Dissertations / Theses on the topic 'Diagram algebra'

This thesis concerns the representation theory of diagram algebras and related problems. In particular, we consider subalgebras and generalisations of the partition algebra. We study the d-tonal partition algebra and the planar d-tonal partition algebra. Regarding the d-tonal partition algebra, a complete description of the J -classes of the underlying monoid of this algebra is obtained. Furthermore, the structure of the poset of J -classes of the d-tonal partition monoid is also studied and numerous combinatorial results are presented. We observe a connection between canonical elements of the d-tonal partition monoids and some combinatorial objects which describe certain types of hydrocarbons, by using the alcove system of some reflection groups. We show that the planar d-tonal partition algebra is quasi-hereditary and generically semisimple. The standard modules of the planar d-tonal partition algebra are explicitly constructed, and the restriction rules for the standard modules are also given. The planar 2-tonal partition algebra is closely related to the two coloured Fuss-Catalan algebra. We use this relation to transfer information from one side to the other. For example, we obtain a presentation of the 2-tonal partition algebra by generators and relations. Furthermore, we present a necessary and sufficient condition for semisimplicity of the two colour Fuss-Catalan algebra, under certain known restrictions.

There has been much interest in algebras which have a basis consisting of diagrams, which are multiplied in some natural diagrammatic way. Examples of these so-called diagram algebras include the partition, Brauer and Temperley-Lieb algebras. These three examples all have the property that the product of two diagram basis elements is always a scalar multiple of another basis element. Motivated by this observation, we find that these algebras are examples of twisted semigroup algebras. Such algebras are an obvious extension of twisted group algebras, which arise naturally in various contexts; examples include the complex numbers and the quaternions, considered as algebras over the real numbers. The concept of a cellular algebra was introduced in a famous paper of Graham and Lehrer; an algebra is called cellular if it has a basis of a certain form, in which case the general theory of cellular algebras allows us to easily derive information about the semisimplicity of the algebra and about its representation theory, even in the non-semisimple case. Many diagram algebras (including the above three examples) are known to be cellular. The aim of this thesis is to deduce the cellularity of these examples (and others) by proving a general result about the cellularity of twisted semigroup algebras. This will extend a recent result of East. In Chapters 2 and 3 we discuss semigroup theory and twisted semigroup algebras, and realise the above three examples as twisted semigroup algebras. Chapters 4 to 7 detail and extend slightly the theory of cellular algebras. In Chapter 8 we state and prove the main theorem, which shows that certain twisted semigroup algebras are cellular. Under the assumptions of the main theorem, we explore the cell representations of twisted semigroup algebras in Chapter 9. Finally in Chapter 10, we apply the theorem to various examples, including the three diagram algebras mentioned above.

Fix R = k[[x]]. Let Q_n be the category whose objects are ((M₁,...,M_n),(f₁,...,f_n-1)) where each M_i is a free R-module and f_i:M_i⟶M_i+1 for each i=1,...,n-1, and in which the morphisms are the obvious ones. Let β_n be the full subcategory of Ω_n in which each map f_i is a monomorphism whose cokernel is a torsion module. It is shown that there is a full dense functor Ω_n⟶β_n. If X is an object of β_n, we say that X diagonalizes if X is isomorphic to a direct sum of objects ((M₁,...,M_n),(f₁,...,f_n-1)) in which each M_i is of rank one. We establish an algorithm which diagonalizes any diagonalizable object X of β_n, and which fails only in case X is not diagonalizable. Let Λ be an artin algebra of finite type. We prove that for a fixed C in mod(Λ) there are only finitely many modules A in mod(Λ) (up to isomorphism) for which a short exact sequence of the form 0⟶A⟶B⟶C⟶0 is indecomposable.
Ph. D.
incomplete_metadata

There has been much interest in algebras which have a basis consisting of diagrams, which are multiplied in some natural diagrammatic way. Examples of these so-called diagram algebras include the partition, Brauer and Temperley-Lieb algebras. These three examples all have the property that the product of two diagram basis elements is always a scalar multiple of another basis element. Motivated by this observation, we find that these algebras are examples of twisted semigroup algebras. Such algebras are an obvious extension of twisted group algebras, which arise naturally in various contexts; examples include the complex numbers and the quaternions, considered as algebras over the real numbers. The concept of a cellular algebra was introduced in a famous paper of Graham and Lehrer; an algebra is called cellular if it has a basis of a certain form, in which case the general theory of cellular algebras allows us to easily derive information about the semisimplicity of the algebra and about its representation theory, even in the non-semisimple case. Many diagram algebras (including the above three examples) are known to be cellular. The aim of this thesis is to deduce the cellularity of these examples (and others) by proving a general result about the cellularity of twisted semigroup algebras. This will extend a recent result of East. In Chapters 2 and 3 we discuss semigroup theory and twisted semigroup algebras, and realise the above three examples as twisted semigroup algebras. Chapters 4 to 7 detail and extend slightly the theory of cellular algebras. In Chapter 8 we state and prove the main theorem, which shows that certain twisted semigroup algebras are cellular. Under the assumptions of the main theorem, we explore the cell representations of twisted semigroup algebras in Chapter 9. Finally in Chapter 10, we apply the theorem to various examples, including the three diagram algebras mentioned above.

Abstract In this project, we searched for new constructions and symmetric presentations of important groups, nonabelian simple groups, their automorphism groups, or groups that have these as their factor groups. My target nonabelian simple groups included sporadic groups, linear groups, and alternating groups. In addition, we discovered finite groups as homomorphic images of progenitors and proved some of their isomorphism type and original symmetric presentations. In this thesis we found original symmeric presentations of M12, J1 and the simplectic groups S(4,4) and S(3,4) on various con- trol groups. Using the technique of double coset enumeration we constucted J2 as a homomorphic image of the permutation progenitor 2∗10 : (10 × 2). From our mono- mial progenitor 11∗4 : (2 : 4) we found a homomorphic image of M11. In the following chapters we will discuss how we went about obtaining homomorphic images, some con- structions of the Cayley Diagrams, and how we solved some extension problems.

Thesis (MSc)--Stellenbosch University, 2011.
ENGLISH ABSTRACT: Some of the diagram lemmas of Homological Algebra, classically known for abelian categories, are not characteristic of the abelian context; this naturally leads to investigations of those non-abelian categories in which these diagram lemmas may hold. In this Thesis we attempt to bring together two different directions of such investigations; in particular, we unify the five lemma from the context of homological categories due to F. Borceux and D. Bourn, and the five lemma from the context of modular semi-exact categories in the sense of M. Grandis.
AFRIKAANSE OPSOMMING: Verskeie diagram lemmata van Homologiese Algebra is aanvanklik ontwikkel in die konteks van abelse kategorieë, maar geld meer algemeen as dit behoorlik geformuleer word. Dit lei op ’n natuurlike wyse na ’n ondersoek van ander kategorieë waar hierdie lemmas ook geld. In hierdie tesis bring ons twee moontlike rigtings van ondersoek saam. Dit maak dit vir ons moontlik om die vyf-lemma in die konteks van homologiese kategoieë, deur F. Borceux en D. Bourn, en vyflemma in die konteks van semi-eksakte kategorieë, in die sin van M. Grandis, te verenig.

In coding theory we wish to find as many codewords as possible, while simultaneously maintaining high distance between codewords to ease the detection and correction of errors. For linear codes, this translates to finding high-dimensional subspaces of a given metric space, where the induced distance between vectors stays above a specified minimum. In this work I describe the recent advances of this problem in the contexts of lexicodes and Ferrers diagram rank-metric codes. In the first chapter, we study lexicodes. For a ring R, we describe a lexicographic ordering of the left R-module Rn. With this ordering we set up a greedy algorithm which sequentially selects vectors for which all linear combinations satisfy a given property. The resulting output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. We describe a generalization of the algorithm to finite principal ideal rings. In the second chapter, we investigate Ferrers diagram rank-metric codes, which play a role in the construction of subspace codes. A well-known upper bound for dimension of these codes is conjectured to be sharp. We describe several solved cases of the conjecture, and further contribute new ones. In addition, probabilities for maximal Ferrers diagram codes and MRD codes are investigated in a new light. It is shown that for growing field size, the limiting probability depends highly on the Ferrers diagram.

In this thesis we study the modular representation theory of diagram algebras, in particular the Brauer and partition algebras, along with a brief consideration of the Temperley-Lieb algebra. The representation theory of these algebras in characteristic zero is well understood, and we show that it can be described through the action of a reflection group on the set of simple modules (a result previously known for the Temperley-Lieb and Brauer algebras). By considering the action of the corresponding affine reflection group, we give a characterisation of the (limiting) blocks of the Brauer and partition algebras in positive characteristic. In the case of the Brauer algebra, we then show that simple reflections give rise to non-zero decomposition numbers. We then restrict our attention to a particular family of Brauer and partition algebras, and use the block result to determine the entire decomposition matrix of the algebras therein.

In this thesis lessons learned from the use of computer algebra systems and machine assisted theorem provers are developed in order to give an insight into both the problems and their solutions. Many algorithms in computational algebra and automated deduction (for example Grobner basis computations and Knuth-Bendix completion) tend to produce redundant facts and can contain more than one proof of any particular fact. This thesis introduces proof diagrams in order to compare and contrast the proofs of facts which such procedures generate. Proof diagrams make it possible to analyse the effect of heuristics which can be used to guide implementations of such algorithms. An extended version of an inference system for Knuth-Bendix completion is introduced. It is possible to see that this extension characterises the applicability of critical pair criteria, which are heuristics used in completion. We investigate a number of executions of a completion procedure by analysing the associated proof diagrams. This leads to a better understanding of the heuristics used to control these examples. Derived rales of inference are also investigated in this thesis. This is done in the formalism of proof diagrams. Rewrite rules for proof diagrams are defined: this is motivated by the notion of a transformation tactic in the Nuprl proof development system. A method to automatically extract 'useful' derived inference rales is also discussed. 'Off the shelf' theorem provers, such as the Larch Prover and Otter, are compared to specialised programs from computational group theory. This analysis makes it possible to see where methods from automated deduction can improve on the tools which group theorists currently use. Problems which can be attacked with theorem provers but not with currently used specialised programs are also indicated. Tietze transformations, from group theory, are discussed. This makes it possible to link ideas used in Knuth-Bendix completion programs and group presentation simplification programs. Tietze transformations provide heuristics for more efficient and effective implementations of these programs.

A semigroup is simply a set with an associative binary operation; computational semigroup theory is the branch of mathematics concerned with developing techniques for computing with semigroups, as well as investigating semigroups with the help of computers. This thesis explores both sides of computational semigroup theory, across several topics, especially in the finite case. The central focus of this thesis is computing and describing maximal subsemigroups of finite semigroups. A maximal subsemigroup of a semigroup is a proper subsemigroup that is contained in no other proper subsemigroup. We present novel and useful algorithms for computing the maximal subsemigroups of an arbitrary finite semigroup, building on the paper of Graham, Graham, and Rhodes from 1968. In certain cases, the algorithms reduce to computing maximal subgroups of finite groups, and analysing graphs that capture information about the regular I-classes of a semigroup. We use the framework underpinning these algorithms to describe the maximal subsemigroups of many families of finite transformation and diagram monoids. This reproduces and greatly extends a large amount of existing work in the literature, and allows us to easily see the common features between these maximal subsemigroups. This thesis is also concerned with direct products of semigroups, and with a special class of semigroups known as Rees 0-matrix semigroups. We extend known results concerning the generating sets of direct products of semigroups; in doing so, we propose techniques for computing relatively small generating sets for certain kinds of direct products. Additionally, we characterise several features of Rees 0-matrix semigroups in terms of their underlying semigroups and matrices, such as their Green's relations and generating sets, and whether they are inverse. In doing so, we suggest new methods for computing Rees 0-matrix semigroups.

In this thesis we present two important closely related examples of Bratteli diagrams: the Pascal triangle and the Eulerian Bratteli diagram. The former is well-known and related to binomial coefficients. The latter, which is the main object of the thesis, is related to the Eulerian numbers. Bratteli diagrams were introduced in 1972 by Ola Bratteli in his study of approximately finite dimensional (AF) C*-algebras. In 1976, George Arthur Elliott associated to an AF C*-algebra or to a corresponding Bratteli diagram an ordered group, he called dimension group. In the first part of the thesis we study the space of infinite paths of the Eulerian diagram, and we realize it as a projective limit of finite permutation groups. In the second part, we study the state space of the dimension group associated to the Eulerian Bratteli diagram. It is a compact convex set and we describe its extremal points. Finally, we use this description to give a necessary and sufficient condition for an element of this dimension group to be positive.

We study the relations among unitarity cuts of a Feynman integral computed via diagrammatic cutting rules, the discontinuity across the corresponding branch cut, and the coproduct of the integral. For single unitarity cuts, these relations are familiar, and we show that they can be generalized to cuts in internal masses and sequences of cuts in different channels and/or internal masses. We develop techniques for computing the cuts of Feynman integrals in real kinematics. Using concrete one- and two-loop scalar integral examples we demonstrate that it is possible to reconstruct a Feynman integral from either single or double unitarity cuts. We then formulate a new set of complex kinematics cutting rules generalising the ones defined in real kinematics, which allows us to define and compute cuts of general one-loop graphs, with any number of cut propagators. With these rules, which are consistent with the complex kinematic cuts used in the framework of generalised unitarity, we can describe more of the analytic structure of Feynman diagrams. We use them to compute new results for maximal cuts of box diagrams with different mass configurations as well as the maximal cut of the massless pentagon. Finally, we construct a purely graphical coproduct of one-loop scalar Feynman diagrams. In this construction, the only ingredients are the diagram under consideration, the diagrams obtained by contracting some of its propagators, and the diagram itself with some of its propagators cut. Using our new definition of cut, we map the graphical coproduct to the coproduct acting on the functions Feynman diagrams and their cuts evaluate to. We finish by examining the consequences of the graphical coproduct in the study of discontinuities and differential equations of Feynman integrals.

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE SUPORTE À PÓS-GRADUAÇÃO DE INSTS. DE ENSINO
PROGRAMA DE EXCELENCIA ACADEMICA
Iniciamos o trabalho com uma revisão da classificação de álgebras de Lie semi-simples sobre corposo algebraicamente fechados de caracteristica zero a traves dos Diagramas de Dyinkin. Posteriormente estudamos sigma - sistemas normais e classificamos eles a traves de diagramas de Satake. Finalmente estudamos a estrutura das formas reais de álgebras de Lie semi-simples complexas, explicitando a conexão com os diagramas de Satake e fornecenendo assim uma classificação das mesmas.
We begin the work with a review of the classification of semisimple Lie algebras over an algebraically field of characteristic zero through the Dyinkin Diagrams. Subsequently we study sigma - normal systems and classify them through Satake diagrams. Finally we study the structure of the real forms of complex semi-simple Lie algebras, explaining the connection with the Satake diagrams and thus providing a classification of them.

Cette thèse se divise en deux grandes parties, la première traite la combinatoire associée à l’ordre normal des opérateurs non-commutatifs et la seconde aborde des distributions symétriques du nombre de croisements et du nombre d’emboîtements, respectivement k-croisements et k-emboîtements, dans des structures combinatoires (partitions, permutations, permutations colorées, …). La première partie étudie l’ordre normal des opérateurs en termes de placements de tours. Nous étudions la forme de l’ordre normal en connectant deux opérateurs non-commutatifs D et U, et des polynômes orthogonaux spéciaux, et établissons des bijonctions entre les coefficients de (D+U)n et le nombre de placements de tours sur un diagramme de Ferrers. Nous donnons également des preuves combinatoires à des conjectures quantiques posées par des physiciens. Dans la seconde partie, nous définissons des statistiques, comme emboîtements et k-emboîtements, sur l’ensemble des permutations du groupe de Coxeter de type B. Nous donnons également des extensions au type B des résultats sur les croisements et les emboîtements, respectivement k-croisements et k-emboîtements dans les permutations de type A, en termes de distributions symétriques. De plus, nous étudions le lien entre les opérateurs non-commutatifs et ces statistiques. D’autres extensions de la distribution de ces statistiques sur les ensembles de partitions colorées et de permutations colorées de types A et B sont ainsi établies
This thesis is divided into two parts, the first deals with the combinatorics associated to the normal ordering form of noncommutative operators and the second addresses the symmetric distributions of the crossing numbers and nesting numbers, respectively k-crossings and k-nestings, in combinatorial structures (partitions, permutations, colored permutations, …). The first part studies the normal order of operators in terms of rook placements. We study the normal ordering form connecting two noncommutative operators D and U, and some special orthogonal polynomials, and establish bijonctions between coefficients of (D+U)n and rook placements in Ferrers diagrams. We also give combinatorial proofs and alternatives to some quantum conjectures posed by physicists. In the second part, we define the notions of statistics, nestings and k-nestings, on the sets of permutations of the Coxeter group of type B. We also give extensions to type B of the results of the crossings and nestings, respectivelu k-crossings and K-nestings in the set of permutations of type A, in terms of symmetric distributions. Likewise, we study the link between non-commutative operators and these statistics. Other extensions of the distribution of these statistics on the sets of colored partitions and colored permutations of type A and B are established

Cette thèse est composée de trois parties principales : les calculs des courbes médiatrices de deux courbes ou d’un point et d’une courbe dans le plan, des surfaces médiatrices de deux surfaces dans R3, et du diagramme de Voronoï d’une famille finie de demi-droites parallèles de même orientation. Ces trois sujets sont étroitement liés et trouvent des applications dans le domaine de la CAO/CGAO et de la géométrie algorithmique. Dans ces trois sujets, nous allons présenter des méthodes algorithmiques pour obtenir une certaine représentation de l’objet qui nous intéresse : la courbe médiatrice, la surface médiatrice ou le diagramme de Voronoï. En utilisant la règle de Cramer généralisée et certaines méthodes d’élimination, nous présentons une nouvelle approche pour déterminer une paramétrisation algébrique exacte (rationnelle ou non rationnelle) de la courbe médiatrice de deux courbes planes rationnelles. L’approche est, ensuite, généralisée pour déterminer une paramétrisation algébrique exacte (rationnelle ou non rationnelle) de la surface médiatrice de deux surfaces rationnelles de petit degré. La méthode est appliquée pour obtenir les paramétrisations de la médiatrice de deux courbes planes rationnelles, dans lesquelles une des courbes est un cercle ou une droite. D’autre part, nous montrons, aussi, comment il est facile d’obtenir les paramétrisations de la médiatrice de paires de surfaces suivantes : plan-quadrique, plan-tore, cylindre circulaire-quadrique non développable, cylindre circulaire-tore, cylindre-cylindre, cylindre-cône et cônecône. Les paramétrisations obtenues sont rationnelles dans la plupart des cas. Dans le reste des cas, les paramétrisations contiennent de racines carrées qui est bien adopté pour determiner une bonne approximation de la médiatrice. Nous présentons aussi une différente approche traitant du problème de la courbe médiatrice plane. Cette nouvelle méthode utilise la couleur dynamique en GeoGebra pour les caractérisations géométrique et numérique de la courbe médiatrice de deux objets géométriques dans le plan (deux courbes, ou une courbe et un point). Même si elle ne fournit pas de représentation algébrique, la méthode peut conduire au calcul d’une représentation approximative de la courbe médiatrice. Le diagramme de Voronoï (VD) est une structure de données fondamentale de la géométrie algorithmique avec des applications très variées dans des domaines théoriques et pratiques. Nous considérons le VD d’un ensemble fini de demi-droites parallèles de même orientation restreint à un domaine compact D0 ⊂ R3 pour la distance euclidienne. Ce nouveau type de VD peut être utilisé pour apporter des réponses efficaces à certains problèmes dans l’industrie de forage, tels que l’hydraulique ou la mine. Nous présentons un algorithme approximatif efficace pour le calcul de tel VD, en utilisant le processus de subdivision produisant un maillage qui représente la topologie de VD dans D0.
Este trabajo consta de tres partes principales : el calculo de las bisectrices de dos curvas o de un punto y una curva en el plano, de la superficie bisectriz de dos superficies en R3, y del diagrama de Voronoi de una familia finita de semirrectas paralelas y con la misma orientación en R3. Estos temas están estrechamente relacionados y tienen aplicaciones en CAD/CAGD y en Geometría Computacional. Se presenta un nuevo método para determinar, utilizando la regla de Cramer generalizada y métodos de eliminación adecuados, una parametrización algebraica exacta (racional o no racional) de la curva bisectriz de dos curvas planas racionales dadas. Este método se generaliza para determinar una parametrización algebraica exacta de la superficie bisectriz de dos superficies racionales de grado bajo. El método se aplica, en particular, para obtener parametrizaciones de la bisectriz de dos curvas planas racionales, cuando una de ellas es una circunferencia o una recta. Por otro lado, se muestra cómo obtener fácilmente una parametrizacin de la bisectriz de los siguientes pares de superficies : plano y cuádrica, plano y toro, cilindro circular y cuádrica no desarrollable, cilindro circular y toro, dos cilindros, cilindro y cono, y dos conos. Estas parametrizaciones son racionales en la mayora de los casos. En los casos restantes, la parametrización contiene una raíz cuadrada, que resulta adecuada para determinar una buena aproximación de la bisectriz. Además, se presenta un enfoque diferente para el problema de la curva bisectriz plana. Este nuevo método utiliza color dinámico en GeoGebra para el cálculo de una caracterización geométrica y numérica de la bisectriz de dos objetos geométricos en el plano (dos curvas, o una curva y un punto). Aunque no proporciona una representación algebraica, el método permite el cálculo de una representación aproximada de la curva bisectriz. El diagrama de Voronoi (DV) es una estructura de datos fundamental en geometría computacional con diversas aplicaciones en distintas áreas teóricas y prácticas. Se estudia el DV de un conjunto de semirrectas paralelas y con la misma orientación, restringidas a un dominio compacto D0 ⊂ R3, con respecto a la distancia euclidiana. Este nuevo tipo de DV se puede utilizar para proporcionar una solución eficiente a algunos problemas relacionados con la perforación, en industrias tales como la hidráulica o la minería. Se presenta un algoritmo eficiente para calcular una aproximación de un DV de esa clase, utilizando un proceso de subdivisión, que produce una malla que representa correctamente la topología del DV.
This thesis has three main parts: computation of the bisectors of two curves or a point and a curve in the plane, of the bisector of two surfaces in R3, and of the Voronoi diagram of a finite family of parallel half lines in R3, with the same orientation. These subjects are closely related, and have applications in CAD/CAGD and Computational Geometry. In each of the three parts, we present algorithmic methods for computing certain representations of the geometric object of interest: the bisector curve, the bisector surface, or the Voronoi diagram. We present a new approach to determine, using the generalized Cramer’s rule and suitable elimination steps, an exact algebraic parameterization (rational or non rational) of the bisector curve of two given planar rational curves. The approach is, then, generalized to determine an exact algebraic parameterization of the bisector surface of two low degree rational surfaces. In particular, we apply the method to obtain parametrizations of the bisector of two rational plane curves, when one of them is a circle or a straight line. On the other hand, we show how to easily obtain parametrizations of the bisector of the following pairs of surfaces: planequadric, plane-torus, circular cylinder-non developable quadric, circular cylindertorus, cylinder-cylinder, cylinder-cone and cone-cone. These parametrizations are rational in most cases. In the remaining cases the parametrization involves one square root which is well-suited to determine a good approximation of the bisector. In addition, a different approach for the bisector curve problem will be presented. This new method uses dynamic color in GeoGebra for the computation of a geometric and numerical characterization of the bisector of two planar geometric objects (two curves, or a curve and a point). Even if it does not provide an algebraic representation, the method could lead to the computation of an approximate representation of the bisector curve. The Voronoï diagram (VD) is a fundamental data structure in computational geometry with various applications in theoretical and practical areas. We consider the VD of a set of parallel half-lines, with the same orientation, constrained to a compact domain D0 ⊂ R3, with respect to the Euclidean distance. This new kind of VD can be used to provide an efficient solution to some problems in the drilling industry, such as hydraulic or mining. We present an efficient approximate algorithm for computing such VD, using a box subdivision process, which produces a mesh representing the topology of the VD in D0.

A dissertação tem como tema as álgebras de Lie. Especificamente álgebras de Lie semi-simples e suas propriedades . Para encontramos essas propriedades estudamos os conceitos básicos da teoria das álgebras de Lie e suas representações. Então fizemos a classificação dessas álgebras por diagramas de Dynkin explicitando quais os possíveis diagramas que são associados a uma álgebra de Lie semi-simples. Por fim, demonstramos vários resultados concernentes a essa classificação, dentre esses, o principal resultado demonstrado foi: os diagramas de Dynkin são um invariante completo das álgebras de Lie semi-simples
The dissertation has the theme Lie algebras. Specifically semi-simple Lie algebras and its properties. To find these properties we studied the basic concepts of the theory of Lie algebras and their representations. Then we did the classification by Dynkin diagrams of these algebras and explaining the possible diagrams that are associated with a semi-simple Lie algebra. Finally, we demonstrate several results related to this classification, among these, the main result demonstrated was: the Dynkin diagrams are a complete invariant of semi-simple Lie algebras

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String diagrams turn algebraic equations into topological moves that have recurring shapes, involving the sliding of one diagram past another. We individuate, at the root of this fact, the dual nature of polygraphs as presentations of higher algebraic theories, and as combinatorial descriptions of "directed spaces". Operations of polygraphs modelled on operations of topological spaces are used as the foundation of a compositional universal algebra, where sliding moves arise from tensor products of polygraphs. We reconstruct several higher algebraic theories in this framework. In this regard, the standard formalism of polygraphs has some technical problems. We propose a notion of regular polygraph, barring cell boundaries that are not homeomorphic to a disk of the appropriate dimension. We define a category of non-degenerate shapes, and show how to calculate their tensor products. Then, we introduce a notion of weak unit to recover weakly degenerate boundaries in low dimensions, and prove that the existence of weak units is equivalent to a representability property. We then turn to applications of diagrammatic algebra to quantum theory. We re-evaluate the category of Hilbert spaces from the perspective of categorical universal algebra, which leads to a bicategorical refinement. Then, we focus on the axiomatics of fragments of quantum theory, and present the ZW calculus, the first complete diagrammatic axiomatisation of the theory of qubits. The ZW calculus has several advantages over ZX calculi, including a computationally meaningful normal form, and a fragment whose diagrams can be read as setups of fermionic oscillators. Moreover, its generators reflect an operational classification of entangled states of 3 qubits. We conclude with generalisations of the ZW calculus to higher-dimensional systems, including the definition of a universal set of generators in each dimension.

This thesis consists of two parts. Part one revolves around a construction for homogeneous Gorenstein ideals and properties of these ideals. Part two focuses on the behavior of the Boij-Söderberg decomposition of lex ideals. Gorenstein ideals are known for their nice duality properties. For codimension two and three, the structures of Gorenstein ideals have been established by Hilbert-Burch and Buchsbaum-Eisenbud, respectively. However, although some important results have been found about Gorenstein ideals of higher codimension, there is no structure theorem proven for higher codimension cases. Kustin and Miller showed how to construct a Gorenstein ideals in local Gorenstein rings starting from smaller such ideals. A modification of their construction in the case of graded rings is discussed. In a Noetherian ring, for a given two homogeneous Gorenstein ideals, we construct another homogeneous Gorenstein ideal and so we describe the resulting ideal in terms of the initial homogeneous Gorenstein ideals. Gorenstein liaison theory plays a central role in this construction. Using liaison properties, we examine structural relations between the constructed homogeneous ideal and the starting ideals. Boij-Söderberg theory is a very recent theory. It arose from two conjectures given by Boij and Söderberg and their proof by Eisenbud and Schreyer. It establishes a unique decomposition for Betti diagram of graded modules over polynomial rings. In the second part of this thesis, we focus on Betti diagrams of lex ideals which are the ideals having the largest Betti numbers among the ideals with the same Hilbert function. We describe Boij-Söderberg decomposition of a lex ideal in terms of Boij-Söderberg decompositions of some related lex ideals.

In 1942, Paul C. Rosenbloom put out a definition of a Post algebra after Emil L. Post published a collection of systems of many–valued logic. Post algebras became easier to handle following George Epstein’s alternative definition. As conceived by Rosenbloom, Post algebras were meant to capture the algebraic properties of Post’s systems; this fact was not verified by Rosenbloom nor Epstein and has been assumed by others in the field. In this thesis, the long–awaited demonstration of this oft–asserted assertion is given. After an elemental history of many–valued logic and a review of basic Classical Propositional Logic, the systems given by Post are introduced. The definition of a Post algebra according to Rosenbloom together with an examination of the meaning of its notation in the context of Post’s systems are given. Epstein’s definition of a Post algebra follows the necessary concepts from lattice theory, making it possible to prove that Post’s systems of many–valued logic do in fact form a Post algebra.

In questa tesi abbiamo studiato le forme reali di algebre e superalgebre di Lie. Il lavoro si suddivide in tre capitoli diversi, il primo è di introduzione alle algebre di Lie e serve per dare le prime basi di questa teoria e le notazioni. Nel secondo capitolo abbiamo introdotto le algebre compatte e le forme reali. Abbiamo visto come sono correlate tra di loro tramite strumenti potenti come l'involuzione di Cartan e relativa decomposizione ed i diagrammi di Vogan e abbiamo introdotto un algoritmo chiamato "push the button" utile per verificare se due diagrammi di Vogan sono equivalenti. Il terzo capitolo segue la struttura dei primi due, inizialmente abbiamo introdotto le superalgebre di Lie con relativi sistemi di radici e abbiamo proseguito studiando le relative forme reali, diagrammi di Vogan e abbiamo introdotto anche qua l'algoritmo "push the button".

In questa tesi studiamo il ruolo dei sistemi di radici nella classificazione delle algebre di Lie e delle superalgebre di Lie. L'interesse per le superalgebre di Lie nasce nei primi anni '70 quando una parte dei fisici si convinse che sarebbe stato più utile e molto più chiaro riuscire ad avere uno schema di riferimento unitario in cui non dovesse essere necessario trattare separatamente particelle fisiche come bosoni e fermioni. Una teoria sistematica sulle superalgebre di Lie fu introdotta da V. Kac nel 1977 che diede la classificazione delle superalgebre di Lie semplici su un campo algebricamente chiuso.

Un quiver è costituito da un insieme di vertici e un insieme di frecce fra di essi.Verrà introdotta la nozione di rappresentazione per un quiver che abbia un numero finito di vertici e frecce. La categoria delle rappresentazioni e loro morfismi è equivalente a quella degli A-moduli destri dove A è l'algebra dei cammini del quiver, ovvero l' algebra che ha come base tutti i cammini ottenibili percorrendo le frecce e come prodotto quello indotto dalla composizione di cammini. Dal punto di vista della teoria delle rappresentazioni lo studio delle algebre finito dimensionali si riduce allo studio di quozienti dell'algebra dei cammini. Lo scopo di questa tesi è arrivare a una classificazione dei quiver che hanno solamente un numero finito di rappresentazioni indecomponibili (ovvero che non si possono scrivere come somma diretta di altre rappresentazioni non banali) a meno di isomorfismo. Il risultato è abbastanza recente ed è stato provato da Gabriel nel 1972. Vi è un numero finito di classi di isomorfismo di rappresentazioni solamente nei casi in cui il grafo associato al quiver è un particolare diagramma di Dynkin. Questo è sorprendente poiché questi diagrammi sono in corrispondenza biunivoca con le algebre di Lie semisemplici su un campo algebricamente chiuso.

Ce manuscrit est consacré à l'étude de la combinatoire de deux algèbres de Hopf d'extraction-contraction. La première est l'algèbre de Hopf de mots tassés WMat introduite par Duchamp, Hoang-Nghia et Tanasa dont l'objectif était la construction d'un modèle de coproduit d'extraction-contraction pour les mots tassés. Nous expliquons certains sous-objets ou objets quotients ainsi que des applications vers d'autres algèbres de Hopf. Ainsi, nous considérons une algèbre de permutations dont le dual gradué possède un coproduit de déconcaténation par blocs et un produit de double battage décalé. Le double battage engendre la commutativité de l'algèbre qui est donc distincte de celle de Malvenuto et Reutenauer. Nous introduisons également une algèbre de Hopf engendrée par les mots tassés de la forme x₁...x₁. Elle est isomorphe à l'algèbre de Hopf des fonctions symétriques non commutatives. Son dual gradé est donc isomorphe à l'algèbre de Hopf des fonctions quasi-symétriques. Nous considérons également une algèbre de Hopf de compositions et donnons son interprétation en termes de coproduit semi-direct d'algèbres de Hopf. Le deuxième objet d'étude est l'algèbre de Hopf de diagrammes de dissection HD introduite par Dupont en théorie des nombres. Nous cherchons des éléments de réponse concernant la nature de sa cogèbre sous-jacente. Est-elle colibre ? La dimension des éléments primitifs de degré 3 ne permet pas de conclure. Le cas du degré 5 permet d'établir la non-coliberté dans le cas où le paramètre de HD vaut - 1. Nous étudions également la structure pré-Lie du dual gradué HD. Nous réduisons le champ de recherche à la sous-algèbre pré-Lie non triviale engendrée par le diagramme de dissection de degré 1. Cette algèbre pré-Lie n'est pas libre
This thesis deals with the study of combinatorics of two Hopf algebras. The first one is the packed words Hopf algebra WMAT introduced by Duchamp, Hoang-Nghia, and Tanasa who wanted to build a coalgebra model for packed words by using a selection-quotient process. We describe certain sub-objects or quotient objects as well as maps to other Hopf algebras. We consider first a Hopf algebra of permutations. Its graded dual has a block deconcatenation coproduct and double shuffle product. The double shuffle product is commutative so the Hopf algebra is different from the Malvenuto and Reutenauer one. We analyze then the Hopf algebra generated by packed words looking like x₁...x₁. This Hopf algebra and non commutative symmetric functions are isomorphic. So its graded dual and quasi-symmetric functions are isomorphic too. Finally we consider a Hopf algebra of compositions an give its interpretation in terms of a semi-direct coproduct structure. The second objet we study is the Hopf algebra of dissection diagrams HD introduced by Dupont in number theory. We study the cofreedom problem. We can't conclude with homogeneous primitive elements of degree 3. With the degree 5 case, we can say that is not cofree with the parameter -1. We study the pre-Lie algebra structure of HD's graded dual too. We consider in particular the sup-pre-Lie algebra generated by the dissection diagram of degree 1. It is not a free pre-Lie algebra

Dans cette thèse, on étudie les invariants quantiques des 3-variétés de Kuperberg, qui sont basées sur les algèbres de Hopf. On montre que, pour les super-algèbres de Hopf involutives, les invariants de Kuperberg s’étendent à la classe, plus générale, des 3-variétés suturées balancées et en particulier aux compléments d’entrelacs. Pour accomplir ceci, on relève plusieurs aspects de la théorie des torsions de Reidemeister au monde des invariants quantiques, tels que la procédure pour tordre des invariants, le calcul de Fox et les structures Spin^c, et on clarifie les aspects de la théorie des algèbres de Hopf auxquels ils correspondent. Quand notre construction est spécialisée au cas d’une algèbre extérieure, on montre qu’elle calcule la torsion de Reidemeister tordue des 3-variétés suturées
In this thesis, we study Kuperberg's Hopf algebra approach to quantum invariants of closed 3-manifolds. We show that, for involutive Hopf superalgebras, Kuperberg invariants extend to the more general class of balanced sutured 3-manifolds, and in particular, to link complements. To achieve this, we bring many aspects of Reidemeister torsion theory into the realm of quantum invar-iants, such as twisting, Fox calculus and Spin^c structures and we make clear to which aspects of Hopf algebra theory these correspond. When our construction is specialized to an exterior algebra, we show that it recovers the twisted Reidemeister torsion of sutured 3-manifolds

Une approche inductive est développée pour la théorie des représentations de la chaîne des algèbres de Hecke cyclotomiques de type G(m,1,n). Cette approche repose sur l'étude du spectre d'une famille commutative maximale, formée par les analogues des éléments de Jucys--Murphy.Les représentations irréductibles, paramétrées par les multi-partitions, sont construites avec l'aide d'une nouvelle algèbre associative, dont l'espace vectoriel sous-jacent est le produit tensoriel de l'algèbre de Hecke cyclotomique avec l'algèbre associative libre engendrée par les multi-tableaux standards.L'analogue de cette approche est présentée pour la limite classique, c'est-à-dire la chaîne des groupes de réflexions complexes de type G(m,1,n).Dans une seconde partie, une base des algèbres de Hecke cyclotomiques est donnée et la platitude de la déformation est montrée sans utiliser la théorie des représentations. Ces résultats sont généralisés aux algèbres de Hecke affines de type A.Ensuite, une procédure de fusion est présentée pour les groupes de réflexions complexes et les algèbres de Hecke cyclotomiques de type G(m,1,n). Dans les deux cas, un ensemble complet d'idempotents primitifs orthogonaux est obtenu par évaluation consécutive d'une fonction rationnelle.Dans une troisième partie, une nouvelle présentation est obtenue pour les sous-groupes alternés de tous les groupes de Coxeter. Les générateurs sont reliés aux arêtes orientées du graphe de Coxeter. Cette présentation est ensuite étendue, pour tous les types, aux extensions spinorielles des groupes alternés, aux algèbres de Hecke alternées et aux sous-groupes alternés des groupes de tresses
An inductive approach to the representation theory of the chain of the cyclotomic Hecke algebras of type G(m,1,n) is developed. This approach relies on the study of the spectrum of a maximal commutative family formed by the analogues of the Jucys--Murphy elements.The irreducible representations, labelled by the multi-partitions, are constructed with the help of a new associative algebra, whose underlying vector space is the tensor product of the cyclotomic Hecke algebra with the free associative algebra generated by the standard multi-tableaux.The analogue of this approach is presented for the classical limit, that is for the chain of complex reflection groups of type G(m,1,n).In a second part, a basis of the cyclotomic Hecke algebras is given and the flatness of the deformation is proved without using the representation theory. These results are extended to the affine Hecke algebras of type A.Then a fusion procedure is presented for the complex reflection groups and the cyclotomic Hecke algebras of type G(m,1,n). In both cases, a complete set of primitive orthogonal idempotents is obtained by successive evaluations of a rational fonction.In a third part, a new presentation is obtained for the alternating subgroups of all Coxeter groups. The generators are related to oriented edges of the Coxeter graph. This presentation is then extended, for all types, to the spinor extensions of the alternating groups, the alternating Hecke algebras and the alternating subgroups of braid groups

Cette thèse est consacrée à l’étude de deux questions très différentes, reliées par les outils que nous utilisons pour les étudier. La première question est celle de la définition des théories de jauge sur un réseau avec un groupe de structure non commutatif. Ici, non commutatif ne signifie pas non Abelian, mais plutôt non commutatif au sens général de la géométrie non commutative. La deuxième question est celle du comportement des diffusions Browniennes sur des groupes matriciels non compacts d’un type spécifique, à savoir des groupes de matrices pseudo-orthogonales, pseudo-unitaires ou pseudo-symplectiques. Dans le premier chapitre, nous étudions des théories de jauge quantiques sur un réseau et leur limite continue sur le plan euclidien ayant une algèbre de Zhang pour groupe de stuc-ture. Les algèbres de Zhang sont des analogues non commutatifs des groupes et contiennent la classe des groupes duaux de Voiculescu. Nous nous intéressons donc aux analogues non commutatifs des champs de jauges quantiques, que nous décrivons par l’holonomie aléatoire qu’ils induisent. Nous proposons une définition générale d’un champ d’holonomies ayant une symétrie de jauge présentant la structure d’une algèbre de Zhang, et construisons un tel champ à partir d’un processus quantique de Lévy sur une algèbre de Zhang. Dans le deuxième chapitre, nous étudions les approximations matricielles des champs maîtres en dimensions supérieures construits dans le chapitre précédent. Ces approximations (en distribution non commutative) sont obtenues en extrayant des blocs d’une diffusion unitaire Brownienne (à coefficients dans les algèbres de nombres réels, complexes ou quaternioniques) et en laissant la dimension de ces blocs tendre vers l’infini. Nous divisons notre étude en deux parties : dans la première, nous extrayons des blocs carrés tandis que dans la seconde, nous autorisons des blocs rectangulaires. Dans les deux derniers chapitres, nous utilisons les outils introduits (algèbres de Zhang et diagrammes de Brauer colorés) dans les deux premiers pour étudier des diffusions sur des groupes de matrices pseudo-unitaires. Nous prouvons la convergence non commutative des mouvements Browniens pseudo-unitaires que nous considérons vers des semi-groupes libres avec amalgamation sous l’hypothèse de convergence de la signature normalisée de la métrique de l’espace sous-jacent. Dans le cas déployé, c’est-à-dire, qu’au moins asymptotiquement, la métrique a autant de directions négatives que de directions positives, la distribution limite est la distribution d’un processus de Lévy, solution d’une équation différentielle stochastique libre. Nous laissons ouverte la question d’une telle réalisation de la distribution limite dans le cas général. De plus, nous présentons des résultats numériques sur la convergence de la distribution spectrale de ces matrices aléatoires et faisons deux conjectures. Dans le dernier chapitre, nous prouvons la normalité asymptotique des fluctuations
This thesis is devoted to the study of two quite different questions, which are related by the tools that we use to study them. The first question is that of the definition of lattice gauge theories with a non-commutative structure group. Here, by non-commutative, we do not mean non-Abelian, but instead non-commutative in the general sense of non-commutative geometry. The second question is that of the behaviour of Brownian diffusions on non-compact matrix groups of a specific kind, namely groups of pseudo-orthogonal, pseudo-unitary or pseudo-symplectic matrices. In the first chapter, we investigate lattice and continuous quantum gauge theories on the Euclidean plane with a structure group that is replaced by a Zhang algebra. Zhang algebras are non-commutative analogues of groups and contain the class of Voiculescu’s dual groups. We are interested in non-commutative analogues of random gauge fields, which we describe through the random holonomy that they induce. We propose a general definition of a holonomy field with Zhang gauge symmetry, and construct such a field starting from a quantum Lévy process on a Zhang algebra. As an application, we define higher dimensional generalizations of the so-called master field. In the second chapter, we study matricial approximations of higher dimensional master fields constructed in the previous chapter. These approximations (in non-commutative distribution) are obtained by extracting blocks of a Brownian unitary diffusion (with entries in the algebras of real, complex or quaternionic numbers) and letting the dimension of these blocks tend to infinity. We divide our study into two parts: in the first one, we extract square blocks while in the second one we allow rectangular blocks. In both cases, free probability theory appears as the natural framework in which the limiting distributions are most accurately described. In the last two chapters, we use tools introduced (Zhang algebras and coloured Brauer diagrams) in the first two ones to study Brownian motion on pseudo-unitary matrices in high dimensions. We prove convergence in non-commutative distribution of the pseudo-unitary Brownian motions we consider to free with amalgamation semi-groups under the hypothesis of convergence of the normalized signature of the metric. In the split case, meaning that at least asymptotically the metric has as much negative directions as positive ones, the limiting distribution is that of a free Lévy process, which is a solution of a free stochastic differential equation. We leave open the question of such a realization of the limiting distribution in the general case. In addition we provide (intriguing) numerical evidences for the convergence of the spectral distribution of such random matrices and make two conjectures. At the end of the thesis, we prove asymptotic normality for the fluctuations

In this thesis I present new insights into aspects of Peter Guthrie Tait's life and work, derived principally from largely-unexplored primary source material: Tait's scrapbook, the Tait–Maxwell school-book and Tait's pocket notebook. By way of associated historical insights, I also come to discuss the innovative and far-reaching mathematics of the elusive Frenchman, C.-V. Mourey. P. G. Tait (1831–1901) F.R.S.E., Professor of Mathematics at the Queen's College, Belfast (1854–1860) and of Natural Philosophy at the University of Edinburgh (1860–1901), was one of the leading physicists and mathematicians in Europe in the nineteenth century. His expertise encompassed the breadth of physical science and mathematics. However, since the nineteenth century he has been unfortunately overlooked—overshadowed, perhaps, by the brilliance of his personal friends, James Clerk Maxwell (1831–1879), Sir William Rowan Hamilton (1805–1865) and William Thomson (1824–1907), later Lord Kelvin. Here I present the results of extensive research into the Tait family history. I explore the spiritual aspect of Tait's life in connection with The Unseen Universe (1875) which Tait co-authored with Balfour Stewart (1828–1887). I also reveal Tait's surprising involvement in statistics and give an account of his introduction to complex numbers, as a schoolboy at the Edinburgh Academy. A highlight of the thesis is a re-evaluation of C.-V. Mourey's 1828 work, La Vraie Théorie des quantités négatives et des quantités prétendues imaginaires, which I consider from the perspective of algebraic reform. The thesis also contains: (i) a transcription of an unpublished paper by Hamilton on the fundamental theorem of algebra which was inspired by Mourey and (ii) new biographical information on Mourey.

by Chun Gu.
Thesis (M.Phil.)--Chinese University of Hong Kong, 1999.
Includes bibliographical references (leaves 69-71).
Abstract also in Chinese.
Chapter 1 --- Introduction --- p.3
Chapter 2 --- Definitions --- p.7
Chapter 3 --- Some Previous Work on OBDD --- p.13
Chapter 3.1 --- The Work of Bryant --- p.13
Chapter 3.2 --- Some Variations of the OBDD --- p.14
Chapter 3.3 --- Previous Work on Variable Ordering of OBDD --- p.16
Chapter 3.3.1 --- The FIH Heuristic --- p.16
Chapter 3.3.2 --- The Dynamic Variable Ordering --- p.17
Chapter 3.3.3 --- The Interleaving method --- p.19
Chapter 4 --- Two Level Logic Function and OBDD --- p.21
Chapter 5 --- DSCF Algorithm --- p.25
Chapter 6 --- Thin Boolean Function --- p.33
Chapter 6.1 --- The Structure and Properties of thin Boolean functions --- p.33
Chapter 6.1.1 --- The construction of Thin OBDDs --- p.33
Chapter 6.1.2 --- Properties of Thin Boolean Functions --- p.38
Chapter 6.1.3 --- Thin Factored Functions --- p.49
Chapter 6.2 --- The Revised DSCF Algorithm --- p.52
Chapter 6.3 --- Experimental Results --- p.54
Chapter 7 --- A Pattern Merging Algorithm --- p.59
Chapter 7.1 --- Merging of Patterns --- p.60
Chapter 7.2 --- The Algorithm --- p.62
Chapter 7.3 --- Experiments and Conclusion --- p.65
Chapter 8 --- Conclusions --- p.67

Diagram drawing is generally accepted as an important heuristic strategy for solving mathematical problems. However, novice problem solvers do not frequently choose to use this strategy. Further, when asked to draw a diagram, their attempts often do not result in a useful representation of the problem. The exploratory study, which used individual interviews with remedial mathematics students at the University of Massachusetts, identified five factors that influence whether a diagram is used and whether its use is successful: (1) Understanding of the mathematics involved in the problem and of basic arithmetic concepts (i.e. fractions, ratio); (2) Diagram drawing skills and experience; (3) Conceptions of mathematics; (4) Self-concept in mathematics; (5) Motivation to solve the problem correctly. The interviews also generated a set of diagram drawing subskills. The main study focused on factor two. It attempted to experimentally verify the importance of the subskills identified in the exploratory study. The list of subskills was translated into a series of external control suggestions for guiding the subjects' work during individual interviews. Subjects were precalculus students at the University of Massachusetts. These suggestions were provided by the experimenter as appropriate. Subjects who received these suggestions drew significantly higher quality diagrams than did subjects in the control group. The enhanced quality was particularly apparent in the area of completeness of the diagram. In addition, the study indicated several important metacognitive skills necessary for successful diagram drawing as well as a number of specific difficulties encountered by the subjects.

The notion of finite type invariants of virtual knots, introduced by Goussarov, Polyak and Viro, leads to the study of the space of diagrams with directed chords mod 6T (also known as the space of arrow diagrams), and weight systems on it. It is well known that given a Manin triple together with a representation we can construct a weight system. In the first part of this thesis we develop combinatorial formulae for weight systems coming from standard Manin triple structures on the classical Lie algebras and these structures' defining representations. These formulae reduce the problem of finding weight systems in the defining representations to certain counting problems. We then use these formulae to verify that such weight systems, composed with the averaging map, give us the weight systems found by Bar-Natan on (undirected) chord diagrams mod 4T. In the second half of the thesis we present results from computations done jointly with Bar-Natan. We compute, up to degree 4, the dimensions of the spaces of arrow diagrams whose skeleton is a line, and the ranks of all classical Lie algebra weight systems in all representations. The computations give us a measure of how well classical Lie algebras capture the spaces of arrow diagrams up to degree 4, and our results suggest that in degree 4 there are already weight systems which do not come from the standard Manin triple structures on classical Lie algebras.

La majorité des courbes et surfaces rencontrées dans la modélisation géométrique sont définies comme l'ensemble des solutions d'un système d'équations et d'inéquations algébriques (ensemble semi-algébrique). De nombreux problèmes dans différentes disciplines scientifiques font appel à des requètes de proximité telles que la recherche du ou des voisins les plus proches ou la quantification du voisinage de deux objets.

Le diagramme de Voronoï d'un ensemble d'objets est une décomposition de l'espace en zones de proximité. La zone de proximité d'un objet est l'ensemble des points plus proches de cet objet que de tout autre objet. Les diagrammes de Voronoï permettent de répondre aux requètes de proximité après avoir identifié la zone de proximité à laquelle le point objet de la requète appartient. Le graphe dual du diagramme de Voronoï est appelé le graphe de Delaunay. Seules les approximations par des coniques peuvent garantir un ordre de continuité approprié au niveau des points de contact, ce qui est nécessaire pour garantir l'exactitude du graphe de Delaunay.

L'objectif théorique de cette thèse est la mise en évidence des propriétés algébriques et géométriques élémentaires de la courbe déplacée d'une courbe algébrique et de réduire le calcul semi-algébrique du graphe de Delaunay à des calculs de valeurs propres. L'objectif pratique de cette thèse est le calcul certifié du graphe de Delaunay pour des ensembles semi-algébriques de faible degré dans le plan euclidien.

La méthodologie associe l'analyse par intervalles et la géométrie algébrique algorithmique. L'idée centrale de cette thèse est qu'un pré-traitement symbolique unique peut accélérer l'évaluation numérique certifiée du détecteur de conflits dans le graphe de Delaunay. Le pré-traitement symbolique est le calcul de l'équation implicite de la courbe déplacée généralisée d'une conique. La réduction du problème semi-algébrique de la détection de conflits dans le graphe de Delaunay à un problème d'algèbre linéaire a été possible grâce à la considération du sommet de Voronoï généralisé (un concept introduit dans cette thèse).

Le calcul numérique certifié du graphe de Delaunay a été éffectué avec une librairie de résolution de systèmes zéro-dimensionnels d'équations et d'inéquations algébriques basée sur l'analyse d'intervalles (ALIAS). Le calcul certifié du graphe de Delaunay repose sur des théorèmes sur l'unicité de racines dans des intervalles donnés (Kantorovitch et Moore-Krawczyk). Pour les coniques, les calculs sont accélérés lorsque l'on ne considère que les équations implicites des courbes déplacées.

Most of the curves and surfaces encountered in geometric modelling are denned as the set of solutions of a system of algebraic equations and inequalities (semialgebraic sets). Many problems from different fields involve proximity queries like finding the (nearest) neighbours or quantifying the neighbourliness of two objects. The Voronoi diagram of a set of sites is a decomposition of space into proximal regions. The proximal region of a site is the locus of points closer to that site than to any other one. Voronoi diagrams allow one to answer proximity queries after locating a query point in the Voronoi zone it belongs to. The dual graph of the Voronoi diagram is called the Delaunay graph. Only approximations by conies can guarantee a proper order of continuity at contact points, which is necessary for guaranteeing the exactness of the Delaunay graph. The theoretical purpose of this thesis is to elucidate the basic algebraic and geometric properties of the offset to an algebraic curve and to reduce the semialgebraic computation of the Delaunay graph to eigenvalues computations. The practical objective of this thesis is the certified computation of the Delaunay graph for low degree semi-algebraic sets embedded in the Euclidean plane. The methodology combines interval analysis and computational algebraic geometry. The central idea of this thesis is that a (one time) symbolic preprocessing may accelerate the certified numerical evaluation of the Delaunay graph conflict locator. The symbolic preprocessing is the computation of the implicit equation of the generalised offset to conies. The reduction of the Delaunay graph conflict locator for conies from a semi-algebraic problem to a linear algebra problem has been possible through the use of the generalised Voronoi vertex (a concept introduced in this thesis). The certified numerical computation of the Delaunay graph has been possible by using an interval analysis based library for solving zero-dimensional systems of equations and inequalities (ALIAS). The certified computation of the Delaunay graph relies on theorems on the uniqueness of a root in given intervals (Kantorovitch, Moore-Krawczyk). For conies, the computations get much faster by considering only the implicit equations of the generalised offsets.
Science, Faculty of
Computer Science, Department of
Graduate

碩士
國立屏東教育大學
數理教育研究所
101
The aim of this research was to find out the fifth grade students’ solving performance of the algebra problems by the teaching strategy of using “line-diagram” representation. The quasi-experimental methodology was adopted and 60 fifth-graders from two classes were chosen as subjects. The researcher taught the experimental group by five steps of “line-diagram” strategy. On the other side, the control group was given the normal teaching strategy eight lessons were implemented in two weeks. This research using the post-test and delay test which researchers designed as assessment tools. Data were analyzed by Analysis of Covariance (ANCOVA) and nonparametric statistic test. According to the students' problem-solving performance, we could find out their strategies using and errors made on problem solving. The result shows that the experimental group students’ performances were significantly higher than the control group on post and delay test. The participants in experimental group have nice learning effects and learning retention effects. Besides, for the middle-achievement students shows more obvious progress on learning. On the problem solving, the control group students were easy to use key-word and functional reciprocal strategy, and it causes the failures on problem solving. However, the experimental group students tried to draw the line segments, and it clearly showed the relations between the numbers. They understood the reciprocal concepts from line segments, so that they can solve the problem successfully. Finally, according to the result, this study brings out some suggestions for teaching and the further research.

Formal methods promise the ultimate quality of software artifacts with mathematicalproof of their correctness. Algebraic specification is one of such methods, providingformal specifications of system components suitable for verification of correctnessof all individual steps in the software development process, and hence of the entiredevelopment process and of the resulting program.In this thesis we propose a new approach to algebraic specifications of software architectures,called diagrams of construction specifications. Constructions, as introducedhere, model parameterised modules, with dependency relation captured directly onsignature symbols. They give a uniform treatment of first- and higher-order parameterisation,and are equipped with a single sum operation which subsumes the moststandard operations on parameterised modules. We introduce specifications for suchconstructions, study their compositionality properties, and define a notion of refinementfor constructor specifications. Diagrams of construction specifications capturedesign and development of modular software architecture, based on decomposition andrefinement of construction specifications.Throughout the thesis we illustrate new concepts and problems discussed by means ofsimple examples; a somewhat longer example is also added to summarize our presentation.
Metody formalne umożliwiają uzyskanie najwyższej jakości procesu wytwarzania oprogramowaniaprzez dostarczenie matematycznych dowodów jego poprawności. Jedną ztakich metod są specyfikacje algebraiczne, które podają sposób formalnej specyfikacjiposzczególnych komponentów systemu informatycznego oraz weryfikacji poprawnościwszystkich kroków procesu wytwarzania oprogramowania i w rezultacie dają możliwośćzapewnienia poprawności zarówno całego procesu, jak i samego wynikowego programu.W niniejszej rozprawie proponuje się nowe podejście do algebraicznych specyfikacji architekturoprogramowania zwane diagramami specyfikacji konstrukcji. Wprowadzonejest pojęcie konstrukcji, które są modelami sparametryzowanych modułów wraz z relacjązależności wyrażoną bezpośrednio na symbolach z ich sygnatur. Konstrukcjezapewniają jednolite podejście do parametryzacji pierwszego i wyższych rzędów. Jedynąoperacją na konstrukcjach jest suma, która z powodzeniem zastępuje większośćstandardowych operacji na modułach sparametryzowanych. W rozprawie przedstawianesą specyfikacje konstrukcji, przeprowadzane są badania ich kompozycjonalnościoraz definiowane jest pojęcie uściślenia (ang. refinement) specyfikacji konstrukcji. Diagramyspecyfikacji konstrukcji pozwalają modelować strukturę i rozwój modularnycharchitektur oprogramowania opartych na dekompozycji i uściślaniu specyfikacji konstrukcji.W całym tekście podawane są liczne małe przykłady wprowadzanych pojęć i dyskutowanychproblemów. Na koniec przytoczony jest nieco dłuższy przykład ilustrującykilka kroków rozwoju architektury prostego systemu informatycznego.

The direct sum decomposition of tensor products for SU(3) has many applications in physics, and the problem has been studied extensively. This has resulted in many decomposition methods, each with its advantages and disadvantages. The description given here is geometric in nature and it describes both the constituents of the direct sum and their multiplicities. In addition to providing decompositions of specific tensor products, this approach is very well suited to studying tensor products as the parameters vary, and drawing general conclusions. After a description and proof of the method, several applications are discussed and proved. The decompositions are also studied further for the special cases of tensor products of an irreducible representation with itself or with its conjugate. In particular, questions regarding multiplicities are considered. As an extension of this diagrammatic method, the repeated tensor product of N copies of the fundamental representation is studied, and a method for its decomposition is provided. Again, questions regarding multiplicities are considered.