实对称五对角矩阵逆特征值问题

1 引 言 对于n阶实对称矩阵A=(aij),r是一个正整数,且1≤r≤n-1,当|i-j|>r时,aij=0(i,j=1,2,…,n),至少有一个i使得ai,i+r≠0,则称矩阵A是带宽为2r+1的实对称带状矩阵.特别地,当r=1时,称A为实对称三对角矩阵;当r=2时,称A为实对称五对角矩阵. 实对称带状矩阵逆特征值问题应用十分广泛,这类问题不仅来自微分方程逆特征值问     

Unconditionally Stable Methods for Hamilton–Jacobi Equations

We present new numerical methods for constructing approximate solutions to the Cauchy problem for Hamilton–Jacobi equations of the form u_t+H(D_xu)=0. The methods are based on dimensional splitting and front tracking for solving the associated (non-strictly hyperbolic) system of conservation laws p_t+D_xH(p)=0, where p=D_xu. In particular, our methods depend heavily on a front tracking method for one-dimensional scalar conservation laws with discontinuous coefficients. The proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as “large-time-step” Godunov-type (or front tracking) methods. We present several numerical examples illustrating the main features of the proposed methods. We also compare our methods with several methods from the literature.     

Neither G9（Q） nor G11（Q） Is a Subgroup of K2（Q）

It is proved that neither G9(Q) nor G11(Q) is a subgroup of K2(Q) which confirms two special cases of a conjecture proposed by Browkin, J. (Lecture Notes in Math., 966, Springer-Verlag, New York, Heidelberg, Berlin, 1982, 1-6).     

A complexity measure for families of binary sequences

In earlier papers finite pseudorandom binary sequences were studied, quantitative measures of pseudorandomness of them were introduced and studied, and large families of “good” pseudorandom sequences were constructed. In certain applications (cryptography) it is not enough to know that a family of “good” pseudorandom binary sequences is large, it is a more important property if it has a “rich”, “complex” structure. Correspondingly, the notion of “f-complexity” of a family of binary sequences is introduced. It is shown that the family of “good” pseudorandom binary sequences constructed earlier is also of high f-complexity. Finally, the cardinality of the smallest family achieving a prescibed f-complexity and multiplicity is estimated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Computing orthogonal polynomials on a triangle by degree raising

We give an algorithm for computing orthogonal polynomials over triangular domains in Bernstein–Bézier form which uses only the operator of degree raising and its adjoint. This completely avoids the need to choose an orthogonal basis (or tight frame) for the orthogonal polynomials of a given degree, and hence the difficulties inherent in that approach. The results are valid for Jacobi polynomials on a simplex, and show the close relationship between the Bernstein form of Jacobi polynomials, Hahn polynomials and degree raising.     

A Comparison-estimate of the second Rauch type for Ricci curvature

We establish a comparison-estimate of the second Rauch type for Ricci curvature. As an application, we get a result of local splitting for Riemannian manifolds. This work is partly supported by the National Natural Science Foundation (10371047) of China.     

C^∞映射芽的Boardman符号的机器算法

本给出了C^∞映射芽的Boardman符号的机器算法，利用具有符号运算的数学软件（如MATLAB）可实现所给算法。     

An extremal property of rectangular distributions

It is proved that the correlation coefficient between the elements of the order statistics is maximal for a rectangularly (uniformly) distributed population.     

Modular forms and differential operators

In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn ≥ 0 a bilinear operation which assigns to two modular formsf andg of weightk andl a modular form [f, g]_n of weightk +l + 2n. In the present paper we study these “Rankin-Cohen brackets” from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure (“RC algebra”) consisting of a graded vector space together with a collection of bilinear operations [,]_n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivationS of degree 2 and an element Φ of degree 4, up to the equivalence relation (∂,Φ) ~ (∂ - ϕE, Φ - ϕ² + ∂(ϕ)) where ϕ is an element of degree 2 andE is the Fuler operator (= multiplication by the degree). Dedicated to the memory of Professor K G Ramanathan     

Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line

We study ratio asymptotics, that is, existence of the limit of P_n+1(z)/P_n(z) (P_n= monic orthogonal polynomial) and the existence of weak limits of p_n² dμ (p_n=P_n/||P_n||) as n→∞ for orthogonal polynomials on the real line. We show existence of ratio asymptotics at a single z₀ with Im(z₀)≠0 implies dμ is in a Nevai class (i.e., a_n→a and b_n→b where a_n,b_n are the off-diagonal and diagonal Jacobi parameters). For μ's with bounded support, we prove p_n² dμ has a weak limit if and only if lim b_n, lim a_2n, and lim a_2n+1 all exist. In both cases, we write down the limits explicitly.