矩阵连分式序列的收敛性

本文研究了矩阵连分式的性质，获得了关于矩阵连分式序列收敛性的一些结果．     

On Sufficient Conditions for Convergence of Two-Dimensional Continued Fractions

By the method of majorant fractions and equivalent transformations, the analogies of leszyski–Pringsheim criteria for two-dimensional continued fractions are obtained.     

有理真分式分解中的系数公式

本文利用导数给出了有理真分式分解为部分分式时的一个简洁的系数公式以及该公式的使用 .     

某些超越连分数的代数无关性(Ⅱ)

本文证明了某些以ａｎｘ（ａｎ为正代数数）为元素的连分式在代数点和超越点上值的代数无关性．特别地,某些关于简单连分数的代数无关性结果被扩充到更广泛的情形。     

有理真分式部分分式分解的证明及系数公式

基于多项式知识给出了有理真分式部分分式分解定理的一个简洁的构造性证明.此外,还对分解系数的计算方法进行总结,给出了赋值法、极限法与导数法的全部计算公式.结果表明,利用极限法与导数法都能求出全部分解系数,且导数法的计算公式更简单、易算.     

对称型插值框架的注记

通过引进新的参数,将对称型插值的一般框架作进一步推广和改进,新的插值框架包含更为丰富的插值格式;给出几种新形式的对称型有理插值格式;最后,将结果推广到向量值及矩阵值情形.     

On Somos' dissection identities

We give proofs of a list of M. Somos' dissection identities. An eta function identity presented by B.C. Berndt and W.B. Hart, a theorem by H.-C. Chan on the congruence property of a(n) with generating function , and a theorem by G.E. Andrews, A. Schilling, and S.O. Warnaar are shown to be related to dissection identities. Several new corollaries are also presented as applications.     

二元插值的一般框架

本文通过引进多参数建立了二元插值的一般框架.这样,许多著名的经典插值格式,如Newton插值、分叉连分式插值、对称连分式插值等均可视为本文的特殊情形．     

A multidimensional continued fraction based on a high-order recurrence relation

The paper describes and studies an iterative algorithm for finding small values of a set of linear forms over vectors of integers. The algorithm uses a linear recurrence relation to generate a vector sequence, the basic idea being to choose the integral coefficients in the recurrence relation in such a way that the linear forms take small values, subject to the requirement that the integers should not become too large. The problem of choosing good coefficients for the recurrence relation is thus related to the problem of finding a good approximation of a given vector by a vector in a certain one-parameter family of lattices; the novel feature of our approach is that practical formulae for the coefficients are obtained by considering the limit as the parameter tends to zero. The paper discusses two rounding procedures to solve the underlying inhomogeneous Diophantine approximation problem: the first, which we call ``naive rounding' leads to a multidimensional continued fraction algorithm with suboptimal asymptotic convergence properties; in particular, when it is applied to the familiar problem of simultaneous rational approximation, the algorithm reduces to the classical Jacobi-Perron algorithm. The second rounding procedure is Babai's nearest-plane procedure. We compare the two rounding procedures numerically; our experiments suggest that the multidimensional continued fraction corresponding to nearest-plane rounding converges at an optimal asymptotic rate.

     

Singular values of the Rogers-Ramanujan continued fraction

Let z∊ C be imaginary quadratic in the upper half plane. Then the Rogers-Ramanujan continued fraction evaluated at q = e ^{2π iz} is contained in a class field of Q(z). Ramanujan showed that for certain values of z, one can write these continued fractions as nested radicals. We use the Shimura reciprocity law to obtain such nested radicals whenever z is imaginary quadratic. 2000 Mathematics Subject Classification Primary—11Y65; Secondary—11Y40     

Integral medial axis and the distance between closest points

In dimension two we prove an inequality that implies a desirable property of the integral medial axis as defined by Hesselink in [1]. In dimension three we conjecture a similar inequality.      

Some general theorems on the explicit evaluations of Ramanujan''s cubic continued fraction

In 2001, Jinhee Yi found many explicit values of the famous Rogers–Ramanujan continued fraction by using modular equations and transformation formulas for theta-functions. In this paper, we use her method to find some general theorems for the explicit evaluations of Ramanujan's cubic continued fraction.     

An analogue of continued fractions in number theory for Nevanlinna theory

We show an analogue of continued fractions in approximation to irrational numbers by rationals for Nevanlinna theory. The analogue is a sequence of points in the complex plane which approaches a given finite set of points and at a given rate in the sense of Nevanlinna theory.

     

Computer verification of the Ankeny-Artin-Chowla Conjecture for all primes less than

Let be a prime congruent to 1 modulo 4, and let be rational integers such that is the fundamental unit of the real quadratic field . The Ankeny-Artin-Chowla conjecture (AAC conjecture) asserts that will not divide . This is equivalent to the assertion that will not divide , where denotes the th Bernoulli number. Although first published in 1952, this conjecture still remains unproved today. Indeed, it appears to be most difficult to prove. Even testing the conjecture can be quite challenging because of the size of the numbers ; for example, when , then both and exceed . In 1988 the AAC conjecture was verified by computer for all . In this paper we describe a new technique for testing the AAC conjecture and we provide some results of a computer run of the method for all primes up to .

     

BIVARIATE RATIONAL INTERPOLANTS WITHRECTANGLE-HOLE-STRUCTURE

1.IntroductionGivenasetofdistinctrealpoints{xi,i~0,1,2,',n:xiER}andasetofcomplexvectordata{d'),i=0,1,2,',n:n)ECd},Graves-Momsshowed[5]thatthevectorvaluedThieletypecontinuedfractioncanservetointerpolatethegivenvectors.TheconstructionprocessiscloselyralatedtotheadoptionoftheSamelsoninverseforvectorswhere7denotesthecomplexconjugateofvector6.ItwasprovedthatS(x)isavectorvaluedrationalfunctionwithnumeratorbeingad-dimensionalpolynomialofdegreenanddenominatorbeingapolynomialofdegree2[n/2],here…     

Two-point Padé expansions for a family of analytic functions

Each member G(z) of a family of analytic functions defined by Stieltjes transforms is shown to be represented by a positive T-fraction, the approximants of which form the main diagonal in the two-point Padé table of G(z). The positive T-fraction is shown to converge to G(z) throughout a domain D(a, b) = [z: z?[?b, ?a]], uniformly on compact subsets. In addition, truncation error bounds are given for the approximants of the continued function; these bounds supplement previously known bounds and apply in part of the domain of G(z) not covered by other bounds. The proofs of our results employ properties of orthogonal $\text{L}$-polynomials (Laurent polynomials) and $\text{L}$-Gaussian quadrature which are of some interest in themselves. A number of examples are considered.     

Explicit invariant measures for products of random matrices

We construct explicit invariant measures for a family of infinite products of random, independent, identically-distributed elements of SL. The matrices in the product are such that one entry is gamma-distributed along a ray in the complex plane. When the ray is the positive real axis, the products are those associated with a continued fraction studied by Letac & Seshadri [Z. Wahr. Verw. Geb. 62 (1983) 485-489], who showed that the distribution of the continued fraction is a generalised inverse Gaussian. We extend this result by finding the distribution for an arbitrary ray in the complex right-half plane, and thus compute the corresponding Lyapunov exponent explicitly. When the ray lies on the imaginary axis, the matrices in the infinite product coincide with the transfer matrices associated with a one-dimensional discrete Schrödinger operator with a random, gamma-distributed potential. Hence, the explicit knowledge of the Lyapunov exponent may be used to estimate the (exponential) rate of localisation of the eigenstates.

     

连幂式插值

根据连续幂指形式的函数,提出了连续幂指形式的函数插值的概念,简称连幂式插值,用构造式方法得到了满足插值条件的连幂式插值函数。最后,通过一个算例与连分式插值函数做了对比。     

关于两种连分式加速收敛方法等价性的一般猜想的证明

本文证明了［２］提出的两种连分式加速收敛方法等价性的一般猜想是正确的。     

Engel连分数中一个例外集的Hausdorff维数

本文研究了Engel连分数展式中部分商以某种速度增长的集合的Hausdorff维数.利用自然覆盖和质量分布原理,得到了集合B(α)={x∈(0,1):lim n→∞ log bn+1(x)/log bn(x)=α}的Hausdorff维数是1/α的结果.