Two piecewise-linearized methods for the numerical solution of nonlinear second-order differential equations which contain the first-order derivative are presented. These methods provide piecewise analytical solutions in open intervals, result in explicit finite difference equations for both the displacement and the velocity or only the displacement, at nodal points, and continuous solutions ev...

In this paper, we investigate the connection between closed Newton-Cotes formulae, trigonometrically-fitted methods, symplectic integrators and efficient integration of the Schr¨odinger equation. The study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry (see the relevant lit...

An explicit trigonometrically-fitted Runge–Kutta–Nyström (ETFRKN) method is constructed in this paper based on Simos technique, which exactly integrates initialvalue problems whose solutions are linear combinations of functions of the form e and e−iwx or equivalently sin(wx) and cos(wx) with w > 0 the principal frequency of the problem. The numerical results show the efficacy of the new method ...

AC-operators were introduced by Berkson and Gillespie as a generalization in the context of well-boundedness of normal operators on Hilbert space. In this paper we explore some of the properties of these operators, such as the uniqueness of their splitting into real and imaginary parts, and their interpolation properties. We also examine the interpolation properties of the important subclass co...

Let E(·) : R → B(X) be the spectral decomposition of a trigonometrically well-bounded operator U acting on the arbitrary Banach space X, and suppose that the bounded function φ : T → C has the property that for each z ∈ T, the spectral integral R [0,2π] φ(e)dEz(t) exists, where Ez(·) denotes the spectral decomposition of the (necessarily) trigonometrically well-bounded operator (zU). We show th...

External fixation correction of angular deformities leads to secondary translation deformities when occurring around an axis located proximal or distal to the center of rotation of angulation (CORA); secondary length deformities result when correction occurs around an axis concave or convex to the CORA. With circular fixation, the hinge axis can be matched to the CORA. With monolateral fixation...

Motivated by problems in the spectral theory of linear operators the authors previously introduced a new concept of variation for functions defined on a nonempty compact subset of the plane. In this paper we examine the algebras of functions of bounded variation one obtains from these new definitions for the case where the underlying compact set is either a rectangle or the unit circle, and com...