Principles Of Algebra Research Articles

Radio Astronomical Polarimetry and Point‐Source Calibration

A mathematical framework is presented for use in the experimental determination of the polarimetric response of observatory instrumentation. Elementary principles of linear algebra are applied to model the full matrix description of the polarization measurement equation by least-squares estimation of non-linear, scalar parameters. The formalism is applied to calibrate the center element of the Parkes Multibeam receiver using observations of the millisecond pulsar, PSR J0437-4715, and the radio galaxy, 3C 218 (Hydra A).

Preserving commutativity

Every commutativity preserving linear map on the algebra of all n× n matrices over an algebraically closed field F with characteristic 0 is either a Jordan automorphism multiplied by a nonzero constant and perturbed by a scalar type operator, or its image is commutative. The assumption of preserving commutativity can be reformulated as preserving zero Lie products. So, this theorem is an extension of the well-known result on the structure of Lie homomorphisms of matrix algebras. We first prove the result for the special case in which F is the complex field and then apply the transfer principle in Model Theoretic Algebra to extend it to the general case.

Some general techniques on linear preserver problems

Several general techniques on linear preserver problems are described. The first one is based on a transfer principle in Model Theoretic Algebra that allows one to extend linear preserver results on complex matrices to matrices over other algebraically closed fields of characteristic 0. The second one concerns the use of some simple geometric technique to reduce linear preserver problems to standard types so that known results can be applied. The third one is about solving linear preserver problems on more general (operator) algebras by reducing the problems to idempotent preservers. Numerous examples will be given to demonstrate the proposed techniques.

Algebraic multi-grid method in two-dimension electrically large problems

In this paper the algebraic multi-grid principle is applied to the multilevel moment method, which makes the new multilevel method easier to implement and more adaptive to structure. Moreover, the error spectrum is analyzed, and the reason why conjugate gradient iteration is not a good relaxation scheme for multi-grid algorithm is explored. The numerical results show that our algebraic block Gauss Seidel multi-grid algorithm is very effective.

An algebraic principle for blind separation of white non-Gaussian sources

An algebraic principle for blind source separation is presented in this paper. This separation principle identifies a (smaller) set of equations whose solutions can blindly extract non-Gaussian signals. The concept of “ Mth-order uncorrelatedness” is introduced and it is proven that for Mth-order uncorrelated source signals, signals with nonzero kth-order cumulant (2< k⩽ M) can always be extracted by setting a small set of kth-order cross-cumulants of output signals to zero. The set of kth-order cross-cumulants specified here is a sub-set of those used by other existing methods. The relationship between the algebraic principle and several existing algorithms is presented. The contributions of this principle are the reduction of the number of cross-cumulants used and the flexibility it affords in designing algorithms for blind source separation.

A Transient Heat and Mass Transfer Model of Residential Attics Used to Simulate Radiant Barrier Retrofits, Part II: Validation and Simulations

A computer program was developed and used to implement the model described on Part I of this paper. The program used an iterative process to predict temperatures and heat fluxes using linear algebra principles. The results from the program were compared to experimental data collected during a three-year period. The model simulated different conditions such as variations in attic ventilation, variations in attic ceiling insulation, and different radiant barrier orientations for summer and winter seasons. It was observed that the model predicted with an error of less than ten percent for most cases. This paper presents model results for nonradiant barrier cases as well as cases for radiant barriers installed horizontally on top of the attic floor (HRB) and for radiant barriers stapled to the attic rafters (TRB). Savings produced by radiant barriers and sensitivity analyses are also presented. The model results supported the experimental trend that emissivity was the single most significant parameter that affected the performance of radiant barriers.

An Algebraic Principle for the Stability of Difference Operators

This paper gives necessary and sufficient algebraic criteria for the stability of the difference systemx(t)=Ax(t−r1)+Bx(t−r2).

Economic aspects of animal breeding

Basic concepts. Basic concepts in quantitative genetics. Basic concepts in economics. Principles of matrix algebra and selection index. Introduction to systems analysis. Economic evaluation of genetic differences. The main elements of returns and costs. Evaluation of genetic differences from profit equations. Evaluation of genetic differences by alternate methods. Long-term considerations. Construction, uses and problems of multitrait selection indices. Selection indices for non-linear profit functions. Comparison of different selection indices. Economic evaluation of breeding programs. Economic evaluation of breeding programs, theory. Comparison of alternative breeding programs. Evaluation of existing breeding programs. Crossbreeding and heterosis. Economic evaluation of heterosis. Choice of the most profitable parental combination. Planned matings together with line breeding. Glossary of symbols. References Index.

Ray vector fields, prentice's equation, and Fourier representation of spherocylindrical lenses.

To illustrate how the concept of ray vector fields can be used to demonstrate the elementary properties of thin lenses. Principles of matrix algebra are used. Using simple geometry we give a generalization of Prentice's rule to include thick astigmatic lenses. Starting from the ray vector fields we show the connection between various representations of spherocylindrical lenses (Fourier representation, expansion in terms of the cross-cylinders). We suggest two clinical areas to which these concepts can be applied.

Improving manufacturing competitiveness through the application of Human Performance Engineering&amp;trade;

This paper describes the advantages of the application of human performance engineering to problems of employee, manager and organizational development. The aim of this paper is to introduce a new method (called Human Performance Engineering™ or HPE) of organizing, developing and challenging the human resources of an organisation, utilizing geometric or algebraic principles as typically found in engineering analysis together with the functional elements of performance. The paper will concentrate on the following areas: an economic evaluation of costs associated with productivity within a hypothetical but typical, profit centred organisation; a mechanical and economic visualization of the relationship within that typical organisation, reflecting human resources change and the corresponding result; the problems related to human resources from top–management to the production floor; a new way of defining jobs, establishing accountabilities, training and organisational development using the practical concepts of Human Performance engineering, including the perception of the typical employee within a typical job; static vs. dynamic job descriptions; economic advantages associated with an organisation to be developed within the tenets of Human Performance engineering. These points will be demonstrated via an overview of the operation and difficulties of Tech–O–Switch. This is a manufacturing company located in a region of the United States where a number of firms compete for capable skilled, semi–skilled and professional people. Conclusions are drawn from this example based upon actual experiences.

A training algorithm for binary feedforward neural networks

The authors present a new training algorithm to be used on a four-layer perceptron-type feedforward neural network for the generation of binary-to-binary mappings. This algorithm is called the Boolean-like training algorithm (BLTA) and is derived from original principles of Boolean algebra followed by selected extensions. The algorithm can be implemented on analog hardware, using a four-layer binary feedforward neural network (BFNN). The BLTA does not constitute a traditional circuit building technique. Indeed, the rules which govern the BLTA allow for generalization of data in the face of incompletely specified Boolean functions. When compared with techniques which employ descent methods, training times are greatly reduced in the case of the BLTA. Also, when the BFNN is used in conjunction with A/D converters, the applicability of the present algorithm can be extended to accept real-valued inputs.

The parallel numerical method of mechanical theorem proving

In this paper, we present results of the work which allow us to prove geometry theorems by the parallel numerical method based on the multi-instance numerical verification of algebraic identity. The algebraic principle of the parallel numerical method is discussed and illustrated intuitively; the advantages of our method are given. It is acceptable on the complexity of both memory and time. It can be used to prove non-trivial geometric theorems by microcomputer, even by hand. We give some examples proved by parallel numerical method, including certain new unexpected results.

Estimation of the Rate Constants in a Data-Sparse Environment: Comparison of a Mathematical Method and Least Squares Analysis

A new method is presented for estimating the rate constants for the one-compartment open model with first-order absorption in a data-sparse environment. It is based on the principles of matrix algebra and system theory and requires only three to four plasma samples drawn at equal time intervals (a minimum of at least one sample in the absorption phase and one sample in the elimination phase). The utility of the technique is illustrated by comparing the parameter estimates from the matrix method with the estimates from a nonlinear computer parameter estimation program. In the preliminary evaluation of the method with both perfect data and data with randomly distributed error, the parameter estimates from the matrix method proved to be as good as the computer estimation for data-sparse systems. In a more realistic comparison with published patient data, the matrix method resulted in 0 to 456% better estimates of Kel, and Ka than the computer estimation. Since the matrix method is mathematically simple and requires only three to four blood samples, it should prove very useful in data-sparse systems (e.g., clinical or small animal situations) where a minimum amount of blood samples can be drawn.

Teaching Modeling and Axiomatization with Boolean Algebra

Traditionally, Boolean algebra is largely taught in connection with computer programming courses, logic, or set theory. Since Boolean algebra arose from George Boole's application of algebraic principles to the study of logic in 1854, this approach would seem natural.

TRIAXIAL NUMBERS WITH APPLICATIONS TO GEODESY

AbstractThe meaning and algebraic principles of triaxial numbers are presented. The mathematics is found to be suited to applications to three-dimensional geodesy and to special trigonometric situations such as the Celestial Triangle.

A new criterion for the pairing of control and manipulated variables

AbstractA criterion is developed for the selection of the best pairing of the control and manipulated variables of a multiloop control system. This criterion is based on the difficulty caused by the interaction terms (the off‐diagonal elements) in finding the inverse of the steady state gain matrix. From an analysis based on the proposition that the most desired or best pairing is that one for which the system most closely resembles a set of independent single‐loop systems, a quantitative measure of the best pairing is obtained. Although the development of the pairing criterion is based on purely algebraic principles, the validity of the pairing criterion is evident from analogous developments obtained within the control framework. Furthermore, it is shown that the pairing criterion may be used to determine the stability of a multiloop control system, thus enhancing the value of the criterion presented.

A Systematic Approach to the Determination of Possible Reaction Mechanisms of Oxygen Reduction on Platinum in Alkaline Medium

The reaction mechanisms of oxygen reduction of platinum in alkaline medium are determined once a set of reacting species and elementary reactions is specified. The technique utilized to generate all the possible reaction mechanisms is based upon the principles of combinatorial analysis and linear algebra along with application of the principle of microscopic reversibility. The development of this method by Happel and Sellers has enabled them to study complex mechanisms for some important industrial reactions in heterogeneous catalysis. A preliminary evaluation analysis of the myriad reaction mechanisms for oxygen reduction is accomplished by invoking the concept of a stoichiometric number and a corresponding rate‐determining elementary reaction. The number of mechanisms which are consistent with a given experimentally derived stoichiometric number is considerably less. The resulting set of mechanisms forms an important basis for further investigations.

Assessing Discrimination: A Boolean Approach

Since the passage of the Civil Rights Act of 1964, various forms of employment discrimination have been prohibited by law. The courts have enforced these laws vigorously, often requiring plaintiffs and defendants to present quantitative analyses of applicant and employee records as evidence. In fact, statistical analyses have become a regular feature of employment discrimination cases. Consequently, there has been an explosion of interest in the production of statistical assessments of discrimination, as more and more social scientists and statisticians have become involved as consultants and expert witnesses on both sides of discrimination cases. This paper discusses current practices in the production of statistical assessments of discrimination and suggests nonstatistical techniques that can be used to aid the interpretation of the results of statistical analyses. Statistical assessments (1) may contradict accumulated social scientific knowledge about decision making, in general, and discrimination, in particular, and (2) may also contradict the spirit of civil rights legislation concerning discrimination. These shortcomings motivate the introduction of a method of analysis that incorporates principles of Boolean algebra, an approach that allows holistic comparison of categories of similarly situated individuals.

The Square Root of a 2 × 2 Matrix

(1980). The Square Root of a 2 × 2 Matrix. Mathematics Magazine: Vol. 53, No. 4, pp. 222-224.

Production planning and scheduling for multi-purpose batch chemical plants

A computer program for solving the general production planning problem in the case of multi-purpose batch chemical plants is presented. The procedure starts with an evolutionary enumeration of alternative production strategies. First, alternative ways of producing one batch of each of the various products are sought. Then, alternative production lines are constructed for each product by assembling types of batches of this product sequentially on the time axis. Finally, production lines of the same, or different products are set in parallel to form alternative campaign candidates. After the alternative production strategies (campaign candidates) have been generated, they are screened by identifying and rejecting non-dominant (i.e. inefficient) strategies. The screenign procedure is based on some principles of linear algebra and uses a duality between points and hyperplanes in a multi-dimensional space. Finally an optimal production plan is obtained by solving an optimization problem in which production time is allocated to the various production strategies.