Genus and Braid Index Associated to Sequences of Renormalizable Lorenz Maps

We describe the Lorenz links generated by renormalizable Lorenz maps with reducible kneading invariant (K(f)(-), = K(f)(+)) = (X, Y) * (S, W) in terms of the links corresponding to each factor. This gives one new kind of operation that permits us to generate new knots and links from the ones corresponding to the factors of the *-product. Using this result we obtain explicit formulas for the genus and the braid index of this renormalizable Lorenz knots and links. Then we obtain explicit formulas for sequences of these invariants, associated to sequences of renormalizable Lorenz maps with kneading invariant (X, Y) * (S,W)*(n), concluding that both grow exponentially. This is specially relevant, since it is known that topological entropy is constant on the archipelagoes of renormalization.

Symbolic dynamics and big bang bifurcation in Weibull-Gompertz-Fréchet's growth models

In this paper, motivated by the interest and relevance of the study of tumor growth models, a central point of our investigation is the study of the chaotic dynamics and the bifurcation structure of Weibull-Gompertz-Fréchet's functions: a class of continuousdefined one-dimensional maps. Using symbolic dynamics techniques and iteration theory, we established that depending on the properties of this class of functions in a neighborhood of a bifurcation point PBB, in a two-dimensional parameter space, there exists an order regarding how the infinite number of periodic orbits are born: the Sharkovsky ordering. Consequently, the corresponding symbolic sequences follow the usual unimodal kneading sequences in the topological ordered tree. We verified that under some sufficient conditions, Weibull-Gompertz-Fréchet's functions have a particular bifurcation structure: a big bang bifurcation point PBB. This fractal bifurcations structure is of the so-called "box-within-a-box" type, associated to a boxe ω1, where an infinite number of bifurcation curves issues from. This analysis is done making use of fold and flip bifurcation curves and symbolic dynamics techniques. The present paper is an original contribution in the framework of the big bang bifurcation analysis for continuous maps.

Symbolic dynamics and renormalization of non-autonomous k periodic dynamical systems

The purpose of this paper was to introduce the symbolic formalism based on kneading theory, which allows us to study the renormalization of non-autonomous periodic dynamical systems.

Tuneable micro- and nano-periodic structures in urethane/urea networks

Micro- and nano-patterned materials are of great importance for the design of new nanoscale electronic, optical and mechanical devices, ranging from sensors to displays. A prospective system that can support a designed functionality is elastomeric polyurethane thin films with nano- or micromodulated surface structures ("wrinkles"). These wrinkles can be induced on different lengthscales by mechanically stretching the films, without the need for any sophisticated lithographic techniques. In the present article we focus on the experimental control of the wrinkling process. A simple model for wrinkle formation is also discussed, and some preliminary results reported. Hierarchical assembly of these tunable structures paves the way for the development of a new class of materials with a wide range of applications, from electronics to biomedicine.

Periodic paths on nonautonomous graphs

We define nonautonomous graphs as a class of dynamic graphs in discrete time whose time-dependence consists in connecting or disconnecting edges. We study periodic paths in these graphs, and the associated zeta functions. Based on the analytic properties of these zeta functions we obtain explicit formulae for the number of n-periodic paths, as the sum of the nth powers of some specific algebraic numbers.

Large-area homogeneous periodic surface structures generated on the surface sputtered boron carbide thin films by femtosecond laser processing

Amorphous and crystalline sputtered boron carbide thin films have a very high hardness even surpassing that of bulk crystalline boron carbide (≈41 GPa). However, magnetron sputtered B-C films have high friction coefficients (C.o.F) which limit their industrial application. Nanopatterning of materials surfaces has been proposed as a solution to decrease the C.o.F. The contact area of the nanopatterned surfaces is decreased due to the nanometre size of the asperities which results in a significant reduction of adhesion and friction. In the present work, the surface of amorphous and polycrystalline B-C thin films deposited by magnetron sputtering was nanopatterned using infrared femtosecond laser radiation. Successive parallel laser tracks 10 μm apart were overlapped in order to obtain a processed area of about 3 mm2. Sinusoidal-like undulations with the same spatial period as the laser tracks were formed on the surface of the amorphous boron carbide films after laser processing. The undulations amplitude increases with increasing laser fluence. The formation of undulations with a 10 μm period was also observed on the surface of the crystalline boron carbide film processed with a pulse energy of 72 μJ. The amplitude of the undulations is about 10 times higher than in the amorphous films processed at the same pulse energy due to the higher roughness of the films and consequent increase in laser radiation absorption. LIPSS formation on the surface of the films was achieved for the three B-C films under study. However, LIPSS are formed under different circumstances. Processing of the amorphous films at low fluence (72 μJ) results in LIPSS formation only on localized spots on the film surface. LIPSS formation was also observed on the top of the undulations formed after laser processing with 78 μJ of the amorphous film deposited at 800 °C. Finally, large-area homogeneous LIPSS coverage of the boron carbide crystalline films surface was achieved within a large range of laser fluences although holes are also formed at higher laser fluences.

Spectral invariants of periodic nonautonomous discrete dynamical systems

For an interval map, the poles of the Artin-Mazur zeta function provide topological invariants which are closely connected to topological entropy. It is known that for a time-periodic nonautonomous dynamical system F with period p, the p-th power [zeta(F) (z)](p) of its zeta function is meromorphic in the unit disk. Unlike in the autonomous case, where the zeta function zeta(f)(z) only has poles in the unit disk, in the p-periodic nonautonomous case [zeta(F)(z)](p) may have zeros. In this paper we introduce the concept of spectral invariants of p-periodic nonautonomous discrete dynamical systems and study the role played by the zeros of [zeta(F)(z)](p) in this context. As we will see, these zeros play an important role in the spectral classification of these systems.

Nonautonomous Graphs and Topological Entropy of Nonautonomous Lorenz Systems

In this work, we associate a p-periodic nonautonomous graph to each p-periodic nonautonomous Lorenz system with finite critical orbits. We develop Perron-Frobenius theory for nonautonomous graphs and use it to calculate their entropy. Finally, we prove that the topological entropy of a p-periodic nonautonomous Lorenz system is equal to the entropy of its associated nonautonomous graph.

The Sticky Information Macro Model: Beyond Perfect Foresight

Sticky information monetary models have been used in the macroeconomic literature to explain some of the observed features regarding inflation dynamics. In this paper, we explore the consequences of relaxing the rational expectations assumption usually taken in this type of model; in particular, by considering expectations formed through adaptive learning, it is possible to arrive to results other than the trivial convergence to a fixed point long-term equilibrium. The results involve the possibility of endogenous cyclical motion (periodic and a-periodic), which emerges essentially in scenarios of hyperinflation. In low inflation settings, the introduction of learning implies a less severe impact of monetary shocks that, nevertheless, tend to last for additional time periods relative to the pure perfect foresight setup.

A influência da fixação na demonstração do glicogénio

A fixação é fundamental para a preservação dos tecidos e é o primeiro passo de uma técnica histológica de rotina cujo objectivo final é obter uma boa visualização das estruturas e, no caso da anatomia patológica, um diagnóstico. Algumas substâncias presentes nos tecidos, como é o caso do glicogénio, necessitam de uma solução fixadora específica capaz de prevenir a dispersão das suas moléculas, sendo a fixação do glicogénio favorecida se essa solução contiver álcool ou ácido pícrico na sua constituição. A preservação do glicogénio é fundamental porque da sua identificação podem depender diagnósticos, utilizando-se colorações histoquímicas como o Periodic Acid Schiff e o Carmim de Best para a sua visualização. Este estudo pretende comparar a qualidade da demonstração do glicogénio com PAS e Carmim de Best, em tecidos fixados com diferentes soluções fixadoras. O estudo foi efectuado a partir de cortes histológicos de fígados de porco, fixados em formol 10%, formol 10% tamponado, formol alcoólico, solução de Bouin e solução de Gendre e procedeu-se à análise estatística das avaliações feitas às lâminas. As soluções fixadoras que apresentaram melhores resultados para ambas as colorações foram o formol 10% tamponado e a solução de Gendre. Foi possível concluir que, numa fixação de 24 horas, o formol 10% tamponado pode ser utilizado com um risco mínimo de comprometer a preservação e demonstração do glicogénio. Em alternativa a esta solução fixadora, utilizada na rotina laboratorial, a solução de Gendre mostrou ser um fixador eficaz.

Multi-scaled femtosecond laser structuring of stationary titanium surfaces

The evolution of the topography of titanium surfaces treated with femtosecond laser radiation in stationary conditions as a function of radiation fluence and number of laser pulses is investigated. Depending on the processing parameters, ripples, microcolumns, wavy or smooth surfaces can be obtained. The ripples predominate for fluences near the damage threshold of titanium (0.2+/-0.1) J/cm(2), while microcolumns form during the first 200 pulses for fluences between (0.6+/-0.2) and (1.7+/-0.2) J/cm(2). A wavy topography develops for fluences and number of pulses higher than (1.7+/-0.2) J/cm(2) and 300, respectively. A bimodal surface topography consisting of surface ripples overlapping a microcolumnar topography can be obtained if the surfaces are firstly treated to create microcolumns followed by laser treatment with a lower fluence near the ablation threshold of the material, in order to generate periodic ripple

Dimensionamento de um passadiço pedonal em estrutura metálica, com fundações em betão armado

A elaboração deste projecto, integrado no âmbito do Trabalho Final de Mestrado, para a obtenção do grau de Mestre em Engenharia Civil, tem como objectivo o dimensionamento de um passadiço pedonal em estrutura metálica, com fundações em betão armado. Este documento inclui quase todos os elementos necessários ao projecto de execução da referida estrutura. Para o dimensionamento do passadiço pedonal procedeu-se à quantificação das acções e posteriormente à verificação da segurança de todos os elementos estruturais tendo por base os critérios e especificações técnicas preconizados nas Normas Europeias relativas ao projecto estrutural (Eurocódigos estruturais). Tratando-se de um passadiço destinado à circulação de peões e cuja estrutura metálica apresenta um certo grau de flexibilidade devido à esbelteza dos elementos estruturais, esta poderá estar sujeita a acções dinâmicas periódicas provocadas pelas pessoas quando percorrem o passadiço, podendo ocasionar certos níveis de vibração que sob o ponto de vista de segurança estrutural serão pouco relevantes, sendo no entanto excessivos do ponto de vista do conforto humano. Foi por isso efectuado um estudo dinâmico, com o objectivo de caracterizar a resposta dinâmica da estrutura quando solicitada a carregamentos de natureza periódica como é o caso da acção do peão, de modo a garantir que a utilização desta estrutura esteja dentro dos parâmetros de conforto aceitáveis. A modelação da estrutura e consequente discretização geral desta, foi feita recorrendo a programa de elementos finitos, SAP2000, versão 14.0.0. O dimensionamento das ligações constitui outros dos aspectos fundamentais no projecto desta estrutura metálica.

Measuring and Controlling the Chaotic Motion of Profits

The study of economic systems has generated deep interest in exploring the complexity of chaotic motions in economy. Due to important developments in nonlinear dynamics, the last two decades have witnessed strong revival of interest in nonlinear endogenous business chaotic models. The inability to predict the behavior of dynamical systems in the presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we study a specific economic model from the literature. More precisely, a system of three ordinary differential equations gather the variables of profits, reinvestments and financial flow of borrowings in the structure of a firm. Firstly, using results of symbolic dynamics, we characterize the topological entropy and the parameter space ordering of kneading sequences, associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. Finally, we show that complicated behavior arising from the chaotic firm model can be controlled without changing its original properties and the dynamics can be turned into the desired attracting time periodic motion (a stable steady state or into a regular cycle). The orbit stabilization is illustrated by the application of a feedback control technique initially developed by Romeiras et al. [1992]. This work provides another illustration of how our understanding of economic models can be enhanced by the theoretical and numerical investigation of nonlinear dynamical systems modeled by ordinary differential equations.

Ferromagnetic order in aged Co-doped TiO2 anatase nanopowders

Oxide based diluted magnetic semiconductor (DMS) materials have been a subject of increasing interest due to reports of room temperature ferromagnetism in several systems and their potential use in the development of spintronic devices. However, concerns on the stability of the magnetic properties of different DMS systems have been raised. Their magnetic moment is often unstable, vanishing with a characteristic decay time of weeks or months, which precludes the development of real applications. This paper reports on the ferromagnetic properties of two-year-aged Ti1-xCoxO2-δ reduced anatase nanopowders with different Co contents (0.03≤x≤0.10). Aged samples retain rather high values of magnetization, remanence and coercivity which provide strong evidence for a quite preserved long-range ferromagnetic order. In what concern Co segregation, some degree of metastability of the diluted Co doped anatase structure could be inferred in the case of the sample with the higher Co content.

Electrical transport properties of CuS single crystals

Electrical resistivity, transverse magnetoresistance and thermoelectric power measurements were performed on CuS high quality single crystals in the range 1.2-300 K and under fields of up to 16 T. The zero field resistivity data are well described below 55 K by a quasi-2D model, consistent with a carrier confinement at lower temperatures, before the transition to the superconducting state. The transverse magnetoresistance develops mainly below 30 K and attains values as large as 470% for a 16 T field at 5 K, this behaviour being ascribed to a band effect mechanism, with a possible magnetic field induced DOS change at the Fermi level. The transverse magnetoresistance shows no signs of saturation, following a power law with field Delta rho/rho(0) proportional to H(1.4), suggesting the existence of open orbits for carriers at the Fermi surface. The thermoelectric power shows an unusual temperature dependence, probably as a result of the complex band structure of CuS.