Semiconjugate Factorizations of Higher Order Linear Difference Equations in Rings

We study linear difference equations with variable coefficients in a ring using a new nonlinear method. In a ring with identity, if the homogeneous part of the linear equation has a solution in the unit group of the ring (i.e., a unitary solution) then we show that the equation decomposes into two linear equations of lower orders. This decomposition, known as a semiconjugate factorization in the nonlinear theory, generalizes the classical operator factorization in the linear context. Sequences of ratios of consecutive terms of a unitary solution are used to obtain the semiconjugate factorization. Such sequences, known as eigensequences are well-suited to variable coefficients; for instance, they provide a natural context for the expression of the classical Poincar\'{e}-Perron Theorem. We discuss some applications to linear difference equations with periodic coefficients and also derive formulas for the general solutions of linear functional recurrences satisfied by the classical special functions such as the modified Bessel and Chebyshev.Comment: Application of nonlinear semiconjugate factorization theory to linear difference equations with variable coefficients in rings; 29 pages, containing the main theory and more than 8 examples worked out in detai

Solution of linear equations and the inversion of matrices

Thesis (M.A.)--Boston UniversityIt has been noted that the writing of a system of linear equations in matrix form quite naturally suggests the computation of an inverse matrix which may then be used to compute solutions to a system of equations in which the coefficients remain the same but the constants have been changed. The elementary properties of matrices and determinants have been reviewed and a general method of matrix inversion based on these properties was considered. The very considerable practical deficiencies of this theoretically attractive method suggested that the method would be almost completely worthless for a system larger than two by two [TRUNCATED

Weighted spanning trees on some self-similar graphs

We compute the complexity of two infinite families of finite graphs: the Sierpi\'{n}ski graphs, which are finite approximations of the well-known Sierpi\'nsky gasket, and the Schreier graphs of the Hanoi Towers group $H^{(3)}$ acting on the rooted ternary tree. For both of them, we study the weighted generating functions of the spanning trees, associated with several natural labellings of the edge sets.Comment: 21 page