The definition of the standard differential operator is extended from integer steps to arbitrary stepsize. The classical, nonrelativistic Hamiltonian is quantized, using these new continuous operators. The resulting Schroedinger type equation generates free particle solutions, which are confined in space. The angular momentum eigenvalues are calculated algebraically. It is shown, that the charm...

Overview It has long been known that the solutions produced by adaptive solvers for ordinary differential (ODE) and differential algebraic (DAE) systems, while generally reliable, are not smooth with respect to perturbations in initial conditions or other problem parameters. Söderlind and Wang [12, 13] have recently developed a digital filter stepsize controller that has a theoretical basis fro...

The existing distributed algorithms for Network Utility Maximization (NUM) problems are mostly constructed using dual decomposition and first-order (gradient or subgradient) methods, which suffer from slow rate of convergence. Recent work [25] proposed an alternative distributed Newton-type algorithm for solving NUM problems with self-concordant utility functions. For each primal iteration, thi...

In this letter, we address the distributed optimization problem with event-triggered communication by notion of input feedforward passivity (IFP). First, analyze continuous-time algorithm over uniformly jointly strongly connected balanced digraphs in an IFP-based framework. Then, propose a mechanism for algorithm. Next, discretize forward Euler method constant stepsize irrelevant to network siz...

In this paper, we workout a detailed mathematical analysis for a new learning algorithm termed Cascade Error Projection (CEP) and a general learning frame work. This frame work can be used to ob(ain the cascade correlation learning algorithm by choosing a particular set of parameters. Furthemlore, CEP learning algorithm is operated only on one layer, whereas the other set of weights can be calc...

Recent years have witnessed the surge of asynchronous parallel (async-parallel) iterative algorithms due to problems involving very large-scale data and a large number of decision variables. Because of asynchrony, the iterates are computed with outdated information, and the age of the outdated information, which we call delay, is the number of times it has been updated since its creation. Almos...

A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the Euler– Lagrange equations is presented. A new class of integrators is defined: the super partitioned additive Runge–Kutta (SP...

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations (SDEs). We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications (section 1), and g...

When numerically tracking implicitly-defined paths, such as is required for homotopy continuation methods, efficiency and reliability are enhanced by using adaptive stepsize and adaptive multiprecision methods. Both efficiency and reliability can be further improved by adapting precision and stepsize simultaneously. This paper presents a strategy for adjusting precision and stepsize together to...