Topics in bifurcation theory

I. Existence and Structure of Bifurcation Branches

The problem of bifurcation is formulated as an operator equation in a Banach space, depending on relevant control parameters, say of the form G(u,λ) = 0. If dimN(G_u(u_O,λ_O)) = m the method of Lyapunov-Schmidt reduces the problem to the solution of m algebraic equations. The possible structure of these equations and the various types of solution behaviour are discussed. The equations are normally derived under the assumption that G^O_λεR(G^O_u). It is shown, however, that if G^O_λεR(G^O_u) then bifurcation still may occur and the local structure of such branches is determined. A new and compact proof of the existence of multiple bifurcation is derived. The linearized stability near simple bifurcation and "normal" limit points is then indicated.

II. Constructive Techniques for the Generation of Solution Branches

A method is described in which the dependence of the solution arc on a naturally occurring parameter is replaced by the dependence on a form of pseudo-arclength. This results in continuation procedures through regular and "normal" limit points. In the neighborhood of bifurcation points, however, the associated linear operator is nearly singular causing difficulty in the convergence of continuation methods. A study of the approach to singularity of this operator yields convergence proofs for an iterative method for determining the solution arc in the neighborhood of a simple bifurcation point. As a result of these considerations, a new constructive proof of bifurcation is determined.

Optimal scaling in ductile fracture

This work is concerned with the derivation of optimal scaling laws, in the sense of matching lower and upper bounds on the energy, for a solid undergoing ductile fracture. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. The solid is assumed to obey deformation-theory of plasticity and, in order to further simplify the analysis, we assume isotropic rigid-plastic deformations with zero plastic spin. When hardening exponents are given values consistent with observation, the energy is found to exhibit sublinear growth. We regularize the energy through the addition of nonlocal energy terms of the strain-gradient plasticity type. This nonlocal regularization has the effect of introducing an intrinsic length scale into the energy. We also put forth a physical argument that identifies the intrinsic length and suggests a linear growth of the nonlocal energy. Under these assumptions, ductile fracture emerges as the net result of two competing effects: whereas the sublinear growth of the local energy promotes localization of deformation to failure planes, the nonlocal regularization stabilizes this process, thus resulting in an orderly progression towards failure and a well-defined specific fracture energy. The optimal scaling laws derived here show that ductile fracture results from localization of deformations to void sheets, and that it requires a well-defined energy per unit fracture area. In particular, fractal modes of fracture are ruled out under the assumptions of the analysis. The optimal scaling laws additionally show that ductile fracture is cohesive in nature, i.e., it obeys a well-defined relation between tractions and opening displacements. Finally, the scaling laws supply a link between micromechanical properties and macroscopic fracture properties. In particular, they reveal the relative roles that surface energy and microplasticity play as contributors to the specific fracture energy of the material. Next, we present an experimental assessment of the optimal scaling laws. We show that when the specific fracture energy is renormalized in a manner suggested by the optimal scaling laws, the data falls within the bounds predicted by the analysis and, moreover, they ostensibly collapse---with allowances made for experimental scatter---on a master curve dependent on the hardening exponent, but otherwise material independent.

Singularities and horizons in the collisions of gravitational waves

This thesis presents a study of the dynamical, nonlinear interaction of colliding gravitational waves, as described by classical general relativity. It is focused mainly on two fundamental questions: First, what is the general structure of the singularities and Killing-Cauchy horizons produced in the collisions of exactly plane-symmetric gravitational waves? Second, under what conditions will the collisions of almost-plane gravitational waves (waves with large but finite transverse sizes) produce singularities?

In the work on the collisions of exactly-plane waves, it is shown that Killing horizons in any plane-symmetric spacetime are unstable against small plane-symmetric perturbations. It is thus concluded that the Killing-Cauchy horizons produced by the collisions of some exactly plane gravitational waves are nongeneric, and that generic initial data for the colliding plane waves always produce "pure" spacetime singularities without such horizons. This conclusion is later proved rigorously (using the full nonlinear theory rather than perturbation theory), in connection with an analysis of the asymptotic singularity structure of a general colliding plane-wave spacetime. This analysis also proves that asymptotically the singularities created by colliding plane waves are of inhomogeneous-Kasner type; the asymptotic Kasner axes and exponents of these singularities in general depend on the spatial coordinate that runs tangentially to the singularity in the non-plane-symmetric direction.

In the work on collisions of almost-plane gravitational waves, first some general properties of single almost-plane gravitational-wave spacetimes are explored. It is shown that, by contrast with an exact plane wave, an almost-plane gravitational wave cannot have a propagation direction that is Killing; i.e., it must diffract and disperse as it propagates. It is also shown that an almost-plane wave cannot be precisely sandwiched between two null wavefronts; i.e., it must leave behind tails in the spacetime region through which it passes. Next, the occurrence of spacetime singularities in the collisions of almost-plane waves is investigated. It is proved that if two colliding, almost-plane gravitational waves are initially exactly plane-symmetric across a central region of sufficiently large but finite transverse dimensions, then their collision produces a spacetime singularity with the same local structure as in the exact-plane-wave collision. Finally, it is shown that a singularity still forms when the central regions are only approximately plane-symmetric initially. Stated more precisely, it is proved that if the colliding almost-plane waves are initially sufficiently close to being exactly plane-symmetric across a bounded central region of sufficiently large transverse dimensions, then their collision necessarily produces spacetime singularities. In this case, nothing is now known about the local and global structures of the singularities.

Sociedade de Geografia do Rio de Janeiro: identidade e espaço nacional (1883-1909)

Criada em 1883 por um grupo de políticos e intelectuais da Corte Imperial, a Sociedade de Geografia do Rio de Janeiro teve como meta sistematizar os conhecimentos geográficos disponíveis. Com o advento da República, a entidade passou por uma série de percalços, uma vez que os novos governantes a identificavam como um reduto monarquista. Apesar das dificuldades, a Sociedade continuou a desenvolver suas atividades. A geografia fornecia o conhecimento necessário para resolver os problemas do Estado, tais como o desconhecimento do território nacional e a necessidade de novas vias de comunicação. Com intuito de resolver tais questões, os seus associados esforçaram-se colocando a ciência a serviço da nação.

A Real-Time Sliding Mode Control for a Wind Energy System Based on a Doubly Fed Induction Generator

In this paper, a real time sliding mode control scheme for a variable speed wind turbine that incorporates a doubly feed induction generator is described. In this design, the so-called vector control theory is applied, in order to simplify the system electrical equations. The proposed control scheme involves a low computational cost and therefore can be implemented in real-time applications using a low cost Digital Signal Processor (DSP). The stability analysis of the proposed sliding mode controller under disturbances and parameter uncertainties is provided using the Lyapunov stability theory. A new experimental platform has been designed and constructed in order to analyze the real-time performance of the proposed controller in a real system. Finally, the experimental validation carried out in the experimental platform shows; on the one hand that the proposed controller provides high-performance dynamic characteristics, and on the other hand that this scheme is robust with respect to the uncertainties that usually appear in the real systems.

Teoria de controle ótimo com aplicações a sistemas biológicos

Neste trabalho apresentamos as etapas para a utilização do método da Programação Dinâmica, ou Princípio de Otimização de Bellman, para aplicações de controle ótimo. Investigamos a noção de funções de controle de Lyapunov (FCL) e sua relação com a estabilidade de sistemas autônomos com controle. Uma função de controle de Lyapunov deverá satisfazer a equação de Hamilton-Jacobi-Bellman (H-J-B). Usando esse fato, se uma função de controle de Lyapunov é conhecida, será então possível determinar a lei de realimentação ótima; isto é, a lei de controle que torna o sistema globalmente assintóticamente controlável a um estado de equilíbrio. Como aplicação, apresentamos uma modelagem matemática adequada a um problema de controle ótimo de certos sistemas biológicos. Este trabalho conta também com um breve histórico sobre o desenvolvimento da Teoria de Controle de forma a ilustrar a importância, o progresso e a aplicação das técnicas de controle em diferentes áreas ao longo do tempo.

Klaxon, Base, Noigrandes : o design das revistas brasileiras de vanguarda

As revistas Klaxon, base e Noigandres são exemplos de revistas de vanguarda em que o design gráfico foi usado como instrumento para a expressão de idéias, compartilhadas pelos grupos artísticos envolvidos em sua publicação. São marcos do design moderno, encaixando-se no conceito de produto erudito (Bourdieu), destinado não ao grande público, mas a grupos de pares, que compartilhavam os valores estéticos e as ambições à condição de vanguarda em seus respectivos domínios artísticos. Estas revistas são expoentes do campo do design gráfico moderno, que só viria a se institucionalizar na década de 60, com a criação da ESDI e o surgimento do movimento concretista nas artes plásticas.

Length-weight relationship of marine fish species off Sao Sebastiao system, Sao Paulo, Southeastern Brazil

The parameters a and b of the length-weight relationship of the form W = a . L super(b) were estimated for 57 fish species sampled in Sao Sebastiao Channel and shelf system in 1997, Sao Paulo, Brazil. The b values ranged from 2.746 to 3.617. The Student's t-test revealed that mot (44) species had b values significantly different from 3. A normal distribution of the calculated LWR exponents (b) was obtained.

Localização eletrônica de sistemas aperiódicos em uma dimensão

Desde a descoberta do estado quasicristalino por Daniel Shechtman et al. em 1984 e da fabricação por Roberto Merlin et al. de uma superrede artificial de GaAs/ AlAs em 1985 com características da sequência de Fibonacci, um grande número de trabalhos teóricos e experimentais tem relatado uma variedade de propriedades interessantes no comportamento de sistemas aperiódicos. Do ponto de vista teórico, é bem sabido que a cadeia de Fibonacci em uma dimensão se constitui em um protótipo de sucesso para a descrição do estado quasicristalino de um sólido. Dependendo da regra de inflação, diferentes tipos de estruturas aperiódicas podem ser obtidas. Esta diversidade originou as chamadas regras metálicas e devido à possibilidade de tratamento analítico rigoroso este modelo tem sido amplamente estudado. Neste trabalho, propriedades de localização em uma dimensão são analisadas considerando-se um conjunto de regras metálicas e o modelo de ligações fortes de banda única. Considerando-se o Hamiltoniano de ligações fortes com um orbital por sítio obtemos um conjunto de transformações relativas aos parâmetros de dizimação, o que nos permitiu calcular as densidades de estados (DOS) para todas as configurações estudadas. O estudo detalhado da densidade de estados integrada (IDOS) para estes casos, mostra o surgimento de plateaux na curva do número de ocupação explicitando o aparecimento da chamada escada do diabo" e também o caráter fractal destas estruturas. Estudando o comportamento da variação da energia em função da variação da energia de hopping, construímos padrões do tipo borboletas de Hofstadter, que simulam o efeito de um campo magnético atuando sobre o sistema. A natureza eletrônica dos auto estados é analisada a partir do expoente de Lyapunov (γ), que está relacionado com a evolução da função de onda eletrônica ao longo da cadeia unidimensional. O expoente de Lyapunov está relacionado com o inverso do comprimento de localização (ξ= 1 /γ), sendo nulo para os estados estendidos e positivo para estados localizados. Isto define claramente as posições dos principais gaps de energia do sistema. Desta forma, foi possível analisar o comportamento autossimilar de cadeias com diferentes regras de formação. Analisando-se o espectro de energia em função do número de geração de cadeias que seguem as regras de ouro e prata foi feito, obtemos conjuntos do tipo-Cantor, que nos permitiu estudar o perfil do calor específico de uma cadeia e Fibonacci unidimensional para diversas gerações

O Processo de esvaziamento da função docente: Bases Políticas e Teóricas

O objetivo precípuo da pesquisa que ora se apresenta consiste em analisar as políticas e reformas da educação no que concerne à Formação de Professores da década de 1990, considerando o intenso barateamento e a precarização, que vem ocorrendo nas últimas décadas, nessa formação. Buscamos entender em que medida as políticas são contingentes e elaboradas para atender aos problemas emergenciais e porque não trazem em seu bojo a construção de ações que visam à transformação da formação e da própria sociedade. Busca, ainda, compreender se a Teoria Reflexiva, cujos expoentes são John Dewey e Donald Schön, que fundamenta o trabalho de alguns professores formadores, é uma extensão das políticas e reformas empreendidas no Brasil nas últimas décadas. De igual modo, visa a analisar se palavras de ordem como flexibilização, polivalência, habilidades e competência, que compõem a agenda neoliberal, foram aderidas ao pacote para a Formação de Professores da Teoria Reflexiva. Pretende contrapor os fundamentos da Teoria Reflexiva àqueles da Pedagogia Histórico-Crítica. Utiliza, nesta etapa da investigação, a pesquisa bibliográfica e documental pertinente e levanta informações sobre o Curso de Pedagogia no Brasil por meio da pesquisa empírica, que ocorreu em três Universidades: Universidade Estadual de Ponta Grossa (UEPG), Universidade Federal do Paraná (UFPR) e Universidade Estadual do Oeste do Paraná (UNIOESTE-Câmpus Cascavel). Analisa o Projeto Político-Pedagógico de cada uma para detectar qual Referencial Teórico pautou sua construção e realiza entrevistas semiestruturadas com professores do Curso de Pedagogia, buscando meios que possibilitou realizar o levantamento do possível Referencial Teórico que pauta o trabalho desses professores

Influence of particle properties on the erosive wear of sintered boron carbide

Sintered boron carbide is very hard, and can be an attractive material for wear-resistant components in critical applications. Previous studies of the erosion of less hard ceramics have shown that their wear resistance depends on the nature of the abrasive particles. Erosion tests were performed on a sintered boron carbide ceramic with silica, alumina and silicon carbide erodents. The different erodents caused different mechanisms of erosion, either by lateral cracking or small-scale chipping; the relative values of the hardness of the erodent and the target governed the operative mechanism. The small-scale chipping mechanism led to erosion rates typically an order of magnitude lower than the lateral fracture mechanism. The velocity exponents for erosion in the systems tested were similar to those seen in other work, except that measured with the 125 to 150 μm silica erodent. With this erodent the exponent was initially high, then decreased sharply with increasing velocity and became negative. It was proposed that this was due to deformation and fragmentation of the erodent particles. In the erosion testing of ceramics, the operative erosion mechanism is important. Care must be taken to ensure that the same mechanism is observed in laboratory testing as that which would be seen under service conditions, where the most common erodent is silica.

Determination of length-weight relationship in Sepia pharaonis and test for sex specific variation

The parameters a and b of the length-weight relationship of Sepia pharaonis of the form of W=a.L was determined. Sex separated size fequency data collected from Karachi fish Harbour was analysed the length-weight equations, separable by male, female and sex combined. The apparent difference in paired values of exponents b1, b2 for any combination i.e. male versus female and male, female versus sex combined was tested for their significant difference. No significant difference was observed for any combination, this indicated no sex specific variation in length-weigh relationship of Sepia pharaonis.

SCALING OF LIMB PROPORTIONS AND LIMB BONE DIAMETERS IN 3 SPECIES OF CHINESE SNUB-NOSED LANGURS (GENUS RHINOPITHECUS)

In this study, aspects of the structural mechanics of the upper and lower limbs of the three Chinese species of Rhinopithecus were examined. Linear regression and reduced major axis (RMA) analyses of natural log-transformed data were used to examine the dimensions of limb bones and other relationships to body size and locomotion. The results of this study suggest that: (1) the allometry exponents of the lengths of long limbs deviate from isometry, being moderately negative, while the shaft diameters (both sagittal and transverse) show significantly positive allometry; (2) the sagittal diameters of the tibia and ulna show extremely significantly positive allometry - the relative enlargement of the sagittal, as opposed to transverse, diameters of these bones suggests that the distal segments of the fore- and hindlimbs of Rhinopithecus experience high bending stresses during locomotion; (3) observations of Rhinopithecus species in the field indicate that all species engage in energetic leaping during arboreal locomotion. The limbs experience rapid and dramatic decelerations upon completion of a leap. We suggest that these occasional decelerations produce high bending stresses in the distal limb segments and so account for the hypertrophy of the sagittal diameters of the ulna and tibia.

Outage probability of the gaussian MIMO free-space optical channel with PPM

Atmospheric effects can significantly degrade the reliability of free-space optical communications. One such effect is scintillation, caused by atmospheric turbulence, refers to random fluctuations in the irradiance and phase of the received laser beam. In this paper we inv stigate the use of multiple lasers and multiple apertures to mitigate scintillation. Since the scintillation process is slow, we adopt a block fading channel model and study the outage probability under the assumptions of orthogonal pulse-position modulation and non-ideal photodetection. Assuming perfect receiver channel state information (CSI), we derive the signal-to-noise ratio (SNR) exponents for the cases when the scintillation is lognormal, exponential and gammagamma distributed, which cover a wide range of atmospheric turbulence conditions. Furthermore, when CSI is also available at the transmitter, we illustrate very large gains in SNR are possible (in some cases larger than 15 dB) by adapting the transmitted power. Under a long-term power constraint, we outline fundamental design criteria via a simple expression that relates the required number of lasers and apertures for a given code rate and number of codeword blocks to completely remove system outages. Copyright © 2009 IEEE.

A coordination-based approach to elasticity of floppy and stiff random networks

We study the role of connectivity on the linear and nonlinear elastic behavior of amorphous systems using a two-dimensional random network of harmonic springs as a model system. A natural characterization of these systems arises in terms of the network coordination relative to that of an isostatic network $\delta z$; a floppy network has $\delta z<0$, while a stiff network has $\delta z>0$. Under the influence of an externally applied load we observe that the response of both floppy and rigid network are controlled by the same critical point, corresponding to the onset of rigidity. We use numerical simulations to compute the exponents which characterize the shear modulus, the amplitude of non-affine displacements, and the network stiffening as a function of $\delta z$, derive these theoretically and make predictions for the mechanical response of glasses and fibrous networks.