Showing papers on "Orthonormal basis published in 1981"
TL;DR: An alternative to the Gilbert construction is presented in this paper, where for any nonnegative, normalized density an arbitrary number of orthonormal orbitals can be constructed with squares which sum, with minimal restrictions on the occupation numbers, to the given density.
Abstract: An alternative to the Gilbert construction is presented. It is shown that for any nonnegative, normalized density an arbitrary number of orthonormal orbitals can be constructed with squares which sum, with minimal restrictions on the occupation numbers, to the given density. In three dimensions substantial freedom remains in the choice of orbitals within the basic scheme. The kinetic-energy density obtained includes the Weizs\"acker term and another term which, in simple cases, is proportional to the cube of the density.
293 citations
TL;DR: In this article, the geometry of canonical variate analysis is described as a two-stage orthogonal rotation, where the first stage involves a principal component analysis of the original variables.
Abstract: The geometry of canonical variate analysis is described as a two-stage orthogonal rotation. The first stage involves a principal component analysis of the original variables. The second stage involves a principal component analysis of the group means for the orthonormal variables from the first-stage eigenanalysis. The geometry of principal component analysis is also outlined. Algebraic aspects of canonical variate analysis are discussed and these are related to the geometrical description. Some practical implications of the geometrical approach for stability of the canonical vectors and variable selection are presented. [Multivariate analysis; canonical variate analysis; discriminant analysis; principal component analysis.]
277 citations
TL;DR: In this paper, a functional analytic formulation of sensitivity theory is extended to include treatment of additional types of responses, such as general operators acting on the system's state vector and parameters, as response.
Abstract: This work extends a recent, functional‐analytic formulation of sensitivity theory to include treatment of additional types of responses. There are physical systems where a critical point of a function that depends on the system’s state vector and parameters defines the location in phase‐space where the response functional is evaluated. The Gâteaux differentials giving the sensitivities of both the functional and the critical point to changes in the system’s parameters are obtained by alternative formalisms. The foward sensitivity formalism is the simpler and more general, but may be prohibitively expensive for problems with large data bases. The adjoint sensitivity formalism, although less generally applicable and requiring several adjoint calculations, is likely to be the only practical approach. Sensitivity theory is also extended to include treatment of general operators, acting on the system’s state vector and parameters, as response. In this case, the forward sensitivity formalism is the same as for functional responses, but the adjoint sensitivity formalism is considerably different. The adjoint sensitivity formalism requires expanding the indirect effect term, an element of a Hilbert space, in terms of elements of an orthonormal basis. Since as many calculations of adjoint functions are required as there are nonzero terms in this expansion, careful consideration of truncating the expansion is needed to assess the advantages of the adjoint sensitivity formalism over the forward sensitivity formalism.
218 citations
TL;DR: In this article, a linear transformation τ on J is an isometry of J onto itself if and only if there are unitary operators U and V on H such that either τ(X) = UXV or τ[X] = UXtV, where Xt denotes the transpose of X with respect to a fixed orthonormal basis of H.
51 citations
01 Jan 1981
TL;DR: In this article, the theory of Hilbert modules over noncommutative C* -algebras is used to establish a general theory of extensions of C*-algesbras that extends results of Brown, Douglas, and Fillmore.
Abstract: In a recent paper of Kasparov [K] the theory of Hilbert modules over noncommutative C* -algebras is used to establish a general theory of extensions of C*-algebras that extends results of Brown, Douglas, and Fillmore [BDF], Fillmore [F], and Pimsner, Popa, and Voiculescu [PPV]. Since the category of Hilbert C (X) -modules is equivalent to the category of Hilbert bundles over X [DD;DG], many questions of topological interest can be recast in terms of Hilbert C(X)-modules which then give rise to questions about general Hilbert modules. In particular, Kasparov’s stability theorem [K] (which plays an essential part in the proof that inverses exist in the general theory of EXT) is the noncommutative extension of a triviality theorem of Dixmier and Douady [DD, Th.4] (which itself provides the existence of classifying maps for arbitrary separable Hilbert bundles over paracompact spaces).
26 citations
TL;DR: In this article, a causal FIR filter may be approximated by an IIR filter of lower order requiring fewer memory elements, and an iterative procedure is given for optimizing the approximation in the discrete time domain, based on representing the IIR function as a sum of orthonormalized complex decaying exponentials.
Abstract: A given causal FIR filter may be approximated by an IIR filter of lower order requiring fewer memory elements. An iterative procedure is given for optimizing the approximation in the discrete time domain, based on representing the IIR function as a sum of orthonormalized complex decaying exponentials.
21 citations
TL;DR: In this paper, a method for analyzing the response of a class of weakly non-linear and lightly damped systems to a separable non-stationary random excitation is presented.
Abstract: A method for analyzing the response of a class of weakly non-linear and lightly damped systems to a separable non-stationary random excitation is presented. The random excitation is represented as the product of a slowly varying modulating deterministic function and a broad-band stationary process. Using an averaging procedure a first order equation governing the time evolution of the response amplitude is derived. The Fokker-Planck equation which describes the diffusion of the probability density function of the response amplitude is considered. A particularly convenient basis of orthonormal functions, as well as, necessary formulae for the determination of an approximate solution of the Fokker-Planck equation by means of the Galerkin technique are presented. Furthermore, based on this solution an equation is given for the determination of the statistical moments of the response amplitude.
18 citations
TL;DR: In this paper, the fitting technique by orthonormal polynomials and Forsythe's recurrence relationship, generalized in order to take into account experimental errors, is discussed.
13 citations
TL;DR: A new computer program for symbolic calculations in general relativity, named ORTOCARTAN, which calculates the curvature quantities from an orthonormal tetrad of forms representing the metric, both the tetrad and the coordinate components.
Abstract: This paper is meant to announce the appearance of a new computer program for symbolic calculations in general relativity, named ORTOCARTAN. The program calculates the curvature quantities from an orthonormal tetrad of forms representing the metric, both the tetrad and the coordinate components. The paper describes the main features of the program's input and output, and compares the program's speed to a few general-purpose systems, notably LAM, ALTRAN, FORMAC, REDUCE, and SYMBAL. A general overview of technical parameters of the program, and conceptual features of the algorithm is given.
11 citations
TL;DR: An analytical formula for the overlap integrals in the case of the non-canonical basis of Bargmann and Moshinsky (1961) has been obtained as discussed by the authors, and the transformation coefficients are tabulated for mu = 0, 1, 2, 3, 4 and lambda > mu.
Abstract: An analytical formula for the overlap integrals in the case of the non-canonical basis of Bargmann and Moshinsky (1961) has been obtained. These integrals are tabulated for mu =0, 1, 2, 3, 4 and lambda > mu . The overlap integrals are used for the construction (by means of the Hilbert-Schmidt procedure) of an orthonormal basis. The transformation coefficients are tabulated for mu =0, 1, 2, 3 and lambda > mu .
10 citations
TL;DR: In this article, the diagonal and symmetric Bianchi type IX models are coupled to a homogeneous spinor field, and an action for the combined fields is constructed, where the orthonormal basis used is given explicitly in terms of the metric.
Abstract: The diagonal and symmetric Bianchi type IX models are coupled to a homogeneous spinor field. An action for the combined fields is constructed, where the orthonormal basis used is given explicitly in terms of the metric. This allows one to vary the action with respect to the metric and the spinor fields only. Next, a Hamiltonian formulation is given, and a qualitative solution for the problem is presented. We also show that the k = +1 FRW (Friedmann–Robertson–Walker) model is not compatible with a homogeneous spinor field, while the more complicated models are.
ARCO1
TL;DR: In this article, the orthonormal lattice and maximum entropy algorithms are compared to the results of deconvolution using the maximum entropy and Levinson algorithms, and the results show that the direct methods have fewer windowing problems, higher resolving power, and are more suited for use in a time-varying manner than the indirect method.
Abstract: The Gram‐Schmidt orthogonalization procedure is simplified under the assumption of stationarity and implemented to perform recursive predictive deconvolution. This process is called the orthonormal lattice filter. The results of deconvolution by this method are compared to the results of deconvolution using the maximum entropy and Levinson algorithms. The orthonormal lattice and maximum entropy algorithms are direct methods and estimate the filter from the data, while the Levinson algorithm is an indirect method and estimates the filter from the autocorrelation function. Results from synthetic and real data show that the direct methods have fewer windowing problems, higher resolving power, and are more suited for use in a time‐varying manner than the indirect method. Results from real data show that optimally weighted space averaging and zero‐phase band‐pass filtering after time‐varying direct deconvolution produces highly resolved and spatially coherent seismic sections.
TL;DR: A general approach to calibration problems is proposed, based on the use of polynomials orthonormal over discrete point sets with arbitrary positive weights, numerically built by means of recurrence relations of the Forsythe-Weisfeld type.
Patent•
28 Apr 1981
TL;DR: In this paper, a method of recognizing different perspective views or images of the same object (i.e., intraclass patterns) is described as an orthonormal basis function expansion, and a single averaged matched spatial filter is produced from a weighted linear combination of these functions.
Abstract: A method of recognizing different perspective views or images (i.e., multivariant views) of the same object (i.e., intraclass patterns). Each intraclass pattern, or different representation of the same object, is described as an orthonormal basis function expansion, and a single averaged matched spatial filter is produced from a weighted linear combination of these functions. The method eliminates the multiple matched spatial filters, and the extensive postprocessing of the matrix output from a multichannel correlator, which are used in the prior art.
TL;DR: In this article, four algorithms for reorthogonalization of erroneous rotation matrices are described: (a) using the orthonormal relations; (b) reducing to Eulerian angles; reducing to a single rotation about an inclined axis; (d) Newton's method.
TL;DR: The relative quality of the (stable) orthonormal and (fast) pick-out procedures is compared, and within this class a new fast algorithm is described.
01 May 1981
TL;DR: It is shown that properties of orthonormal excitations of the circuit that was used in [1] for deriving sensitivity can as well be used as a mathematical tool for the determination of element values of the analog circuits.
Abstract: In this paper it is shown that properties of orthonormal excitations of the circuit that was used in [1] for deriving sensitivity can as well be used as a mathematical tool for the determination of element values of the analog circuits. Simple methods are then presented for the computation of the circuit elements if all the nodes are accessible.
TL;DR: In this article, a nonlinear filtering problem is formulated where the state evolves in a compact Riemannian manifold and the observations evolve in a non-compact symmetric space, and absolute continuity methods are used to obtain an expression for the conditional probability.
01 Mar 1981
TL;DR: For every orthonormal complete system in L 2 (0, 1) there exists a set A, of measure arbitrarily close to 1, which carries no nonzero function with Fourier transform in lp, for every p < 2 as discussed by the authors.
Abstract: It is proved that for every orthonormal complete system in L2(0, 1) there exists a set A, of measure arbitrarily close to 1, which carries no nonzero function with Fourier transform in lp, for every p < 2.
01 Jul 1981
TL;DR: In this paper, the authors define the Haar and Walsh systems on a triangle domain, prove the orthogonality and completeness in L sub 2, and show uniform convergence of the Walsh-Fourier series.
Abstract: : A number of papers have been concerned with developing the theories of discontinuous orthonormal systems and their applications. In particular, the Haar and Walsh systems are presently the most important examples of nonsinusoidal functions, and have proved most useful in communication. Some authors have studied the properties of approximation from the mathematical point of view. It seems interesting and helpful for both theory and practice to investigate the Haar and Walsh functions for a multivariate setting. In fact, many signals in communications and other functions are of several variables (for instance, TV signals have two space variables and the time variable). If the domain of definition of the system is tensor product, then the existing systems are readily extended to several variables. The problem is how to construct an orthonormal system on a triangular domain in the plane, or more generally, on a simplex in n-dimensional space. This paper defines the Haar and Walsh system on a triangle domain, proves the orthogonality and completeness in L sub 2. Also the uniform convergence for the Haar-Fourier series, uniform convergence by group for the Walsh-Fourier series are studied. All of these results can be generalized easily to n dimensions.
TL;DR: In this article, the convergence or divergence of the Haar system of functions on the interval was investigated. But the results were not conclusive for the general Haar model. And they were only applicable to the case where or with.
Abstract: Let be an orthonormal system of functions on the interval , and let the function . We investigate the question of the convergence or divergence (depending on the smoothness of the function ) of series of the form where or with .It is shown that in a certain sense, the assertions obtained are definitive for the Haar system.Bibliography: 14 titles.
TL;DR: The concept of the order of informativeness of a system is introduced as the minimal number of functionals that can be retained without making the approximation error worse.
Abstract: The problem of approximating a linear functional from approximate values of a system of functionals is considered. The concept of the order of informativeness of a system is introduced as the minimal number of functionals that can be retained without making the approximation error worse. For approximation with respect to an orthonormal system of functionals in a Hilbert space the order of informativeness, the best approximation method, and its error are determined. Bibliography: 4 titles.
01 Jan 1981
TL;DR: In this article, the existence, linear and multilinear cases, approximation, and stability of abstract nonlinear wave equations are discussed, and the stability of the zero solution is investigated.
Abstract: Publisher Summary This chapter discusses the existence, linear and multilinear cases, approximation, and stability of abstract nonlinear wave equations. It is extremely useful to have an explicit expression for the functions. It is found that the nonlinearity is reproducing relative to the sequence. The idea is that much nonlinearity has computable expansion coefficients, when applied to a suitable complete orthonormal system. An explicit example is also given. The second remark concerns the case when the nonlinearity M(u) is the gradient of a functional where R denotes terms of higher order in h. This holds when M(u) is a cyclically monotone operator. The approximations have the form of a conservative Hamiltonian system of classical mechanics. The stability of the zero solution is investigated. The analysis makes essential use of the results of Rutkowski, who use the reproducing property of the nonlinearity to obtain explicit expansion coefficients.
TL;DR: In this article, a number of products of arrays can be defined by using various functional relations to define the indexes of the resulting array in terms of the corresponding index of the component arrays.
Abstract: A number of products of arrays can be defined by using various functional relations to define the indexes of the resulting array in terms of the corresponding indexes of the component arrays. Some examples of such products are proposed in this paper which give an orthonornal matrix of higher order in terms of component orthonormal matrices of lower order. The technique developed in this paper can be used in the annlysis and design of linear systems with longer inputs in terms of component systems with smaller inputs.
TL;DR: In this paper, a new technique of obtaining a complete orthonormal transform based on a particular interpretation of DFT is developed, which can be deduced from the knowledge of the permutation properties of the two component transforms.
Abstract: A new technique of obtaining a complete orthonormal transform based on a particular interpretation of DFT is developed. One such transform, namely, the Fourier-twiddled H-DF transform, has been discussed in detail. The permutation properties of such transforms can be deduced from the knowledge of the permutation properties of the two component transforms.
TL;DR: In particular, it was shown in this paper that every orthonormal system contains a Sidon subsystem, and the problem of extracting lacunary subsystems from general sequences of functions is NP-hard.
Abstract: This paper considers the problem of extracting lacunary subsystems from general sequences of functions In particular, it is shown that every orthonormal system contains a Sidon subsystemBibliography: 7 titles
01 Jan 1981
TL;DR: In this article, it was shown that if one wants the N-order H-matrix to exist, then N should be the multiple of 4. But up till now, it is not clear whether this is a sufficient condition.