Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1979. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Bibliography: leaves 163-165. / Ph.D.

A álgebra simpléctica $sp(2)$ é realizada em termos de operadores bosônicos e sua ação local acontece numa porção de um extit{jet-space} associado com as variáveis independentes. Entretanto as derivadas da variável dependente, que é mantida fixa, se transformam sob a ação dos campos vetoriais prolongados. A existência de um extit{coframe} invariante neste extit{jet-space} nos permite construir operadores diferenciais invariantes que produzem invariantes diferenciais através de sua ação em invariantes de ordem menor. Apresentamos explicitamente neste trabalho invariantes diferenciais de segunda ordem para $sp(2n), n=1,2,3$. Todos invariantes de ordem maior podem ser obtidos mediante diferenciação. Estes invariantes diferenciais assim obtidos constituem uma base funcional explícita para equaç ões diferenciais parciais invariantes pela ação local do grupo simpléctico. Esta nova classe de equações diferenciais parciais com simetria pré-determinada não somente oferece seu cardápio usual de benefícios operacionais relacionados com a simetria carregada, mas restringe o formato que um problema variacional com tal simetria pode apresentar.

A álgebra simpléctica $sp(2)$ é realizada em termos de operadores bosônicos e sua ação local acontece numa porção de um extit{jet-space} associado com as variáveis independentes. Entretanto as derivadas da variável dependente, que é mantida fixa, se transformam sob a ação dos campos vetoriais prolongados. A existência de um extit{coframe} invariante neste extit{jet-space} nos permite construir operadores diferenciais invariantes que produzem invariantes diferenciais através de sua ação em invariantes de ordem menor. Apresentamos explicitamente neste trabalho invariantes diferenciais de segunda ordem para $sp(2n), n=1,2,3$. Todos invariantes de ordem maior podem ser obtidos mediante diferenciação. Estes invariantes diferenciais assim obtidos constituem uma base funcional explícita para equaç ões diferenciais parciais invariantes pela ação local do grupo simpléctico. Esta nova classe de equações diferenciais parciais com simetria pré-determinada não somente oferece seu cardápio usual de benefícios operacionais relacionados com a simetria carregada, mas restringe o formato que um problema variacional com tal simetria pode apresentar.

We give an overview of the basic properties of Banach Algebras. After that we specialize to the case of commutative Banach Algebras and study the Gelfand Map. We study the main characteristic of that map, and work on some applications.

Ce mémoire comporte deux parties distinctes. La première partie concerne une étude d'algèbres n-aires. Une algèbre n-aire est un espace vectoriel sur lequel est définie une multiplication sur n arguments. Classiquement les multiplications sont binaires, mais depuis l'utilisation en physique théorique de multiplications ternaires, comme les produits de Nambu, de nombreux travaux mathématiques se sont focalisés sur ce type d'algèbres. Deux classes d'algèbres n-aires sont essentielles: les algèbres n-aires associatives et les algèbres n-aires de Lie. Nous nous intéressons aux deux classes. Concernant les algèbres n-aires associatives, on s'intéresse surtout aux algèbres 3-aires partiellement associatives, c'est-à-dire dont la multiplication vérifie l'identité ((xyz)tu)+(x(yzt)u)+(xy(ztu))=0 Ce cas est intéressant car les travaux connus concernant ce type d'algèbres ne distinguent pas les cas n pair et n-impair. On montre dans cette thèse que le cas n=3 ne peut pas être traité comme si n était pair. On étudie en détail l'algèbre libre 3-aire partiellement associative sur un espace vectoriel de dimension finie. Cette algèbre est graduée et on calcule précisément les dimensions des 7 premières composantes. On donne dans le cas général un système de générateurs ayant la propriété qu'une base est donnée par la sous famille des éléments non nuls. Les principales conséquences sont L'algèbre libre 3-aire partiellement associative est résoluble. L'algèbre libre commutative 3-aire partiellement associative est telle que tout produit concernant 9 éléments est nul. L'opérade quadratique correspondant aux algèbres 3-aires partiellement associatives ne vérifient pas la propriété de Koszul. On s'intéresse ensuite à l'étude des produits n-aires sur les tenseurs. L'exemple le plus simple est celui d'un produit interne sur des matrices non carrées. Nous pouvons définir le produit 3aire donné par A . ^tB . C. On montre qu'il est nécessaire de généraliser un peu la définition de partielle associativité. Nous introduisons donc les produits -partiellement associatifs où  est une permutation de degré p. Concernant les algèbres de Lie n-aires, deux classes d'algèbres ont été définies: les algèbres de Fillipov (aussi appelées depuis peu les algèbres de Lie-Nambu) et les algèbres n-Lie. Cette dernière notion est très générale. Cette dernière notion, très important dans l'étude de la mécanique de Nambu-Poisson, est un cas particulier de la première. Mais pour définir une approche du type Maurer-Cartan, c'est-à-dire définir une cohomologie scalaire, nous considérons dans ce travail les algèbres de Fillipov comme des algèbres n-Lie et développons un tel calcul dans le cadre des algèbres n-Lie. On s'intéresse également à la classification des algèbres n-aires nilpotentes. Le dernier chapitre de cette partie est un peu à part et reflète un travail poursuivant mon mémoire de Master. Il concerne les algèbres de Poisson sur l'algèbre des polynômes. On commence à présenter le crochet de Poisson sous forme duale en utilisant des équations de Pfaff. On utilise cette approche pour classer les structures de Poisson non homogènes sur l’algèbre des polynômes à trois variables . Le lien avec les algèbres de Lie est clair. Du coup on étend notre étude aux algèbres de Poisson dont l'algèbre de Lie sous jacent est rigide et on applique les résultats aux algèbres enveloppantes des algèbres de Lie rigides. La partie 2 concerne l'arithmétique des intervalles. Cette étude a été faite suite à une rencontre avec une société d'ingénierie travaillant sur des problèmes de contrôle de paramètres, de problème inverse (dans quels domaines doivent évoluer les paramètres d'un robot pour que le robot ait un comportement défini). [...] / This thesis has two distinguish parts. The first part concerns the study of n-ary algebras. A n-ary algebra is a vector space with a multiplication on n arguments. Classically the multiplications are binary, but the use of ternary multiplication in theoretical physic like for Nambu brackets led mathematicians to investigate these type of algebras. Two classes of n-ary algebras are fundamental: the associative n-ary algebras and the Lie n-ary algebras. We are interested by both classes. Concerning the associative n-ary algebras we are mostly interested in 3-ary partially associative 3-ary algebras, that is, algebras whose multiplication satisfies ((xyz)tu)+(x(yzt)u)+(xy(ztu))=0. This type is interesting because the previous woks on this subject was not distinguish the even and odd cases. We show in this thesis that the case n=3 can not be treated as the even cases. We investigate in detail the free partially associative 3-ary algebra on k generators. This algebra is graded and we compute the dimensions of the 7 first components. In the general case, we give a spanning set such as the sub family of non zero vector is a basis. The main consequences are the free partially associative 3-ary algebra is solvable. In the free commutative partially associative 3-ary algebra any product on 9 elements is trivial. The operad for partially associative 3-ary algebra do not satisfy the Koszul property. Then we study n-ary products on the tensors. The simplest example is given by a internal product of non square matrices. We can define a 3-ary product by taking A . ^tB . C. We show that we have to generalize a bit the definition of partial associativity for n-ary algebras. We then introduce the products -partially associative where  is a permutation of the symmetric group of degree n. Concerning the n-ary algebras, two classes have been defined: Filipov algebras (also called recently Lie-Nambu algebras) and some more general class, the n-Lie algebras. Filipov algebras are very important in the study of the mechanic of Nambu-Poisson, and is a particular case of the other. So to define an approach of Maurer-Cartan type, that is, define a scalar cohomology, we consider in this work Fillipov as n-Lie algebras and develop such a calculus in the n-Lie algebras frame work. We also give some classifications of n-ary nilpotent algebras. The last chapter of this part concerns my work in Master on the Poisson algebras on polynomials. We present link with the Lie algebras is clear. Thus we extend our study to Poisson algebras which associated Lie algebra is rigid and we apply these results to the enveloping algebras of rigid Lie algebras. The second part concerns intervals arithmetic. The interval arithmetic is used in a lot of problems concerning robotic, localization of parameters, and sensibility of inputs. The classical operations of intervals are based of the rule : the result of an operation of interval is the minimal interval containing all the result of this operation on the real elements of the concerned intervals. But these operations imply many problems because the product is not distributive with respect the addition. In particular it is very difficult to translate in the set of intervals an algebraic functions of a real variable. We propose here an original model based on an embedding of the set of intervals on an associative algebra. Working in this algebra, it is easy to see that the problem of non distributivity disappears, and the problem of transferring real function in the set of intervals becomes natural. As application, we study matrices of intervals and we solve the problem of reduction of intervals matrices (diagonalization, eigenvalues, and eigenvectors).

We construct an algorithm for getting a reduced expression for any element in a complex reflection group G(m, 1, r) by sorting the element, which is in the form of a sequence of complex numbers, to the identity. Thus, the algorithm provides us a set of reduced expressions, one for each element. We establish a one-one correspondence between the set of all reduced expressions for an element and a set of certain sorting sequences which turn the element to the identity. In particular, this provides us with a combinatorial method to check whether an expression is reduced. We also prove analogues of the exchange condition and the strong exchange condition for elements in a G(m, 1, r). A Bruhat order on the groups is also defined and investigated. We generalize the Geck-Pfeiffer reducibility theorem for finite Coxeter groups to the groups G(m, 1, r). Based on this, we prove that a character value of any element in an Ariki-Koike algebra (the Hecke algebra of a G(m, 1, r)) can be determined by the character values of some special elements in the algebra. These special elements correspond to the reduced expressions, which are constructed by the algorithm, for some special conjugacy class representatives of minimal length, one in each class. Quasi-parabolic subgroups are introduced for investigating representations of Ariki- Koike algebras. We use n x n arrays of non-negative integer sequences to characterize double cosets of quasi-parabolic subgroups. We define an analogue of permutation modules, for Ariki-Koike algebras, corresponding to certain subgroups indexed by multicompositions. These subgroups are naturally corresponding, not necessarily one-one, to quasi-parabolic subgroups. We prove that each of these modules is free and has a basis indexed by right cosets of the corresponding quasi-parabolic subgroup. We also construct Murphy type bases, Specht series for these modules, and establish a Young's rule in this case.

Thesis (M.S.)--Brigham Young University. Dept. of Chemical Engineering, 2005. / Includes bibliographical references (p. 159-161).

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Successful methods for a large class of nonlinear convex optimization problems have recently been developed. This class, known as self-scaled optimization problems, has been defined by Nesterov and Todd in 1994. As noticed by Guler in 1996, this class is best described using an algebraic structure known as Euclidean Jordan algebra, which provides an elegant and powerful unifying framework for its study. Euclidean Jordan algebras are now a popular setting for the analysis of algorithms designed for self-scaled optimization problems : dozens of research papers in optimization deal explicitely with them. This thesis proposes an extensive and self-contained description of Euclidean Jordan algebras, and shows how they can be used to design and analyze new algorithms for self-scaled optimization. Our work focuses on the so-called spectral functions on Euclidean Jordan algebras, a natural generalization of spectral functions of symmetric matrices. Based on an original variational analysis technique for Euclidean Jordan algebras, we discuss their most important properties, such as differentiability and convexity. We show how these results can be applied in order to extend several algorithms existing for linear or second-order programming to the general class of self-scaled problems. In particular, our methods allowed us to extend to some nonlinear convex problems the powerful smoothing techniques of Nesterov, and to demonstrate its excellent theoretical and practical efficiency.