Showing papers on "Auxiliary function published in 1996"

Generalized regular variation of second order

Abstract: Assume that for a measurable funcion f on (0, ∞) there exist a positive auxiliary function a(t) and some γ ∈ R such that . Then f is said to be of generalized regular variation. In order to control the asymptotic behaviour of certain estimators for distributions in extreme value theory we are led to study regular variation of second order, that is, we assume that exists non-trivially with a second auxiliary function a1(t). We study the possible limit functions in this limit relation (defining generalized regular variation of second order) and their domains of attraction. Furthermore we give the corresponding relation for the inverse function of a monotone f with the stated property. Finally, we present an Abel-Tauber theorem relating these functions and their Laplace transforms.

Efficient recursive computation of molecular integrals for density functional methods

Abstract: The expansion of the electronic density and the exchange‐correlation potential in auxiliary functions is a successful technique to reduce the computational effort in linear‐combination‐of‐ Gaussian‐type‐orbitals density functional theory (LCGTO‐DFT) methods. A new approach for the efficient calculation of the necessary three‐center overlap and electron repulsion integrals for this technique is presented. A new set of recurrence relations is derived, which take advantage of the special structure of the auxiliary function sets (primitive functions with shared exponents). Pathway diagrams from uncontracted integrals over s functions to any given class of target integrals are presented. The efficiency of different paths is discussed on the basis of floating‐point operations (FLOPs). The FLOP counting indicates that the new method represents a substantial improvement for the calculation of three‐center overlap and electron repulsion integrals.

A Sieve Auxiliary Function

Abstract: In the sieve theories of Rosser-Iwaniec and Diamond-Halberstam-Richert, the upper and lower bound sieve functions (F and f, respectively) satisfy a coupled system of differential-difference equations with retarded arguments. To aid in the study of these functions, Iwaniec introduced a conjugate difference-differential equation with an advanced argument, and gave a solution, q, which is analytic in the right half-plane. The analysis of the bounding sieve functions, F and f, is facilitated by an adjoint integral inner-product relation which links the local behaviour of F — f with that of the sieve auxiliary function, q. In addition, q plays a fundamental role in determining the sieving limit of the combinatorial sieve, and hence in determining the boundary conditions of the sieve functions, F and f. The sieve auxiliary function, q, has been tabulated previously, but these data were not supported by numerical analysis, due to the prohibitive presence of high-order partial derivatives arising from the numerical quadrature methods used [15, 17]. In this paper, we develop additional representations of q. Certain of these representations are amenable to detailed error analysis. We provide this error analysis, and as a consequence, we indicate how q-values guaranteed to at least seven decimal places can be tabulated.

On Sprindzuk's classification of transcendental numbers.

Abstract: In 1962 Sprindzuk [13] proposed a classification of the complex numbers, in connection with that of Mahler [9], by dividing them into four classes and he called the numbers in these classes Ä-, S-, T-, and {7-numbers (cf. also [14], p. 140 and [15], p. 1 3). The purpose of this paper is to show the existence of numbers with certain properties related to this classification, and especially to show the existence of -numbers. We first recall the definition of Sprindzuk' s classification and the several known results concerning this classification, then we state our main results.

On the convergence of a trust region SQP algorithm for nonlinearly constrained optimization problems

Abstract: In (Boggs, Tolle and Kearsley, 1994) the authors introduced an effective algorithm for general large scale nonlinear programming problems. In this paper we describe the theo-retical foundation for this method. The algorithm is based on a trust region, sequential quadratic programming (SQP) technique and uses a special auxiliary function, called a merit function or line-search function, for assessing the steps that are generated. A global convergence theorem for a basic version of the algorithm is stated and its proof is outlined.

Continuity of filled-in Julia sets and the closing lemma

Abstract: In this paper we discuss the continuity of filled-in Julia sets of functions meromorphic in the complex plane, i.e. rational or transcendental functions, or polynomials. The Main Theorem is: The filled-in Julia set depends continuously on the function provided the function in question has no Baker domain, wandering domain or parabolic cycle (theorem 3.1). The proofs are based on homotopy arguments and do not require any assumption on the number of singular values, actually, they simultaneously work for rational and transcendental functions. By examples we show the Main Theorem to be sharp. In order to illustrate the usage of filled-in Julia sets, applications to (relaxed) Newton's method are described. Using the continuity result a closing lemma for polynomials and entire transcendental functions is proven.