Multiderivative Runge-Kutta processes for two-point boundary value problems

A class of finite difference schemes for the solution of a nonlinear system of first order differential equations with two point boundary conditions which shares properties with Runge-Kutta processes and gap schemes is discussed. The order conditions for the coefficients of these processes, techniques for reducing these order conditions in number and the symmetry conditions are given. A symmetricA-stable eight order process which has second, fourth and sixth orderA-stable processes embedded in it is given as an example.Research supported in part by the United States Air Force under contract AFOSR-89-0383.     

Estimation and inference for dependence in multivariate data

In this paper, a new measure of dependence is proposed. Our approach is based on transforming univariate data to the space where the marginal distributions are normally distributed and then, using the inverse transformation to obtain the distribution function in the original space. The pseudo-maximum likelihood method and the two-stage maximum likelihood approach are used to estimate the unknown parameters. It is shown that the estimated parameters are asymptotical normally distributed in both cases. Inference procedures for testing the independence are also studied.     

Local dependence functions for some families of bivariate distributions and total positivity

The purpose of this paper is to investigate a very useful application of a certain local dependence function γ_f(x,y), which was considered recently by Holland and Wang [20]. An interesting property of γ_f(x,y) is that the underlying joint density f(x,y) is TP₂ (that is, totally positive of order 2) if and only if . This gives an elegant way to investigate the TP₂ property of any bivariate distribution. For the Saramanov family, the Ali-Mikhail-Haq family of bivariate distributions and the family of bivariate elliptical distributions, we derive the local dependence function and obtain conditions for f(x,y) to be TP₂. These families are quite rich and include many other large classes of bivariate distributions as their special cases. Similar conditions are obtained for bivariate distributions with exponential conditionals and bivariate distributions with Pareto conditionals.     

On the maximum number of diagonals of a circuit in a graph

For a graph G, let ?(G) denote the maximum number k such that G contains a circuit with k diagonals.Theorem. For any graph G with minimum valency$\text{n? 3, ?(G) ?}\text{1}\text{2}\text{(n+1)(n-2)}$.If the equality holds and G is connected, then either G is isomorphic to K_n+1 or G is separable and each of its terminal blocks is isomorphic to K_n+1, or K_n+1 with one edge subdivided.     

On the Distribution of the Sum of n Non-Identically Distributed Uniform Random Variables

The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the characteristic function, we derive explicit formulae for the distribution of the sum of n non-identically distributed uniform random variables in both the continuous and the discrete case. The results, though involved, have a certain elegance. As examples, we derive from our general formulae some special cases which have appeared in the literature.     

Some properties of the bivariate lognormal distribution for reliability applications

In this paper, we study the bivariate lognormal distribution from a reliability point of view. The conditional distribution of X given Y > y is found to be log‐skew normal. The monotonicity of the hazard rates of the univariate as well as the conditional distributions is discussed. Clayton's association measure is obtained in terms of the hazard gradient, and its value in the case of our model is derived. The probability distributions, in the case of series and parallel systems, are derived, and the monotonicity of their failure rates is discussed. Three real applications of the bivariate lognormal distribution are provided, two from financial economics and one from reliability. Copyright © 2012 John Wiley & Sons, Ltd.     

First-order definability and algebraicity of the sets of annihilating and generating collections of elements for some relatively free solvable groups

We study the first-order definable, Diophantine, and algebraic subsets in the set of all ordered sets generating a group or generating a group as a normal subgroup for some relatively free solvable groups.     

Novel evolutionary models and applications to sequence alignment problems

In this paper, we present a novel graph-theoretical approach for representing a wide variety of sequence analysis problems within a single model. The model allows incorporation of the operations “insertion”, “deletion”, and “substitution”, and various parameters such as relative distances and weights. Conceptually, we refer the problem as the minimum weight common mutated sequence (MWCMS) problem. The MWCMS model has many applications including multiple sequence alignment problem, the phylogenetic analysis, the DNA sequencing problem, and sequence comparison problem, which encompass a core set of very difficult problems in computational biology. Thus the model presented in this paper lays out a mathematical modeling framework that allows one to investigate theoretical and computational issues, and to forge new advances for these distinct, but related problems. Through the introduction of supernodes, and the multi-layer supergraph, we proved that MWCMS is -complete. Furthermore, it was shown that a conflict graph derived from the multi-layer supergraph has the property that a solution to the associated node-packing problem of the conflict graph corresponds to a solution of the MWCMS problem. In this case, we proved that when the number of input sequences is a constant, MWCMS is polynomial-time solvable. We also demonstrated that some well-known combinatorial problems can be viewed as special cases of the MWCMS problem. In particular, we presented theoretical results implied by the MWCMS theory for the minimum weight supersequence problem, the minimum weight superstring problem, and the longest common subsequence problem. Two integer programming formulations were presented and a simple yet elegant decomposition heuristic was introduced. The integer programming instances have proven to be computationally intensive. Consequently, research involving simultaneous column and row generation and parallel computing will be explored. The heuristic algorithm, introduced herein for multiple sequence alignment, overcomes the order-dependent drawbacks of many of the existing algorithms, and is capable of returning good sequence alignments within reasonable computational time. It is able to return the optimal alignment for multiple sequences of length less than 1500 base pairs within 30 minutes. Its algorithmic decomposition nature lends itself naturally for parallel distributed computing, and we continue to explore its flexibility and scalability in a massive parallel environment.     

Simultaneous approximation for the Bézier variant of Baskakov-Beta operators

In the present paper, we study the rate of convergence in simultaneous approximation for the Bézier variant of the Baskakov-Beta operators by using the decomposition technique of functions of bounded variation.