Mesh selection for stiff two-point boundary value problems

This paper is concerned with the mesh selection algorithm of COLSYS, a well known collocation code for solving systems of boundary value problems. COLSYS was originally designed to solve non-stiff and mildly stiff problems only. In this paper we show that its performance for solving extremely stiff problems can be considerably improved by modifying its error estimation and mesh selection algorithms. Numerical examples indicate the superiority of the modified algorithm.Dedicated to John Butcher on the occasion of his sixtieth birthday     

A fast computer algorithm for finding the permanent of adjacency matrices

A fast computer algorithm is described which brings computation of the permanents of sparse matrices, specifically, chemical adjacency matrices, within the reach of a desktop computer. Examples and results are presented, along with a discussion of the relationship of the permanent to the Kekulé structure count. Also presented is a C-language implementation which was deliberately written for ease of translation into other high-level languages.     

A high order method for the numerical solution of two-point boundary value problems

A one step finite difference scheme of order 4 for the numerical solution of the general two-point boundary value problemy=f(t,y),a t b, withg(y(a),y(b))=0 is presented. The global discretization error of the scheme is shown, in sufficiently smooth cases, to have an asymptotic expansion containing even powers of the mesh size only. This justifies the use of Richardson extrapolation (or deferred correction) to obtain high orders of accuracy. A theoretical examination of the new scheme for large systems of equations shows that for a given mesh size it generally requires about twice as much work as the Keller box scheme. However, the expectation of higher accuracy usually justifies this extra computational effort. Some numerical results are given which confirm these expectations and show that the new scheme can be generally competitive with the box scheme.     

A fluorogenic 1,3-dipolar cycloaddition reaction of 3-azidocoumarins and acetylenes

Copper(I)-catalyzed 1,3-dipolar cycloaddition reaction of nonfluorescent 3-azidocoumarins and terminal alkynes afforded intense fluorescent 1,2,3-triazole products. The mild condition of this reaction allowed us to construct a large library of pure fluorescent coumarin dyes. Since both azide and alkyne are quite inert to biological systems, this reaction has potential in bioconjugation and bioimaging applications. [reaction: see text]     

A stability result for general linear methods with characteristic function having real poles only

It is shown that there exist A-stable multistep formulae, with a characteristic function havings poles, all of which are real, with orderp satisfyingp>s+1. This contradicts the widely held belief thatp=s+1 is the maximum possible order of such a method.     

An algorithm for calculating the independence and vertex-cover polynomials of a graph

The independence polynomial, ω(G,x)=∑w_kx^k, of a graph, G, has coefficients, w_k, that enumerate the ways of selecting k vertices from G so that no two selected vertices share an edge. The independence number of G is the largest value of k for which w_k≠0. Little is known of less straightforward relationships between graph structure and the properties of ω(G,x), in part because of the difficulty of calculating values of w_k for specific graphs. This study presents a new algorithm for these calculations which is both faster than existing ones and easily adaptable to high-level computer languages.     

A chemically bonded liquid crystal as a stationary phase for high performance liquid chromatography. Synthesis on silica via an organochlorosilane pathway

Chromatographia - A compound which becomes a liquid crystal when bonded to a polysiloxane was reacted with dimethylchlorosilane to produce a reagent suitable for bonding to a silica surface. After...     

Blended extended linear multistep methods for the accurate numerical integration of stiff initial value problems

A class of blended extended linear multistep methods suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is described. These methods are formulated as a result of combining the second derivative extended backward differentiation formulae of Cash and the blended linear multistep methods of Skeel and Kong. The new methods combine a high order or accuracy with good stability properties and, as a direct consequence, they are often suitable for the numerical integration of stiff differential systems when high accuracy is requested. In the first part of the present paper we consider the derivation of these new blended methods and give the coefficients and stability regions for formulae of order up to and including 10. In the second half we consider their practical implementation. In particular we describe a variable order/variable step package based on these blended formulae and we evaluate the performance of this package on the well known DETEST test set. It is shown that the new code is reliable on this test set and is competitive with the well known second derivative method of Enright.