Showing papers on "Auxiliary function published in 1988"

Ab initio calculations on large molecules: The multiplicative integral approximation

Abstract: In the multiplicative integral approximation (MIA), two-electron integrals are evaluated using an expansion of a product of two Gaussians in terms of auxiliary functions. An estimator of the error introduced by the approximation is incorporated in the self-consistent field (SCF) calculations and the integrals for which the error estimate is larger than a preset value are systematically corrected. In this way the results of a MIA-assisted calculation have the same accuracy as a conventional calculation. The full exploitation of the expansion technique while constructing the Fock-matrix allows important time savings. Results are presented for a number of test cases.

Homogeneous denumerable Markov processes

Abstract: I Construction Theory of Sample Functions of Homogeneous Denumerable Markov Processes.- I The First Construction Theorem.- 1.1 Introduction.- 1.2 Definition of transformation gn.- 1.3 Convergence of the sequence X(n)(?) (n?1).- 1.4 Further properties of X(n)(?) (n?1).- 1.5 The first construction theorem.- II The Second Construction Theorem.- 2.1 Introduction.- 2.2 The mapping Tmn.- 2.3 The mapping Wn.- 2.4 Constructing auxiliary functions.- 2.5 The second construction theorem.- 2.6 Summary.- 2.7 Two notes.- II Theory of Minimal Nonnegative Solutions for Systems of Nonnegative Linear Equations.- III General Theory.- 3.1 Introduction.- 3.2 Definition of a system of nonnegative linear equations and definition, existence and uniqueness of its minimal nonnegative solution.- 3.3 Comparison theorem and linear combination theorem.- 3.4 Localization theorem.- 3.5 Connecting property of the minimal nonnegative solution.- 3.6 Limit theorem.- 3.7 Matrix representation.- 3.8 Dual theorem.- IV Calculation.- 4.1 Some lemmas.- 4.2 Reduction of the problems.- 4.3 Ordinary systems of strictly nonhomogeneous equations with dimension n.- V Systems of 1-Bounded Equations.- 5.1 Introduction.- 5.2 First-type leading-outside systems of equations.- 5.3 First-type consistent systems of equations.- 5.4 Tailed random systems of strictly nonhomogeneous equations.- 5.5 Regular systems of equations.- 5.6 Pseudo-normal systems of equations.- 5.7 Pseudo-normal systems of equations of finite dimension.- 5.8 Second-type regular systems of equations.- III Homogeneous Denumerable Markov Chains.- VI General Theory.- 6.1 Introduction.- 6.2 Transition probabilities.- 6.3 Distribution and moments of the first passage time.- 6.4 Distribution and moments of the first passage time of a homogeneous finite Markov chain.- 6.5 Distribution and moments of the times of passage.- 6.6 Criteria for recurrence.- 6.7 Distribution and moments of additive functionals.- 6.8 Derived Markov chains and criteria for atomic almost closed sets.- VII Martin Exit Boundary Theory.- 7.1 Introduction.- 7.2 Decomposition for Markov chains.- 7.3 Limit behaviour of excessive functions.- 7.4 Green functions and Martin kernels.- 7.5 h-chains.- 7.6 Limit theorem for Martin kernels.- 7.7 Martin boundaries.- 7.8 Distribution of x?.- 7.9 Martin expressions of excessive functions.- 7.10 Exit space.- 7.11 Uniqueness theorem.- 7.12 Minimal excessive functions.- 7.13 Terminal random variables.- 7.14 Criteria for potentials and excessive functions, Riesz decomposition.- 7.15 Criteria for minimal harmonic functions, minimal potentials and minimal excessive functions.- 7.16 Atomic exit spaces and nonatomic exit spaces.- 7.17 Blackwell decomposition of the state space.- VIII Martin Entrance Boundary Theory.- 8.1 Introduction.- 8.2 The first group of lemmas.- 8.3 Properties of finite excessive measures.- 8.4 The second group of lemmas.- 8.5 Entrance boundary.- 8.6 Entrance space and the expressions of excessive measures.- IV Homogeneous Denumerable Markov Processes.- IX Minimal Q-Processes.- 9.1 Introduction.- 9.2 Transition probabilities.- 9.3 Distribution and moments of the first passage time.- 9.4 Criterion for the positive recurrence.- 9.5 Distribution and moments of integral-type functionals.- 9.6 Distribution and moments of integral-type functionals on pseudo-translatable sets.- 9.7 Extensions of the results in 9.3.- X Q-Processes of Order One.- 10.1 Introduction.- 10.2 Transition probabilities.- 10.3 Distribution and moments of the first passage time.- XI Arbitrary Q-Processes.- 11.1 Strengthening of the first construction theorem.- 11.2 Transition probability.- 11.3 Decomposition theorems for excessive measures and excessive functions.- V Construction Theory of Homogeneous Denumerable Markov Processes.- XII Criteria for the Uniqueness of Q-Processes.- 12.1 Introduction.- 12.2 Lemmas.- 12.3 Proof of the main theorem.- 12.4 The case of diagonal type.- 12.5 The bounded case.- 12.6 The case when E is finite.- 12.7 The case of a branch Q-matrix.- 12.8 Another criterion and the finite and nonconservative case.- 12.9 Independence of the two conditions in Theorem 12.1.1.- 12.10 Probability interpretation of Condition (i) in Theorem 12.1.1.- XIII Construction of Q-Processes.- 13.1 Construction theorem.- 13.2 Specifications of all the Q-processes.- 13.3 Expression of $$\left\{ {Q,\,{\Pi _{{{\left( {\partial X} \right)}_{e,}}\,x\,E}}} \right\}$$-processes.- 13.4 Discussion.- XIV Qualitative Theory.- 14.1 Introduction.- 14.2 Statement of results.- 14.3 Reduction of the construction problem of B-type Q-processes, Doob processes.- 14.4 Reduction of the construction problem of B?F-type Q-processes.- 14.5 Proofs of Theorems 14.2.1-14.2.3.- 14.6 Proof and examples of applications of Theorem 14.2.4.- 14.7 Proofs of Theorems 14.2.5-14.2.10.

Generalization of the Fourier transform: Implications for inverse scattering theory.

Abstract: An infinite number of ways are developed for representing a function in terms of the eigenfunctions of a three-dimensional scattering problem and simple known auxiliary functions. The utility of the new expansions, which generalize both the Fourier and Radon transforms, is shown by derivation of a new representation of the scatterer for the near- (far-) field inverse problem. Further, the scattering amplitude and potential are shown to be a generalized Fourier-transform pair.

Generalized eigenfunction expansions for scattering in inhomogeneous three-dimensional media

Abstract: An infinite number of ways are developed for representing a function in terms of the (generalized) eigenfunctions of a three‐dimensional scattering problem and simple known auxiliary functions. The freedom represented by this variety of expansions arises from the causal nature of the wave equations considered. The new expansions are shown to generalize both the Fourier and Radon transforms. An application of the new expansions to the inverse scattering problem is given. It is shown (under some restrictions) that the scattering amplitude and potential are related via one of the generalized transforms.

Complex Dynamical System for Toda Lattice

Abstract: Extending an independent variable into complex and introducing an auxiliary function, we investigate nonlinear interactions between solitons for the Toda lattice by observing behavior of zeros of the function The Newton's method calculating them is identified with a complex dynamical system We present numerical results of the Fatou set on the dynamics According to motion of solitons, the set changes surprisingly Since soliton solutions include the exponential function, the Fatou set is different from that of the polynomial and the rational functions

Auxiliary function controller for nc machine

Abstract: PURPOSE:To attain the overall processing of auxiliary functions for plural sets of NC machines, by providing an input means which receives a signal outputted from an overall controller, an auxiliary function decoding circuit which inputs a received auxiliary function code, and an execution circuit which executes a decoded auxiliary function. CONSTITUTION:An arithmetic processing circuit 3 performs an arithmetic processing on a set auxiliary function code so as to be the same code as the auxiliary function from the NC machine 14 which belongs to the NC machine 10. Respective NC machine 10 receives the signal from the overall controller 1 by the input means 11, and supplies it to the auxiliary function decoding circuit 12. The auxiliary function decoding circuit 12 performs decoding as if the auxiliary function code is supplied from the NC machine 14, because the auxiliary function of the circuit is the same as that from the NC machine 14, and supplies at bit of information to the auxiliary function execution circuit 13. The auxiliary function execution circuit 13 operates the actuator of a mechanical constituting parts 15 based on a supplied bit of information. In this way, it is possible to perform the auxiliary function operation of respective NC machine 10 from the overall controller 1 side.

Limit-point criteria for symmetric and j-symmetric quasi-differential expressions of even order with a positive definite leading coefficient

Abstract: A limit-point criterion is proved by the method of inequalities for differential expressions of the form where the coefficients are matrix-valued and L is symmetric or J-symmetric. In the symmetric case the assumptions imply that some or all real polynomials in L are also limit-point. One of the main theorem is indeed formulated as an interval type criterion, but it is neither assumed that the sub-intervals are mutually disjoint nor that certain auxiliary functions vanish at the endpoints of the subintervals. Applications of the main results yield generalizations of a umber of known criteria, for example sow of those obtained by Hinton [15,16], Atkinson [2] and Evans and Zettl [6,7]. Also some kind of integrable perturbations of the coefficients can be dealt with and so some of the results of [25], which were proved by asymptotic methods, follow from the main theorem. Finally a criterion with positive supporting coefficients is derived. Such results were proved up to now by perturbation method on...

On the asymptotic stability of systems with impulses by the direct method of Lyapunov

Abstract: In this paper the asymptotic and globally asymptotic stability of the zero solution of systems with impulses is investigated. For this purpose piecewise continuous auxiliary functions are used which are an analogue to Lyapunov's functions. The theorem of Marachkov on the asymptotic stability of systems without impulses is generalized. The results obtained are formulated in four theorems.