $W = V_q(r)$ be the $r$-dimensional irreducible type 1 representation of $U_q(gl_2(\mathbb{C}))$. In the usual basis $\{v_0, \ldots, v_r\}$, the action ...
0
votes
0answers
37 views

$L^\infty$-contractive semigroups

Let $L^\infty(\mathbb T)$ be the space of $2\pi$-periodic and bounded measurable functions and $\mathcal P$ be a pseudo-differential operator defined on $\mathcal D(\mathcal P)\subset L^\infty(\...
3
votes
0answers
76 views

Curves embedding in plane

Given two closed simple(no self-intersection point) curves $C_1,C_2$ in the plane $\mathbb R^2$, is there a good way to judge whether one curve could be embedded inside into the other one, here ...
1
vote
1answer
90 views

a space isomorphic to $S^{p+q}$

I've asked this question few days ago in MathStackExchange, but there was no result. So, I decided to ask it there. In one of the paper I have met that $$\mathbb{S}^{p+q} \cong \mathbb{S}^p \times \...
4
votes
1answer
198 views

Is the assignment of a root system to a complex semisimple Lie algebra functorial?

As described here, we have a category of root systems, where a morphism from a root system $\Phi$ in a Euclidean space $E$ to a root system $\Phi'$ in $E'$ is given by a linear map $f: E \to E'$ such ...
0
votes
0answers
42 views

Is it possible for a quantum group algebra $U_{q}\left(\mathfrak{g}\right)$ to have a diagonal universal $R$-matrix?

I am writing a research paper and have shown that in the special case when a quantum group algebra $U_{q}\left(\mathfrak{g}\right)$ with the quantum group parameter $q$ not a root of unity has a ...
1
vote
0answers
36 views

Concentration of functional of Gaussian random variable

Suppose I have two Gaussian distributions $p(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_p|^{1/2}}\exp(-\frac{1}{2}x^\top \Sigma_p^{-1} x)$ and $q(x) = \frac{1}{(2\pi)^{d/2}|\Sigma_q|^{1/2}}\exp(-\frac{1}{2}x^\...
6
votes
1answer
98 views

Chevalley restriction theorem for non-split Cartan

Let $G$ be a reductive group over a field $k$ with maximal torus $H$. Let $\mathfrak{g}$ and $\mathfrak{h}$ denote the corresponding Lie algebra. If $k$ is algebraically closed, we have a theorem of ...
2
votes
0answers
45 views

Average range of Motzkin path

Motzkin path are paths from (0,0) to (n,0) in $\mathbb{Z}^2$ such that we are allowed to move SE, E and NE. More on this is here https://en.wikipedia.org/wiki/Motzkin_number I would like to know if ...
0
votes
0answers
27 views

Recast a finite-dimensional multiparameter SDE as an infinite dimensional SDE

In another question, I've asked how we can derive a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories. More concretely, I want to obtain a SDE of type as ...
8
votes
1answer
193 views

Groups whose word problem can be solved in constant time

Given a finitely generated group $G$, define an encoding of $G$ to be a one-to-one function $\Phi:G\to \bigcup_n \{0,1\}^n$ that sends each group element to a unique finite word. For $a,b\in G$, ...
5
votes
0answers
36 views

Exponential approximation for 3F2 hypergeometric function with repeated indices

In my research I have run across the hypergeometric function $${}_3F_2(d,d,d;d+1,d+1;z)$$ where d is a positive integer and 0≤z≤1. When I plot this as a function of d on a semilog plot, it appears to ...
0
votes
0answers
31 views

Making a multivariate polynomial monic in one of its variables

I apologise in advance for the general nature of this question. Suppose we have a non-commutative ring $R$ that is relatively well-behaved as non-commutative rings go (I was thinking of $R$ being the ...
6
votes
1answer
119 views

q-analog of a combinatorial identity involving binomial coefficients

Using, e.g., properties of iterated finite differences it is easy to show that for any pair of integers $n$ and $m$ with $n>\!>m$ one has the identity $$ \sum_{k=0}^m(-1)^{k-m} {n-k\choose m}{m\...
5
votes
1answer
366 views

Uniqueness of sums of roots of unity

Let $\zeta:=e^{\frac{2\pi i}{n}}$, with $n\geq4$, and let $2\leq k\leq n-2$. Let us suppose that the prime factorization of $n$ is $n=p_1^{\alpha_1}\cdot\dots\cdot p_s^{\alpha_s}$, with $\alpha_i>...
-1
votes
0answers
58 views

Random walk on d-dimensional torus

I am reading the following paper: http://arxiv.org/pdf/1602.03849v2.pdf I will explain the general setup below. Let $x\in X=\mathbb{T}^d$, where $\mathbb{T}^d$ is the d dimensional torus. Let $\rho$ ...
-2
votes
0answers
197 views

About consistency of New foundations

Add a one place function symbol $F$ to the first order language of set theory. Define $T$ as follows: $T = Z + \forall n,m (F(n)=F(m) \iff\ n=m) + \exists V:$ $\forall S ((\forall y \in S \exists a,...
-1
votes
0answers
51 views

FInd a straight line, which goes through 2 points [on hold]

I am new to analytical geometry and excuse me for my notations. We have four lines: ...
1
vote
1answer
128 views

Polynomial ring operations on $\mathbb{Z}$

I have asked this on Math Stack Exchange but without answers: The usual ring operations on $\mathbb{Z}$ can be defined via polynomials in $\mathbb{Z}[a,b]$ (when viewing $a,b$ as variables): ...
4
votes
0answers
75 views

Symbol of differential operator and change of variables [on hold]

Recently I posted the following question on stack exchange, but it remained with no answer http://math.stackexchange.com/questions/1863658/symbol-of-differential-operator-and-change-of-coordinates I ...
2
votes
0answers
28 views

Automorphic forms annihilated by $I_1$ but not $I_2$ for finite codim ideals $I_1 \subsetneq I_2$

Suppose $G$ is a connected real reductive Lie group, $\mathfrak{g} = \text{Lie}(G)$, and $\mathcal{Z} = \mathcal{Z}[U(\mathfrak{g})]$ the center of the universal enveloping algebra of $\mathfrak{g}$. ...
0
votes
1answer
43 views

Isometry of the Gordon-Webb-Wolpert's problem in $2$-dimensions [on hold]

I showed the problem of Gordon-Webb-Wolpert in $2$-dimensions is isospectral, but I have some doubts how to show that they are not isometric. Could anyone be able to explain to me rigorously and in ...
3
votes
1answer
145 views

What kind of set theory is obtained from the canonical models of K?

Consider the minimal normal modal logic $K$ (axioms = classical propositional logic + $(\Box(p\land q)\leftrightarrow\Box p\land\Box q)$ + $(\Box\top)$, nothing else). Its canonical model with no ...
5
votes
0answers
120 views

Classifying map for a surface bundle

Let $E\longrightarrow X$ be a surface (with holes) bundle. The structure group is then $M_{g, s}$, the mapping class group of the fiber. It follows from the famous work of Penner that the classifying ...
1
vote
1answer
122 views

Spinor bundle of line bundle

Given any complex line bundle over a manifold $L\to M$, we know this admits a canonical Hermitian spinor bundle $S$. Suppose we know the first Chern class of the line bundle, i.e. $c_1(L)$ is known. ...
1
vote
1answer
57 views

Compactness and omega models

If $T$ is a first order set theory having finitely many axioms, suppose the consistency of $T$ is already known and that $T$ proves existence of naturals, now suppose that $S$ is a schema and that $T+...
3
votes
0answers
31 views
+100

Matrix semigroups in which a weighted average of eigenvalues is multiplicative

A problem in fractal geometry requires me to consider matrix semigroups with the following curious property. For a $d \times d$ real matrix $A$ let $\lambda_1(A) \geq \lambda_2(A) \geq \cdots \geq \...
-2
votes
0answers
83 views

What are your favourite correspondences? [on hold]

For example, I like the Curry-Howard correspondence. But I'm interested in gathering up a bunch of interesting examples from various fields.
0
votes
0answers
21 views

About Cahn - Hilliard equation solution uniqueness

The uniqueness of the solutions of the Cahn - Hilliard nonlinear PDE $$\dfrac{\partial c}{\partial t}=\nabla\dot{}(M\nabla\mu)$$ has been proved for many form of the chemical potential $\mu$. What ...
7
votes
1answer
188 views

Reference for push-pull formula in cohomology

I would like a precise reference for the following fact. Assume that $$ \begin{array}{ccc} M\times_B N & \stackrel{f'}{\to} & N \newline \quad\downarrow g' & & \quad\downarrow g \...
-2
votes
0answers
72 views

If we find compactification of dense subset of a topological spaces, then what can we say about compactification of original space? [on hold]

Let $(X,T)$ be a topological space and $F$ a dense subset of $X$. Suppose that we have compactification of $(F,T)$, $(f,\beta F)$. Does $(X,T)$ possess a compactification?

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