Showing papers on "Basis (linear algebra) published in 1970"

On the Nonorthogonality Problem

Abstract: Publisher Summary This chapter discusses three orthonormalization procedures, such as successive orthonormalization, symmetric orthonormalization, and canonical orthonormalization The simplest way of orthonormalizing a finite set of functions is by the classical Schmidt procedure, in which each member of the set in order is orthogonalized against all the previous members and subsequently normalized In solid-state theory, one could probably construct orthonormal combinations of the atomic orbitals of the system, which would still preserve the natural symmetry In such an approach, it would be necessary to treat the given functions ϕ = {ϕ 1 , ϕ 2 …, ϕ n } simultaneously, on an equivalent basis instead of successively as in the Schmidt procedure In molecular and solid-state theory, there are cases when also the symmetric orthonormalization procedure will break down, depending on the fact that, even if the basis ϕ = {ϕ 1 , ϕ 2 …, ϕ n } is linearly independent from the mathematical point of view, it may be approximately linearly dependent from the computational point of view This phenomenon causes a great many complications and may lead to very misleading results, since the associated secular equations may be almost identically vanishing Unfortunately, it seems as if many of the conventionally used basic systems are strongly affected by approximate linear dependencies In order to systematize this problem, it is convenient to study the metric matrix

On the Foundations of Combinatorial Theory IV Finite Vector Spaces and Eulerian Generating Functions

Abstract: : The paper studies combinatorial aspects of the lattice of subspaces of a vector space over a finite field and its use in deriving classical and new q-identities. Set theoretic interpretations of these identities are given in terms of the enumeration of vector spaces and linear transformations. The incidence algebra of a partially ordered set is shown to be a true generalization of the notion of a generating function and Eulerian generating functions are applied to count a variety of vector space objects. Combinatorial interpretations are provided for general q-difference equations. (Author)

Model to Explain Large Changes in the Electronic Density of States with Atomic Ordering in V 3 Au

Abstract: The superconducting transition temperature of ${\mathrm{V}}_{3}$Au has been experimentally shown to vary by a factor of 300, depending on the degree of atomic ordering. This has previously been attributed to a change in the electronic density of states $n(E)$ with ordering. A model is proposed here which provides a theoretical basis for this change in $n(E)$ with ordering.

On the problem of a finite basis of identities in groups

Abstract: This paper contains a proof that the set of varieties of groups has the cardinality of the continuum. This implies the existence of an infinite system of group identities not equivalent to any finite system.

Symmetry Adaptation to Sequences of Finite Groups

Abstract: Publisher Summary This chapter provides an introduction to symmetry adaptation and the assignment of group theoretical quantum numbers For symmetry adaptation with respect to a finite group, the matric basis of the Frobenius (group) algebra may be employed Wedderburn's construction of a matric basis is reviewed For symmetry adaptation with respect to a group sequence, a special matric basis called the sequence adapted matric basis may be employed The chapter also presents the modification of Wedderburn's method to yield a constructive proof of the existence of a sequence adapted matric basis Developments of the properties of the sequence adapted matric basis are shown The sequence adapted matric basis for special types of group sequences is considered; formulas for the corresponding sequence adapted irreducible representation matrix elements are presented Further, two methods for obtaining a sequence adapted irreducible representation from one that is not sequence adapted is provided followed by a description of application to the symmetric group The chapter also presents examples from the symmetric group

Magnetism and crystal symmetry of α-Mn

Abstract: A systematic method of magnetic structure analysis is developed and applied to the case of α-Mn. Scalar, vector, and tensor quantities in a phenomenological thermodynamical potential are expanded in complete sets of basis functions which form irreducible representations of the space group, and on this basis, a possible magnetic structure for α-Mn is discussed. A non-collinear magnetic structure containing thirteen parameters is proposed. The magnetic point group is isomorphic with either D 2 d of C 3 v depending on whether the principal axis is along either a or a , respectively. Possible modes of magnetostrictive atom-displacement are also discussed.

Schauder decompositions in locally convex spaces

Abstract: A decomposition of a topological vector space E is a sequence of non-trivial subspaces of E such that each x in E can be expressed uniquely in the form , where yi∈Ei for each i. It follows at once that a basis of E corresponds to the decomposition consisting of the one-dimensional subspaces En = lin{xn}; the theory of bases can therefore be regarded as a special case of the general theory of decompositions, and every property of a decomposition may be naturally denned for a basis.

A new method for phase determination. The `maximum determinant rule'

Abstract: The joint probability of a set of structure factors involved in a Karle–Hauptman determinant is evaluated. For equal atoms, under conditions to be specified, the theory leads to the conclusion; among all combinations of phases compatible with the condition of non-negativity of a Karle–Hauptman determinant, the most probable combination is that which maximizes this determinant. This `maximum determinant rule' can be used as a basis for the practical determination of phases. Special cases and further properties of the determinants are also obtained from the main expression for the joint probability.

Sum Rule Relating Optical Properties to the Charge Distribution

Abstract: A sum rule is constructed which relates the third moment of the imaginary part of the dielectric function at zero wave vector to an integral of the product of the Laplacian of the crystal potential and the fluctuation of the electron density. This sum rule, though rigorous, will be of real use only when the real solid can be replaced by a solid of pseudoatoms having only valence electrons. A method is found for obtaining an "experimental" third moment from other known moments. In this approximation some of the recent empiricism about bonding in crystals can be given a more rigorous basis. A quantitative application of the theorem to the prediction of the dielectric constant of GaAs is sketched.

In what spaces is every closed normal cone regular

Abstract: It is known ( 13 , p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Frechet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.

Stochastic linear programming with chance constraints

Abstract: An ordinary linear programming model is said to be chance-constrained if its linear constraints are associated with a set of measures indicating the extent of violation of the constraints. The CCP approach usually assumes the resource vector to be normally and mutually independently distributed and then derives a deterministic concave programming problem. In the SLP approach the tolerance measure for the linear constraints is not preassigned by the decision maker and the approach seeks to derive the statistical distribution of the optimal solution vector and also of the optimal objective function under the assumption that the set (A, b, c) of parameters contains elements with known probability distributions. Some basic differences of the CCP and the SLP approaches may be noted at the outset. First, the CCP approach utilizes the distribution properties of relevant random variables statisfying preassigned tolerance limits to specify a deterministic nonlinear program, whereas the SLP approach starts from a deterministic linear program (e.g., a program where all random elements are replaced by their expected values) and admits the random variations around its optimal basis to derive the probability distribution of the optimal solution satisfying (if necessary at a later stage) some tolerance measures if and when feasible. Second, nonlinearities are introduced in both approaches, although the initial problem in both cases is a linear programming problem. Third, the CCP approach restricts decision rules within a certain class (e.g.,

Equivalent Linearization for Random Vibration

Abstract: The concept of equivalence between linear and nonlinear systems undergoing random vibration is investigated, and the advantages and limitations of each of several definitions of equivalence are analyzed. The particular nonlinear system considered is the bilinear hysteretic single-degree-of-freedom oscillator, and the linear systems considered include one- and two-degree-of-freedom oscillators. The general characteristics of the stationary response of both the linear and nonlinear systems to random excitation are summarized. Various definitions of equivalence are then proposed on the basis of matching various response statistics of the two systems. The response statistics considered are rms displacement and velocity levels and power spectral density. Experimental values for the response statistics of the bilinear system are used with analytical results for the linear systems to obtain values for the equivalent linear system parameters under each definition of equivalence. The relationship of the proposed definitions and their results to previous work for similar systems with harmonic excitation and to previously suggested approximate analytical methods for random vibration is presented.

Markuschevich bases and duality theory

Abstract: Several duality theorems concerning Schauder bases in locally convex spaces have analogues in the theory of Markuschevich bases. For example, a locally convex space with a Markuschevich basis is semireflexive iff the basis is shrinking and boundedly complete. The strong existence Theorem 111.1 for Markuschevich bases allows us to show that a separable Banach space is isomorphic to a conjugate space iff it admits a boundedly complete Markuschevich basis, and that a separable Banach space has the metric approximation property iff it admits a Markuschevich basis which is a generalized summation basis in the sense of Kadec.

Solution of the Hartree–Fock problem by expansion onto nested bases

Abstract: A method for solving the Hartree–Fock problem in a finite basis set is derived, which permits each orbital to be expanded in a different basis. If the basis set for each orbital ϕi contains the basis functions for the preceding orbitals, ϕi−1, ϕi−2,… ϕ1, then the ϕi form an orthonormal set. One advantage over the standard Hartree–Fock method is that a different long range behavior for each orbital, as for example is required in the Hartree–Fock-Slater method, can be forced. A calculation on the ground state of beryllium is performed using the nested procedure. Very little energy is lost because of nesting, and the node in the 1s orbital disappears.

Analyses of the solution to a linear fractional functionals programming

Abstract: In this paper the effect of changing one constant of the Linear Fractional Functionals Programming problem has been discussed under the condition that the optimal basis for the original problem remains unaffected. In the changed form the optimal solution and new value of the objective functions are obtained.

Matrix elements in harmonic oscillator representations I. General solution and application to one dimension

Abstract: General expressions have been obtained for the matrix elements of linear or radial perturbations in the representations of the one-, two-, and three-dimensional harmonic oscillators The solutions are also general for the two types of matrix element where the two wave functions belong to the same basis set or to two different basis sets In this paper these expressions are applied to one- dimensional problems

Defect Chemistry and Non-Stoichiometric Compounds

Abstract: Crystallographic theory treats the structure of crystals in terms of the space lattice — a set of nodes or points in real space, related by unit translation vectors and symmetry operations that define the geometry and dimensions of the unit cell. Each point of the space lattice is associated with a group of atoms — the basis group — that defines the composition and detailed structure of the crystal. The effect of the translational and symmetry operations on this basis group is an identity operation: the space lattice concept implies that the environment of any atom in the basis group is identical throughout the crystal. There is no place in lattice theory for deviations from the ideal order or for variations in compositions and atomic positions from one unit cell to another.

Preprocessing on the basis of frequency of occurrence

Abstract: The release from PI methodology was used to test Oldfield's (1966) suggestion that human memory is organized in part on the basis of frequency of occurrence. An observed release from PI identified frequency of occurrence as an encoding category and provided support for the underlying assumptions of Oldfield's model.

On the optimal control of linear systems with incomplete information

Abstract: : The control of linear systems with incomplete information is considered where the unknown disturbances and/or random parameters are assumed to satisfy some statistical laws. The observer theory for linear systems is developed which generalizes the concepts due to Kalman and Luenberger pertaining to the design of linear systems which estimate the state of a linear plant on the basis of both noise-free and noisy measurements of the output variables. The separation theorem for linear system is then extended for such observers- estimators. The problem of controlling a linear system with unknown gain is then considered. An open-loop-feedback-optimal control algorithm is developed which seems to be computationally feasible. Existence of such suboptimal control scheme is proved under the assumption that the uncertainties in the unknown gail will not grow in time. Convergence of such suboptimal control system to the truly optimal control system is considered. A computer program is developed to study the control of a variety of third order systems with known poles but unknown zeroes. The experimental results serve to provide more insights into the structure and behavior of the open-loop-feedback-optimal control systems.

Widths of Rotational Lines of an Asymmetric‐Top Molecule SO2 I. Broadening by a Rare Gas

Abstract: Widths of six rotational lines of SO2 with Ji ranging from 3 to 16 have been measured. Linewidths have been calculated theoretically on the basis of the Anderson theory, and an assessment of the relative importance of different interactions has been made. Calculations have also been done on the basis of the Murphy–Boggs theory. The theoretical and experimental linewidths are found to agree well.

Atomic Structure Calculations in a Factorized Shell

Abstract: A method is described for calculating matrix elements of operators in a basis in which the spin-up and spin-down spaces of an atomic l shell are both split into separate parts. Instead of elaborate tables of fractional parentage coefficients, only a few reduced matrix elements are required. These are tabulated for d, f, g and h electrons. The transformation coefficients relating the states in the new basis to the familiar LS states are given for f electrons. An operator e h that separates and classifies repeating terms of the configuration h N (in analogy to the separation achieved by Racah for f N ) is shown to take a simpler form if parts off diagonal with respect to N are added to it, thereby accounting for some puzzling coincidences in the eigenvalues of e n .

Bases and α -dimensions of countable vector spaces with recursive operations

Abstract: This paper is based on the notions originally described by Dekker [2], [3], and the reader is referred to these for explanation of notation etc. Briefly, we are concerned with a countably infinite dimensional countable vector space Ū with recursive operations, regarded as being coded as a set of natural numbers. Necessarily, then, Ū must be a vector space over a field which itself is in some sense recursively enumerable and has recursive operations.

Symmetry Properties of the Kinetic Energy Matrix and Their Applications to Problems of Molecular Vibrations

Abstract: In the present paper it is shown that G, the inverse of the kinetic energy matrix in the internal coordinates space, provides all the possible information on the symmetry of molecular vibrations. Namely, once the G matrix of a molecule is known, the point group G of the molecule, the coordinates which provide a basis for a completely reduced representation of G (symmetry coordinates), and the minimal set of independent force constants which must be chosen can be derived. Corresponding computer programs have been written which provide an automatic way for the factorization of the dynamical matrices of large and/or highly symmetrical molecules.