On CAP Representations for Even Orthogonal Groups Ⅰ:A Correspondence of Unramified Representations

The authors prove the local unramified correspondence for a new type of construction of CAP representations of even orthogonal groups by a generalized automorphic descent method. This method is expected to work for all classical groups.     

Nonsymmetric Macdonald polynomials and matrix coefficients for unramified principal series

We show how a certain limit of the nonsymmetric Macdonald polynomials appears in the representation theory of semisimple groups over p-adic fields as matrix coefficients for the unramified principal series representations. The result is the nonsymmetric counterpart of a classical result relating the same limit of the symmetric Macdonald polynomials to zonal spherical functions on groups of p-adic type.     

Algorithms for Computing Generalized U(<Emphasis Type="Italic">N</Emphasis>) Racah Coefficients

Algorithms are developed for computing generalized Racah coefficients for the U(N) groups. The irreducible representations (irreps) of the U(N) groups, as well as their tensor products, are realized as polynomials in complex variables. When tensor product irrep labels as well as a given irrep label are specified, maps are constructed from the irrep space to the tensor product space. The number of linearly independent maps gives the multiplicity. The main theorem of this paper shows that the eigenvalues of generalized Casimir operators are always sufficient to break the multiplicity. Using this theorem algorithms are given for computing the overlap between different sets of eigenvalues of commuting generalized Casimir operators, which are the generalized Racah coefficients. It is also shown that these coefficients are basis independent. Mathematics Subject Classifications (2000) 22E70, 81R05, 81R40.     

Introducing technology in algebra in Hong Kong: addressing issues in learning

A review of recent studies in Hong Kong suggests that possible problems in secondary school algebra may be due to the procedural paradigm orientation in the curriculum and the conventional style of teaching in the classroom which do not provide sufficient opportunities for students to develop conceptual understanding. Based on the works of a number of projects in the West, it is hypothesized that the introduction of technology in lessons which embody a cognitive model in their design and delivery will provide a viable alternative for enhancing algebraic thinking. The key features of the cognitive model are concrete preparation, cognitive conflict, construction, metacognition and bridging, imbedded in a ‘mediation’ style of teaching. These features are exemplified by detailed accounts of classroom episodes from a Hong Kong lesson supported by the use of graphics calculators. Results indicate that while the approach places particular demands on teachers (a tension between a transmission style of teaching and mediating) there is real potential for supporting more dynamic student constructions.     

有关Lucas与Babbage同余式的推广

利用一个素数模意义上的整数分拆的结果,证明了高斯二项式系数上的Lucas同余式,并且分别研究了Lucas同余式和Babbage同余式在广义二项式系数上的情况,得到了相应的形式简洁的同余式.     

lp(1≤p<∞)商空间的Maluta常数

为研究介于正规结构和一致正规结构之间的Banach空间的结构性质，Maluta[1]给出Banach空间X的Maluta常数D（X）的定义，该文给出当M是X的具Schur性的可余闭子空间时有D（X／M）＝D（X），并计算出lp商空间的Maluta常数．     

Homology of Orthogonal Groups: A Quadratic Algebra

We prove that a graded-commutative algebra, composed of certain homology groups of orthogonal groups with twisted coefficients, is a quadratic algebra.     

几类二阶变系数微分方程的求解

给出了可化为常系数或可降阶的变系数二阶微分方程的条件及在此条件下求变系数微分方程的解.     

The evaluation of connection coefficients on an interval

A connection coefficient is an interval of products of wavelet basis functions, their derivatives and their translates. It can be used to solve PDEs numerically by the wavelet-Galerkin method. This paper presents an exact method for evaluating connection coefficients on an interval.     

Negation of binomial coefficients

We extend the concept of a binomial coefficient to all integer values of its parameters. Our approach is purely algebraic, but we show that it is equivalent to the evaluation of binomial coefficients by means of the Γ-function. In particular, we prove that the traditional rule of “negation” is wrong and should be substituted by a slightly more complex rule. We also show that the “cross product” rule remains valid for the extended definition.     

Bredon-Illman cohomology with actions of totally disconnected groups

We study the Bredon-Illman cohomology with local coefficients for a G-space X in the case of G being a totally disconnected, locally compact group. We prove that any short exact sequence of equivariant local coefficients systems on X gives a long exact sequence of the associated Bredon-Illman cohomology groups with local coefficients.     

A free boundary problem describing S–K–T competition ecological model with cross-diffusion

In this paper we investigate a free boundary problem describing S–K–T competition ecological model with two competing species and with cross-diffusion and self-diffusion in one space dimension, where one species is made up of two groups separated by a free boundary, and the other has a single group. The system under consideration is strongly coupled and the coefficients of the equations are allowed to be discontinuous. We first show the global existence and uniqueness of the solutions for the corresponding diffraction problem by approximation method, Galerkin method and Schauder fixed point theorem, and then prove the local existence of the solutions for the free boundary problem by Schauder fixed point theorem.     

imaNon-Lipschitz条件的随机微分方程的逼近定理(英文)

本文给出了Non Lipschitz条件下的随机微分方程的一个逼近定理 .     

Prime power divisors of binomial coefficients

It is known that for sufficiently large n and m and any r the binomial coefficient (ⁿ_m) which is close to the middle coefficient is divisible by p^r where p is a ‘large’ prime. We prove the exact divisibility of (ⁿ_m) by p^r for p> c(n). The lower bound is essentially the best possible. We also prove some other results on divisibility of binomial coefficients.     

Recursive method for building Arnold’s triangle

We prove the propositions stated in Part I [Funct Anal Other Math 1(2):93–109 ([2006])] and give the explicit formulas for X _p (a,b).      

Rudin-Shapiro-like polynomials in

We examine sequences of polynomials with coefficients constructed using the iterations , where is the degree of and is the reciprocal polynomial of . If these generate the Rudin-Shapiro polynomials. We show that the norm of these polynomials is explicitly computable. We are particularly interested in the case where the iteration produces sequences with smallest possible asymptotic norm (or, equivalently, with largest possible asymptotic merit factor). The Rudin-Shapiro polynomials form one such sequence.

We determine all of degree less than 40 that generate sequences under the iteration with this property. These sequences have asymptotic merit factor 3. The first really distinct example has a of degree 19.

     

Fast evaluation of the Gaunt coefficients

Addition theorems for vector spherical harmonics require the determination of the Gaunt coefficients that appear in a linearization expansion of the product of two associated Legendre functions. This paper presents an algorithm for the efficient calculation of these coefficients through solving the most appropriate (lower triangular) linear system and derives all relevant recurrence relations needed in the calculation. This algorithm is also applicable to the calculation of the Clebsch-Gordan coefficients that are closely related to the Gaunt coefficients and are frequently encountered in the quantum theory of angular momentum. The new method described in this paper reduces the computing time to , compared to the existing formulation that is widely used. This new method can be applied to the calculation of both low- and high-degree coefficients, while the existing formulation works well only for low degrees.

     

M-bonomial coefficients and their identities

In this note, we introduce M-bonomial coefficients or (M-bonacci binomial coefficients). These are similar to the binomial and the Fibonomial (or Fibonacci–binomial) coefficients and can be displayed in a triangle similar to Pascal's triangle from which some identities become obvious.     

半变系数模型PLS估计的渐近正态性

半变系数模型在统计建模中具有重要的应用．最近几年，人们提出了许多方法来估计其常系数和函数系数，但是估计的渐近性质还没有被系统的研究．本文介绍了半变系数模型的PLS估计，在Fan和Huang对常系数渐近性质研究的基础上，给出了函数系数的渐近正态性。