0
votes
0answers
51 views

Generalized weight space

In their paper Lepowsky and Mcmollum sketch theory of weights in a more general setting. Here is their definition of a weight space: If $A$ is a subset of $\mathfrak g$ and $\lambda$ is a function ...
0
votes
0answers
44 views

Green's function of the Ornstein-Uhlenbeck operator

Consider $\mathbb R^d$ with the Gaussian measure $d\gamma(x) = e^{\frac{1}{4}|x|^2}\,dx$. The Ornstein-Uhlenbeck operator $L$ is given by $$ Lu = \Delta u- \frac{1}{2}x\cdot \nabla u. $$ Is there a ...
1
vote
0answers
42 views

Ozsvath-Szabo orientation convention for Seifert fibred spaces

I am confused by the orientation convention that Ozsvath and Szabo use in On Heegaard Floer homology and Seifert Fibered Surgeries and would appreciate if someone clarifies this for me. On page 15 ...
0
votes
0answers
42 views

Which of the following is true? [on hold]

$f(x),g(x)$ are defined on $[-1,1]$, $f'(0),g'(0)$ exist, $f(0)=g(0)$, and $f(x)\ge g(x)$ holds for an open interval containing $0$. Then which of the following is correct: I, $f(x)$ and $g(x)$ have ...
2
votes
0answers
58 views

Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
2
votes
1answer
53 views

Limit-circle and limit-point at endpoints

I was wondering if the following holds: If you have an ODE $$-y''(x) + q(x) y(x) = \lambda y(x)$$ on a finite interval $(a,b)$ and you know that this equation is limit-circle or limit-point at the ...
2
votes
1answer
117 views

A question about running MMP with scaling

Let $\pi:X \to U$ be a projective morphism, and $(X, \Delta = A + B)$ be a KLT pair, where $A$ is a general ample divisor and $B$ is effective. Suppose $K_X + \Delta$ is not nef (over $U$) and there ...
0
votes
0answers
24 views

question about Baer sum of extensions

Let $E_1$ and $E_2$ be extensions of $\mu_p$ by $\mathbb{Z}/{p\mathbb{Z}}$. Assume that $E$ contains $E_1$ and $E_2$ both, and $E_1 \cap E_2 = \mathbb{Z}/{p\mathbb{Z}}$. Then, does $E$ contain their ...
1
vote
0answers
128 views

Is there any progress on Problem 13 (from Schoen and Yau)?

This is closed related to the question asked here. I wonder if there is any progress on Problem 13 from the "Problem Section" in Schoen and Yau, page 281, problem 13, which asks: Let $M_1$ and $M_2$ ...
3
votes
1answer
120 views

Real points of zero-dimensional real algebraic varieties

There have been a number of discussions of zeros of random polynomials here (the most recent being: Why do roots of polynomials tend to have absolute value close to 1?). Here is a closely related ...
1
vote
0answers
33 views

Classification properties of fusion rings

Fusion rings have so many classification properties (I checked the literature a bit) that my head hurts. For practical reasons I define the following three new properties (which might coincide with ...
-3
votes
0answers
32 views

Does $(n + 2)$ have a multiplicative inverse mod $(n - 1)$ over $GF(5)$? [on hold]

I have been stuck on understanding this for hours. The reason I am confused is that I thought over $GF(k)$, only constants have inverses. Also, how would one go about applying EGDC to figure this out? ...
-2
votes
0answers
75 views

Closure in Hilbertspace [on hold]

I know that i asked this question already on stackexchange.com (http://math.stackexchange.com/questions/983377/closure-in-a-hilbertspace) Define for a pure contraction $S$ (remember: $\|S\|\leq1$ and ...
8
votes
0answers
143 views

Mixed Hodge structure on configuration spaces

Let $X$ be a smooth complex projective variety. Let $F(X,n)$ be the configuration space parametrizing $n$ distinct ordered points in $X$. The cohomology groups $H^k(F(X,n),\mathbf Q)$ carry a mixed ...
0
votes
0answers
43 views

Estimation of growth rate of spectral radius

I have following problem: Let the spectral radius of $S=(a_{ij})_{n\times n}$ be $\lambda>1$, where each $a_{i,j}$ is a positive integer, then we have that $$\lim_{k\to ...
3
votes
1answer
152 views

Chances for a cosine polynomial to be positive at a point

Let $k_1,\ldots,k_n$ be distinct integers. Let $s_n(t)=\cos (k_1t)+\cdots+\cos (k_nt)$ be a trigonometric sum. Consider any interval $I\subset [-\pi,\pi)$ of length $\delta=\delta(n)$. Let $\,U$ be a ...
1
vote
0answers
27 views

How to define Product of Conditional Measures?

I have been wondering how to define the product of conditional measures as defined by Renyi-Popper. I spell the details below. If $(X,\Sigma)$ is a measurable space, then the function $\mu : ...
2
votes
0answers
52 views

Number of maximal chains in Bruhat order

Is there a formula for the number of maximal chains between two permutation in the (strong) Bruhat order?
5
votes
0answers
72 views

Picard scheme of varieties over imperfect fields

Let $k$ be a field and $X$ a proper $k$-scheme. It is a theorem of Murre and Oort that the Picard functor is representable by a $k$-group scheme $\operatorname{Pic}_{X/k}$ which is locally of finite ...
3
votes
0answers
40 views

Reference for existence results for 2D forced viscous Burgers equation

I am looking for results concering the following parabolic PDE $$u\cdot\nabla u + \Delta u = F(x),$$ where $$u\colon\Omega\to\mathbb{R}^2,$$ and $\Omega\subset\mathbb{R}^2$ is a 2D domain (bounded or ...
1
vote
2answers
142 views

Calculating Exterior Distance from Measurements of Inner Geometry

Gauss has proven in his famous Theorema Egregium, that it is possible, to calculate the gaussian curvature from measuring angles and distances on the surface, irrespective of how the surface is ...
1
vote
1answer
62 views

Integrals involving trigonometric functions and polynomes

Let $P(x)$ be a real polynome. Specify all such $P(x)$ that one of the next integrals converge: $$ \int_0^{\infty} sin(P(x))dx, \int_0^{\infty} cos(P(x))dx ? $$ Among special cases are such ...
1
vote
1answer
79 views

On the Saito Kurokawa representation

I know Saito-Kurokawa(SK) representation is the famous non-tempered representation of $SO(5)$. But since the tempered or non-tempered terms are concerned with local phenomenon, I am wondering that ...
3
votes
1answer
148 views

A number array related to colored necklaces and the primes

I stumbled upon entry OEIS-A208535 on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the Moreau ...
4
votes
1answer
124 views

Reference request: Invariant sets of dynamical systems

(I should preface this with the disclaimer that this is a slightly elaborated version of a question that I posted onto math se recently to which I received no responses, and have since deleted the ...
-1
votes
1answer
42 views

differential Proper Maps [on hold]

If $K$ is a subset of $M$ we write $M_{K}(M,M)$ for the set of diffferential maps of $M$ into $M$ with support in $K$.If $K$ is compact,then $M_{K}(M,M)$ consists of maps for which the preimages of ...
0
votes
0answers
29 views

Questions about some special tensor transformation

Suppose tensor $U_{i\alpha\beta}$ with dimension $M*N*N$ satisfy following condition: $$U_{i\beta\alpha}=W^1_{\alpha\alpha'}W^2_{\beta\beta'}U_{i\alpha'\beta'}$$ where $W^1$ and $W^2$ are $N*N$ ...
-4
votes
0answers
51 views

Lemma on Polish Spaces [on hold]

I asked the following question on StackExchange earlier, but received no replies yet. Let $(X_1,\Sigma_1,\mu_1)$ and $(X_2,\Sigma_2,\mu_2)$ be probability spaces. Suppose the following conditions ...
1
vote
0answers
31 views

Anti-Invariant Polynomials of the Dihedral group

I'm interested in the one-dimensional irreducible representations of $D_{2n}$ acting on $\mathbb{R}[x,y]$. I have found that the trivial representations for an algebra freely generated by $x^2+y^2$ ...
0
votes
0answers
51 views

How to prove $\lim _{ \delta\rightarrow {0}^{+}}\int_{a}^{b} F_{\delta}(x)dx=0 ? $ [on hold]

This question comes from http://math.stackexchange.com/questions/982231/a-function-fx-that-riemann-integrable-on-a-b Define a function $f(x)$ that Riemann integrable on $[a,b]$. Let ...
2
votes
0answers
87 views

Invariant Theory over finite adeles

Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$. I am ...
-2
votes
0answers
35 views

estimate angle between two lines y = 1000x and y = 999x [on hold]

How to estimate the angle between line y = 1000 x and y = 999 x? I use the calculator and get 10^(-6) but how to approximate it by hand. Does it relate to Taylor Expansion?
26
votes
4answers
1k views

Why is it so hard to compute $\pi_n(S^n)$?

Of course it isn't really that hard - nowhere near as hard as $\pi_k(S^n)$ for $k>n$, for instance. The hardness that I'm referring to is based on the observation that apparently nobody knows how ...
3
votes
0answers
50 views

The 4-th generator of $K_1$ group for 3-dimensional NC tori algebra

An $n$-dimensional NC torus algebra $A_\theta^{(n)}$ is defined for any antisymmetric $n\times n$ matrix $\theta$ of real numbers as the universal $C^*$-algebra, generated by unitaries ...
2
votes
0answers
74 views

The behavior of series involving special subsets of the prime numbers

It is well known that the series $\sum_{p\in \mathbb{P}} \frac{1}{p}$ diverges where $\mathbb{P}$ denotes the set of primes. Brun proved that $\sum_{p\in \mathbb{P_2}} \frac{1}{p}$ converges where $ ...
8
votes
1answer
173 views

A curious Gauss-Sum type identity

Let $q=e^{2\pi i/m}$, $a\in\mathbb{R}$ and $1\leq j\leq m-1$. I would like to prove that: $$(a-1)\sum_{n=0}^{m-1} q^n\frac{\prod_{k=0}^{j-2} (q^{n+k+1}-a)}{\prod_{k=0}^{j} (aq^{n+k}-1)}=0.$$ For ...
-8
votes
0answers
89 views

Is there a site where I should post questions about mathematics for which I seek a solution? [on hold]

Is there a site where I should post questions about mathematics for which I seek a solution, without risk that it will be closed for not being "research level"?
0
votes
0answers
18 views

Is there effective algorithm for finding “minimal discovery time” for large graphs?

Consider a large, probably sparse graph with Markovian random walkers on it. Define discovery time as time to first reach a vertex by random walk from uniform start. Are there effective ways to find ...
0
votes
0answers
26 views

Isotropic correlation function for a vector valued random field

I'm having trouble with some of the implications of the following theorem. Let $\mathbf{T} (\mathbf{x})$ be a mean-square continuous vector valued random field on $\mathbb{R}^3$ satisfying conditions ...
6
votes
0answers
89 views

What is known about the chromatic number for minimum-distance graphs in higher dimensions?

For a set of points in $\mathbb{R}^d$ with minimum distance $a$, the minimum-distance graph connect two points iff they are at distance $a$. We can also view it as the tangency graph for a set of ...
-5
votes
0answers
85 views

Which univariate function satisfies $e^{g(x)} + e^{-g(x)} = \alpha x$ for $x>0$ and some constant $\alpha>0$? [on hold]

Which univariate function $x \mapsto g(x)$ satisfies $$e^{g(x)} + e^{-g(x)} = \alpha x$$ for $x>0$ and some constant $\alpha>0$? How can it be computed? What does it look like? How can it be ...
-2
votes
0answers
84 views

toledo's lecture on cartwirght-steger surface [on hold]

I am interested in Toledo's lecture given in IAS workshop. I want to find some related reference about his lecture. While actually i am not able to find much. Is someone also interested in this and ...
0
votes
0answers
95 views

A probability application question

Suppose there are two possible states $H$ and $L$, with prior probability $p$ and $1-p$ respectively. There are infinite rounds with a discount factor $ d$. In round 1, you could choose a value ...
1
vote
0answers
27 views

About the $C^{1,1} $regularity of the boundary of a set

I am studying a paper that uses the following property : Consider $U$ and $V$ open, bounded and convex domains in $R^n$ with $U \supset \overline{V}$ and suppose that $min_{y \in V} |x - y| = \lambda ...
2
votes
1answer
117 views

Quantitative stability: Hausdorff distance between subdifferentials

Suppose I have two convex functions $f$ and $g$ mapping $\mathbb{R}^{n}\to\mathbb{R}$ (so they are 'more' than proper). Suppose $\|f-g\|_{\infty}<\epsilon$, the sup-norm. In particular, the ...
1
vote
1answer
120 views

Which $\frak{sl}_2$-Representations Arise From Hermitian Metrics

Recal that $\frak{sl}_2$ is the Lie algebra with basis elements $e,f,h$, and bracket $$ [e,f] = h, ~~~ [h,e] = 2e, ~~~ [h,f] = -2f. $$ For $M$ a $2n$-complex manifold, the Lefschetz identities tell us ...
3
votes
0answers
139 views

Monte Carlo variant of Hilbert's Tenth Problem

Let $k \in \mathbb{N}$. Given an algorithm $\mathcal{A}$ which takes as argument a polynomial $P \in \mathbb{Z}[x_1,\dots,x_k]$ and either returns true or false, we say that $\mathcal{A}$ works for ...
1
vote
1answer
166 views

Can (how) one distinguish germs of continuous functions by a countable set of params?

Continuous functions can be distinguished by their values at say rational points of [0 1]. Germs of analytic functions can be distinguished by derivatives at a point. So in both cases we see ...
-2
votes
0answers
41 views

identity of equation [on hold]

We have the equation ($\partial_{\mu}\partial_{\nu}$-$\eta_{\mu\nu}\Box$)$\phi=0$, where $\phi$ is a scalar field, $\Box=\partial_{\mu}\partial^{\mu}$ is a standart Dalamber operator, $\eta_{\mu\nu}$ ...
-3
votes
0answers
271 views

College - Ecuation to solve in 3 variables (p,q,r) in [1,2] [on hold]

This is the image containing the ecuation

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