Showing papers on "Basis (linear algebra) published in 1972"

The Exact Finite Sample Properties of the Estimators of Coefficients in the Error Components Regression Models

Abstract: Wallace and Hussain (1969) considered the use of an error components regression model in the analysis of time series of cross-sections and developed an Aitken estimator of the coefficient vector based on an estimated variance-covariance matrix of error terms. In this paper, we have shown that under the set of assumptions adopted by Wallace and Hussain there are an infinite number of estimators which have the same asymptotic variancecovariance matrix as the Wallace-Hussain estimator and also that it is not possible to choose an estimator on the basis of asymptotic efficiency. We have developed an alternative estimator of the variance-covariance matrix of error terms and have used this estimator in developing a feasible "Aitken" type estimator for the coefficient vector. We have derived some small sample properties of this estimator and have compared them with those of other estimators of the coefficient vector.

Computation of large molecules with the hartree-fock model.

Abstract: The usual way to compute Hartree-Fock type functions for molecules is by an expansion of the one-electron functions (molecular orbitals) in a linear combination of analytical functions (LCAO-MO-SCF, linear combination of atomic orbitals—Molecular Orbital—Self Consistent field). The expansion coefficients are obtained variationally. This technique requires the computation of several multicenter two-electron integrals (representing the electron-electron interaction) proportional to the fourth power of the basis set size. There are several types of basis sets; the Gaussian type introduced by S. F. Boys is used herein. Since it requires from a minimum of 10 (or 15) Gaussian-type functions to about 25 (or 30) Gaussian functions to describe a second-row atom in a molecule, the fourth power dependency of the basis set has been the de facto bottleneck of quantum chemical computations in the last decade. In this paper, the concept is introduced of a “dynamical” basis set, which allows for drastic computational simplifications while retaining full numerical accuracy. Examples are given that show that computational saving in computer time of more than a factor of one hundred is achieved and that large basis sets (up to the order of several hundred Gaussian functions per molecule) can be used routinely. It is noted that the limitation in the Hartree-Fock energy (correlation energy error) can be easily computed by use of a statistical model introduced by Wigner for solid-state systems in 1934. Thus, large molecules can now be simulated by computational techniques without reverting to semi-empirical parameterization and without requiring enormous computational time and storage.

Decay of Order in Classical Many-Body Systems. I. Introduction and Formal Theory

Abstract: In this work we briefly review the Ornstein-Zernike prediction for the decay of correlation functions, extend it to treat the decay of correlation near surfaces, and then contrast this prediction with the exactly known results for the two-dimensional Ising model. We develop the transfer-matrix approach to classical statistical mechanics in sufficient generality for its use in later papers in this series, where it is employed to derive general forms for the decay of correlation functions in Ising models away from the critical point, which provide a clear explanation of the failure of the Ornstein-Zernike theory for the two-dimensional Ising model. In particular, we show that the thermodynamic behavior of a classical system with short-range interactions reduces, when the system becomes infinite in at least one dimension, to the calculation of the largest eigenvalue of the transfer matrix. Using the Perron-Fr\"obenius theorem, we show that for a system infinite in no more than one dimension, an arbitrary correlation function defined on the system decays at least exponentially fast. One is able to predict whether the decay of correlation is monotone or oscillatory on the basis of the largest few eigenvalues of the transfer matrix.

On pricing and backward transformation in linear programming

Abstract: In this paper we re-examine some of the available methods for pricing out the columns in the simplex method and point out their potential advantages and disadvantages. In particular, we show that a simple formula for updating the pricing vector can be used with some advantage in the standard product form simplex algorithm and with very considerable advantage in two recent developments: P.M.J. Harris' dynamic scaling method and the Forrest—Tomlin method for maintaining triangular factors of the basis.

Integral representation of excessive measures and excessive functions

Abstract: One of the central results of classical potential theory is the theorem on the representation of an arbitrary non-negative superharmonic function in the form of a sum of a Green's potential and a Poisson integral. We obtain similar integral representations for the excessive measures and functions connected with an arbitrary Markov transition function. Many authors have studied the homogeneous excessive measures connected with a homogeneous transition function. We begin with the inhomogeneous case and then reduce the homogeneous case to it. The method proposed gives a considerable gain in generality. The investigation is carried out in the language of convex measurable spaces and in contrast to previous papers no topological arguments are used. Our basis are the results obtained in [3] (also without topology) on the integral representation of Markov processes with a given transition function. For the reduction of the homogeneous case to the inhomogeneous we use a theorem from the theory of dynamical systems due to Yu. I. Kifer and S. A. Pirogov (see the Appendix at the end of this paper).

Variational Estimates and Generalized Perturbation Theory for the Ratios of Linear and Bilinear Functionals

Abstract: Variational functionals are presented which provide an estimate of ratios of linear and bilinear functionals of the solutions of the direct and adjoint equations (inhomogeneous and homogeneous) governing linear systems. These variational functionals are used as the basis for a generalized perturbation theory for estimating the effects of changes in system parameters upon these ratios of linear and bilinear functionals. The relation of the present theory to the variational theory of Pomraning and to the generalized perturbation theory of Usachev and Gandini is discussed. Potential applications of the theory to nuclear reactor physics are outlined.

The probabalistic basis of stereology

Abstract: The geometric probability basis of the theory of estimating the properties of the distribution of one material in another by investigating the observed patterns on plane and line intersects is reviewed. Various measures of particle size and the estimation of size distribution of spheres are described. The problem of ellipsoids and particles of other shapes is discussed and a review given of most of the standard theories of the random division of space into cells. The theory of linear probes and the estimation of spatial properties on a plane section are also considered.

Geometric aspects of primary lattices

Abstract: Introduction* The classical correspondence between vector spaces, projective spaces and complemented modular lattices was extended to finitely generated modules over completely primary and uniserial rings and primary lattices by Baer [5], Inaba [7] and, recently, by Jόnsson and Monk [8]. In these extensions, however, an analogue to the classical projective space is missing. It is shown in the present paper, that the appropriate concept is that of a Hjelmslev space as defined by Klingenberg [9], [10] and by Luck [11]. To be correct, this is only shown for the case of a plane geometry, namely Hjelmslev planes of level n, corresponding to primary lattices with homogeneous basis of three ^-cycles, and to free modules R\ Also, we have the complete correspondence only in the desarguesian case. The restriction to this case is justified, as the author believes, by the fact it is well known to be typical for higher dimensional spaces in the classical theory. In the non desarguesian case, there is a coordinatization theory for Hjelmslev planes of level n given by Drake [6], but this does not seem to lead to a construction of a lattice from the plane. Every primary lattice with a homogeneous basis of three π-cycles, however, leads to a Hjelmslev plane of level n (Theorem 2.13). Planes of level 1 (ordinary projective planes) and of level 2 (uniform Hjelmslev planes) can be shown to be obtainable from lattices. For uniform planes, this was done by the author in [2]. A combination of Theorem 2.13 with results of [4] shows that a desarguesian Hjelmslev plane £έ?(&) is of level n if and only if & is completely primary and uniserial of rank n. 0* Definitions*

Skyrme's interaction in the asymptotic basis

Abstract: HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Skyrme’s interaction in the asymptotic basis P. Quentin

The Weyl Basis of the Unitary Group U(k)

Abstract: Weyl's method for the construction of irreducible tensors of the unitary group is used to construct a basis for any irreducible representation of U(k) or GL(k) in terms of Bose creation operators. A simple way is indicated to select a complete but not over complete basis from the functions obtained. The basis obtained can be useful in nuclear or molecular calculations, as well as in some mathematical problems.

Reduction of the Poincaré Group with Respect to the Lorentz Group

Abstract: The representations of the Poincare Group for spinless particles, reduced with respect to the Lorentz subgroup, are investigated. They involve the principal series of representations of SL(2, C) and use is made of a basis, introduced by Gel'fand in which the states are labeled by a complex number z. The transformation matrices relating to the Wigner basis are derived. The matrix elements of the momentum operators are obtained. The general form of the S matrix in the new basis is discussed. This basis may be relevant for a field theoretical description of the Veneziano model.

A fundamental problem in linear inequalities with applications to the travelling salesman problem

Abstract: We consider a system ofm linearly independent equality constraints inn nonnegative variables:Ax = b, x ≧ 0. The fundamental problem that we discuss is the following: suppose we are given a set ofr linearly independent column vectors ofA, known asthe special column vectors. The problem is to develop an efficient algorithm to determine whether there exists a feasible basis which contains all the special column vectors as basic column vectors and to find such a basis if one exists. Such an algorithm has several applications in the area of mathematical programming. As an illustration, we show that the famous travelling salesman problem can be solved efficiently using this algorithm. Recent published work indicates that this algorithm has applications in integer linear programming. An algorithm for this problem using a set covering approach is described.

Boundary-Condition Constants of the Lane-Robson Calculable Theory

Abstract: It is demonstrated that the $S$ matrix derived from the Lane-Robson calculable theory of nuclear reactions is independent of the choice of the boundary-condition constants even if a truncated set of the basis states is used.

Time-dependent statistics of binary linear lattices

Abstract: The time-dependent statistics of binary linear lattices is investigated on the basis of a master equation at the microscopic level. It is assumed that the kinetics may be formulated as transformations of specified sequences of clusters ofA units andB units into other specified sequences. On the basis of aStosszahlansatz, a master equation at the macroscopic level is derived. In the limit of a large system, the densities of clusters of all types satisfy rate equations similar to the equations of chemical kinetics. AnH-theorem is proven and the nonequilibrium thermodynamics of the system is studied. The theory has application to the kinetics of the helix-coil phase transition in biopolymers.

Combined Basis Function Method for Energy Band Calculations

Abstract: A method to calculate relativistic energy bands and wave functions is presented, in which rapidly converging basis sets are created from linearly combined RAPW functions. By constructing other basis sets, convergence of the band structure of complex materials (e.g. compounds) can be studied with a matrix of moderate size. Results of an application to fcc praesodymium show the usefulness of the method and indicate a considerable potential for future developments.

A comparison of lobe and cartesian gaussian basis sets for molecular calculations

Abstract: Lobe gaussian and cartesian gaussian basis sets, of approximately minimal basis Slater accuracy, have been compared for molecular calculations. The basis sets were constructed so that they only differed in the representation of the angular dependence of the p function. Calculation of total energy and several one-electron properties for a series of nine molecules shows that, for molecular calculations, the lobe and cartesian gaussian representations are equivalent.

Bases of the Space of Continuous Functions

Abstract: In this paper, new bases for the space of continuous functions are constructed, similar to the Schauder basis but having better differentiability properties. The bases constructed are applied to the problem of the order of growth of the degrees of a polynomial basis of the space . It is proved that for any nondecreasing sequence of natural numbers satisfying the condition it is possible to construct a polynomial basis with order of growth , .Bibliography: 16 items.

Nonlinear Representations of the Poincaré Group as a Possible Basis of Particle Dynamics

Abstract: Nonlinear representations of the Poincar\'e group are studied as a possible basis for hadron dynamics. In particular we study whether interacting fields are, in general, equally well described by nonlinear representations of the Poincar\'e group as free fields are by linear representations. The fields are identified with group parameters of the appropriate group cosets, thus forming local coordinate systems on the group manifold. On this basis we derive possible interaction Lagrangians, expressed by invariant metrics of the coset spaces. In contrast to the success of nonlinear representations of internal-symmetry groups (chiral groups), we conclude that recently suggested nonlinear representations of the Poincar\'e group are inadequate for describing interacting fields and for deriving particle interactions, mainly because of physically unacceptable features inherent in those representations.

Application of weighted phase-sum formulae for phase determination

Abstract: The simple `weighted sum formula' has been studied as regards its applicability to phase refinements. These studies show that proper weighting of the phase relations significantly improves the phase estimates from phase-sum formulae. The evaluation of weights used in the `variance-weighted sum formula' is discussed, together with the problem of the 2π ambiguity. A simple procedure for selecting efficient basis sets is also given. The procedures described have been programmed and used to solve twelve unknown structures. Applications to somewhat complicated cases are described.

Λ(α)-Bases and Nuclear Spaces

Abstract: IN A STUDY of the properties of bases in nuclear Frechet spaces, Dynin and Mitiagin [5] proved that in such spaces every Schauder basis is an absolute basis; another proof of this interesting result was given by Mitiagin [7]. Replacing the sequence space l 1 in the definition of nuclear maps by the nuclear sequence space Λ(α) of power series the second author initiated, in [10], a study of Λ(α)- nuclear spaces and this study is presented in greater depth in a recent paper of the authors [3]. In this paper we introduce the notion of Λ(α)-basis in locally convex spaces.

Expansion in eigenfunctions of integral operators of convolution on a finite interval with kernels whose fourier transforms are rational. “weakly” nonselfadjoint regular kernels

Abstract: A study is made of the asymptotic behavior of the eigenvalues and of expansions in the root vectors of the class of integral operators specified in the title. If some natural conditions, ensuring "regularity" of the asymptotic behavior of the spectrum, are imposed on the kernel, the root vectors form a basis in Lp(0,T) (1 < p < ∞) and a Riesz basis in L2(0,T).

Singular matrix representations of point groups

Abstract: Starting from a comparison of the properties of cartesian coordinate functions and plane waves as basis functions, a discussion is given of the use of contravariant and covariant components in setting up representation matrices for group operators. The use of covariant components together with a linearly dependent basis is shown to lead to singular matrix representations of a point group. A general theory of such singular representations is given, together with some examples.

Comments on Social Field Theory

Abstract: : The topics developed are: What is social field theory; interpretation of the basic formulae; the dynamics of the spaces problem; the time and space frame problem; the basis problem; model II and the notion of theory; the concept of distance and its implications; the use and interpretation of canonical correlation; and the development of status-field theory.

A Simple Method for Calculating Nongray Radiation

Abstract: A very simple, low-frequency set of differential equations can be integrated to provide the information needed to construct the direction cosine transition matrix. This matrix need be constructed only as required by the navigation or control functions. The evaluation of one square root, one sine and one cosine, together with some algebraic manipulations, are needed for each attitude update. The equations possess a singularity, but it is a trivial singularity. If magnitude of the three-angle vector is zero, then the vector derivative is equal to the inertial body-rate vector. Simultaneous three-axis control can be easily mechanized if the navigation loop uses the universal three-angle basis.