Orientador: Plamen Emilov Koshlukov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatística e Computação Científica / Made available in DSpace on 2018-08-14T14:50:24Z (GMT). No. of bitstreams: 1 Freitas_JoseAntonioOliveirade_D.pdf: 1578135 bytes, checksum: a3352669dd5077f0f5949766026e7bb1 (MD5) Previous issue date: 2009 / Resumo: Nesta tese estudamos identidades polinomiais graduadas para certas álgebras. Inicialmente, estudamos identidades satisfeitas pelo produto tensorial Z2-graduado. Este estudo foi motivado pelo trabalho de Regev e Seeman com produtos tensoriais Z2-graduados. Eles provaram vários casos nos qual tal produto tensorial é PI equivalente a certas álgebras T-primas. Também conjeturaram que isto sempre ocorre. Trabalhamos com os demais casos e conseguimos provar que tal conjetura e verdadeira. Alêm disso provamos que para certas álgebras, quando consideramos corpos de característica positiva, o produto tensorial graduado ainda se comporta como o não graduado. Consideramos também o produto tensorial-graduado e suas identidades. Provamos que o Teorema A B de Regev continua válido no caso do produto tensorial-graduado quando as álgebras são graduadas por grupos abelianos nitos, e é um bicaracter antissimétrico. Também estudamos a PI equivalência do produto tensorial-graduado de álgebras T-primas. Em seguida estudamos identidades graduadas, descrevemos um conjunto de geradores para as identidades Z-graduadas da álgebra de Lie W1. A álgebra W1 é a álgebra das derivações do anel de polinômios K[t], e é conhecida como a álgebra de Witt. Provamos que se a característica do corpo for 0, então as identidades Z-graduadas de W1 são geradas por um conjunto de identidades de grau 2 e 3. Mais ainda, provamos que não é possível obter um conjunto nito de geradores para as identidades Z-graduadas de W1. / Abstract: In this PhD thesis we study graded polynomial identities for certain types of algebras. First, we study polynomial identities satised by the Z2-graded tensor products. This research was motivated by the paper of Regev and Seeman about the Z2-graded tensor products. They proved that in a series of cases such tensor products are PI equivalent to T-prime algebras. Then they conjectured that this is always the case. We deal here with the remaining cases and thus conrm Regev and Seeman's conjecture. Furthermore, we prove that for some algebras we can remove the restriction on the characteristic of the base eld, and we show that the behaviour of the corresponding graded tensor products is quite similar to that for the usual ungraded tensor products. We consider too the graded tensor products and their identities where is a skew symmetric bicharacter. We show that Regev's A B theorem holds for graded tensor products whenever the gradings are by nite abelian groups. Furthermore we study the PI equivalence of -graded tensor products of T-prime algebras. Afterwards we study the graded identities of the Lie algebra W1. We describe a set of generators of the Z-graded identities of W1. The algebra W1 is the algebra of derivation of the polynomial ring K[t], and it is known as the Witt algebra. We prove that if K is a eld of characteristic 0, then the Z-graded identities of W1 are consequences of a collection of polynomials of degree 2 and 3. Furthermore we prove that the Z-graded identities for W1 do not admit a nite basis. / Doutorado / Algebra / Doutor em Matemática

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Markov chains offer a rigorous mathematical framework to describe systems that exhibit stochastic behaviour, as they are supported by a plethora of methodologies to analyse their properties. Stochastic process algebras are high-level formalisms, where systems are represented as collections of interacting components. This compositional approach to modelling allows us to describe complex Markov chains using a compact high-level specification. There is an increasing need to investigate the properties of complex systems, not only in the field of computer science, but also in computational biology. To explore the stochastic properties of large Markov chains is a demanding task in terms of computational resources. Approximating the stochastic properties can be an effective way to deal with the complexity of large models. In this thesis, we investigate methodologies to approximate the stochastic behaviour of Markovian process algebra models. The discussion revolves around two main topics: approximate state-space aggregation and stochastic simulation. Although these topics are different in nature, they are both motivated by the need to efficiently handle complex systems. Approximate Markov chain aggregation constitutes the formulation of a smaller Markov chain that approximates the behaviour of the original model. The principal hypothesis is that states that can be characterised as equivalent can be adequately represented as a single state. We discuss different notions of approximate state equivalence, and how each of these can be used as a criterion to partition the state-space accordingly. Nevertheless, approximate aggregation methods typically require an explicit representation of the transition matrix, a fact that renders them impractical for large models. We propose a compositional approach to aggregation, as a means to efficiently approximate complex Markov models that are defined in a process algebra specification, PEPA in particular. Regarding our contributions to Markov chain simulation, we propose an accelerated method that can be characterised as almost exact, in the sense that it can be arbitrarily precise. We discuss how it is possible to sample from the trajectory space rather than the transition space. This approach requires fewer random samples than a typical simulation algorithm. Most importantly, our approach does not rely on particular assumptions with respect to the model properties, in contrast to otherwise more efficient approaches.

An algebra is a typical object of study in pure mathematics. Take a collection of numbers (for example, all whole numbers or all decimal numbers). Inside, you can add and multiply, but with respect to these operations different collections can behave differently. Here is an example of what I mean by this. The collection of whole numbers is called Z. Starting anywhere in Z you can get to anywhere else by adding other members of the collection: 9 + (-3) + (-6) = 0. This is not true with multiplication; to get from 5 to 1 you would need to multiply by 1/5 and 1/5 doesn’t exist in the restricted universe of Z. Enter R, the collection of all numbers that can be written as decimals. Now, if you start anywhere—apart from 0—you can get to anywhere else by multiplying by members of R—if you start at zero you’re stuck there. By adjusting what you mean by ‘add’ and ‘multiply’, you can add and multiply other things too, like polynomials, transformations or even symmetries. Some of these collections look different, but behave in similar ways and some look the same but are subtly different. By defining an algebra to be any collection of things with a rule to add and multiply in a sensible way, all of these examples (and many more you can’t imagine) can be treated in general. This is the power of abstraction: proving that an arbitrary algebra, A, has some property implies that every conceivable algebra (including Z and R) has that property too. In order to start navigating this universe of algebras it is useful to group them together by their behaviour or by how they are constructed. For example, R belongs to a class called simple algebras. There are mental laboratories full of machinery used to construct new and interesting algebras from old ones. One recipe, invented by Ivan Cherednik in 1993, produces Cherednik algebras. Attached to each algebra is a collection of modules (also called representations). As shadows are to a sculpture, each module is a simplified version of the algebra, with a taste of its internal structure. They are not algebras in their own right: they have no sense of multiplication, only addition. Being individually simple, modules are often much easier to study than the algebra itself. However, everything that is interesting about an algebra is captured by the collective behaviour of its modules. The analogy fails here: for example, shadows encode no information about colour. Sometimes the interplay between its modules leads to subtle and unexpected insights about the algebra itself. Nobody understands what the modules for Cherednik algebras look like. One first step is to simplify the problem by only considering modules which behave ‘nicely’. This is what is referred to as category O. Being Koszul is a rare property of an algebra that greatly helps to describe its behaviour. Also, each Koszul algebra is mysteriously linked with another called its Koszul dual. One of the main results of the thesis is that, in some cases, the modules in category O behave as if they were the modules for some Koszul algebra. It is an interesting question to ask, what the Koszul dual might be and what this has to do with Cherednik’s recipe. Geometers study tangled, many-dimensional spaces with holes. In analogy with the algebraic world, just as algebraists use modules to study algebras, geometers use sheaves to study their spaces. Suppose one could construct sheaves on some space whose behaviour is precisely the same as Cherednik algebra modules. Then, for example, theorems from geometry about sheaves could be used to say something about Cherednik algebra modules. One way of setting up this analogy is called localisation. This doesn’t always work in general. The last part of the thesis provides a rule for checking when it does.

No abstract / Mathematical Sciences / (D. Phil. (Mathematics))

Let G be a finite group. Given a contravariant, product preserving functor F:G-sets → AB, we construct a Green-functor A(F):G-sets → CRNG which specializes to the Burnside ring functor when F is trivial. A(F) permits a natural addition and multiplication between elements in the various groups F(S), S ∈ G-sets. If G is the Galois group of a field extension L/K, and SEP denotes the category of K-algebras which are isomorphic with a finite product of subfields of L, then any covariant, product preserving functor ρ:SEP → AB induces a functor Fᵨ:G → AB, and thus the Green-functor Aᵨ may be obtained. We use this observation for the case ρ = Br, the Brauer group functor, and show that Aᵦᵣ(G/G) is free on K-algebra isomorphism classes of division algebras with center in SEP. We then interpret the induction theory of Mackey-functors in this context. For a certain class of functors F, the structure of A(F) is especially tractable; for these functors we deduce that (DIAGRAM OMITTED), where the product is over isomorphism class representatives of transitive G-sets. This allows for the computation of the prime ideals of A(F)(G/G), and for an explicit structure theorem for Aᵦᵣ, when G is the Galois group of a p-adic field. We finish by considering the case when G = Gal(L/Q), for an arbitrary number field L.

published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

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