Showing papers on "Auxiliary function published in 2002"

A fully direct RI-HF algorithm: Implementation, optimised auxiliary basis sets, demonstration of accuracy and efficiency

Abstract: A direct HF algorithm using the resolution of identity for Coulomb and exchange integrals (RI-HF) was implemented within the program system TURBOMOLE. A variational procedure for the optimisation of auxiliary functions is presented as well as optimised auxiliary basis sets for large basis sets up to Br. The accuracy of RI-HF energies and of MP2 energies based on RI-HF wave functions is demonstrated for a large test set of molecules. Accuracy of first order properties is documented for selected cases. The size dependency of the RI errors and the efficiency of the method are investigated for closo-boranes [BnHn]2− (n = 4–12).

Integral Characterizations of Uniform Asymptotic and Exponential Stability with Applications

Abstract: We present integral characterizations of uniform asymptotic stability and uniform exponential stability for differential equations and inclusions. These characterizations are used to establish new results on concluding uniform global asymptotic stability when uniform global stability is already known and uniform convergence must be established by additional arguments. In one case we generalize Matrosov's theorem on the use of a differentiable auxiliary function. In another case we draw conclusions from a system related to the original by suitable output injection.

On the calculation of arbitrary multielectron molecular integrals over Slater-type orbitals using recurrence relations for overlap integrals. III. Auxiliary functions and

Abstract: The auxiliary functions and which are used in our previous paper [Guseinov, I. I.; Mamedov, B. A. Int J Quantum Chem 2001, 81, 117] for the computation of multicenter electron-repulsion integrals over Slater-type orbitals (STOs) are discussed in detail, and the method is given for their numerical computation. The present method is suitable for all values of the parameters pa, p, and pt. Three- and four-center electron-repulsion integrals are calculated for extremely large quantum numbers using relations for auxiliary functions obtained in this paper. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001

Inverse spectral theory of finite Jacobi matrices

Abstract: We solve the following physically motivated problem: to determine all finite Jacobi matrices J and corresponding indices i, j such that the Green's function is proportional to an arbitrary prescribed function f(z). Our approach is via probability distributions and orthogonal polynomials. We introduce what we call the auxiliary polynomial of a solution in order to factor the map (J,i,j) → [ ] (where square brackets denote the equivalence class consisting of scalar multiples). This enables us to construct the solution set as a fibration over a connected, semi-algebraic coordinate base. The end result is a wealth of explicit constructions for Jacobi matrices. These reveal precise geometric information about the solution set, and provide the basis for new existence theorems.

Duality and Auxiliary Functions for Bregman Distances (revised)

Abstract: : In this paper, the authors formulate and prove a convex duality theorem for minimizing a general class of Bregman distances subject to linear constraints. The duality result is then used to derive iterative algorithms for solving the associated optimization problem. Their presentation is motivated by the recent work of Collins, Schapire, and Singer (2001), who showed how certain boosting algorithms and maximum likelihood logistic regression can be unified within the framework of Bregman distances. In particular, specific instances of the results given here are used by Collins et al. (2001) to show the convergence of a family of iterative algorithms for minimizing the exponential or logistic loss. Following an introduction, Section 2 recalls the standard definitions from convex analysis that will be required, and presents the technical assumptions made on the class of Bregman distances that the authors work with. They also introduce some new terminology, using the terms Legendre-Bregman conjugate and Legendre-Bregman projection to extend the classical notion of the Legendre conjugate and transform to Bregman distances. Section 3 contains the statement and proof of the duality theorem that connects the primal problem with its dual, showing that the solution is characterized in geometrical terms by a Pythagorean equality. Section 4 defines the notion of an auxiliary function, which is used to construct iterative algorithms for solving constrained optimization problems. This section shows how convexity can be used to derive an auxiliary function for Bregman distances based on separable functions. The last section summarizes the main results of the paper.

Aitken-Steffensen-type methods for nonsmooth functions (II)

Abstract: We present some new conditions which assure that the Aitken-Steffensen method yields bilateral approximation for the solution of a nonlinear scalar equation. The auxiliary functions appearing in the method are constructed under the hypothesis that the nonlinear application is not differentiable on an interval containing the solution.

Ideal construction and irrationality measures of roots of algebraic numbers

Abstract: This paper addresses the problem of determining the best results one can expect using the Thue-Siegel method as developed by Bombieri in his equivariant approach to effective irrationality measures to roots of high order of algebraic numbers, in the non-archimedean setting. As an application, we show that this method, under a nonvanishing assumption for the auxiliary polynomial which replaces the appeal to Dyson’s Lemma type arguments and together with a version of Siegel’s Lemma due to Struppeck and Vaaler, yields a result comparable to the best results obtained to date by transcendence methods.

Ritz method for approximate solution of boundary value problems in the nonlinear theory of thin shells

Abstract: The application of the Ritz method is justified in relation to the boundary value problems of the theory of geometrically and physically nonlinear thin elastic non-shallow shells. A distinctive feature of the paper is that we use some special function space, other than the usual spaces of displacements and stresses, and express the displacements and strains through some auxiliary functions. Boundary value problems for non-shallow shells whose middle surface is a surface of revolution or a convex surface having an involute, were considered in [1-4]. In this paper, we propose a method for the investigation of arbitrary non-closed non-shallow shells of a nonzero Gaussian curvature. The shells are supposed to be rigidly fixed along the entire boundary. We obtain equations for which the following conditions hold: there exists at least one point of absolute minimum for the total energy functional for the shell-external forces system; approximate solutions converge to a minimizer of this problem; the minimizer is a generalized solution of the equation obtained from the variational Lagrange principle.

A sequential convexification method for continuous global optimization

Abstract: A new method for continuous global minimization problems, acronymed SCM, is introduced. This method gives a simple transformation to convert the objective function to an auxiliary function with gradually ‘fewer’ local minimizers. All Local minimizers except a prefixed one of the auxiliary function are in the region where the function value of the objective function is lower than its current minimal value. Based on this method, an algorithm is designed which uses a local optimization method to minimize the auxiliary function to find a local minimizer at which the value of the objective function is lower than its current minimal value. The algorithm converges asymptotically with probability one to a global minimizer of the objective function. Numerical experiments on a set of standard test problems with several problems' dimensions up to 50 show that the algorithm is very efficient compared with other global optimization methods.