Linear Vector Space Theory Research Articles

Alignment of time series with use of the generalized Legendre's transformation and methods of idempotenty mathematics

Alignment of time series [time-series smoothing] identification of the main tendency of development (временнго a trend) by "cleaning" of a time series of the accidental deviations distorting this tendency. At a research of time series of economy (bioinformation science) apply for detection of patterns [1-3]. 
 In this work it is offered to use for this purpose Legendre's transformation well-known in physics and mathematics. Its direct application to poorly regular objects is difficult therefore in work its idempotent analog is defined previously and on its basis the concept of the TRACK for a time series is defined. 
 In recent years within the international center "Cuofus Li" the new field of mathe-matics idempotent or "tropical" mathematics gained intensive development that is reflected in works of the academician V.P. Maslov and his pupils: G.L. Litvinov, A.N. Sobolevsky, etc. 
 The purpose of this work to be beyond duality of the theory of linear vector spaces, using similar concepts of duality of conformally flat Riemannian geometry and of idempotent algebra.
 By analogy with the polar transformation of a conformally flat Riemannian metrics entered in E.D. Rodionov and V.V. Slavsky's works the abstract idempotent analog of transformation of Legendre is under construction. In the MATLAB system the program complex for calculation the TRACK of a time series is created. It is in-vestigated its opportunities for digital processing of time series.

On the convergence of MoM and mode matching solutions for infinite array and waveguide problems

It is shown that the method of moment (MoM) fails to analyze an infinite array of strip discontinuities if the number of basis functions and the number of expansion modes are set equal, however large they individually may be. Using spectral analysis and linear vector space theory it is found that for a given number of basis functions, there exists a minimum number of expansion modes required for a reasonable accuracy. Subsequently, for a given number of expansion modes, there is an upper limit for the number of basis functions. The mode matching (MM) analysis is viewed as a special case of the MoM analysis and the accuracy of the analysis is discussed with respect to the number of modes in the waveguides.

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.

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>

Perpendicular Projections and Elementary Statistics

Abstract Beginning statistics at the undergraduate level can be taught by using a few geometric principles of linear vector space theory. Even formulas for simple sample means and variances can be derived with these principles by assuming a univariate linear statistical model. The least squares estimator of the sample mean is found by a perpendicular projection. The analogy of the bivariate model to the univariate model is indicated, and an analogous perpendicular projection solution for it is shown. Vector geometric diagrams illustrate the basic concepts. Once the basic technique is understood, the appropriate application or perpendicular projections can be used to illustrate the problems of multicollinearity and tests of hypotheses in regression models. The translation of the geometric concepts into concrete algebraic equations is shown. The emphasis is on geometric thinking as a means of visualizing and thereby improving an understanding of methods of data analysis.

Moment Methods in Electromagnetics for Undergradcuates

Moment methods have recently emerged as one of the most important tools for the solution of electromagnetic fields problems. This approach is conceptually simple and easily usable by the practicing engineer. However, the formal presentation of moment method is founded in the theory of linear vector spaces which makes it beyond the scope of most undergraduate electrical engineering curricula This paper documents the experience in undergraduate teaching of moment methods in electromagnetics through the solution of integral equations with a "point matching" or collocation approach at the University of Mississippi. The main purpose of the paper is to provide a guide for teaching moment methods for the educator who is involved with undergraduate fields courses, particularly, the non-field theorist. Static and time-harmonic examples are presented with enough detail for in-course use.

The Linear Vector Space Theory of Unfolding

Abstract : The problem of unfolding a spectrum is treated by the methods of linear vector space theory. For a system with a finite number of measurements consisting of the outputs from detectors with known responses, some part of the unknown spectrum cannot be inferred. One may approximate the unknown spectrum with a finite expansion in terms of some arbitrary set of functions w sub n (E). The relationships between the various solutions obtained by using different sets of w sub n (E) functions are explored, and it is shown that the solution obtained by allowing the w sub n (E) to be the response functions themselves displays some unique and desirable properties. Among these properties is an insensitivity to small errors in the measured outputs from the detectors and in the analytic response functions used to describe the true detector responses. (Author)

Experimental Study of Moment Method for Undergraduates in Electromagnetic

The experimental study and Moment Method with is programmed in Mathematica are presented here to facilitate introductory instruction on numerical methods for undergraduate electrical education students. The Method of Moments (MOM) is introduced at the beginning level to prepare students for subsequent advanced topics in complex matrix method and linear vector space theory. The study is carried out with parallel plate capacitor. The different capacitor (dielectric) materials are tried and coincidental program of MOM are studied. Ill. 4, bibl. 9, tabl. 1 (in English; abstracts in English and Lithuanian).http://dx.doi.org/10.5755/j01.eee.109.3.184

Study of symmetrical and related components through the theory of linear vector spaces

The paper proposes a study of symmetrical and related components, based on the theory of linear vector spaces. Using the concept of equivalence, the transformation matrixes of Clarke, Kimbark, Concordia, Boyajian and Koga are shown to be column equivalent to Fortescue&apos;s symmetrical-component transformation matrix. With a constraint on power, criteria are presented for the choice of bases for voltage and current vector spaces. In particular, it is shown that, for power invariance, either the same orthonormal (self-reciprocal) basis must be chosen for both voltage and current vector spaces, or the basis of one must be chosen to be reciprocal to that of the other. The original α, β, 0 components of Clarke are modified to achieve power invariance. For machine analysis, it is shown that invariant transformations lead to reciprocal mutual inductances between the equivalent circuits. The relative merits of the various components are discussed.