Showing papers on "Basis (linear algebra) published in 1974"

Stochastic theory of minimal realization

Abstract: In this paper it is shown that a natural representation of a state space is given by the predictor space, the linear space spanned by the predictors when the system is driven by a Gaussian white noise input with unit covariance matrix. A minimal realization corresponds to a selection of a basis of this predictor space. Based on this interpretation, a unifying view of hitherto proposed algorithmically defined minimal realizations is developed. A natural minimal partial realization is also obtained with the aid of this interpretation.

Approach to the description of atoms using hyperspherical coordinates

Abstract: We formulate and develop an expansion of N‐electron atomic wavefunctions in hyperspherical coordinates. The expansion basis is a complete set of functions of 3N − 1 angle variables which describe the relative configuration of the electrons. The system is scaled in size by one extensive variable, the hyperradius r. All angular momentum and Pauli antisymmetry properties of the system are contained in the hyperspherical basis. A new, total angular momentum, antisymmetry‐adapted basis is derived and used to construct an expansion of the total electronic wavefunction which has all of these properties term by term. These basis functions are called configurational normal modes and provide a new element of atomic structure to replace the electron orbital. Several methods of implementing calculations with this expansion are discussed.

Lie theory and separation of variables. 4. The groups SO (2,1) and SO (3)

Abstract: Winternitz and coworkers have shown that the eigenfunction equation for the Laplacian on the hyperboloid x02−x12−x22=1 separates in nine orthogonal coordinate systems, associated with nine symmetric quadratic operators L in the enveloping algebra of SO (2,1). Corresponding to each of the operators L, we employ the standard one‐variable model for the principal series of representations of SO (2,1) and compute explicitly an L basis for the Hilbert space as well as the unitary transformations relating different bases. We also compute the associated results for realizations of these representations on the hyperboloid. Three of our bases are related to well‐known subgroup reductions of SO (2,1). Of the remaining six, one is related to Bessel functions, two to Legendre functions, and three to Lame functions. We show that there is virtually a perfect correspondence between the known theory of the Lame functions and the representation theory of SO (2,1) and SO (3).

Existence of a basis in the space of functions analytic in the disk, and some properties of franklin's system

Abstract: It is shown that there is a basis in the space of functions analytic in the unit disk and continuous in the closed unit disk. This answers a question posed by Banach. It is further shown that the Franklin system is an unconditional basis in the spaces Lp(0, 1) for 1 < p < ∞. Bibliography: 8 items.

A Method for the Solution of the Distribution Problem of Stochastic Linear Programming

Abstract: A method is proposed for finding a closed form expression for the cumulative distribution function (CDF) of the maximum value of the objective function in a stochastic linear programming problem in the case where either the objective function coefficients or the right-hand side coefficients are given by known probability distributions. If the objective function coefficients are random, the CDF is obtained by integrating over the space for which a given feasible basis remains optimal. A transformation is presented using the Jacobian theorem which simplifies the regions of integration and often simplifies the integration itself. A similar analysis is presented in the case where the right-hand side coefficients are random. An example is given to illustrate the technique.

A note on metric-fine spaces

Abstract: The coreflection into metric-fine spaces X is explicitly described, and it is shown that metric-fine proximally fine spaces are just the spaces X such that f: X Y is uniformly continuous whenever the pre-images under f of zero sets are zero sets. Basic properties of metric-fine and (separable metric)-fine spaces are derived, and the corresponding coreflections are described using apparently a new concept of a complete projective family with respect to a class of spaces. In conclusion the uniform spaces determined by cozero sets are identified as metric-fine proximally fine spaces. The notion of metric-fine spaces (introduced by A. Hager [1]) seems to be very useful in the theory of uniform spaces. If we add to metric-fine some property we usually get a very interesting notion. One example is given in this note. More examples are given in subsequent papers; e.g., measurable uniform spaces are just hereditarily metric-fine spaces (see Frolik [3]). Let F be a coreflection of a category &, and let K be a class of objects of the category &. A K F object is an object A such that any morphism of A into an object K in X factorizes (uniquely) through the coreflection FK K. By a theorem of J. !Tilimovsky [1], the class of all K F objects is coreflective provided that C1 fulfils the natural conditions. This note concerns the category of uniform spaces. In general we use the terminology and notation of E. Cech [1]. Following Isbell [1], denote by a( the usual fine coreflection, and denote by e the usual reflection onto separable uniform space. Thus (xX is the underlying set X of X endowed with the (uniformly finest) uniformity topologically equivalent to X, and eX is X endowed with the uniformity which has all countable uniform covers of X for a basis of uniform covers. Received by the editors March 2, 1973. AMS (MOS) subject classifications (1970). Primary 54E10, 54E15.

Basis Factorization for Block-Angular Linear Problems: Unified Theory of Partitioning and Decomposition using the Simplex Method

Abstract: A General Block-Angular Basis Factorization is developed to represent the inverse of the basis of block-angular linear problems in factorized form. This factorization takes advantage of the structure of the matrix and can be efficiently updated when one column is replaced by another. Partitioning and Decomposition methods (excluding Dantzig-Wolfe decomposition) for block-angular linear problems with coupling constraints, or coupling variables, or both, are shown to be variants of a Simplex Method using this General Block-Angular Basis Factorization form of the inverse, with various criteria as to the vector pair selected to enter and to leave the basis. By considering other criteria new algorithms are obtained. In particular, algorithms are presented for which at each iteration only a subset of the terms in the factorization needs to be used or to be updated. Preliminary experimental results with such an algorithm for block-angular linear problems with coupling constraints are included. Results are extended to the case when embedded in the block-angular structures there are blocks which themselves are of block-angular form. Applications to the solution of dynamic linear programs (staircase structure) are developed.

Identification of parameters in an inhomogeneous aquifer by use of the maximum principle of optimal control and quasi-linearization

Abstract: Optimal identification of aquifer parameters in a distributed system is formulated as an optimal control problem. The dynamics of the head is governed by a second-order nonlinear partial differential equation. The numerical example presented considers that the parameters to be identified are functions of the space variable. Observations on head variations are available at several observation wells distributed within the system. Spatial discretization is first used to transform the distributed system to a lumped system. The least squares criterion function is then established. After introducing the Lagrange multipliers, the maximum principle is applied to obtain the set of necessary conditions that is optimal. These conditions are expressed in terms of a set of canonic equations of two-point boundary value type that is easily solved by the technique of quasi-linearization. Thus aquifer parameters are directly identified on the basis of observational data taken at observation stations. The maximum principle formulation is inherently more accurate and stable, since it minimizes the least squares error over the whole time and space domains. Computationally, it is extremely efficient. The numerical example presented demonstrates simultaneous identification of 11 parameters defined at discretized points along the space variable. Quadratic convergence is also demonstrated by numerical experimentation.

The non-additive ligand field

Abstract: The semi-empirical ligand field is a perturbation operator whose consequences are taken to first order using a basis set ofl functions. Since the basis spans an irreducible representation of the 3-dimensional rotation-inversion groupR 3i it is useful to express the operator as a sum of components of irreducible tensor operators with respect to this group. IfR 3i is reduced with respect to the molecular subgroup the electronic factor of each term in the sum must be totally symmetrical within this group. This choice of operator leads to thecrystal field parameterization without implying an electrostatic model. Alternatively a shift operator withinl space may be chosen as the essential part of the perturbation operator. This leads to theligand field parameterization. Between the two parameterizations there exists a one to one relationship, whose coefficients are proportional to 3l symbols. This relationship is given together with a brief discussion of the reasons for the proposed parameterizations.

A unified formulation of the variational methods in scattering theory

Abstract: The Schwinger and Kohn variational methods are derived in a unified manner from the approximation theory of linear operators in Hilbert space. The analytic basis behind their relationship with the method of the Pade approximants is clarified. The present formulation has enabled us to derive some known results in variational methods and the Pade approximants relatively briefly and simply.

APOSS: a partial order in the solution space of bivalent programs

Abstract: Order relations are constructed in the solution space of a linear 0-1 program and they are used to obtain information about the problem (infeasibility, forced values for certain variables, equality or non-equality of certain pairs of variables, etc.). On this basis an algorithm is described for solving 0-1 programs. Good computational experience is reported with the algorithm.

Solutions of the Dirac equation in finite de Sitter space (representations ofSO 4.1)

Abstract: All the solutions of the Dirac equation in the finite space de Sitter universe are found and it is shown that they form a basis for a representation of the group of motions for this universe,SO 4,1. The action of the generators ofSO 4.1 on these solutions is explicitly given as a linear superposition of solutions.

An Introduction to Adaptive Arrays

Abstract: : A tutorial introduction to adaptive arrays is presented via analysis of linear arrays with adaptive control loops of the Applebaum analog type, which derive weighting adjustment control from the correlations between element signals, i.e., on the basis of the covariance matrix of the set of system inputs. Phase conjugacy, cross-correlation interferometers, and the IF phase- cancellation principle were reviewed, and a simple two-element array with a single adaptive control loop was analyzed. The analysis was based on reduction to a type-O follower servo equivalent circuit and includes transient behavior, bandwidth effects, and the Applebaum hard-limiter modification. The analysis then proceeded to a K-element linear array with K adaptive control loops. The system analysis consisted of a Q-matrix transformation into orthonormal eigenvector space, and interpreting the transformation in terms of an orthogonal beam-forming network, similar in principle to a Butler matrix network. The system was thus converted into an equivalent 'orthonormal,' adaptive control- loop network to which the type-O follower servo analysis can again be applied. The Q-matrix transformation network produced a set of K orthogonal, normalized, eigenvector beams whose output powers are proportional to the eigenvalues of the covariance matrix. These beams were used as the basis of a convenient expression for calculating the time-dependent output pattern function for the array.

Graph-theoretic considerations on the invariance of return difference

Abstract: Contrary to the common assumption, it is shown that the general return difference and the general null return difference are variant under the general transformations between a system of basis circuits and a system of basis cutsets. Necessary and sufficient conditions are presented for the invariance of these functions. Formulation of these functions in terms of the primary systems of equations and illustrative examples are also given.

Linear time dependent systems

Abstract: In studies of linear open-loop systems, the assumption of time invariance is often tacitly made. In this paper we show how a time dependent system can arise quite naturally. The estimation of time dependent transfer function, on the basis of a single realization, when the input/output processes are nonstationary is considered. We also consider the problem of testing a given open-loop system for time dependence. The tests described here make use of the concept of the "evolutionary cross spectra," and rests essentially on testing the "uniformity" of a set of vectors, whose components consist of the "evolutionary gain spectra" and "evolutionary phase spectra." Using a logarithmic transformation on the evolutionary gain spectra, we show that the mechanics of the tests are formally equivalent to a two-factor multivariate analysis of variance (MANOVA) procedure. Numerical illustrations, from the real and simulated data, of the proposed tests are included.

Classification of stationary space-times

Abstract: A systematic approach to the geometric structure of stationary gravitational fields is presented. The algebraic type of the trace-free Ricci tensor together with the propagation properties of the eigenrays in the background 3-space defined by the Killing trajectories serve as a basis for classifying the solutions of the stationary field equations. The eigenrays are the integral curves belonging to the solutions ξA of the eigenvalue problemGABξB=μξA,GABspinor representing the gravitational field in the background space. Many of the already known stationary metrics can be derived in the present scheme but new solutions of the field equations are also obtained. The possible types of the vacuum and electrovac fields are discussed in their connection with the corresponding exact solutions.

A Unified Theory of the Algebraic Topological Methods for the Synthesis of Switching Systems

Abstract: A set of logic operators, first proposed by Roth [3], based on an algebraic topological interpretation of a function table, have provided the basis for the subsequent development of algorithms for the computer aided synthesis of combinatorial networks.

$B$-convexity and reflexivity in Banach spaces

Abstract: A proof of James that uniformly nonsquare spaces are reflexive is extended in part to B-convex spaces. A condition sufficient for non-B-convexity and related conditions equivalent to non-B-convexity are given. The following theorem is proved: A Banach space is B-convex if each subspace with basis is B-convex. 0. Introduction. The notion of a B-convex Banach space was introduced by A. Beck [1], [21 as a characterization of those Banach spaces X having the property that a certain strong law of large numbers holds for X valued random variables. Definition. Let k be a positive integer and e a positive number. X is said to be k, t-convex if for any 1x1, ...* Xkl, Kxi 1 < 19 i = 1, * * * , k, there is some choice of signs 61' * * k so that xk_ 1| < k(l c). X is said to be B-convex if it is k, t-convex for some k and c. Further study of B-convex spaces has been done by R. C. James [61, [71, D. P. Giesy [51 and C. A. Kottman [8]. Giesy showed that B-convex spaces have many of the properties of reflexive spaces. James conjectured that all B-convex spaces are reflexive, and proved the conjecture true for 2, E-convex spaces. Both James and Giesy proved the conjecture true for B-convex spaces having an unconditional basis. Kottman extended James' 2, E-convex proof to a larger subclass, P-convex spaces. Examples are known of spaces which are reflexive but not B-convex. ?1 of this paper adopts a part of James' 2, E-convex theorem to all non-B-convex spaces, presents a condition sufficient for non-B-convexity, and gives related characterizations of non-B-convex spaces, though the conjecture of James remains open. ?2 proves a theorem on B-convexity and subspaces with basis analogous to a theorem of Pekczyn'ski on reflexivity and subspaces with basis. For a Banach space X, U(X) will denote the closed unit ball lx: |lxii < 11 of X. I. Non-B-convexity. In James' proof [6] that 2, E-convex spaces are reflexive, he defines for a Banach space X a sequence of numbers Kn, and shows that if X Received by the editors July 7, 1971. AMS (MOS) subject classifications (1970). Primary 46B10; Secondary 46B05, 46B15.

Extension theory for operators and spaces with indefinite metric

Abstract: We show that the classical extension theory for isometric operators cannot be automatically extended to J-isometric and J-Hermitian operators in J-spaces with infinite rank. We construct a single extension theory which includes both the isometric and Hermitian operators in a Hilbert space and the J-isometric and J-Hermitian operators in a J-space with arbitrary indefinite rank. The basis of the construction is a scheme for extension of a neutral subspace of a J-space to a maximal or hypermaximal subspace.

Equilibrium Situations in Games with a Hierarchical Structure of the Vector of Criteria

Abstract: Among models of decision making in n-person games with vector criteria it is possible to distinguish one class of models in which the vector of criteria have a so-called hierarchical structure. Multi-level economic and social systems are used as the basis of such models. Each of these “large” systems are sub-divided into a series of groups, which in their turn are again sub-divided, etc., until some elementary unit remains — a player, constituting the “subject” for making a decision.

The Structure of Integer Programs under the Hermitian Normal Form

Abstract: This paper discusses the structure of integer programming under the Hermitian normal form. It shows that this formulation is akin to the simplex technique for linear programming in that there is a basis and a basic solution associated with each Hermitian normal matrix, and that this Hermitian basis forms a set of natural cutting planes; these cutting planes are strong in that they provide facets for at least one of the corner polyhedra associated with a linear programming basis B. In addition, the cofactors of the Hermitian basis are group elements under a homomorphism of the original group structure. The Hermitian basis also allows one to characterize the values of the right-hand side for which the present solution is feasible. Finally, for an optimal Hermitian basis, one can perform sensitivity analysis on the cost coefficients.

$P$-convexity and $B$-convexity in Banach spaces

Abstract: Two properties of B-convexity are shown to hold for P-convexity: (1) Under certain conditions, the direct sum of two P-convex spaces is P-convex. (2) A Banach space is P-convex if each subspace having a Schauder decomposition into finite dimensional subspaces is P-convex. 0. Introduction. In the previous paper [1] the question of whether all B-convex spaces are reflexive was discussed. The concept of a P-convex space was introduced by C. Kottman [4] as follows: Definition. For a positive integer n, let P(n, X) be the supremum of all numbers r such that there is a set of n disjoint closed balls of radius r inside U(X)= fx: llxll < 11. X is said to be P-convex if P(n, X) < Y2 for some n. Kottman showed that all P-convex spaces are both B-convex and reflexive. Therefore the question "Is there a B-convex space that is not P-convex?" is of interest. Many properties of B-convex spaces are not known for P-convex spaces. In this paper we consider two of these properties and prove partial analogs of them for P-convex spaces: The first property is that direct sums of B-convex spaces are B-convex [2]. The proof of this fact for B-convex spaces rests on the invariance of B-convexity under isomorphism, but it is not known whether P-convexity possesses this invariance. Two partial analogs of the direct sum property are obtained, Theorems 1.3 and 1.5, using Ramsey's theorem of combinatorics. The second property is that a space is B-convex if each subspace having a basis is B-convex [1]. A partial analog of this is proved, Theorem 2.1, using one of the direct sum results. We will use the following characterization of P-convexity from Remark 1.4 of [4]: Let a set of n elements be called 5 separated of order n provided the distance between any two elements of the set is at least &. Then a space X is P-convex if and only if for some positive integer n and some positive number e < 2 there is no 2 e separated set of order n in U(X). Received by the editors July 7, 1971. AMS (MOS) subject classifications (1970). Primary 46B10; Secondary 46B05, 46B15, 05A05.