Showing papers on "Auxiliary function published in 1982"

Bäcklund generated solutions of Liouville’s equation and their graphical representations in three spatial dimensions

Abstract: The Backlund–Bianchi method is employed to generate, in three spatial dimensions, the following multiple solutions of Liouville’s equation ∇2α = exp α: The three‐wave interaction function α3 and the five‐wave interaction function α5. It is verified numerically that α3 satisfies Liouville’s equation to an accuracy of one part in 1014, while α5 satisfies it to one part in 106. The construction of α5 is conditional upon solving ten nonlinear constraint equations. We analyze the complicated structures of α3 and α5 with the help of a three‐dimensional plotting routine. It is found that α3 is, surprisingly enough, only characterized by a single ring singularity, while α5 exhibits three ring singularities. It is speculated that the function tanh α3 represents a ring soliton whose shape appears to be preserved in the nonlinear superposition of similar ring solitons. The derivation of Liouville’s solutions α3 and α5 is intimately connected with the auxiliary functions β2 and β4 which solve Laplace’s equation. The latter are also derived and plotted in the paper.

Two Theorems for Time-Dependent Dynamical Systems

Abstract: There is a growing aggregation of papers related to the search for invariants, or first integrals, of time-dependent dynamical systems. The citation of all such work since the seminal papers by LewisIl•) would be prohibitively lengthy in a short note. Two short review papers by Ray3),.) contain many references to recent literature on this subject. We will refer here only to key papers selected to serve the main points of the present paper. Our purpose is to prove two theorems which pertain to the equations of motion of dynamical systems recently discussed in the literature. The proof of the theorems will show that an auxiliary variable is naturally associated with these equations of motion, will reveal the origin of useful rescaling variables and will lead to direct determination of an associated invariant of the motion, The discussion of time-dependent systems is frequently presented in terms of an auxiliary function p(t), say, where t denotes the independent variable time. For many such systems the function p (t) is any solution to the specific nonlinear differential equation

A variational principle for the Heitler-London intermolecular interaction energy

Abstract: An extension of the Delves variational principle is used to find an expression for the Heitler-London (exchange) interaction energy which differs from the exact results by second order terms. The auxiliary functions so introduced are found by relating the variational principle to Symmetry-Adapted Double Perturbation theory and, with neglect of a term likely to be small, this expression is the same as that usually used. From these considerations two criteria for the optimization of wavefunctions can be found. The criteria are applied to the Helium-Helium and Neon-Neon interactions using minimal basis wavefunctions and, by allowing the parameter values to vary with internuclear distance, significant improvements in the interaction energies are obtained, in particular those for Helium-Helium agreeing with accurate SCF results.

On Auxiliary Functions in Molecular Integrals

Abstract: A brief review of work on an analytical solution to three center nuclear attraction integrals and relationship among selected auxiliary functions is presented. The review is preceded by an outline of a few concepts from graph theory -- the subject of current intensive interest of the author. Graphs already play useful role in molecular calculations, even in some problems involving molecular integrals and graph theory deserves a better exposure. In the spirit of graph theoretical tradition, the problem of solving of four center integrals has been here exalted to a conjecture that such integrals can be evaluated analytically. While believers will try to prove the conjecture, now there is also a burden on non-believers to disprove it! In a more solemn direction, the contribution suggests that overlap integrals play a role of auxiliary functions and some hope is expressed that additional molecular integrals will be expressed in terms of general overlap integrals or as some function of overlap integrals.

Real-Time Solution of Linear Least-Squares Estimation Problem with Semi-Degenerate Covariance

Abstract: On making use of an invariant imbedding, an initial-value solution of the recursive Hopf type integral equations with semi-degenerate asymmetric kernels is presented for the linear least-squares filtering problems with time-variant properties. Furthermore, it is shown that the real-time solution of the optimal estimate is reduced to a Cauchy system for an integral of the stochastic processes weighted with an auxiliary function and also that in the case of displacement kernels the Sobolev resolvent equation is rigorously obtained with the aid of Busbridge's combined operations method.