# All Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $M_2$ ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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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 $$\lim_{k\to ... 1answer 156 views ### Chances for a cosine polynomial to be positive at a point Let k_1,\ldots,k_n be distinct integers. Let s_n(t)=\cos (k_1t)+\cdots+\cos (k_nt) be a trigonometric sum. Consider any interval I\subset [-\pi,\pi) of length \delta=\delta(n). Let \,U be a ... 0answers 27 views ### How to define Product of Conditional Measures? I have been wondering how to define the product of conditional measures as defined by Renyi-Popper. I spell the details below. If (X,\Sigma) is a measurable space, then the function \mu : ... 0answers 52 views ### Number of maximal chains in Bruhat order Is there a formula for the number of maximal chains between two permutation in the (strong) Bruhat order? 0answers 73 views ### Picard scheme of varieties over imperfect fields Let k be a field and X a proper k-scheme. It is a theorem of Murre and Oort that the Picard functor is representable by a k-group scheme \operatorname{Pic}_{X/k} which is locally of finite ... 0answers 43 views ### Reference for existence results for 2D forced viscous Burgers equation I am looking for results concering the following parabolic PDE$$u\cdot\nabla u + \Delta u = F(x),$$where$$u\colon\Omega\to\mathbb{R}^2,$$and \Omega\subset\mathbb{R}^2 is a 2D domain (bounded or ... 2answers 149 views ### Calculating Exterior Distance from Measurements of Inner Geometry Gauss has proven in his famous Theorema Egregium, that it is possible, to calculate the gaussian curvature from measuring angles and distances on the surface, irrespective of how the surface is ... 1answer 62 views ### Integrals involving trigonometric functions and polynomes Let P(x) be a real polynome. Specify all such P(x) that one of the next integrals converge:$$ \int_0^{\infty} sin(P(x))dx, \int_0^{\infty} cos(P(x))dx ? $$Among special cases are such ... 1answer 80 views ### On the Saito Kurokawa representation I know Saito-Kurokawa(SK) representation is the famous non-tempered representation of SO(5). But since the tempered or non-tempered terms are concerned with local phenomenon, I am wondering that ... 1answer 153 views ### A number array related to colored necklaces and the primes I stumbled upon entry OEIS-A208535 on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the Moreau ... 1answer 129 views ### Reference request: Invariant sets of dynamical systems (I should preface this with the disclaimer that this is a slightly elaborated version of a question that I posted onto math se recently to which I received no responses, and have since deleted the ... 1answer 42 views ### differential Proper Maps [on hold] If K is a subset of M we write M_{K}(M,M) for the set of diffferential maps of M into M with support in K.If K is compact,then M_{K}(M,M) consists of maps for which the preimages of ... 0answers 29 views ### Questions about some special tensor transformation Suppose tensor U_{i\alpha\beta} with dimension M*N*N satisfy following condition:$$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 ... 0answers 52 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 ... 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 ... 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 ... 0answers 93 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 ... 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? 4answers 2k 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 ... 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 ... 0answers 75 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  ... 1answer 177 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.$$For ... 0answers 89 views ### Is there a site where I should post questions about mathematics for which I seek a solution? [closed] Is there a site where I should post questions about mathematics for which I seek a solution, without risk that it will be closed for not being "research level"? 0answers 18 views ### Is there effective algorithm for finding “minimal discovery time” for large graphs? Consider a large, probably sparse graph with Markovian random walkers on it. Define discovery time as time to first reach a vertex by random walk from uniform start. Are there effective ways to find ... 0answers 26 views ### Isotropic correlation function for a vector valued random field I'm having trouble with some of the implications of the following theorem. Let \mathbf{T} (\mathbf{x}) be a mean-square continuous vector valued random field on \mathbb{R}^3 satisfying conditions ... 0answers 89 views ### What is known about the chromatic number for minimum-distance graphs in higher dimensions? For a set of points in \mathbb{R}^d with minimum distance a, the minimum-distance graph connect two points iff they are at distance a. We can also view it as the tangency graph for a set of ... 0answers 85 views ### Which univariate function satisfies e^{g(x)} + e^{-g(x)} = \alpha x for x>0 and some constant \alpha>0? [closed] Which univariate function x \mapsto g(x) satisfies$$e^{g(x)} + e^{-g(x)} = \alpha x$$for x>0 and some constant \alpha>0? How can it be computed? What does it look like? How can it be ... 0answers 84 views ### toledo's lecture on cartwirght-steger surface [on hold] I am interested in Toledo's lecture given in IAS workshop. I want to find some related reference about his lecture. While actually i am not able to find much. Is someone also interested in this and ... 0answers 96 views ### A probability application question Suppose there are two possible states H and L, with prior probability p and 1-p respectively. There are infinite rounds with a discount factor  d. In round 1, you could choose a value ... 0answers 27 views ### About the C^{1,1} regularity of the boundary of a set I am studying a paper that uses the following property : Consider U and V open, bounded and convex domains in R^n with U \supset \overline{V} and suppose that min_{y \in V} |x - y| = \lambda ... 1answer 117 views ### Quantitative stability: Hausdorff distance between subdifferentials Suppose I have two convex functions f and g mapping \mathbb{R}^{n}\to\mathbb{R} (so they are 'more' than proper). Suppose \|f-g\|_{\infty}<\epsilon, the sup-norm. In particular, the ... 1answer 120 views ### Which \frak{sl}_2-Representations Arise From Hermitian Metrics Recal that \frak{sl}_2 is the Lie algebra with basis elements e,f,h, and bracket$$ [e,f] = h, ~~~ [h,e] = 2e, ~~~ [h,f] = -2f. $$For M a 2n-complex manifold, the Lefschetz identities tell us ... 0answers 144 views ### Monte Carlo variant of Hilbert's Tenth Problem Let k \in \mathbb{N}. Given an algorithm \mathcal{A} which takes as argument a polynomial P \in \mathbb{Z}[x_1,\dots,x_k] and either returns true or false, we say that \mathcal{A} works for ... 1answer 173 views ### Can (how) one distinguish germs of continuous functions by a countable set of params? Continuous functions can be distinguished by their values at say rational points of [0 1]. Germs of analytic functions can be distinguished by derivatives at a point. So in both cases we see ... 0answers 42 views ### identity of equation [closed] We have the equation (\partial_{\mu}\partial_{\nu}-\eta_{\mu\nu}\Box)\phi=0, where \phi is a scalar field, \Box=\partial_{\mu}\partial^{\mu} is a standart Dalamber operator, \eta_{\mu\nu} ... 0answers 272 views ### College - Ecuation to solve in 3 variables (p,q,r) in [1,2] [closed] This is the image containing the ecuation 1answer 41 views ### orthonormal basis or Parseval frame for Sobolev spaces Consider the uni-variate Sobolev space of order m:$$ \mathcal{W}_2^m=\{ f:[0,1] \rightarrow \mathbb{R}|f,f^{(1)},\dots ,f^{(m-1)}\, \text{are absolutely continuous and}\, f^{(m)}\in L^2\}. It is ...
Let $k$ be any field. Let $r$ and $n$ be two positive integers. Consider the functor $F$ from the category of $k$-schemes to the category of sets which sends a $k$-scheme $T$ to the set of matrices ...