The inverse of circulant matrix

A direct method is proposed to get the inverse matrix of circulant matrix that find important application in engineering, the elements of the inverse matrix are functions of zero points of the characteristic polynomial g(z) and g′(z) of circulant matrix, four examples to get the inverse matrix are presented in the paper.     

一个特殊乘积的逆的矩阵不等式的加强

将两个正定矩阵的Khatri-Rao乘积的矩阵不等式(A*B)^-1≤A^-1*B^-1推广为(A*B)^-1≤（A^-1(α）^-1*B(α））^-1 (A(α′）*B^-1(α′）^-1)^-1≤（A^-1（α）*B（α）^-1) (A(α′）^-1*B^-1(α′））≤A^-1*B^-1,其中A(α）是A的顺序主子矩阵，而A（α′）是A（α）的余子矩阵，同时还给出了其等式成立的充分必要条件。     

A matrix continued fraction approach to multiserver retrial queues

We consider basic M/M/c/c (c≥1) retrial queues where the number of busy servers and that of customers in the orbit form a level-dependent quasi-birth-and-death (QBD) process with a special structure. Based on this structure and a matrix continued fraction approach, we develop an efficient algorithm to compute the joint stationary distribution of the numbers of busy servers and retrial customers. Through numerical experiments, we demonstrate that our algorithm works well even for M/M/c/c retrial queues with large value of c.     

Divisors of convergents of a continued fraction

By applying the theory of linear forms in logarithms, effective and sharpened versions of the results of Erdös and Mahler on the arithmetical properties of the convergents of a continued fraction are given.     

On the Rogers-Ramanujan continued fraction

In the “Lost” note book, Ramanujan had stated a large number of results regarding evaluation of his continued fraction for certain values of τ. It is shown that all these results and many more have their source in the Kronecker limit formula.     

On a continued fraction of order six

In this paper, we derive certain identities for the following continued of order six:     

On a continued fraction of order 12

We present some new relations between a continued fraction U(q) of order 12 (established by M. S. M. Naika et al.) and U(q ⁿ ) for n = 7, 9, 11, 13:     

Explicit evaluations of a Ramanujan-Selberg continued fraction

This paper gives explicit evaluations for a Ramanujan-Selberg continued fraction in terms of class invariants and singular moduli.

     

On the approximation by three consecutive continued fraction convergents

Denote by _pn/_qn,n=1,2,3,…$p_n/q_n,n=1,2,3,\dots$, the sequence of continued fraction convergents of the real irrational number x$x$. Define the sequence of approximation coefficients by _θn:=_qn|_qnx−_pn|,n=1,2,3,…${\theta }_n:=q_n|q_{n}x-p_n|,n=1,2,3,\dots$. A laborious way of determining the mean value of the sequence |_θn+1−_θn−1|,n=2,3,…$|{\theta }_{n+1}-{\theta }_{n-1}|,n=2,3,\dots$, is simplified. The method involved also serves for showing that for almost all x$x$ the pattern _θn−1<_θn<_θn+1${\theta }_{n-1}<{\theta }_n<{\theta }_{n+1}$ occurs with the same asymptotic frequency as the pattern _θn+1<_θn<_θn−1${\theta }_{n+1}<{\theta }_n<{\theta }_{n-1}$, namely 0.12109?$0.12109?$. All the four other patterns have the same asymptotic frequency 0.18945?$0.18945?$. The constants are explicitly given.     

On the approximation of an appell hypergeometric function by a branched continued fraction

We study the question of the boundedness of the approximants to the branched continued fraction expansion of a ratio of Appell hypergeometric functions. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 65–69.     

Itinerary of a discontinuous map from the continued fraction expansion

A piecewise linear, discontinuous one-dimensional map is analyzed combinatorically. The quasi-periodic dynamics generated by iterations are completely characterized by successive convergents of a continued fraction associated with slopes of the map.     

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     

On Worpitzky-like theorems for a two-dimensional continued fraction

Using estimates for the remainders of a two-dimensional continued fraction, the relation for the difference between two approximants of such a fraction in terms of these remainders, and the majorant method, we have proposed generalizations of the Worpitzky convergence theorem. For fractions satisfying the conditions of generalized Worpitzky theorems, we have also obtained estimates for their convergence rate.     

Estimates for the inverse of a matrix

The problem of finding bounds for the elements of the inverse of a matrix satisfying various known regularity conditions is considered. The results are obtained in such a way as to apply automatically to partitioned matrices.     

Dynamics of a Ramanujan-type continued fraction with cyclic coefficients

We study several generalizations of the AGM continued fraction of Ramanujan inspired by a series of recent articles in which the validity of the AGM relation and the domain of convergence of the continued fraction were determined for certain complex parameters (Borwein et al., Exp. Math. 13, 275–286, 2004, Ramanujan J., in press, 2004; Borwein and Crandall, Exp. Math. 12, 287–296, 2004). A study of the AGM continued fraction is equivalent to an analysis of the convergence of certain difference equations and the stability of dynamical systems. Using the matrix analytical tools developed in 2004, we determine the convergence properties of deterministic difference equations and so divergence of their corresponding continued fractions. Russell Luke’s work was supported in part by a postdoctoral fellowship from the Pacific Institute for the Mathematical Sciences at Simon Fraser University.