Showing papers on "Basis (linear algebra) published in 1969"

Comments on bases in dependence structures

Abstract: Dependence structures (in the finite case, matroids) arise when one tries to abstract the properties of linear dependence of vectors in a vector space. With the help of a theorem due to P. Hall and M. Hall, Jr concerning systems of distinct representatives of families of finite sets, it is proved that if B1 and B2 are bases of a dependence structure, then there is an injection σ: B1 → B2 such that (B2 / {σ(e)}) ∩ {e} is a basis for all e in B1. A corollary is the theorem of R. Rado that all bases have the same cardinal number. In particular, it applies to bases of a vector space. Also proved is the fact that if B1 and B2 are bases of a dependence structure then given e in B1 there is an f in B2 such that both (B1 / {e}) ∩ {f} and (B2 / {f}) ∩ {e} are bases. This is a symmetrical kind of replacement theorem.

On the method of matched asymptotic expansions

Abstract: The method of matched (or of ‘inner and outer’) asymptotic expansions is reviewed, with particular reference to two general techniques which have been proposed for ‘matching’; that is, for establishing a relationship between the inner and outer expansions, to finite numbers of terms, of an unknown function. It is shown that the first technique, which uses the idea of overlapping of the two expansions, can be difficult and laborious in some applications; while the second, which is the ‘asymptotic matching principle’ in the form stated by Van Dyke(13) can be incorrect. Two different sets of conditions sufficient for the validity of the asymptotic matching principle are then established, on the basis of assumptions about the structure of expressions which approximate to the desired function f(x,∈) for all relevant values of x. Finally, it is noted that in four classes of singular-perturbation problems for which complete and rigorous asymptotic theories exist, uniform approximations to the solutions have a structure which is a particular form of the general one assumed in this paper.

The intrinsic dimensionality of signal collections

Abstract: In view of the trend toward the representation of signals as physical observables characterized by vectors in an abstract signal space, rather than as time or frequency functions, it is desirable that dimensionality be defined in a manner independent of the choice of basis on which the vectors are projected. The intrinsic dimensionality of a collection of signals is defined to be equal to the number of free parameters required in a hypothetical signal generator capable of producing a close approximation to each signal in the collection. Thus defined, the dimensionality becomes a relationship between the vectors representing the signals. This relationship need not be a linear one and does not depend on the basis onto which the vectors are projected in the signal-measuring process. A digital computer program for estimating this dimensionality from the signal coefficients on an arbitrary basis has been developed. The program makes use of some results obtained from a multi-dimensional scaling problem in experimental psychology and utilizes an inverse relationship between the variance in interpoint distances within a hypersphere and the dimensionality of the hypersphere. Using this method, the results are believed to be independent of the choice of orthogonal basis, and no prior knowledge of the analytical form of the signals is assumed. The validity of the program is tested and verified by using it to estimate the dimensionality of signals of known structure.

Lifting projections of convex polyhedra.

Abstract: If τ is a projection of a closed convex polyhedron P onto a convex polyhedron Q, then a lifting of Q into P is defined to be a single-valued inverse τ* of τ such that τ*(Q) is the union of closed faces of P The main result of this paper, designated the Lifting Theorem, asserts that there always exists a lifting r*, provided only that there exists at least one face of P on which τ acts one-to-one The lifting theorem represents a unifying generalization of a number of results in the theory of convex polyhedra and should prove useful as an investigative as well as a conceptual tool In the course of the proof, a special case of the Lifting Theorem is translated into linear programming terms and stated as the Basis Decomposition Theorem, which summarizes the behavior of a linear program as a function of its right-hand sides In particular, the fact that a lifting is necessarily a piecewise linear homeomorphism is reflected in the Basis Decomposition Theorem as the observation that the optimal solution of a linear program can always be chosen as a continuous function of the right-hand sides

A Cycle Generation Algorithm for Finite Undirected Linear Graphs

Abstract: When the algorithms of J. T. Welch, Jr. were implemented it was discovered that they did not perform as described. The generation of all cycles from a basis is faulty. The generation of the basis is apparently correct. A modified version of Welch's Algorithm 3 is presented. The reasons for modifying Welch's algorithms are presented with examples.

Gaussian-Orbital Basis Sets for the First-Row Transition-Metal Atoms

Abstract: Contracted Gaussian-type function basis sets have been produced for the first-row transition-metal atoms Sc through Cu for use in molecular SCF calculations. The basis functions consist of 15 s-type, eight p-type, and five d-type Gaussian primitives contracted to a four s-type, two p-type, and one d-type basis set and are exponent and coefficient optimized after contraction. It is shown that this basis set is of single-zeta quality for the core functions and of double-zeta quality for the valence-electron functions. Relaxation of the degree of contraction results in a significant lowering of the total energy.

Finite Transformations of SU3

Abstract: The properties of SU3 finite transformations are investigated. These transformations on the defining three‐dimensional complex space are parameterized in a form employing three factors, two of which are the Euler parameterization of an SU2 subgroup. The irreducible representations of the factored parameterization are found explicitly. The volume element is calculated and the orthogonality relation is verified. Spherical harmonic basis states are derived as a specialization of the transformation matrix. Another result is a definition of triality and a simple proof that it is additive modulus three.

Letters to the Editor---Extension of Dantzig's Algorithm to Finding an Initial Near-Optimal Basis for the Transportation Problem

Abstract: This note presents a new method for finding a starting basis for transportation problems; it produces a near-optimal basis, and appears to be superior to present methods.

GF calculation with minimal and extended basis sets for H2O, NH3, CH4 and H2O2

Abstract: A method is proposed for applying the theory of generalized group functions to SCF-GF calculations with large basis sets. A simple procedure for localising the SCF-MO's resulting from a standard SCF calculation is described, with applications to H2O, NH3, CH4 and H2O2. Results compare quite favourably with those obtained by the usual GF method. It is shown that when basis functions are the SCF-MO's and there are only two functions per group, the GF approach is practically equivalent to a configuration interaction treatment where only double excitations within the groups are considered.

Turbulence Theory with a Time‐Varying Wiener‐Hermite Basis

Abstract: A theory of homogeneous turbulence in an incompressible fluid is formulated. The field variable v(r, t) is expanded in terms of a complete set of an ideal random basis, the first term of which is Gaussian. Since the ideal basis follows the actual fluid motion, there should be no incorrect relation between large and small eddies.

Quadratic Filter Theory and Partially Coherent Optical Systems

Abstract: Quadratically nonlinear systems may be analyzed and synthesized by linear methods by exchanging an N-dimensional nonlinear problem for a 2N-dimensional linear formulation. This paper describes the basis for such linearization of a quadratic functional, and applies the method to partially coherent transilluminated optical systems.

On ω-models which are not β-models

Abstract: Publisher Summary This chapter proves a theorem stating that β -models for the second-order arithmetic cannot be distinguished from ω- models by elementary sentences. It determines a similar result for models of the Zermelo–Fraenke1 set theory and gives a solution of a problem concerning the existence of models which are א τ -standard but are not א τ+1 -standard. The chapter presents abbreviations and definitions and formulates a few theorems that can be proved in the basis of defined axioms. It defines semantical notions of satisfaction, model, elementary extension, reduct, and diagram. It describes the pigeon-hole principle, which states that if many objects are put into a small number of drawers, then at least one drawer contains many objects. A new family of non-standard models is constructed for set theory in the chapter.

Apparatus and methods for determining imbalance

Abstract: Radial variations about the periphery of an element are the basis for determining the imbalance of such member. Such variations are resolved into vector components; and integrated over a single traversal of such periphery. The technique is used to determine web roll imbalance; and dimensional growth of the roll is accommodated by use of a positional pickoff. Positioning of the pickoff is used to weight the effect of the vector components.

The Universe of Set Theory

Abstract: Since Cohen’s discovery of forcing, many problems in set theory have been proved to be independent of ZF-set theory just as in the case of the parallel postulate in plane geometry. In plane geometry, only the independence of the parallel postulate was considered, but in set theory it seems that infinitely many problems can be proved to be mutually independent. The consideration of many set theories might not be of advantage to us because set theory is a basis of mathematics and working mathematicians cannot believe that both “yes” and “no” are equally reasonable answers to their problems in natural numbers, real numbers or Hubert spaces.

Finite Transformations and Basis States of SU(n)

Abstract: A convenient parameterization for finite transformations of SU(n) is developed which explicitly exhibits the special unitary subgroups. This is also used to parameterize the defining space. Higher‐dimensional representations are discussed. The question of which representations carry the trivial representation of SU(n − 1) is considered, as well as the parameterization of these states. Application is made to the transformations and basis states of SU(3).

A Space-Dependent Reactor-Noise Formulation Utilizing Modal Expansions

Abstract: A space-dependent noise formulation is developed on the basis of the modal-analysis technique Application of the method is illustrated by numerical calculations carried out for a one-dimensional m

Stochastic Signal Representation

Abstract: The representation of signals in the time domain requires the decomposition of functions of time into linear combinations of a finite number of basis functions. The purpose of this paper is to introduce a stochastic approximation algorithm, which, given an ensemble of signals \{f(t)\} , selects from a set S that subset S_{m} of m basis functions that best represents the elements of the ensemble. In this context the optimum representation is defined as the basis set S_{m} , which minimizes the expected value of a suitable performance index Q , defined as a function of both the basis S_{m} and the random signal f(t) . In particular, Q is here assumed to be the least-square error that may be achieved when the signal f(t) is approximated by a linear combination of the elements of S_{m} . The procedure is analyzed in detail for the case when the basis functions are one-sided complex exponentials. The convergence properties of the algorithm are discussed for this case. An illustrative example of application of the proposed method is also presented.

An algebraic solution of the state estimation problem.

Abstract: The general linear state estimation problem is considered here from a purely algebraic viewpoint without resorting to the concepts of probability theory. A least-squares parameter estimation formula is first established and then extended to permit estimation on a sequential basis. By combining the relations derived with the properties of the state transition matrix, the solutions to the problem of filtering, smoothing, and prediction for systems with both forcing and measurement noise are found.

Geometrical Basis of Lagrange Multipliers and System Constraints in Mechanics

Abstract: The effects of constraints imposed on mechanical systems may be analyzed when the constraints are recognized as geometrical surfaces. Each component constraint restricts the motion due to the applied active forces; taken together the constraint surfaces intersect, forming a new dimensionally reduced surface and thus a single constraint. The gradient vectors on each of the component constraint surfaces must sum to the gradient on the reduced surface. Lagrange multipliers are then just scalars which adjust the magnitudes and senses of the gradient vectors on each component geometrical surface so that the gradient on their intersection has the proper magnitude and is directed along the resultant reaction force. Since the resultant gradient lies in a subspace with a basis formed by the component gradients, these component gradients must be linearly independent. The existence of a nonsingular Jacobian transformation from the subspace to the constraint surfaces guarantees its existence with the required dimension, and thus, the validity of the constraints is guaranteed.

Construction d'une base antisymétrique dans l'espace idéal

Abstract: An antisymmetric basis has been set up in the ideal space as a combination of one boson and one ideal quasi-particle state; in this way a three quasi-particle system may be studied. The knowledge of solutions of linearized equations for two quasi-particles is a requisite for the setting up of this basis. The exclusion principle is thus introduced correctly. The spurious states coming from the non-conservation of the number of particles are separated out and can be rejected easily. The method can be used only in QTD approximation. The one-shell case is discussed as a verification of the method.

Biorthogonal Systems in lp -Spaces

Abstract: Our aim in this paper is to generalize certain ideas and results of Bary (1) on biorthogonal systems in separable Hilbert spaces to their counterparts in separable lp -spaces, 1 < p.The main result of Bary is to characterize a natural generalization of the concept of orthonormal basis for a Hilbert space. That of this paper is to characterize the concept of a Bary basis which is a generalization of the idea of standard basis of an lp -space. The result is interesting for lp -spaces because of the paucity of standard bases in these spaces. Before summarizing our results, we shall introduce some notation and recall a few pertinent definitions and facts. The symbols and denote mutually conjugate lp -spaces, where is the space lt and the space ls with 1 < r <2 and 2 < s = r/(r – 1).

A Theory of Fluctuation Based on the Hierarchy Equation

Abstract: The kinetic theory of fluctuation and correlation in non-equilibrium ideal gases is developed in the μ-s space on the basis of the B-B-G-K-Y procedure, and its evolution equation in ( x , t ) space is derived which is somewhat similar to the Orr-Sommerfeld equation.