Showing papers on "Basis (linear algebra) published in 1989"

Universal Gaussian basis sets for an optimum representation of Rydberg and continuum wavefunctions

Abstract: A universal Gaussian basis set concept for the calculation of Rydberg and continuum states by pure L2 methods is presented It is based on the generation of optimised sequences of Gaussian exponents by maximising the overlap with a series of Slater-type functions characterised by a constant exponent and a variable principal quantum number In this way linear combinations of Gaussian basis functions can be found which are ideally suited to imitate Laguerre-Slater functions It is thus possible to obtain optimum representations of Rydberg orbitals or of complete orthonormal systems of Laguerre functions playing an important role in the L2 expansion of continuum functions The basis sets are tested with the hydrogen atom The effectiveness of the basis is illustrated by the calculation of quantum defects associated with the s, p and d Rydberg series of the alkali metal atoms Li and Na The phaseshifts determined in the ionisation continuat of these systems nicely fit the series below the ionisation limit as is finally demonstrated by an Edlen plot

Direct self control (DSC) of inverter-fed induction machine: a basis for speed control without speed measurement

Abstract: Mechanical speed measurement used to be an undesired necessity for inverter-fed induction machines. A possible way of calculating the exact speed without mechanical devices is presented. In addition, a simplified algorithm that is applicable for drives that do not demand high dynamic speed control performance (e.g., fans and pumps) is described. The control scheme is based on the analog basic version of direct self control (DSC) of inverter-fed induction machines and needs only a few additional operational amplifiers for signal processing. The performance, including start-up of an already rotating drive, is documented. >

The Reduced Basis Method for Incompressible Viscous Flow Calculations

Abstract: The reduced basis method is a type of reduction method that can be used to solve large systems of nonlinear equations involving a parameter. In this work, the method is used in conjunction with a standard continuation technique to approximate the solution curve for the nonlinear equations resulting from discretizing the Navier–Stokes equations by finite–element methods. This paper demonstrates that the reduced basis method can be implemented to approximate efficiently solutions to incompressible viscous flows. Choices of basis vectors, issues concerning the implementation of the method, and numerical calculations are discussed. Two fluid flow calculations are considered, the driven cavity problem and flow over a forward facing step.

On the theory of the electrocardiogram

Abstract: The biophysical basis for understanding the electrocardiogram is set forth. Bioelectric sources arise from electrical activity in the heart at the cellular level. The relation of these sources, which can be formally represented as impressed currents, to potentials involves solution of the volume conductor problem. This solution is based on Green's theorem. Sources are related to the transmembrane action potential through a bidomain model of heart muscle. Microscopic and macroscopic aspects of the bidomain model are developed. Various transformations of the source are considered, including multipoles, multiple dipoles, and replacement of the volume distribution with distributions on the heart surface. Time integrals of the waveform are related to excitation time and action potential duration. The theoretical results form the basis of a computer model of the electrocardiogram that relates skin potentials to the spatial and temporal distribution of action potentials in the heart. >

Reconstructions from zero crossings in scale space

Abstract: In computer vision, the one-parameter family of images obtained from the Laplacian-of-a-Gaussian-filtered version of the image, parameterized by the width of the Gaussian, has proved to be a useful data structure for the extraction of feature data. In particular, the zero crossings of this so-called scale-space data are associated with edges and have been proposed by D. Marr (1982) and others as the basis of a representation of the image data. The question arises as to whether the representation is complete and stable. The authors survey some of the studies and results related to these questions as well as several studies that attempt reconstructions based on this or related representations. They formulate a novel method for reconstruction from zero crossings in scale space that is based on minimizing equation error, and they present results showing that the reconstruction is possible but can be unstable. They further show that the method applies when gradient data along the zero crossings are included in the representation, and they demonstrate empirically that the reconstruction is then stable. >

Closed form two-sided bounds for probabilities that at least r and exactly r out of n events occur

Abstract: In two previous papers Prekopa (Prekopa, A. 1986a. Boole-Bonferroni inequalities and linear programming. Oper. Res. 36 145–162; Prekopa, A. 1986b. Sharp bounds on probabilities using linear programming. To appear in Oper. Res.) gave algorithms to approximate probabilities that at least r and exactly r out of n events occur (1 ≤ r ≤ n). Primal and dual linear programming problems were formulated and solved by dual type algorithms. The purpose of the present paper is to give closed forms for the basis inverse and the corresponding dual vector in case of an arbitrary basis, furthermore to give closed forms for the lower and upper bounds, approximating the probability in question, in case of a dual feasible basis. In the case when the probability that at least one out of n events occurs is approximated, it is shown that the absolute values of the components of any dual vector form a monotonically decreasing sequence. The paper improves the method of inclusion-exclusion, proves new probability inequalities and...

Definability with bounded number of bound variables

Abstract: A theory satisfies the k-variable property if every first-order formula is equivalent to a formula with at most k bound variables (possibly reused). Gabbay has shown that a model of temporal logic satisfies the k -variable property for some k if and only if there exists a finite basis for the temporal connectives over that model. We give a model-theoretic method for establishing the k -variable property, involving a restricted Ehrenfeucht-Fraisse game in which each player has only k pebbles. We use the method to unify and simplify results in the literature for linear orders. We also establish new k -variable properties for various theories of bounded-degree trees, and in each case obtain tight upper and lower bounds on k . This gives the first finite basis theorems for branching-time models of temporal logic.

Assembly strategies for chamferless parts

Abstract: The authors illustrate the need for planning in assembly and describe a set of modeling and planning techniques developed to generate robust force-control strategies for a certain class of assemblies. Specifically, they develop strategies for the chamferless insertion of a planar peg into a hole and the insertion of a three-dimensional rectangular peg into a rectangular hole. The complexity of applying these techniques in three dimensions is also discussed. It is seen from the analysis of the insertion of a planar peg into a chamferless hole that there are cases where an assembly may not be reliably carried out if the contact configurations between parts are not constrained. In such cases it is necessary to a priori specify the configurations through which the assembly must pass, and guarantee that only those configurations are encountered. From the analysis for the insertion of a three-dimensional rectangular peg into a chamferless hole, it is seen that for moderately complex assemblies involving may possible configurations, the specification of a set of configurations through which parts must pass is considerably more difficult than for the planar case. However, by considering only a subset of the possible configurations chosen on the basis of a set of heuristics, successful assembly strategies can be generated. >

First and second order convex approximation strategies in structural optimization

Abstract: In this paper, various methods based on convex approximation schemes are discussed, that have demonstrated strong potential for efficient solution of structural optimization problems. First, theconvex linearization method (CONLIN) is briefly described, as well as one of its recent generalizations, themethod of moving asymptotes (MMA). Both CONLIN and MMA can be interpreted as first order convex approximation methods, that attempt to estimate the curvature of the problem functions on the basis of semi-empirical rules. Attention is next directed toward methods that use diagonal second derivatives in order to provide a sound basis for building up high quality explicit approximations of the behaviour constraints. In particular, it is shown how second order information can be effectively used without demanding a prohibitive computational cost. Various first and second order approaches are compared by applying them to simple problems that have a closed form solution.

AI-based versus geometry-based design or Why design cannot be supported by geometry alone

Abstract: The reasons why the methodology and techniques underlying CAD systems cannot be used as the basis for engineering design support systems required to integrate computer-based support of the engineering design process are given. An argument is put forward that a combination of artificial intelligence techniques and geometric modelling techniques can meet these requirements.

An algorithm to design finite field multipliers using a self-dual normal basis

Abstract: The concept of using a self-dual normal basis to design the Massey-Omura finite-field multiplier is presented. An algorithm is given to locate a self-dual normal basis for GF(2/sup m/) for odd m. A method to construct the product function for designing the Massey-Omura multiplier is developed. It is shown that the construction of the product function based on a self-dual basis is simpler than that based on an arbitrary normal basis. >

On the operator bases underlying Wigner's, Kirkwood's and Glauber's phase space functions

Abstract: The Hermitian operators behind Wigner's phase space function (1932) are recognised to be simple ordered exponentials of the dynamical variables. This operator basis is highly symmetric; it supplies a perfectly unbiased formulation of operator equations in terms of phase space functions, which is as close to classical physics as it possibly can be. The author demonstrates how the symmetry properties of the basis can be exploited to simplify the computation of Wigner functions enormously. The ordered-operator methods are also applicable to the phase space functions of Kirkwood (1933) and Glauber (1963) type. Both kinds are briefly discussed with emphasis on how they differ from Wigner's description.

Application of cholesky-like matrix decomposition methods to the evaluation of atomic orbital integrals and integral derivatives

Abstract: When viewed as a square two-indexed matrix, the array of atomic orbital-based, two-electron integrals (ij|kl) is a positive semidefinite array. Beebe and Linderberg showed, in 1977, that actual or near linear dependencies often exist within the types of atomic orbital basis sets employed in conventional quantum chemical calculations. In fact, large (i.e., higher quality) bases were shown to be substantially more redundant than smaller or more spatially separated bases. In situations where there exists significant basis near redundancy, the rank (r) of the (ij|kl) ≡ Vl,J matrix of integrals will be significantly smaller than the matrix dimension M. When this occurs, it proves computationally tractable to decompose the M-dimensional matrix V into components L (V = LLT) which contain all of the information needed to form the full V matrix. The Cholesky algorithm allow such a decomposition to be carried out and forms the basis of the work described here. The method is found to be highly successful in reducing the number of integrals and integral derivatives that must actually be calculated. In particular, results on the C2 molecule indicate that the algorithm can be superior to traditional methods of integral derivative generation if the orbital basis is large enough to contain appreciable near redundancy. In contrast, results on benzene with a more spatially delocalized basis show that conventional methods are preferred whenever substantial basis (near) redundancy is not present.

Modular representations as a possible basis of finite physics

Abstract: The approach in which physical systems are described by the elements of a linear space over a finite field, and operators of physical quantities by linear operators in this space, is discussed. The mathematical formulation of the correspondence between such a description and the conventional one is given for a case when the characteristic of the finite field is sufficiently large. The above correspondence is considered in the examples of finite analogs of representations of so(2,3), so(1,4), and osp(1,4) algebras. Finite analogs of representations of infinite‐dimensional algebras are briefly discussed.

Mechanized derivation of linear invariants.

Abstract: Linear invariants, discovered by Lake, promise to provide a versatile way of inferring phylogenies on the basis of nucleic acid sequences (the method that he cahed “evolutionary parsimony”). A semigroup of Markov transition matrices embodies the assumptions underlying the method, and alternative semigroups exist. The set of all linear invariants may be derived from the semigroup by using an algorithm described here. Under assumptions no stronger than Lake’s, there are >50 independent linear invariants for each of the 15 rooted trees linking four species.

New crash procedures for large systems of linear constraints

Abstract: Many algorithms for solving linearly constrained optimization problems maintain sets of basic variables. The calculation of the initial basis is of great importance as it determines to a large extent the amount of computation that will then be required to solve the problem. In this paper, we suggest a number of simple methods for obtaining an initial basis and perform tests to indicate how they perform on a variety of real-life problems.

Device and method for determining target value of controllable variable associated with vehicle running state

Abstract: Device and apparatus for determining a target value of a controllable variable such as a drive torque associated with a running state of a motor vehicle, so that the variable is controlled so as to coincide with the determined target value. Initially, a basic value used as a basis for determining the target value is determined depending upon at least a currently required output of the vehicle engine which is represented by an accelerator pedal operating amount, for example. Then, control rules for compensating the basic value are defined based on at least one parameter indicative of a running condition of the vehicle, such as a vehicle weight and a road gradient. Finally, a compensation factor for compensating the basic value to obtain the target value is determined based on the control rules according to a fuzzy set theory.

Interpolation between bases and the shuffle exchange network

Abstract: Let u1. . . , un and v1, . . . , vn be bases of a vector space (the interesting case, when the underlying field is finite). Then there exist vectors w1, . . . , wn-1, such that every n consecutive vectors in the sequence u1, . . . , un, W1, . . . , Wn-l, v1, . . . , vn from a basis. Similar statements hold in structures other then vector spaces. The case of a free Boolean algebra is shown equivalent to an open problem in switching network theory.

Exact wave packet propagation using time‐dependent basis sets

Abstract: The use of a time‐dependent basis set for solving the time‐dependent Schrodinger equation is proposed. The basis set is a generalization of the harmonic oscillator functions. The proposed method is similar to the approaches formulated by Lee and Heller and by Coalson and Karplus. However, in the present method the time dependence of the basis functions is determined differently. Numerical studies on the quartic oscillator and on the Morse potential are carried out to investigate the performance of the various methods.

Invariant pattern recognition, symmetry, and Radon transforms

Abstract: A framework for a large class of invariant-pattern-recognition problems, is suggested. The proposed approach combines the properties of uniqueness, registration, and optimality with the ability to use natural symmetries of input patterns. The approach is based on cross correlation. The cross-correlation function with respect to a group of transformations is defined. The properties of this function and methods of its calculation are discussed. As a special case, it is shown formally that the cross correlation with respect to the motion group is the basis of such techniques as Radon transforms and the Hough technique. The roles of symmetry and parameters are examined. A technique for decomposition of pictures into sums of separable functions associated with groups of transformations is applied to the calculation of the cross-correlation function. Several examples of applications of the proposed approach are given.

Basis sets for manipulator inertial parameters

Abstract: An approach to the problem of finding the minimum number of inertial parameters of the robot manipulator dynamic equations of motion is presented. Based on the energy difference equation, it is equally applicable to serial-link graph-structured manipulators. The authors prove that the minimum number of parameters can be obtained directly from an energy difference equation rather than the dynamic equations of motion. They present a method for the identification of the minimum number of basis vectors which span the vector space containing the components of inertial parameters present in the equations of motion. Simulations are presented and the method is evaluated. The method is conceptually simple, computationally efficient and easy to implement. In particular, the manipulator kinematics and the joint positions and velocities are the only inputs to the algorithm. >

On the accuracy of Dirac-Hartree-Fock calculations using analytic basis sets

Abstract: It is shown that basis set approximation techniques provide a method for solving the Dirac-Hartree-Fock equations which is comparable in accuracy to finite difference approaches, provided attention is paid to the analytic form of the basis functions and to their completeness properties. Numerical examples are presented for Ar and Ar-like ions.

BAND-LIMITED FUNCTIONS: Lp -CONVERGENCE

Abstract: We consider the set BP(Q) (functions of LP(Tl) whose Fourier spectrum lies in (-I2, +I2) ). We prove that the prolate spheroidal wave func- tions constitute a basis of this space if and only if 4/3 < p < 4. The result is obtained as a consequence of the analogous problem for the spherical Bessel functions. The proof rely on a weighted inequality for the Hubert transform.