Besov spaces and the Laplacian on fractal H-sets

Nesta tese são estudados espaços de Besov de suavidade generalizada em espaços euclidianos, numa classe de fractais designados conjuntos-h e em estruturas abstractas designadas por espaços-h. Foram obtidas caracterizações e propriedades para estes espaços de funções. Em particular, no caso de espaços de Besov em espaços euclidianos, foram obtidas caracterizações por diferenças e por decomposições em átomos não suaves, foi provada uma propriedade de homogeneidade e foram estudados multiplicadores pontuais. Para espaços de Besov em conjuntos-h foi obtida uma caracterização por decomposições em átomos não suaves e foi construído um operador extensão. Com o recurso a cartas, os resultados obtidos para estes espaços de funções em fractais foram aplicados para definir e trabalhar com espaços de Besov de suavidade generalizada em estruturas abstractas. Nesta tese foi também estudado o laplaciano fractal, considerado a actuar em espaços de Besov de suavidade generalizada em domínios que contêm um conjunto-h fractal. Foram obtidos resultados no contexto de teoria espectral para este operador e foi estudado, à custa deste operador, um problema de Dirichlet fractal no contexto de conjuntos-h.

Main eigenvalues and (κ, τ)-regular sets

A (κ, τ)-regular set is a subset of the vertices of a graph G, inducing a κ-regular subgraph such that every vertex not in the subset has τ neighbors in it. A main eigenvalue of the adjacency matrix A of a graph G has an eigenvector not orthogonal to the all-one vector j. For graphs with a (κ, τ)-regular set a necessary and sufficient condition for an eigenvalue be non-main is deduced and the main eigenvalues are characterized. These results are applied to the construction of infinite families of bidegreed graphs with two main eigenvalues and the same spectral radius (index) and some relations with strongly regular graphs are obtained. Finally, the determination of (κ, τ)-regular sets is analyzed. © 2009 Elsevier Inc. All rights reserved.

Cumulus momentum fluxes in cloud-resolving model simulations of toga-coare

O objectivo deste trabalho científico é o estudo do transporte vertical de momento linear horizontal (CMT) realizado por sistemas de nuvens de convecção profunda sobre o oceano tropical. Para realizar este estudo, foram utilizadas simulações tridimensionais produzidas por um modelo explícito de nuvens (CRM) para os quatro meses de duração da campanha observacional TOGA COARE que ocorreu sobre as águas quentes do Pacífico ocidental. O estudo foca essencialmente as características estatísticas e à escala da nuvem do CMT durante um episódio de fortes ventos de oeste e durante um período de tempo maior que incluí este evento de convecção profunda. As distribuições verticais e altitude-temporais de campos atmosféricos relacionados com o CMT são avaliadas relativamente aos campos observacionais disponíveis, mostrando um bom acordo com os resultados de estudos anteriores, confirmando assim a boa qualidade das primeiras e fornecendo a confiança necessária para continuar a investigação. A sensibilidade do CMT em relação do domínio espacial do model é analisada, utilizando dois tipos de simulações tridimensionais produzidas por domínios horizontais de diferente dimensão, sugerindo que o CMT não depende da dimensão do domínio espacial horizontal escolhido para simular esta variável. A capacidade da parameterização do comprimento de mistura simular o CMT é testada, destacando as regiões troposféricas onde os fluxos de momento linear horizontal são no sentido do gradiente ou contra o gradiente. Os fluxos no sentido do gradiente apresentam-se relacionados a uma fraca correlação entre os campos atmosféricos que caracterizam esta parameterização, sugerindo que as formulações dos fluxos de massa dentro da nuvem e o fenómeno de arrastamento do ar para dentro da nuvem devem ser revistos. A importância do ar saturado e não saturado para o CMT é estudada com o objectivo de alcançar um melhor entendimento acerca dos mecanismos físicos responsáveis pelo CMT. O ar não saturado e saturado na forma de correntes descendentes contribuem de forma determinante para o CMT e deverão ser considerados em futuras parameterizações do CMT e da convecção em nuvens cumulus. Métodos de agrupamento foram aplicados às contribuições do ar saturado e não saturado, analisando os campos da força de flutuação e da velocidade vertical da partícula de ar, concluindo-se a presença de ondas gravíticas internas como mecanismo responsável pelo ar não saturado. A força do gradiente de pressão dentro da nuvem é também avaliada, utilizando para este efeito a fórmula teórica proposta por Gregory et al. (1997). Uma boa correlação entre esta força e o produto entre efeito de cisalhamento do vento e a perturbação da velocidade vertical é registada, principalmente para as correntes ascendentes dentro da nuvem durante o episódio de convecção profunda. No entanto, o valor ideal para o coeficiente empírico c*, que caracteriza a influência da força do gradiente de pressão dentro da nuvem sobre a variação vertical da velocidade horizontal dentro da nuvem, não é satisfatoriamente alcançado. Bons resultados são alcançados através do teste feito à aproximação do fluxo de massa proposta por Kershaw e Gregory (1997) para o cálculo do CMT total, revelando mais uma vez a importância do ar não saturado para o CMT.

Polyhedral study of mixed integer sets arising from inventory problems

“Branch-and-cut” algorithm is one of the most efficient exact approaches to solve mixed integer programs. This algorithm combines the advantages of a pure branch-and-bound approach and cutting planes scheme. Branch-and-cut algorithm computes the linear programming relaxation of the problem at each node of the search tree which is improved by the use of cuts, i.e. by the inclusion of valid inequalities. It should be taken into account that selection of strongest cuts is crucial for their effective use in branch-and-cut algorithm. In this thesis, we focus on the derivation and use of cutting planes to solve general mixed integer problems, and in particular inventory problems combined with other problems such as distribution, supplier selection, vehicle routing, etc. In order to achieve this goal, we first consider substructures (relaxations) of such problems which are obtained by the coherent loss of information. The polyhedral structure of those simpler mixed integer sets is studied to derive strong valid inequalities. Finally those strong inequalities are included in the cutting plane algorithms to solve the general mixed integer problems. We study three mixed integer sets in this dissertation. The first two mixed integer sets arise as a subproblem of the lot-sizing with supplier selection, the network design and the vendor-managed inventory routing problems. These sets are variants of the well-known single node fixed-charge network set where a binary or integer variable is associated with the node. The third set occurs as a subproblem of mixed integer sets where incompatibility between binary variables is considered. We generate families of valid inequalities for those sets, identify classes of facet-defining inequalities, and discuss the separation problems associated with the inequalities. Then cutting plane frameworks are implemented to solve some mixed integer programs. Preliminary computational experiments are presented in this direction.

Calculus of variations on time scales and applications to economics

We consider some problems of the calculus of variations on time scales. On the beginning our attention is paid on two inverse extremal problems on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variation functional that attains a local minimum at a given point of the vector space. Furthermore, we prove a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. Afterwards, we prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems. Next we investigate the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange equations in integral form, transversality conditions, and necessary optimality conditions for isoperimetric problems, on an arbitrary time scale, are proved. In the end, two main issues of application of time scales in economic, with interesting results, are presented. In the former case we consider a firm that wants to program its production and investment policies to reach a given production rate and to maximize its future market competitiveness. The model which describes firm activities is studied in two different ways: using classical discretizations; and applying discrete versions of our result on time scales. In the end we compare the cost functional values obtained from those two approaches. The latter problem is more complex and relates to rate of inflation, p, and rate of unemployment, u, which inflict a social loss. Using known relations between p, u, and the expected rate of inflation π, we rewrite the social loss function as a function of π. We present this model in the time scale framework and find an optimal path π that minimizes the total social loss over a given time interval.

Relations between (κ, τ)-regular sets and star complements

Let G be a finite graph with an eigenvalue μ of multiplicity m. A set X of m vertices in G is called a star set for μ in G if μ is not an eigenvalue of the star complement G\X which is the subgraph of G induced by vertices not in X. A vertex subset of a graph is (k ,t)-regular if it induces a k -regular subgraph and every vertex not in the subset has t neighbors in it. We investigate the graphs having a (k,t)-regular set which induces a star complement for some eigenvalue. A survey of known results is provided and new properties for these graphs are deduced. Several particular graphs where these properties stand out are presented as examples.

Determination of (0,2)-regular sets in graphs and applications

In this paper, relevant results about the determination of (k,t)-regular sets, using the main eigenvalues of a graph, are reviewed and some results about the determination of (0,2)-regular sets are introduced. An algorithm for that purpose is also described. As an illustration, this algorithm is applied to the determination of maximum matchings in arbitrary graphs.

Integer-valued APARCH processes in the analysis of time series of counts

The Asymmetric Power Arch representation for the volatility was introduced by Ding et al.(1993) in order to account for asymmetric responses in the volatility in the analysis of continuous-valued financial time series like, for instance, the log-return series of foreign exchange rates, stock indices or share prices. As reported by Brannas and Quoreshi (2010), asymmetric responses in volatility are also observed in time series of counts such as the number of intra-day transactions in stocks. In this work, an asymmetric power autoregressive conditional Poisson model is introduced for the analysis of time series of counts exhibiting asymmetric overdispersion. Basic probabilistic and statistical properties are summarized and parameter estimation is discussed. A simulation study is presented to illustrate the proposed model. Finally, an empirical application to a set of data concerning the daily number of stock transactions is also presented to attest for its practical applicability in data analysis.

Fuzzy Ensembles for Embedding Adaptive Behaviours in Semi-Autonomous Avatars in 3D Virtual Worlds

Semi-autonomous avatars should be both realistic and believable. The goal is to learn from and reproduce the behaviours of the user-controlled input to enable semi-autonomous avatars to plausibly interact with their human-controlled counterparts. A powerful tool for embedding autonomous behaviour is learning by imitation. Hence, in this paper an ensemble of fuzzy inference systems cluster the user input data to identify natural groupings within the data to describe the users movement and actions in a more abstract way. Multiple clustering algorithms are investigated along with a neuro-fuzzy classifier; and an ensemble of fuzzy systems are evaluated.

Estimating fuzzy membership functions parameters by the levenberg-marquardt algorithm

In previous papers from the authors fuzzy model identification methods were discussed. The bacterial algorithm for extracting fuzzy rule base from a training set was presented. The Levenberg-Marquardt algorithm was also proposed for determining membership functions in fuzzy systems. In this paper the Levenberg-Marquardt technique is improved to optimise the membership functions in the fuzzy rules without Ruspini-partition. The class of membership functions investigated is the trapezoidal one as it is general enough and widely used. The method can be easily extended to arbitrary piecewise linear functions as well.

Genetic programming and bacterial algorithm for neural networks and fuzzy systems design

In the field of control systems it is common to use techniques based on model adaptation to carry out control for plants for which mathematical analysis may be intricate. Increasing interest in biologically inspired learning algorithms for control techniques such as Artificial Neural Networks and Fuzzy Systems is in progress. In this line, this paper gives a perspective on the quality of results given by two different biologically connected learning algorithms for the design of B-spline neural networks (BNN) and fuzzy systems (FS). One approach used is the Genetic Programming (GP) for BNN design and the other is the Bacterial Evolutionary Algorithm (BEA) applied for fuzzy rule extraction. Also, the facility to incorporate a multi-objective approach to the GP algorithm is outlined, enabling the designer to obtain models more adequate for their intended use.

B-spline and neuro-fuzzy models design with function and derivative equalities

The design of neuro-fuzzy models is still a complex problem, as it involves not only the determination of the model parameters, but also its structure. Of special importance is the incorporation of a priori information in the design process. In this paper two known design algorithms for B-spline models will be updated to account for function and derivatives equality restrictions, which are important when the neural model is used for performing single or multi-objective optimization on-line.

Bacterial algorithm applied for fuzzy rule extraction

This paper presents a method of using the so-colled "bacterial algorithm" (4,5) for extracting a fuzzy rule base from a training set. The bewly proposed bacterial evolutionary algorithm (BEA) is shown. In our application one bacterium corresponds to a fuzzy rule system.

Anytime information processing based on fuzzy and neural network models

In modern measurement and control systems, the available time and resources are often not only limited, but could change during the operation of the system. In these cases, the so-called anytime algorithms could be used advantageously. While diflerent soft computing methods are wide-spreadly used in system modeling, their usability in these cases are limited.