Review of Pertubation Methods for Engineers and Scientists

This is an introduction to perturbation methods, at the beginning graduate level, suitable for courses focusing on methods rather than justification. Boundary layers and fluid flow are emphasized much more than nonlinear oscillations, but given this constraint, this book may have some advantages over its nearest competitor, Nayfeh’s Introduction to Perturbation Techniques. The author introduces perturbation expansions with a few examples, such as motion with small friction, roots of polynomials, and integration by parts. This leads to a second chapter on order symbols, asymptotic expansions, and uniformity. The next four chapters are each devoted to one of the basic classes of perturbation methods for differential equations, strained coordinates, multiple scales, matching, and WKB. (Strained coordinates are handled mostly by renormalization, that is, computing a nonuniform straightforward expansion and then rendering it uniform by straining.) A final chapter concerns asymptotic evaluation of integrals. The chapters on strained coordinates, multiple scales, and matching each have lengthy sections treating a serious physical application at a depth that is unusual in an introductory book; for strained coordinates and matching, these concern fluid flow, while for multiple scales the application is to lubricated bearings

On a conjecture of Wilf

Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum \sum_{j=0}^{n} (-1)^{j} S(n,j) is nonzero for all n>2. We prove this conjecture for all n not congruent to 2 and not congruent to 2944838 modulo 3145728 and discuss applications of this result to graph theory, multiplicative partition functions, and the irrationality of p-adic series.Comment: 18 pages, final version, accepted for publication in the Journal of Combinatorial Theory, Series

Higher order matching polynomials and d-orthogonality

We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials -- the Chebyshev, Hermite, and Laguerre polynomials -- can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogonality for polynomials and we use sign-reversing involutions to show that the higher-order Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are d-orthogonal. We also investigate the moments and find generating functions of those polynomials.Comment: 21 pages, many TikZ figures; v2: minor clarifications and addition

A unified approach to polynomial sequences with only real zeros

We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials, matching polynomials, Narayana polynomials and Eulerian polynomials. We also settle certain conjectures of Stahl on genus polynomials by proving them for certain classes of graphs, while showing that they are false in general.Comment: 19 pages, Advances in Applied Mathematics, in pres