Showing papers on "Auxiliary function published in 1990"
TL;DR: In this article, the exponential stability of singularly perturbed systems with impulse effect is investigated and the comparison method and piecewise continuous auxiliary functions which are analogues of Lyapunov's functions are used.
16 citations
TL;DR: In this paper, the Neumann expansion for the electron-electron interaction term is used to expand and integrate the standard auxiliary functions for the two-center integral over Slater functions, and the resulting simple analytic formula for the total integral is completely general, requiring only the Slater function quantium numbers and exponents and the internuclear separation.
Abstract: For the general two-electron two-center integral over Slater functions, use of the Neumann expansion for the electron-electron interaction term yields the standard auxiliary functions. These are expanded and integrated explicitly by two independent methods. The resulting simple analytic formula for the total integral is completely general, requiring only the Slater function quantium numbers and exponents and the internuclear separation. Hence all two-electron hybrid, coulomb, exchange, and one-center integrals are considered. The efficiency of calculation of this expression is compared with those of other methods, indicating an order of magnitude improvement in speed over recursion for the exchange integral
15 citations
TL;DR: In this paper, the authors discuss molecular orbital theories: density driven self-consistent field theory and density driven Fomalism followed by the construction of one-particle wavefunctions.
Abstract: Publisher Summary This chapter discusses molecular orbital theories: density driven self- consistent field theory; and density driven self- consistent field theory (fomalism) followed by the construction of one- particle wavefunctions The molecular orbital theories, such as self-consistent field, configuration interaction and multi-configurational, constitute minimization of the energy functional The formalism presented in the chapter can be easily extended to other molecular orbital theories One can obtain explicit expressions for the exclusion potential and the associated variational equations for the auxiliary functions and the variational coefficients This allows approximating the exact pseudo- Schrodinger equation for the density amplitude up to any desired degree of accuracy Because the density amplitude is a function of only three Cartesian coordinates, the equation can be solved numerically The auxiliary functions can be approximated by finite basis sets These bases can be of a significantly lower quality that the bases currently used in an ab initio calculations Another possibility offered by the present formalism is the analysis of the results of molecular orbital calculations in terms of the Pauli and exchange-correlation potentials
14 citations
TL;DR: The paper presents various classes of EAF for a non linear programming problem, which has a saddle point of Lagra~ge function, and provides sufficient conditions for exact equivalence of the constrained minimization problem and minimization of Eaf.
Abstract: A new concept of an exact auxiliary function (EAF) is introduced. A function is said to be EAF,if the set of global minimizers of this function coincides with the global solu tion set of the initial optimization problem. Sufficient conditions for exact equivalence of the constrained minimization problem and minimization of EAF are provided. The paper presents various classes of EAF for a non linear programming problem, which has a saddle point of Lagra~ge function.
8 citations
TL;DR: An approach to constructing methods for solving the non-linear programming problem in which direction choosing in the original space is reduced by constraint approximation to choosing the direction of descent of an auxiliary function in a space of smaller dimension is proposed.
Abstract: An approach to constructing methods for solving the non-linear programming problem in which direction choosing in the original space is reduced by constraint approximation to choosing the direction of descent of an auxiliary function in a space of smaller dimension is proposed.
4 citations
Patent•
22 Oct 1990
TL;DR: In this article, the authors propose to shorten a time when practical cutting is not executed as far as possible, and to shorten the whole processing time as a result by preparing numerical control information in a format to control axial shift in parallel with the control of an auxiliary function during uncutting.
Abstract: PURPOSE:To shorten a time when practical cutting is not executed as far as possible, and to shorten the whole processing time as a result by preparing numerical control information in a format to control axial shift in parallel with the control of an auxiliary function during the uncutting CONSTITUTION:Auxiliary function control information SD outputted from an auxiliary function control information preparing part 4 is inputted to an auxiliary function control information separating part 10 The inputted auxiliary function control information SD is divided into practically cutting time auxiliary function control information SI and uncutting time auxiliary function control information SH The practically cutting time auxiliary function control information SI is outputted to a numerical control information output part 7, and the uncutting time auxiliary function control information SH is outputted to a control information synthesizing part 12 Further axial shift control information in terms of an axial shift function prepared in the same way is divided into the information in terms of the function during the practical cutting and the information in terms of the function during uncutting The auxiliary function control information is synthesized with the axial shift control information, and thus the numerical control information, which controls the axial shift in parallel with the control of the auxiliary function during uncutting, is prepared Thus the unnecessary operating time during uncutting is deleted, and the whole processing time can be shortened
3 citations
TL;DR: In this article, the question of boundedness of the solutions of systems of differential equations with impulses in terms of two measures is considered, and piecewise continuous auxiliary functions are used which are an analogue of the classical Lyapunov's functions.
Abstract: In the present paper the question of boundedness of the solutions of systems of differential equations with impulses in terms of two measures is considered. In the investigations piecewise continuous auxiliary functions are used which are an analogue of the classical Lyapunov's functions. The ideas of Lyapunov's second method are combined with the newest ideas of the theory of stability and boundedness of the solutions of systems of differential equations.
1 citations
01 Jan 1990
TL;DR: In this paper, a device based on the idea of Valentine's auxiliary functions is introduced to automatically take care of the assigned conditions in structural optimization problems, where the optimal distribution of thickness turns out to be 0 at some interior points in the domain of optimization Ω.
Abstract: Frequently, in structural optimization one may assign to the design parameters constraints that are independent of either the state function or the basic functional. For example, in the search for an optimal distribution of thickness h in a beam, plate, or shell we assign the constraint h ≥0, which is a direct consequence of the physical meaning of the specific design variable. In optimization problems in which the optimal distribution of thickness turns out to be 0 at some interior points in the domain of optimization Ω, this constraint may cause some difficulties. For this reason it is convenient to use a device based on an idea of Valentine, namely to introduce auxiliary functions that “automatically” take care of the assigned conditions. For example, if we introduce a new design variable φ related to h by the equality h =φ 2, then obviously, the inequality h ≥0 is satisfied for arbitrary real values of φ.
TL;DR: Brownawell-Waldschmidt as discussed by the authors obtained the theorem Theorem (theorem) of the field of complex, rational, and all algebraic numbers by adjoining an arbitrary transcendental number.
Abstract: Let C.Q. and A be, respectively, the fields of complex, rational, and all algebraic numbers; let Z be the ring of integers. Let Q, be an extension of field Q obtained by adjoining an arbitrary transcendental number e e C, and let Q~ be an algebraic extension of finite degree of the field Q1Waldschmidt [i, 2] and Brownawell [3], applying a method of Gel'fond [4, 5], independently obtained the following result: THEOREM (Brownawell-Waldschmidt). Let Xi, X2 e C. as well as Xi, X2 ~ C , be linearly independent over Q. If Ecau e~,~',, e~a*~A , then the extension of field Q , obtained by adjoining the numbers