Theory Of Vector Spaces Research Articles

Character sums and intersection cohomology complexes associated to the space of square matrices

Let χ be a multiplicative character of a finite field F q . We study by an algebro-geometric method the finite Fourier transforms of χ(det x), where x runs over the square matrices over F q (untwisted case), or the Hermitian matrices (twisted case). We take up these examples as a first step to develop Intersection Cohomology Method in the theory of prehomogeneous vector spaces.

Theory of Prehomogeneous Vector Spaces, II, A Supplement

This paper is a supplement of [Gl]. In [Gl, §§2 and 3], we have mainly studied Z>-modules Df* generated by a complex power of a regular function, especially a relative invariant of a prehomogeneous vector space. Here we modify the argument so that we can include a more general Z)-modules such as D(fu), where u is a section of a regular holonomic D-module. The main results are (6.20)-(6.22). In (6.20), we determine the Fourier transform of D(fu), assuming that / is a relative invariant of a prehomogeneous vector space, and that Du is an integrable connection of rank one satisfying certain additional assumptions. As its corollary, we get (6.21) and (6.22). The latter will be used in a study of character sums associated to prehomogeneou s vector spaces over a finite field. Convention and Notation. We denote by Z the rational integer ring, and by C the complex number field. As for ^-modules, we shall work in the algebraic category unless otherwise stated. We define the de Rham functor DR( —) so that DR(@X) = CX, where Gx is the structure sheaf. For a morphism F:X-+ Y between varieties, and for an $ y-module M, F* denotes the usual 0-module pull-back; F*Jt = 0X®F-^Y F'^Jt. We shall refer to [Gl, (a,b,c)] etc. simply as (a,b,c) etc.

Molecular symmetry andab initio calculations. II. Symmetry-matrix and symmetry-supermatrix in the Dirac-Fock method

The concepts of symmetry-matrix and symmetry-supermatrix introduced in article I [J. Comput. Chem.,10, 957 (1989)] can be generalized to the Dirac-Fock method. By using the semidirect product decomposition of Oh and the linear vector space theory, the irreducible representation basis of Oh for any molecular system (Oh or its subgroups) can be deduced analytically in the nonorthonormal Cartesian Gaussian basis. This method is extended to discuss the double-valued representations of Oh* in the complex Cartesian Gaussian spinor basis. In the double-valued irreducible representation basis of D2*, the matrix of kinetic operator c(OVERLINE)σ(/OVERLINE)·(OVERLINE)p(/OVELINE) in the Dirac-Fock equation can be reduced into a real symmetric and can be grouped into classes under the operations in D3d. Therefore, the symmetry-matrix and symmetry-supermatrix can also be used in the Dirac-Fock method to reduce the storage of two electron integrals and calculations of Fock matrix during iterations by a factor of ca. g2 (g is the order of the molecular symmetry group). In addition, a method to deal with the nonorthonormal space is presented. © 1996 by John Wiley & Sons, Inc.

THEORETICAL AND NUMERICAL ASSESSMENT OF STRAIN PATTERN ANALYSIS

The Strain Pattern Analysis (SPA) method was conceived at the RAE in the 1970s as a means of estimating the displacement shape of a helicopter rotor blade by using only strain gauge data, but no attempt was made to provide theoretical justification for the procedure. In this paper, the SPA method is placed on a firm mathematical basis by the use of vector space theory. It is shown that the natural normwhich underlies the SPA projection is the strain energy functionalof the structure under consideration. The natural norm is a weightedversion of the original SPA norm. Numerical experiments on simple flexure and coupled flexure-torsion systems indicate that the use of the natural norm yields structural deflection estimates of significantly greater accuracy than those obtained from the original SPA procedure and that measurement error tolerance is also enhanced. Extensive numerical results are presented for an emulation of the SPA method as applied to existing mathematical models of the main rotor of the DRA Lynx ZD559 helicopter. The efficacy of SPA is demonstrated by using a quasi-linear rotor model in the frequency domain and a fully non-linear, kinematically exact model in the time domain: the procedure based on the natural (or weighted) norm is again found to be superior to that based on the original SPA method, both in respect of displacement estimates and measurement error tolerance.

Meta level in the teaching of unifying and generalizing concepts in mathematics

Certain concepts in mathematics were not invented only to solve new problems; their aim was mainly to find general methods to solve different problems with the same tools. Such concepts, as those of the axiomatic theory of vector spaces or groups or the modern definition of limit, will be called in this paper “unifying and generalizing concepts”. I will point out some epistemological specificities of these concepts and subsequently analyze their influence on teaching. I will explain the reasons which led me to the conclusion that it is necessary to introduce some “meta” aspects into the teaching of unifying and generalizing concepts, and I will present the theoretical framework I adopted for my purpose, in relation to other theoretical approaches. I will then present and analyze one example, from which I will draw conclusions about theoretical questions of evaluation in a long term experiment which includes a meta dimension for the teaching of unifying and generalizing concepts in mathematics.

Avoiding the Exchange Lemma

In finite-dimensional vector space theory, before defining dimension (as the size of a basis) it is necessary to show that all bases have the same number of members. This is usually done by appealing either to the exchange lemma or to the fact that a system of more than n homogeneous linear equations in n unknowns has a nontrivial solution. When the underlying field has zero characteristic, however, the following more direct approach is available. Let V be avector space over a field of characteristic zero. A set {u1, u2, . . ., un} of vectors is a basis for V if each v E V can be written uniquely in the form v = Lin=1Aiui, where the Ai are scalars.

Pontryagin duality in the theory of topological vector spaces

Pontryagin duality in the theory of topological vector spaces

Introducing undergraduates to the moment method

Four application problems are presented which have been found to facilitate introductory instruction on numerical methods for undergraduate electrical engineering students. The method of moments (MOM) is introduced at the level of first principles, preparing students for subsequent advanced topics in matrix methods and developments through linear vector space theory. The problems are such that the associated computer programming assignments are straightforward, minimizing distraction from the mathematical concepts. Numerical solutions may be obtained, or verified, with a pocket calculator in some cases. Computer programs to obtain more sophisticated graphics output and to expedite solution of equations with large matrices can be progressively added as supplementary student exercises.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A general outline of the genesis of vector space theory

The following article presents a general outline of the genesis of the elementary concepts of vector space theory. It presents the main works that contributed to the development of these basic elements and analyzes how they developed and how they influenced each other. The study of systems of linear equations and the search for an intrinsic geometric analysis were the two main sources which gave rise to the theory of linearity. The fact of going beyond the third dimension in geometry in the middle of the 19th century, as well as the dialectical development between algebra and geometry from the creation of analytical geometry on, brought about the development of an initial unification of linear questions around the concept of determinant. This framework was generalized to the countably infinite dimension following work on functional analysis. Axiomatization, which was carried out at the end of the 19th century, although only really put to use after 1920, is a wider process which is part of the general development of mathematics in the beginning of the 20th century. I will analyze how this phenomenon came into existence and how it finally established its influence. L'article qui suit, presente une vue générale de la genèse des concepts élémentaires de la théorie des espacos vectoriels. Il présente les principaux travaux qui ont œuvré dans ce sens en analysant leurs interactions et les grandes lignes de développement. L'étude des systèmes d'équations linéaires et la recherche d'un calcul géométrique intrinsèque sont les deux principales sources de constitution d'une théorie de la lnéarité. Le dépassement de la dimension 3 en géométrie au milieu du XIXème siécle, ainsi qu'un développement dialectique entre algèbre et géométrie depuis la création de la géométrie analytique ont amené la constitution d'une premiére unification des questions linéaires autour de la notion de déterminant. Ce cadre fut généralisé à la dimension infinie dénombrable lors de travaux d'analyse fonctionnelle. L'axiomatisation réahsée à la fin du XIXème siecle, mais vraiement utilisée seulement après 1920, est un processus plus large qui s'inscrit dans un développement général des mathématiques au début du XXème siecle. J'analyserai comment ce phénomène a pu naìtre et comment il a fini par s'imposer. El artículo que sigue presenta una visión general sobre la génesis de los conceptos elementales de la teoría de espacios vectoriales. Aqui se presentan los principales trabajos que se ban desarrollado en este sentido, analizando sus interacciones y las grandes líneas de desarrollo. El estudio de sistemas de ecuación linear y la investigación de un cálculo geométrico intrinseco son las dos principales fuentes de constitutión de una teoria de la linearidad. La superación de la dimension 3 en geometría, a mediados del siglo XIX, asi como el desarrollo dialectico entre el algebra y la geometrií, despues de la creación de la geometria analitica, ban llevado a la constitution de una primera unification des los aspectos líneares alrededor de la notion de determinante. Este cuadro fue generalizado a la dimensión infinita enumerable con los trabajos de analisis funcional. La axiomatisación realizada a fines del siglo XX, pero realmente utilisada después de 1920, es un proceso más largo que se inscribe en un desarrollo general de la matemática a comienzos del siglo XX. Yo analizare como ese fenómeno pudo nacer y como él a terminado por imponerse.

Convergence of matrix powers

We provide an elementary and short proof of the following important result: Let A be an n x nreal or complex matrix. Then {Am} converges if and only the following two conditions hold: (a) if A is an eigenvalue of A, then either A = 1 or A lies in the open unit disk of the complex plane; and (b) if 1 is an eigenvalue of A, then its algebraic multiplicity equals its geometric multiplicity. Unlike the standard proofs of this result, our proof requires no knowledge of Jordan canonical forms. In addition, the proof provides students with an example of the interplay between matrix techniques and vector space theory.

Theory of Prehomogeneous Vector Spaces without Regularity Condition

Theory of Prehomogeneous Vector Spaces without Regularity Condition

Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato’s lecture from Shintani’s note

The purpose of this paper is to introducea-functions andb-functions of prehomogeneous vector spaces in the original way of M. Sato and give a proof of the structure theorem of them. All the results were obtained by M. Sato when he constructed the theory of prehomogeneous vector spaces in 60’s. However he did not write a paper on his outcomes at that time. His theory was distributed through his lectures and informal seminars. Only small number of people could know it. The only publication left for us is a mimeographed note of his lecture [Sa-Sh1] written by T. Shintani in Japanese.

Fractal basin boundaries in a two-degree-of-freedom nonlinear system

The final state for nonlinear systems with multiple attractors may become unpredictable as a result of homoclinic or heteroclinic bifurcations. The fractal basin boundaries due to such bifurcations for a four-well, two-degree-of-freedom, nonlinear oscillator under sinusoidal forcing have been studied, based on a theory of homoclinic bifurcation inn-dimensional vector space developed by Palmer. Numerical simulation is used as a means of demonstrating the consequences of the system dynamics when the bifurcations occur, and it is shown that the basin boundaries in the configuration space (x, y) become fractal near the critical value of the heteroclinic bifurcations.

Comparison of the Craig-Bampton and residual flexibility methods of substructure representation

A theoretical and numerical comparison is made between the fixed-interface Craig-Bampton method and the free-interface methods of MacNeal and Rubin for component substructure representation. The static and dynamic equivalence of the methods is investigated for a restrained substructure. Vector space theory is used to derive a relation which must be satisfied for dynamic equivalence of the Craig-Bampton and Rubin substructure representations. Numerical comparisons are made using an unrestrained substructure. Synthesized system modes for each of the representations investigated are compared with a reference solution using a mode correlation constant. The effect of the inclusion of rotational degrees of freedom at the component interface is also investigated.

Basic sequences and non locally convex topological vector spaces

The paper is concerned with some application of basic sequences in the theory of non-locally convex topological vector spaces. Some results of Kalton, Peck, Gregory and Shapiro are extended.

The dimension of Euclidean subspaces of quasi-normed spaces

The purpose of this article is to extend certain results which are known to hold for convex bodies to a class of non-convex bodies occurring in the theory of topological vector spaces. In the first section after this introduction an analogue of F. John's Theorem on the distance of a finite dimensional space from Euclidean space is obtained, and the result is shown to be best possible. The Dvoretzky-Rogers Lemma on the points of contact of a symmetric convex body with the ellipsoid of maximum volume contained within it is discussed for certain non-convex bodies. In the next part the Dvoretzky Theorem on the existence of ellipsoidal sections is shown to hold with the best possible estimate for the dimension of the sections. It follows from estimates involving cotype constants that the finite dimensional subspaces of Lp (0 &lt; p &lt; 1) possess large almost Hilbertian subspaces. The final section extends the theorem of S. Szarek relating the volume of a body to the existence of ellipsoidal sections.

The Statistical Analysis for Incomplete Diallel Cross Series. Fixed Model Analysis of Variance

AbstractThis paper presents modern statistical methods used for quantitative genetics and plant breeding. It contains biometrioal analysis principles of fixed model for an identical, incomplete diallel cross series. The theory of vector space is used for estimation of parameters assumed in the model and for performance of the analysis of variance. As a result, the estimation of all groups of parameters treated in the model, is independent. Theoretical results are illustrated by numerical examples from winter wheat breeding.

Bijective methods in the theory of finite vector spaces

Bijective methods in the theory of finite vector spaces

Inversion of matrices over a commutative semiring

It is a well-known consequence of the elementary theory of vector spaces that if A and B are n-by-n matrices over a field (or even a skew field) such that AB = 1, then BA = 1. This result remains true for matrices over a commutative ring, however, it is not, in general, true for matrices over noncommutatives rings. In this paper we show that if A and B are n-by-n matrices over a commutative semiring, then the equation AB = 1 implies BA = 1. We give two proofs: one algebraic in nature, the other more combinatorial. Both proofs use a generalization of the familiar product law for determinants over a commutative semiring.

Some model theory of modules. III. On infiniteness of sets definable in modules

This is the third and last part of an investigation in several topics of first order model theory of modules which I began in Some model theory of modules. I. On total transcendence of modules (this Journal, vol. 48 (1983), pp. 570–574) and continued in Some model theory of modules. II. On stability and categoricity of flat modules (this Journal, vol. 48 (1983), pp. 970–985). Throughout I refer to these papers as “Part I” and “Part II”.Although these parts are only loosely connected, I will tacitly use the notation and preliminaries already introduced in the preceding ones. Further details are given in §0. Concerning Part II, the reader is assumed to be familiar with most of §1, a very small part of §4, and the criterion for elementary equivalence of modules over regular rings given in §3 (Lemma 20) of that part.Using those tools in this paper I consider completeness of the (elementary) theory of all modules (and of some of its extensions) (§2), and eliminability of cardinality quantifiers in the elementary theories of modules (§3).§2 is almost entirely devoted to a new short proof based on the technique developed in Part II of a theorem of Tukavkin which seems to be the first relevant result concerning the completeness of the theory of all modules (after the simple observation that any theory of infinite vector spaces is complete).