Ramanujan-Göllnitz-Gordon Continued Fraction

We derive many new identities involving the Ramanujan-Göllnitz-Gordon continued fraction H(q). These include relations between H(q) and H(q ⁿ ) , which are established using modular equations of degree n. We also evaluate explicitly H(q) at for various positive integers n. Using results of M. Deuring, we show that are units for all positive integers n.     

m-continued Fraction Expansions of Multi-Laurent Series

The simple continued fraction expansion of a single real number gives the best solution to its rational approximation problem. A multidimensional generalization of the simple continued fraction expanding procedure is the Jacobi-Perron algorithm (JPA). This algorithm and its     

Simplex—Karyon Algorithm of Multidimensional Continued Fraction Expansion

A simplex–karyon algorithm for expanding real numbers α = (α₁,..., α _d ) in multidimensional continued fractions is considered. The algorithm is based on a (d + 1)-dimensional superspace S with embedded hyperplanes: a karyon hyperplane K and a Farey hyperplane F. The approximation of numbers α by continued fractions is performed on the hyperplane F, and the degree of approximation is controlled on the hyperplane K. A local ?(r)-strategy for constructing convergents is chosen, with a free objective function ?(r) on the hyperplane K.     

The Continued Fraction Solution of the Matrix Equation AX-XB=C

Let K[λ] denote the polynomial ring over the number field K.Suppose thatf(λ)and g(λ)are coprime in K[λ] and deg f(λ)≥1.By the Euclidean algorithmwe obtain the continued fraction     

Ideas from Continued Fraction Theory Extended to Padé Approximation and Generalized Iteration

This is a survey of some basic ideas in the convergence theory for continued fractions, in particular value sets, general convergence and the use of modified approximants to obtain convergence acceleration and analytic continuation. The purpose is to show how these ideas apply to some other areas of mathematics. In particular, we introduce {w _k }-modifications and general convergence for sequences of Padé approximants.     

The Proportional Mean Residual Life Regression Model with Cure Fraction and Auxiliary Covariate

As biological studies become more expensive to conduct, it is a frequently encountered question that how to take advantage of the available auxiliary covariate information when the exposure variable is not measured. In this paper, we propose an induced cure rate mean residual life time regression model to accommodate the survival data with cure fraction and auxiliary covariate, in which the exposure variable is only assessed in a validation set, but a corresponding continuous auxiliary covariate...     

An M/M/1 Driven Fluid Queue – Continued Fraction Approach

This paper presents complete solutions of the stationary distributions of buffer occupancy and buffer content of a fluid queue driven by an M/M/1 queue. We assume a general boundary condition when compared to the model discussed in Virtamo and Norros [Queueing Systems 16 (1994) 373–386] and Adan and Resing [Queueing Systems 22 (1996) 171–174]. We achieve the required solutions by transforming the underlying system of differential equations using Laplace transforms to a system of difference equations leading to a continued fraction. This continued fraction helps us to find complete solutions. We also obtain the buffer content distribution for this fluid model using the method of Sericola and Tuffin [Queueing Systems 31 (1999) 253–264].     

The Bootstrap Methodology in Statistics of Extremes—Choice of the Optimal Sample Fraction

The main objective of statistics of extremes is the prediction of rare events, and its primary problem has been the estimation of the tail index , usually performed on the basis of the largest k order statistics in the sample or on the excesses over a high level u. The question that has been often addressed in practical applications of extreme value theory is the choice of either k or u, and an adaptive estimation of . We shall be here mainly interested in the use of the bootstrap methodology to estimate adaptively, and although the methods provided may be applied, with adequate modifications, to the general domain of attraction of G, , we shall here illustrate the methods for heavy right tails, i.e. for > 0. Special relevance will be given to the use of an auxiliary statistic that is merely the difference of two estimators with the same functional form as the estimator under study, computed at two different levels. We shall also compare, through Monte Carlo simulation, these bootstrap methodologies with other data-driven choices of the optimal sample fraction available in the literature.