Showing papers on "Basis (linear algebra) published in 1975"
TL;DR: It is shown how minimal bases can be used to factor a transfer function matrix G in the form $G = ND^{ - 1} $, where N and D are polynomial matrices that display the controllability indices of G and its controller canonical realization.
Abstract: A minimal basis of a vector space V of n-tuples of rational functions is defined as a polynomial basis such that the sum of the degrees of the basis n-tuples is minimum. Conditions for a matrix G to represent a minimal basis are derived. By imposing additional conditions on G we arrive at a minimal basis for V that is unique. We show how minimal bases can be used to factor a transfer function matrix G in the form $G = ND^{ - 1} $, where N and D are polynomial matrices that display the controllability indices of G and its controller canonical realization. Transfer function matrices G solving equations of the form $PG = Q$ are also obtained by this method; applications to the problem of finding minimal order inverse systems are given. Previous applications to convolutional coding theory are noted. This range of applications suggests that minimal basis ideas will be useful throughout the theory of multivariable linear systems. A restatement of these ideas in the language of valuation theory is given in an Ap...
743 citations
TL;DR: Three old bases for representations of SU (3) reduced according to 0(3), the Bargmann-Moshinsky, Elliott and stretched bases are discussed, together with one new one, the antistretched basis, related to the analytic bases.
186 citations
TL;DR: In this paper, a class of N coupled equations based on a hypothesis concerning biological control by Monod and Jacob is derived, where transitions between volumes in concentration space for these equations are represented as directed edges on N cubes (hypercubes in N dimensions).
Abstract: Combinatorial and topological techniques are developed to classify nonlinear chemical reaction networks in terms of their qualitative dynamics. A class of N coupled equations, based on a hypothesis concerning biological control by Monod and Jacob is derived. Transitions between volumes in concentration space for these equations are represented as directed edges on N cubes (hypercubes in N dimensions). A classification of the resulting state transition diagrams for N=2,3 is given. A version of a topological theorem by Poincare and Hopf is derived which is appropriate for application to chemical systems. This theorem is used to predict the existence of critical points in continuous nonlinear equations with oscillation and bistability on the basis of their state transition diagrams. A large number of nonlinear kinetic equations proposed in previous studies by other authors are classified in terms of their state transition diagrams.
183 citations
96 citations
TL;DR: In this article, a simple method is proposed to calculate the matrix elements of two-body local interactions using a harmonic oscillator basis (HOF), which is shown that any local potential can be replaced by a simple series for the purpose of calculating matrix elements.
73 citations
TL;DR: In this article, the authors proved necessary and sufficient conditions on the reduced-density operator restricted to a functional subspace that ensures the separability of the wave function of a fermion system.
Abstract: A theorem is proved which gives the necessary and sufficient conditions on the $p$-particle reduced-density operator restricted to a functional subspace that ensures the separability of the wave function of a fermion system. This theorem is the theoretical basis of a general method of analyzing atomic and molecular wave functions, a method which is able to reveal the existence of somehow chemically discernible subsystems of electrons.
72 citations
TL;DR: Pople's extended Gaussian type basis, usually termed 4−31G, was adapted for use with the model potential method as mentioned in this paper, and molecular calculations were performed successfully for N2, H2O, CH4, NH3, HCN, PH3, H 2S, and ClF.
Abstract: Pople’s extended Gaussian‐type basis, usually termed 4‐31G, was adapted for use with the model potential method. Molecular calculations were performed successfully for N2, H2O, CH4, NH3, HCN, PH3, H2S, and ClF. Substantial savings were achieved in computing cost.
59 citations
TL;DR: The analysis of the results shows that basis sets approximating the optimum total energy very well can still be markedly improved for the prediction of one‐electron properties and for smaller basis sets, this improvement does not warrant the necessary expense.
Abstract: Expressions are given for calculating the energy gradient vector in the exponent space of Gaussian basis sets and a technique to optimize orbital exponents using the method of conjugate gradients is described. The method is tested on the (9s5p) Gaussian basis space and optimum exponents are determined for the carbon atom. The analysis of the results shows that the calculated one‐electron properties converge more slowly to their optimum values than the total energy converges to its optimum value. In addition, basis sets approximating the optimum total energy very well can still be markedly improved for the prediction of one‐electron properties. For smaller basis sets, this improvement does not warrant the necessary expense.
40 citations
01 Nov 1975
TL;DR: In this article, a topological sequence space has the property of Toeplitz sectional convergence (TK) if and only if the unit sequences of the sequence form a Toe-plitz basis.
Abstract: The concept of sectional convergence (AK) in FK-spaces was investigated by Zeller in (20). In (5) and (6), Garling investigated convergent and bounded sections in more general topological sequence spaces. Many of the results hold for Toeplitz sections in sequence spaces. A topological sequence space has the property of Toeplitz sectional convergence (TK) if and only if the unit sequences form a Toeplitz basis. In section 3, we present characterizations of Toeplitz sectional boundedness (TB) and functional Toeplitz sectional convergence (FTK) in terms of βT- and γT-duality. In section 4, we apply our results to summability fields. These results are related to the Hardy-Bohr property of multipliers for Cesaro summable sequences of positive order. In section 5, we characterize the properties TK and TB in FK-spaces by factorization statements.
27 citations
TL;DR: A space-variant imaging model is proposed, whose behavior is characterized on the basis of orthonormal polynomials, and the intrinsic variables then become available, allowing the clustering of the data (considered as belonging to statistical classes).
Abstract: Optical-pattern-recognition techniques are generally unable to provide an efficient approach to the classification of optical data, because of the linearity of the Fourier transform. A space-variant imaging model is proposed, whose behavior is characterized on the basis of orthonormal polynomials. The optical data are described in an orthonormal space based upon these polynomials. Proper-axis rotation and dimensionality reduction are supplied by the Karhunen–Loeve transform. The intrinsic variables then become available, allowing the clustering of the data (considered as belonging to statistical classes). This work is illustrated by examples of handwriting recognition and classification.
Journal Article•
TL;DR: In this article, it was shown that the Weyl basis formed by the canonical symmetrization of an n-dimensional, p-rank tensor space with canonical projection operators of Sp is a Gel'fand basis of U(n).
Abstract: It is shown that the Weyl basis formed by the canonical symmetrization of an n‐dimensional, p‐rank tensor space with canonical projection operators of Sp is a Gel’fand basis of U(n). This basis may easily be generated using standard projection operator techniques.
01 Jan 1975
TL;DR: In this article, it was shown that any maximal asymptotic nonbasis of order 2 that satisfies a certain finiteness condition is a subset of a maximal non-congruence non-basis.
Abstract: Let A be a set of nonnegative integers. If all but a finite number of positive integers can be written as a sum of h elements of A, then A is an asymptotic basis of order h. Otherwise, A is an asymptotic nonbasis of order h. A class of maximal asymptotic nonbases is constructed, and it is proved that any asymptotic nonbasis of order 2 that satisfies a certain finiteness condition is a subset of a maximal asymptotic nonbasis of order 2. Let A be a set of nonnegative integers containing 0. The h-fold sum of A, denoted hA, is the set of all sums of h not necessarily distinct elements of A. If hA contains all but a finite number of positive integers, then A is an asymptotic basis of order h. The set A is a minimal asymptotic basis of order h'if A is an asymptotic basis of order h, but A\ a1 is not an asymptotic basis of order h for every a c A. Examples of minimal asymptotic bases were constructed in [1], and also an example of an asymptotic basis which contains no subset that is a minimal asymptotic basis. The set A is an asymptotic nonbasis of order h if A is not an asymptotic basis of order h. If A is an asymptotic nonbasis of order h, but AUta1 is an asymptotic basis of order h for every nonnegative integer a ,' A, then A is a maximal asymptotic nonbasis of order h. Maximal asymptotic nonbases were constructed in [1] by taking finite unions of the nonnegative parts of congruence classes. In this paper we construct a new class of maximal asymptotic nonbases that are not unions of congruence classes, and we prove that every asymptotic nonbasis of order 2 that satisfies a certain finiteness condition is a subset of a maximal asymptotic nonbasis of order 2. We do not know whether every asymptotic nonbasis is a subset of a maximal asymptotic nonbasis, nor whether there exist maximal asymptotic nonbases with zero den sity. Let [a, b] denote the set of integers n such that a.< n < b. Received by the editors February 4, 1974. AMS (MOS) subject classifications (1970). Primary 1OL05, LOL10 OJ99.
TL;DR: In this paper, a simple approximate reanalysis technique is presented for use in the automated (optimum) design of large and complex structures, based on the use of the reduced basis method in conjunction with substructuring approach.
Abstract: A simple approximate reanalysis technique is presented for use in the automated (optimum) design of large and complex structures. The technique is based on the use of the reduced basis method in conjunction with substructuring approach. The displacement vector of the modified structure is approximated by a linear combination of a normalized set of s first-order sensitivity analysis vectors, where s is the number of design variables. Consideration is focused on the potential advantages of the reduced basis - substructuring technique for use in automated design systems. The high accuracy of the proposed technique for sizable changes in the design variables is demonstrated by means of a numerical example of a large transmission tower. Also, comparison is made with Taylor expansion method based on the reciprocal variables.
TL;DR: In this article, a basis is developed for the systematic design of linear multivariable discrete-time tracking systems, where the required asymptotic behavior may be achieved by a controller incorporating a series of discrete time vector integrators.
Abstract: In this paper a basis is developed for the systematic design of linear multivariable discrete-time tracking systems. It is shown that the required asymptotic behaviour may be achieved by a controller incorporating a series of discrete-time vector integrators. The controllability conditions for such systems are established and the theory is illustrated by a numerical example.
TL;DR: In this paper, a general scheme for dealing with the problem of estimating the contribution of unknown functions is presented, where the essence of the method is to apply an average energy approximation to the unknown functions and to determine this energy by enforcing the gauge invariance requirement.
TL;DR: The linear hull of a Tchebyshev system is called a Haar-space, and a Markov basis is defined in this article as a basis f 1,fn of an n-dimensional Haar space.
Abstract: The linear hull of a Tchebyshev system is called a Haar-space. A basis f1,...,fn of an n-dimensional Haar-space is called a Markov basis if f1,...,fi form a Tchebyshev system for each i=l,...,n. It is shown by suitable examples that for all n≥3 there exist Haar-spaces without a Markov basis.
TL;DR: In this article, the authors constructed the basis of the irreducible representation D of the exceptional Lie group G 2 corresponding to the reduction of this group to the subgroup SU 3.
TL;DR: In this paper, singular perturbation methods are used to provide a basis for the analysis of the asymptotic controllability of multivariable linear systems containing small " parasitic " elements.
Abstract: Singular perturbation methods are used to provide a basis for the analysis of the asymptotic controllability of multivariable linear systems containing small ‘ parasitic ’ elements, and for the design of feedback controllers which will effect asymptotic eigenvalue assignment in such systems.
01 Jan 1975
TL;DR: The results of R. C. James on characterizations of reflexivity of Banach spaces with an unconditional basis in terms of c and 11 are extended to arbitrary Banach space as discussed by the authors.
Abstract: The results of R. C. James on characterizations of reflexivity of Banach spaces with an unconditional basis in terms of c and 11 are extended to arbitrary Banach spaces. Some consequences are obtained. R. C. James has proved [4, Theorem 2] that a Banach space E with an unconditional basis is reflexive if and only if E contains no subspace isomorphic to co or 11. This result has been shown to remain valid for any subspace E of a space with an unconditional basis (of any power) by C. Bessaga and A. Pekczyn'ski [1], [2], who have also proved [2] that such a space E is reflexive if and only if E* contains no subspace isomorphic to
01 Jan 1975
TL;DR: In this article, the degree of a matrix of rational functions is obtained in a simplified way, which enables them to be generalized to matrices whose elements are not necessarily rational functions.
Abstract: The properties of the degree of a matrix of rational functions are obtained in a simplified way, which enables them to be generalized to matrices whose elements are not necessarilyrational functions. On the basis of these results a theory of realizations is developed, which similarly generalizes the theory of state space realizations of a matrix of rational functions.
01 Feb 1975
TL;DR: This letter discusses a fast digital technique for decomposing signals which are linear superpositions of components having the same form but different widths, a generalization of a method used by Gardner et al. to analyze multicomponent exponential decays.
Abstract: This letter discusses a fast digital technique for decomposing signals which are linear superpositions of components having the same form but different widths. This technique, based on the work of Titschmarsh, is a generalization of a method used by Gardner et al. to analyze multicomponent exponential decays.
TL;DR: On the basis of a new definition of filters, the order of the minimal-order optimal filters for general discrete-time linear stochastic systems is obtained and a method is established for constructing a minimal- order optimal filter.
Abstract: On the basis of a new definition of filters, the order of the minimal-order optimal filters for general discrete-time linear stochastic systems is obtained. A method is established for constructing a minimal-order optimal filter.
TL;DR: In this paper, an explicit representation of the SO (4,1) group in both SO(4) spinor space and four−dimensional Euclidean space has been found.
Abstract: An explicit representation of the SO (4,1) group in both SO (4) spinor space and four−dimensional Euclidean space has been found. The two spaces are related by linearly transforming the variables. It has been shown that the four−dimensional hyperspherical harmonics in both four−dimensional polar coordinate systems transform in accordance with the W = 0, Q = 2 representation of the SO (4,1) group. A set of new recursion relations is derived for the SO (3) group reduced rotational matrices, together with a set of standard recursion relations for the Legendre and Gegenbauer polynomials, all of which are obtained by transforming both forms of the SO (4) group basis states in spinor space into four−dimensional space. The matrix elements of the noncompact SO (4,1) group generators are given in the (j,m) basis.
TL;DR: New labeling techniques are provided for accelerating the basis exchange step of specialized linear programming methods for network problems that substantially reduce the amount of computation involved in updating operations.
Abstract: : New labeling techniques are provided for accelerating the basis exchange step of specialized linear programming methods for network problems. These techniques substantially reduce the amount of computation involved in updating operations.
TL;DR: In this article, it was shown that every super-reflexive space with an unconditional basis is isomorphic to a complemented subspace of a symmetric super-reflective space.
Abstract: Every super-reflexive space with an unconditional basis is isomorphic to a complemented subspace of a super-reflexive space with a symmetric basis.
01 Jan 1975
TL;DR: The product form algorithm using contracted transformation vectors given by Zoutendijk has been extended so as to be able to represent, and update at each iteration, the inverse of the reduced basis as the product of a sequence of transformation matrices.
Abstract: This paper presents a product form version of the reduced basis algorithm for the simplex method. The reduced basis is a sub-matrix of the coefficient matrix formed by the effective constraints and the basic original variables. The product form algorithm using contracted transformation vectors given by Zoutendijk [7], has been extended so as to be able to represent, and update at each iteration, the inverse of the reduced basis as the product of a sequence of transformation matrices.