Showing papers on "Basis (linear algebra) published in 1992"

A novel discrete variable representation for quantum mechanical reactive scattering via the S-matrix Kohn method

Abstract: A novel discrete variable representation (DVR) is introduced for use as the L2 basis of the S‐matrix version of the Kohn variational method [Zhang, Chu, and Miller, J. Chem. Phys. 88, 6233 (1988)] for quantum reactive scattering. (It can also be readily used for quantum eigenvalue problems.) The primary novel feature is that this DVR gives an extremely simple kinetic energy matrix (the potential energy matrix is diagonal, as in all DVRs) which is in a sense ‘‘universal,’’ i.e., independent of any explicit reference to an underlying set of basis functions; it can, in fact, be derived as an infinite limit using different basis functions. An energy truncation procedure allows the DVR grid points to be adapted naturally to the shape of any given potential energy surface. Application to the benchmark collinear H+H2→H2+H reaction shows that convergence in the reaction probabilities is achieved with only about 15% more DVR grid points than the number of conventional basis functions used in previous S‐matrix Kohn...

Multiresolution analysis. Haar bases, and self-similar tilings of R/sup n/

Abstract: Orthonormal bases for L/sup 2/(R/sup n/) are constructed that have properties that are similar to those enjoyed by the classical Haar basis for L/sup 2/(R). For example, each basis consists of appropriate dilates and translates of a finite collection of 'piecewise constant' functions. The construction is based on the notion of multiresolution analysis and reveals an interesting connection between the theory of compactly supported wavelet bases and the theory of self-similar tilings. >

A sampling theorem for wavelet subspaces

Abstract: The classical Shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory. Under weak hypotheses, these subspaces are first shown to have a Riesz basis formed from the reproducing kernels. These in turn are used to construct the sampling sequences. Examples are given. >

Steerable-scalable kernels for edge detection and junction analysis

Abstract: Families of kernels that are useful in a variety of early vision algorithms may be obtained by rotating and scaling in a continuum a ‘template’ kernel. These multi-scale multi-orientation family may be approximated by linear interpolation of a discrete finite set of appropriate ‘basis’ kernels. A scheme for generating such a basis together with the appropriate interpolation weights is described. Unlike previous schemes by Perona, and Simoncelli et al. it is guaranteed to generate the most parsimonious one. Additionally, it is shown how to exploit two symmetries in edge-detection kernels for reducing storage and computational costs and generating simultaneously endstop- and junction-tuned filters for free.

A new analytical diode model including tunneling and avalanche breakdown

Abstract: An analytical model describing reverse and forward DC characteristics is presented. It serves as a basis for a compact model for circuit simulation purposes. The model is based on the solution of the hole continuity equation in the depletion layer of a p-n junction and incorporates the following physical mechanisms: band-to-band tunneling, trap-assisted tunneling (both under forward and reverse bias), Shockley-Read-Hall recombination, and avalanche breakdown. It contains seven parameters which can be determined at one temperature. No additional parameters are needed to describe the temperature dependence. From comparisons with both numerical simulations and measurements it is found that the model gives an adequate description of the DC characteristics in both forward and reverse modes. >

A comparison of text retrieval models

Abstract: Many retrieval models have been proposed as the basis of text retrieval systems. The three main classes that have been investigated are the exact-match, vector space and probabilistic models. The retrieval effectiveness of strategies based on these models has been evaluated experimentally, but there has been little in the way of comparison in terms of their formal properties. In this paper we introduce a recent form of the probabilistic model based on inference networks, and show how the vector space and exact-match models can be described in this framework. Differences between these models can be explained as differences in the estimation of probabilities, both in the initial search and during relevance feedback.

The Generalized Basis Reduction Algorithm

Abstract: Let Fx be a convex function defined in Rn, which is symmetric about the origin and homogeneous of degree 1, and let L be the lattice of integers Zn. A definition of a reduced basis, b1, ', bn, of the lattice with respect to the distance function F is presented, and we describe an algorithm which yields a reduced basis in polynomial time, for fixed n. In the special case in which the bodies {x: Fx ≤ t} are ellipsoids, the definition of a reduced basis is identical with that given by Lenstra, Lenstra and Lovasz 1982 and the algorithm is the well-known basis reduction algorithm. We show that the basis vector b1, in a reduced basis, is an approximation to a shortest nonzero lattice point with respect to F and relate the basis vectors bi to Minkowski's successive minima. The results lead to an algorithm for integer programming which executes in polynomial time for fixed n, but which avoids the ellipsoidal approximations required by Lenstra's algorithm. We also discuss the properties of a Korkine-Zolotarev basis for the lattice.

A Linear Relaxation Heuristic for the Generalized Assignment Problem

Abstract: We examine the basis structure of the linear relaxation of the generalized assignment problem. The basis gives a surprising amount of information. This leads to a very simple heuristic that uses only generalized network optimization codes. Lower bounds can be generated by cut generation, where the violated inequalities are found directly from the relaxation basis. An improvement heuristic with the same flavor is also presented.

Some Aspects of Radial Basis Function Approximation

Abstract: This paper deals with three basic aspects of radial basis approximation. A typical example of such an approximation is the following. A function f in C (ℝn) is to be approximated by a linear combination of ‘easily computable’ functions g 1,…, g m For these functions the simplest choice in the radial basis context is to define g i by x ↦ ∥x − x i∥2 for x ∈ ℝn and i = l,2,…,m. Here ∥ · ∥2 is the usual Euclidean norm on ℝn. These functions are certainly easily computable, but do they form a flexible approximating set? There are various ways of posing the question of flexibility, and we consider here three possible criteria by which the effectiveness of such approximations may be judged. These criteria are labelled density, interpolation and order of convergence in the exposition.

Crossover behavior of the susceptibility and the specific heat near a second-order phase transition

Abstract: Simple crossover equations for the susceptibility and the specific heat in zero field have been obtained on the basis of the renormalization-group method and e-expansion. The equations contain the Ginzburg number as a parameter. At temperatures near the critical temperature, scaling behavior including the first Wegner corrections is reproduced. At temperatures far away from the critical temperature the classical Landau expansion with square-root corrections is recovered. For small values of the Ginzburg number the crossover equations approach a universal form. The equations are applied to represent experimental specific heat data for CH 4 , C 2 H 6 , Ar, O 2 and CO 2 along the critical isochore in a universal form.

Quantitative NDE technique for assessing damages in beam structures

Abstract: A physical model using a massless rotational spring to represent the crack-induced local flexibility is adopted as a basis for developing a quantitative nondestructive evaluation (NDE) technique for assessing the damage in a beam structure. Expressions relating changes in eigenfrequencies of the structure and the quantitative damage index S\I\de\N are formulated via an influence matrix h. The coefficients in the h matrix can be determined from the knowledge of the modal shapes of the undamaged structure. The damage index matrix S\I\de\N can be solved by the resulting system of linear algebraic equations using the measured eigenfrequencies of existing structure. Methods of solution for the system of equations are discussed along with special cases where a pseudo-inverse technique is necessary to solve the system of equations. Series of case studies involving simply supported, cantilevered, and continuous beams are presented to demonstrate the validity of the formulation.

Linear and cyclic isomers of C4. A theoretical study with coupled‐cluster methods and large basis sets

Abstract: The ground electronic states of linear and rhombic C4 have been studied by high level ab initio quantum chemical techniques. Geometries, harmonic vibrational frequencies, infrared intensities, and other quantities have been determined using 4s3p2d1f correlation consistent basis sets and coupled‐cluster methods including triple excitations. The linear–rhombic isomer energy difference has been investigated with a range of basis sets, including a 5s4p3d2f1g correlation consistent set. The linear–rhombic energy difference is influenced significantly by basis set, presence of triple excitations, and the choice of reference function for the open‐shell linear isomer. The effect of basis set variation is complex, but once a reasonable quality of basis set has been achieved, further extensions favor the rhombic isomer. The inclusion of triple excitations also favors the rhombic isomer. The use of a restricted Hartree–Fock reference function for the linear isomer yields higher energies at the coupled‐cluster level ...

On the Structure of Hv0(D) and hv0(D)

Abstract: Weighted spaces of harmonic and holomorphic functions on the unit disc are discussed. It is shown that these spaces are always subspaces of c0. Moreover, for many weights, it is shown that the weighted space of holomorphic functions has a basis.

Genetic algorithm technique for designing neural networks

Abstract: A generic algorithm search is applied to determine an optimum set of values (e.g., interconnection weights in a neural network), each value being associated with a pair of elements drawn from a universe of N elements, N an integer greater than zero, where the utility of any possible set of said values may be measured. An initial possible set of values is assembled, the values being organized in a matrix whose rows and columns correspond to the elements. A genetic algorithm operator is applied to generate successor matrices from said matrix. Matrix computations are performed on the successor matrices to generate measures of the relative utilities of the successor matrices. A surviving matrix is selected from the successor matrices on the basis of the metrics. The steps are repeated until the metric of the surviving matrix is satisfactory.

A finite basis‐discrete variable representation calculation of vibrational levels of planar acetylene

Abstract: A methodology for the calculation of high energy vibrational eigenstates of S0 acetylene is described. Acetylene is modeled as a 5D planar molecule. The discrete variable representation (DVR) is employed for radial coordinates with a finite basis representation (FBR) for the angular coordinates. Symmetry adaptation of the primitive basis (dimension 2.7 × 106) coupled with a two level diagonalization/truncation scheme maintains relatively small basis sets (< 2500 functions) in all diagonalizations. Eigenvalues up to nearly 3700 cm−1 above the ground state are reported.

Hyperspherical Strumian basis functions

Abstract: A Sturmian basis set is a set of solutions to the Schrodinger equation, with the potential scaled in such a way that all the members of the set correspond to the same value of the energy. We discuss, in particular, the set of Sturmian basis functions corresponding to solutions of the d-dimensional hydrogenlike wave equation. These hydrogenlike Sturmian functions are expressed in terms of Laguerre polynomials and hyperspherical harmonics. When they are used as a basis for solving the many-particle Schrodinger equation, the secular equations take on a simple form [Eq. (59)]. The necessary integrals are evaluated explicitly, and the possibility of combining the hyperspherical technique with dimensional scaling is discussed.

Correlation VIA linear sequential circuit approximation of combiners with memory

Abstract: Correlation properties of a general binary combiner with an arbitrary number of memory bits are analyzed. It is shown that there exists a pair of certain linear functions of the output and input, respectively, that produce correlated binary sequences. An efficient procedure, based on a linear sequential circuit approximation, is developed for finding such pairs of linear functions. The result may be a basis for a divide and conquer correlation attack on a stream cipher generator consisting of several linear feedback shift registers combined by a combiner with memory.

Pattern classification system

Abstract: A pattern classification system includes a plurality of classification sections Each of the classification sections includes a device for storing information of N coefficients W representing a reference pattern, a device for calculating an evaluation value V on the basis of N input signals S and the N coefficients W, the N input signals representing an input pattern, the evaluation value V representing a relation between the input pattern and the reference pattern, a device for storing information of a fixed threshold value R, and a device for comparing the evaluation value V and the threshold value R and for outputting an estimation signal depending on a result of the comparing, the estimation signal including a category signal P which represents a category A selection section is operative for selecting one of categories in response to the category signals outputted from the classification sections and for outputting a signal Px representing the selected one of the categories An adjustment section is operative for adjusting parameters in the classification sections in response to the output signal Px of the selection section and a teacher signal T, the parameters including the coefficients W

Karhunen Loeve Feature Extraction for Neural Handwritten Character Recognition

Abstract: The optimality of the Karhunen Loeve (KL) transform is well known. Since its basis is the eigenvector set of the covariance matrix, a statistical, not functional, representation of the variance in pattern ensembles is generated. By using the KL transform coefficients as a natural feature representation of a character image, the eigenvector set can be regarded as an unsupervised biological feature extractor for a (neural) classifier. The covariance matrix and its eigenvectors are obtained from 76,753 handwritten digits. This operation is a unique expense; once the basis set is calculated it forms a linear first layer of a three weight layer feed forward network. The subsequent nonlinear perceptron layers are trained using a scaled conjugate gradient algorithm that typically affords an order of magnitude reduction in computation over the ubiquitous back-propagation algorithm. In conjunction with a massively parallel computer, training is expedited such that tens of initially different random weight sets are trained and evaluated. Increase in training set size (up to 76,755 patterns) gives less accurate learning but improved generalization on the fixed disjoint test set. A neural classifier is realized that recognizes 96.1% of 15,000 handwritten digits from 944 different writers. This recognition is attributed to the energy compaction optimality of the KL transform.

Application of an optimization technique to the discretized version of the Griffin–Hill–Wheeler–Hartree–Fock equations

Abstract: Application of the SIMPLEX optimization method to define the mesh of the discretized version of the Griffin–Hill–Wheeler–Hartree–Fock (GHWHF) equations was studied. Improved discretization parameters with respect to the original method were obtained for atomic systems with two or four electrons and for the H2 molecule. For the atomic systems, the following correlations between the discretization parameters and the total energy were found: N = a · In(ΔE) + b; Ω0 = a′ · In(ΔΩ) + a″; and In(ΔΩ) = b′ · ln(N) + b″. These equations provide a systematic procedure to reach a desired degree of accuracy in the energy for the atomic systems studied as well as to fix the basis set to be employed. These equations are similar to those found earlier for eventempered basis sets and permit the establishment of a relationship between the two methods. The even-tempered method is also an approximate solution of the GHWHF equations. The optimized integral discretized basis is more efficient in representing small basis sets for atoms and the basis for the hydrogen molecule in comparison to the even-tempered one. The optimization procedure was successfully applied to generate the universal basis for the atomic systems studied.

Lie Groups in Electromagnetic Wave Propagation and Scattering

Abstract: The theory of Lie algebras and their classification based on Dynkin diagrams is reviewed and application of the theory to vector wave scattering and propagation is outlined. The main applications arise through the matrix exponential function and associated matrix representations for the underlying groups. By projecting experimental measurements onto the eigenvectors of these basis elements, clear physical significance may be attached to the propagation operators. A new method for dealing with partially coherent propagators is outlined and an example for backscatter from a cloud of dipoles is used to illustrate the technique.

Formulation and efficient computation of inverse dynamics of space robots

Abstract: A free-flying space robot for the construction and maintenance of space structures is considered. Such a robotic system has kinematic and dynamic features that differ from those fixed on the earth mainly because of the momentum constraints that govern its motion. The solution to the inverse dynamics problem of a space robotic system in the presence of external generalized forces is presented. The computations for the inverse kinematics are considered simultaneously, and both computations are developed on the basis of momentum constraints. An efficient computational scheme for the inverse dynamics problem is then established. An intrinsic feature of space robotic systems that can be utilized to reduce the computational time by parallel recursion is discussed, as is the importance of the role of momentum constraints in the solution of inverse dynamics. >

Observations on the Proper Orthogonal Decomposition

Abstract: The Proper Orthogonal Decomposition (P.O.D.), also known as the Karhunen-Loeve expansion, is a procedure for decomposing a stochastic field in an L2 optimal sense. It is used in diverse disciplines from image processing to turbulence. Recently the P.O.D. is receiving much attention as a tool for studying dynamics of systems in infinite dimensional space. This paper reviews the mathematical fundamentals of this theory. Also included are results on the span of the eigenfunction basis, a geometric corollary due to Chebyshev’s inequality and a relation between the P.O.D. symmetry and ergodicity.