Lectures in abstract algebra.

v. 1. Basic concepts.--v. 2. Linear algebra.--v. 3. Theory of fields and Galois theory.

A decision method for elementary algebra and geometry

"This report...was written while the project was a part of Douglas Aircraft Co., inc."

Recursively enumerable sequences.

Mimeographed.

Class Notes for Math 901/902: Abstract Algebra, Instructor Tom Marley

Topics include: Free groups and presentations; Automorphism groups; Semidirect products; Classification of groups of small order; Normal series: composition, derived, and solvable series; Algebraic field extensions, splitting fields, algebraic closures; Separable algebraic extensions, the Primitive Element Theorem; Inseparability, purely inseparable extensions; Finite fields; Cyclotomic field extensions; Galois theory; Norm and trace maps of an algebraic field extension; Solvability by radicals, Galois' theorem; Transcendence degree; Rings and modules: Examples and basic properties; Exact sequences, split short exact sequences; Free modules, projective modules; Localization of (commutative) rings and modules; The prime spectrum of a ring; Nakayama's lemma; Basic category theory; The Hom functors; Tensor products, adjointness; Left/right Noetherian and Artinian modules; Composition series, the Jordan-Holder Theorem; Semisimple rings; The Artin-Wedderburn Theorem; The Density Theorem; The Jacobson radical; Artinian rings; von Neumann regular rings; Wedderburn's theorem on finite division rings; Group representations, character theory; Integral ring extensions; Burnside's paqb Theorem; Injective modules.

Determinant: Old algorithms, new insights (Extended Abstract)

In this paper we approach the problem of computing the characteristic polynomial of a matrix from the combinatorial viewpoint. We present several combinatorial characterizations of the coefficients of the characteristic polynomial, in terms of walks and closed walks of different kinds in the underlying graph. We develop algorithms based on these characterizations, and show that they tally with well-known algorithms arrived at independently from considerations in linear algebra.

Some Problems In Algebra And Topology A Study Of Fuzzy Convexity With Special Reference To Separation Properties

This thesis is a study of abstract fuzzy convexity spaces and fuzzy topology fuzzy convexity spaces No attempt seems to have been made to develop a fuzzy convexity theoryin abstract situations. The purpose of this thesis is to introduce fuzzy convexity theory in abstract situations

Quasiconformal mappings in octonionic algebra: A graphical and analytical comparison

In this article, we present quasiconformal mappings related to octonionic algebra. Based on the metric definition of quasiconformal mappings and using transformations of the type f(z)=zn, we compare the graphical and analytic results. © 2009 Pushpa Publishing House.

Improved static analysis and verification of energy consumption and other resources via abstract interpretation

Resource analysis aims at inferring the cost of executing programs for any possible input, in terms of a given resource, such as the traditional execution steps, time ormemory, and, more recently energy consumption or user defined resources (e.g., number of bits sent over a socket, number of database accesses, number of calls to particular procedures, etc.). This is performed statically, i.e., without actually running the programs. Resource usage information is useful for a variety of optimization and verification applications, as well as for guiding software design. For example, programmers can use such information to choose different algorithmic solutions to a problem; program transformation systems can use cost information to choose between alternative transformations; parallelizing compilers can use cost estimates for granularity control, which tries to balance the overheads of task creation and manipulation against the benefits of parallelization. In this thesis we have significatively improved an existing prototype implementation for resource usage analysis based on abstract interpretation, addressing a number of relevant challenges and overcoming many limitations it presented. The goal of that prototype was to show the viability of casting the resource analysis as an abstract domain, and howit could overcome important limitations of the state-of-the-art resource usage analysis tools. For this purpose, it was implemented as an abstract domain in the abstract interpretation framework of the CiaoPP system, PLAI.We have improved both the design and implementation of the prototype, for eventually allowing an evolution of the tool to the industrial application level. The abstract operations of such tool heavily depend on the setting up and finding closed-form solutions of recurrence relations representing the resource usage behavior of program components and the whole program as well. While there exist many tools, such as Computer Algebra Systems (CAS) and libraries able to find closed-form solutions for some types of recurrences, none of them alone is able to handle all the types of recurrences arising during program analysis. In addition, there are some types of recurrences that cannot be solved by any existing tool. This clearly constitutes a bottleneck for this kind of resource usage analysis. Thus, one of the major challenges we have addressed in this thesis is the design and development of a novel modular framework for solving recurrence relations, able to combine and take advantage of the results of existing solvers. Additionally, we have developed and integrated into our novel solver a technique for finding upper-bound closed-form solutions of a special class of recurrence relations that arise during the analysis of programs with accumulating parameters. Finally, we have integrated the improved resource analysis into the CiaoPP general framework for resource usage verification, and specialized the framework for verifying energy consumption specifications of embedded imperative programs in a real application, showing the usefulness and practicality of the resulting tool.---ABSTRACT---El Análisis de recursos tiene como objetivo inferir el coste de la ejecución de programas para cualquier entrada posible, en términos de algún recurso determinado, como pasos de ejecución, tiempo o memoria, y, más recientemente, el consumo de energía o recursos definidos por el usuario (por ejemplo, número de bits enviados a través de un socket, el número de accesos a una base de datos, cantidad de llamadas a determinados procedimientos, etc.). Ello se realiza estáticamente, es decir, sin necesidad de ejecutar los programas. La información sobre el uso de recursos resulta muy útil para una gran variedad de aplicaciones de optimización y verificación de programas, así como para asistir en el diseño de los mismos. Por ejemplo, los programadores pueden utilizar dicha información para elegir diferentes soluciones algorítmicas a un problema; los sistemas de transformación de programas pueden utilizar la información de coste para elegir entre transformaciones alternativas; los compiladores paralelizantes pueden utilizar las estimaciones de coste para realizar control de granularidad, el cual trata de equilibrar el coste debido a la creación y gestión de tareas, con los beneficios de la paralelización. En esta tesis hemos mejorado de manera significativa la implementación de un prototipo existente para el análisis del uso de recursos basado en interpretación abstracta, abordando diversos desafíos relevantes y superando numerosas limitaciones que éste presentaba. El objetivo de dicho prototipo era mostrar la viabilidad de definir el análisis de recursos como un dominio abstracto, y cómo se podían superar las limitaciones de otras herramientas similares que constituyen el estado del arte. Para ello, se implementó como un dominio abstracto en el marco de interpretación abstracta presente en el sistema CiaoPP, PLAI. Hemos mejorado tanto el diseño como la implementación del mencionado prototipo para posibilitar su evolución hacia una herramienta utilizable en el ámbito industrial. Las operaciones abstractas de dicha herramienta dependen en gran medida de la generación, y posterior búsqueda de soluciones en forma cerrada, de relaciones recurrentes, las cuales modelizan el comportamiento, respecto al consumo de recursos, de los componentes del programa y del programa completo. Si bien existen actualmente muchas herramientas capaces de encontrar soluciones en forma cerrada para ciertos tipos de recurrencias, tales como Sistemas de Computación Algebraicos (CAS) y librerías de programación, ninguna de dichas herramientas es capaz de tratar, por sí sola, todos los tipos de recurrencias que surgen durante el análisis de recursos. Existen incluso recurrencias que no las puede resolver ninguna herramienta actual. Esto constituye claramente un cuello de botella para este tipo de análisis del uso de recursos. Por lo tanto, uno de los principales desafíos que hemos abordado en esta tesis es el diseño y desarrollo de un novedoso marco modular para la resolución de relaciones recurrentes, combinando y aprovechando los resultados de resolutores existentes. Además de ello, hemos desarrollado e integrado en nuestro nuevo resolutor una técnica para la obtención de cotas superiores en forma cerrada de una clase característica de relaciones recurrentes que surgen durante el análisis de programas lógicos con parámetros de acumulación. Finalmente, hemos integrado el nuevo análisis de recursos con el marco general para verificación de recursos de CiaoPP, y hemos instanciado dicho marco para la verificación de especificaciones sobre el consumo de energía de programas imperativas embarcados, mostrando la viabilidad y utilidad de la herramienta resultante en una aplicación real.

Abstract specification in Object-Z and CSP

A number of integrations of the state-based specification language Object-Z and the process algebra CSP have been proposed in recent years. In developing such integrations, a number of semantic decisions have to be made. In particular, what happens when an operation's precondition is not satisfied? Is the operation blocked, i.e., prevented from occurring, or can it occur with an undefined result? Also, are outputs from operations angelic, satisfying the environment's constraints on them, or are they demonic and not influenced by the environment at all? In this paper we discuss the differences between the models, and show that by adopting a blocking model of preconditions together with an angelic model of outputs one can specify systems at higher levels of abstraction.

Components of learning and assessment in linear algebra

Linear algebra provides theory and technology that are the cornerstones of a range of cutting edge mathematical applications, from designing computer games to complex industrial problems, as well as more traditional applications in statistics and mathematical modelling. Once past introductions to matrices and vectors, the challenges of balancing theory, applications and computational work across mathematical and statistical topics and problems are considerable, particularly given the diversity of abilities and interests in typical cohorts. This paper considers two such cohorts in a second level linear algebra course in different years. The course objectives and materials were almost the same, but some changes were made in the assessment package. In addition to considering effects of these changes, the links with achievement in first year courses are analysed, together with achievement in a following computational mathematics course. Some results that may initially appear surprising provide insight into the components of student learning in linear algebra.