$$(O( V \oplus F), O( V))$$ is a Gelfand pair for any quadratic space V over a local field F

Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let with the form Q extending q with Q(e) = 1. Consider the standard embedding and the two-sided action of on . In this note we show that any -invariant distribution on is invariant with respect to transposition. This result was earlier proven in a bit different form in van Dijk (Math Z 193:581–593, 1986) for , in Aparicio and van Dijk (Complex generalized Gelfand pairs. Tambov University, 2006) for and in Bosman and van Dijk (Geometriae Dedicata 50:261–282, 1994) for p-adic fields. Here we give a different proof. Using results from Aizenbud et al. (arXiv:0709.1273 (math.RT), submitted), we show that this result on invariant distributions implies that the pair (O(V), O(W)) is a Gelfand pair. In the archimedean setting this means that for any irreducible admissible smooth Fréchet representation (π, E) of we have A stronger result for p-adic fields is obtained in Aizenbud et al. (arXiv:0709.4215 (math.RT), submitted).     

Multipoint Padé approximants to complex Cauchy transforms with polar singularities

We study diagonal multipoint Padé approximants to functions of the form

where R is a rational function and λ is a complex measure with compact regular support included in , whose argument has bounded variation on the support. Assuming that interpolation sets are such that their normalized counting measures converge sufficiently fast in the weak-star sense to some conjugate-symmetric distribution σ, we show that the counting measures of poles of the approximants converge to , the balayage of σ onto the support of λ, in the weak^* sense, that the approximants themselves converge in capacity to F outside the support of λ, and that the poles of R attract at least as many poles of the approximants as their multiplicity and not much more.     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

Universality limits for random matrices and de Branges spaces of entire functions

We prove that de Branges spaces of entire functions describe universality limits in the bulk for random matrices, in the unitary case. In particular, under mild conditions on a measure with compact support, we show that each possible universality limit is the reproducing kernel of a de Branges space of entire functions that equals a classical Paley-Wiener space. We also show that any such reproducing kernel, suitably dilated, may arise as a universality limit for sequences of measures on [−1,1].     

Projected iterative algorithms for complex symmetric systems arising in magnetized multicomponent transport

We investigate iterative algorithms for solving complex symmetric constrained singular systems arising in magnetized multicomponent transport. The matrices of the corresponding linear systems are symmetric with a positive semi-definite real part and an imaginary part with a compatible nullspace. We discuss well posedness, the symmetry of generalized inverses and Cholesky methods. We investigate projected stationary iterative methods as well as projected orthogonal residuals algorithms generalizing previous results on real systems. As an application, we consider the linear systems arising from the kinetic theory of gases and providing transport coefficients of partially ionized gas mixtures subjected to a magnetic field.     

Bootstrap for estimating the MSE of the Spatial EBLUP

This work assumes that the small area quantities of interest follow a Fay–Herriot model with spatially correlated random area effects. Under this model, parametric and nonparametric bootstrap procedures are proposed for estimating the mean squared error of the empirical best linear unbiased predictor (EBLUP). A simulation study based on the Italian Agriculture Census 2000 compares bootstrap and analytical estimates of the MSE and studies their robustness to non-normality. Results indicate lower bias for the non-parametric bootstrap under specific departures from normality.      

The Orthogonal Subcategory Problem and the Small Object Argument

A classical result of P. Freyd and M. Kelly states that in “good” categories, the Orthogonal Subcategory Problem has a positive solution for all classes of morphisms whose members are, except possibly for a subset, epimorphisms. We prove that under the same assumptions on the base category and on , the generalization of the Small Object Argument of D. Quillen holds—that is, every object of the category has a cellular -injective weak reflection. In locally presentable categories, we prove a sharper result: a class of morphisms is called quasi-presentable if for some cardinal λ every member of the class is either λ-presentable or an epimorphism. Both the Orthogonal Subcategory Problem and the Small Object Argument are valid for quasi-presentable classes. Surprisingly, in locally ranked categories (used previously to generalize Quillen’s result), this is no longer true: we present a class of morphisms, all but one being epimorphisms, such that the orthogonality subcategory is not reflective and the injectivity subcategory Inj is not weakly reflective. We also prove that in locally presentable categories, the injectivity logic and the Orthogonality Logic are complete for all quasi-presentable classes. Financial support by Centre for Mathematics of University of Coimbra and by School of Technology of Viseu is acknowledged by the third author.     

基于品牌手机未来销量预测

本文是用灰色预测GM（1,1）模型,探讨了手机销售总数量动态变化,为手机生产提供参考。同时用灰色关联度,来计算总销售量对需要预测的品牌影响,以关联度作为权重,来预测目标品牌。结果表明,对手机销售数量的历史趋势拟合程度较高,所以用此预测模型预测,具有一定的参考价值。     

Functoriality of automorphic L-functions through their zeros

Let E be a Galois extension of Q of degree , not necessarily solvable. In this paper we first prove that the L-function L(s,π) attached to an automorphic cuspidal representation π of GLm(EA) cannot be factored nontrivially into a product of L-functions over E. Next, we compare the n-level correlation of normalized nontrivial zeros of L(s,π1)···L(s,πk), where πj, j = 1,...,k, are automorphic cuspidal representations of GLmj(QA), with that of L(s,π). We prove a necessary condition for L(s,π) having a factoriz...