High-order one-step P-stable methods for the numerical integration of periodic initial value problems

Using Lobatto nodes, one-step methods of order six and eight have been obtained for the second-order differential equation y″ = f(x, y), y(x₀) = y₀, y′(x₀) = y′₀. The methods are shown to be P-stable. If

, then at each integration step a system of dimension 3s, 4s, respectively, has to be solved. The numerical results, for two problems, obtained by using these methods are given in the end.     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

The minimal number of basic elements in a multiset antichain

Let M be a finite set consisting of k_i elements of type i, i = 1, 2,…, n and let S denote the set of subsets of M or, equivalently, the set of all vectors x = (x₁, x₂,…,x_n) with integral coefficients x_i satisfying 0 ? x_i ? k_i, i = 1, 2,…, n. An antichain

is a subset of S in which there is no pair of distinct vectors x and y such that x is contained in y (that is, there is no pair of distinct vectors x and y such that the inequalities x_i ? y_i, i = 1, 2,…, n all hold). Let $\text{∥Y}\textcolor{red}{}\text{∥}$ denote the number of vectors in S which are contained in at least one vector in

and let $\text{∥B}\textcolor{red}{}\text{∥=∑}{}_{\mathrm{x\in }}\text{(X}{}_1\text{+X}{}_2\text{+?+X}{}_{\mathrm{n}}\text{)}$, the number of basic elements in

. For given m we give procedures for calculating min $\text{∥Y}\textcolor{red}{}\text{∥}$ and min $\text{∥B}\textcolor{red}{}\text{∥}$, where the minima are taken over all m-element antichains

in S.     

Regularities in the distribution of special sequences

For a given sequence ω = (x(k))_k=0^∞ in U^s, U = $\text{R/Z}$, let S(ω) denote the set of all boxes I in U^s with bounded discrepancy function, i.e.,

l_I the characteristic function of I. In this paper the set S(ω) is studied for a type of sequences generalizing the Halton-sequences. S(ω) is completely determined in the one-dimensional case. In higher dimensions the subset

of S(ω) is determined and a necessary condition on the elements of S(ω) is proved. The methods used in the proofs belong to ergodic theory.     

On moment-discretization and least-squares solutions of linear integral equations of the first kind

Let K(s, t) be a continuous function on [0, 1] × [0, 1], and let $\text{K}$ be the linear integral operator induced by the kernel K(s, t) on the space $\text{L}$₂[0, 1]. This note is concerned with moment-discretization of the problem of minimizing 6K_x?y6 in the $\text{L}$₂-norm, where y is a given continuous function. This is contrasted with the problem of least-squares solutions of the moment-discretized equation: ∝₀¹K(s_i, t) x(t) dt = y(s_i), i = 1, 2,h., n. A simple commutativity result between the operations of “moment-discretization” and “least-squares” is established. This suggests a procedure for approximating $\text{K2y}$ (where $\text{K}$² is the generalized inverse of $\text{K}$), without recourse to the normal equation $\text{K1Kx = K1y}$, that may be used in conjunction with simple numerical quadrature formulas plus collocation, or related numerical and regularization methods for least-squares solutions of linear integral equations of the first kind.     

An interpolant defined by subdivision and the analysis of the error

Given a set of points x_i, i=0,…,n on [−1,1] and the corresponding values y_i, i=0,…,n of a 2-periodic function y(x), supplied in some way by interpolation or approximation, we describe a simple method that by doubling iteratively this original set, produces in the limit a smooth function. The analysis of the interpolation error is given.We show that if $\text{y∈}\text{C}{}^4$ then the error in the p-norm, $\text{p=1,}\text{2}$ and ∞ depends on the magnitude of the fourth derivative of the function y(x) and on a function α(x) which is even, concave and bounded on [−1,1].