On some difference schemes for singular singularly-perturbed boundary value problems

Summary Singularly perturbed boundary value ordinary differential problems are considered, where the problem defining the reduced solution is singular. For numerical approximation, families of symmetric difference schemes, which are equivalent to certain collocation schemes based on Gauss and Lobatto points, are used. Convergence results, previously obtained for the regular singularly perturbed case, are extended. While Gauss schemes are extended with no change, Lobatto schemes require a small modification in the mesh selection procedure. With meshes as prescribed in the text, highly accurate solutions can be obtained with these schemes for singular singularly perturbed problems at a very reasonable cost. This is demonstrated by examples.This research was completed while the author was visiting the Department of Applied Mathematics, Weizmann Inst., Rehovot, Israel. The author was supported in part under NSERC grant A4306     

On some sampling schemes for estimating the parameters of a continuous time series

Summary In estimating the mean of certain stationary processes it is shown that it is better to sample at fixed equi-spaced time intervals than to sample randomly according to a renewal process. On the other hand it is shown that the estimation of autocorrelation is sometimes better accomplished by random sampling.     

On Bayesian group sequential sampling procedures

A general group sequential statistical decision model is investigated. Using a Bayesian approach and Bayesian dynamic programming, we examine structural properties of the cost functions and of optimal sampling procedures. In particular, we consider variable-sample-size-sequential probability ratio tests and show that the so-called onion-skins conjecture is false.     

On some finite difference schemes for solution of hyperbolic heat conduction problems

We consider the accuracy of two finite difference schemes proposed recently in [Roy S., Vasudeva Murthy A.S., Kudenatti R.B., A numerical method for the hyperbolic-heat conduction equation based on multiple scale technique, Appl. Numer. Math., 2009, 59(6), 1419–1430], and [Mickens R.E., Jordan P.M., A positivity-preserving nonstandard finite difference scheme for the damped wave equation, Numer. Methods Partial Differential Equations, 2004, 20(5), 639–649] to solve an initial-boundary value problem for hyperbolic heat transfer equation. New stability and approximation error estimates are proved and it is noted that some statements given in the above papers should be modified and improved. Finally, two robust finite difference schemes are proposed, that can be used for both, the hyperbolic and parabolic heat transfer equations. Results of numerical experiments are presented.     

Sequential sampling schemes and other statistical techniques in quality of service business surveys

While there is a general consensus in the business community regarding the importance of quality service surveys in improving the company's business results, relatively little attention was devoted to the theoretical aspects of data analysis and modelling which can increase substantially the effectiveness of those surveys. Using the data from a real‐life project, we present here the developments and improvements achieved in the three stages that ultimately yield the conclusions from the survey: the analysis of the historical data, the design of the sampling method for the current survey and the development of specific indices for differentiating among the units of the financial institution. The statistical techniques fitted to historical data helped both in assessing the relative contributions of the various aspects of service, the overall perceived quality of service, as well as in providing the theoretical grounds for reducing the size of the questionnaire and thus increasing the reliability of the received replies. The sequential sampling scheme designed for this survey was particularly suitable to address the managements concerns. Finally, the classes of quality service based on the quality indices developed in the project contributed substantially to the ability of the management to rank and reward differentially the branches of the financial institution. Copyright © 2002 John Wiley & Sons, Ltd.     

On some sequential simultaneous confidence intervals procedures

Summary The purpose of this paper is two-fold: (i) to extend the simultaneous confidence intervals procedures (SCIP) of Healy [7] along the lines of Chow and Robbins [3] and (ii) to develop certain robust non-parametric SCIP based on the results of Sen [10] and Sen and Ghosh [11]; the allied efficiency results are also presented. Work supported by the National Institutes of Health, Grant GM-12868.     

On some simplicial elimination schemes for chordal graphs

We introduced here an interesting tool for the structural study of chordal graphs, namely the Reduced Clique Graph. Using some of its combinatorial properties we show that for any chordal graph we can construct in linear time a simplicial elimination scheme starting with a pending maximal clique attached via a minimal separator maximal under inclusion among all minimal separators.     

On summing permutations and some statistical properties

A closed form expression is obtained for the sum of all permutations of n objects taken r at a time. The average and variance of the permutations are derived and are shown to be proportional to the average and variance of the objects themselves. The proportionality constant is a function of only r, n and the base b and is independent of the actual objects considered. Previous results aimed at determining the sum of permutations are shown to be very specific cases of the current development.