A singularly perturbed linear two-point boundary-value problem

We consider the following singularly perturbed linear two-point boundary-value problem:

Ly(x) ≡ Ω(ε)D_xy(x) - A(x,ε)y(x) = f(x,ε) 0≤x≤1 (1a)

By ≡ L(ε)y(0) + R(ε)y(1) = g(ε) ε → 0^+ (1b)

Here Ω(ε) is a diagonal matrix whose first m diagonal elements are 1 and last m elements are ε. Aside from reasonable continuity conditions placed on A, L, R, f, g, we assume the lower right mxm principle submatrix of A has no eigenvalues whose real part is zero. Under these assumptions a constructive technique is used to derive sufficient conditions for the existence of a unique solution of (1). These sufficient conditions are used to define when (1) is a regular problem. It is then shown that as ε → 0^+ the solution of a regular problem exists and converges on every closed subinterval of (0,1) to a solution of the reduced problem. The reduced problem consists of the differential equation obtained by formally setting ε equal to zero in (1a) and initial conditions obtained from the boundary conditions (1b). Several examples of regular problems are also considered.

A similar technique is used to derive the properties of the solution of a particular difference scheme used to approximate (1). Under restrictions on the boundary conditions (1b) it is shown that for the stepsize much larger than ε the solution of the difference scheme, when applied to a regular problem, accurately represents the solution of the reduced problem.

Furthermore, the existence of a similarity transformation which block diagonalizes a matrix is presented as well as exponential bounds on certain fundamental solution matrices associated with the problem (1).

Universal quantum computation with trapped ions in thermal motion by adiabatic passage

We propose a universal quantum computation scheme for trapped ions in thermal motion via the technique of adiabatic passage, which incorporates the advantages of both the adiabatic passage and the model of trapped ions in thermal motion. Our scheme is immune from the decoherence due to spontaneous emission from excited states as the system in our scheme evolves along a dark state. In our scheme the vibrational degrees of freedom are not required to be cooled to their ground states because they are only virtually excited. It is shown that the fidelity of the resultant gate operation is still high even when the magnitude of the effective Rabi frequency moderately deviates from the desired value.

The steady-state response of multidegree-of-freedom systems with a spatially localized nonlinearity

This thesis is concerned with the dynamic response of a General multidegree-of-freedom linear system with a one dimensional nonlinear constraint attached between two points. The nonlinear constraint is assumed to consist of rate-independent conservative and hysteretic nonlinearities and may contain a viscous dissipation element. The dynamic equations for general spatial and temporal load distributions are derived for both continuous and discrete systems. The method of equivalent linearization is used to develop equations which govern the approximate steady-state response to generally distributed loads with harmonic time dependence.

The qualitative response behavior of a class of undamped chainlike structures with a nonlinear terminal constraint is investigated. It is shown that the hardening or softening behavior of every resonance curve is similar and is determined by the properties of the constraint. Also examined are the number and location of resonance curves, the boundedness of the forced response, the loci of response extrema, and other characteristics of the response. Particular consideration is given to the dependence of the response characteristics on the properties of the linear system, the nonlinear constraint, and the load distribution.

Numerical examples of the approximate steady-state response of three structural systems are presented. These examples illustrate the application of the formulation and qualitative theory. It is shown that disconnected response curves and response curves which cross are obtained for base excitation of a uniform shear beam with a cubic spring foundation. Disconnected response curves are also obtained for the steady-state response to a concentrated load of a chainlike structure with a hardening hysteretic constraint. The accuracy of the approximate response curves is investigated.

Discounting the value of natural resources in costbenefit analysis: a case study for policy making

4 p.

The Equivalency Principle for Discounting the Value of Natural Assets: An Application to an Investment Project in the Basque Coast

25 p.

Efficient methods for stochastic optimal control

The Hamilton Jacobi Bellman (HJB) equation is central to stochastic optimal control (SOC) theory, yielding the optimal solution to general problems specified by known dynamics and a specified cost functional. Given the assumption of quadratic cost on the control input, it is well known that the HJB reduces to a particular partial differential equation (PDE). While powerful, this reduction is not commonly used as the PDE is of second order, is nonlinear, and examples exist where the problem may not have a solution in a classical sense. Furthermore, each state of the system appears as another dimension of the PDE, giving rise to the curse of dimensionality. Since the number of degrees of freedom required to solve the optimal control problem grows exponentially with dimension, the problem becomes intractable for systems with all but modest dimension.

In the last decade researchers have found that under certain, fairly non-restrictive structural assumptions, the HJB may be transformed into a linear PDE, with an interesting analogue in the discretized domain of Markov Decision Processes (MDP). The work presented in this thesis uses the linearity of this particular form of the HJB PDE to push the computational boundaries of stochastic optimal control.

This is done by crafting together previously disjoint lines of research in computation. The first of these is the use of Sum of Squares (SOS) techniques for synthesis of control policies. A candidate polynomial with variable coefficients is proposed as the solution to the stochastic optimal control problem. An SOS relaxation is then taken to the partial differential constraints, leading to a hierarchy of semidefinite relaxations with improving sub-optimality gap. The resulting approximate solutions are shown to be guaranteed over- and under-approximations for the optimal value function. It is shown that these results extend to arbitrary parabolic and elliptic PDEs, yielding a novel method for Uncertainty Quantification (UQ) of systems governed by partial differential constraints. Domain decomposition techniques are also made available, allowing for such problems to be solved via parallelization and low-order polynomials.

The optimization-based SOS technique is then contrasted with the Separated Representation (SR) approach from the applied mathematics community. The technique allows for systems of equations to be solved through a low-rank decomposition that results in algorithms that scale linearly with dimensionality. Its application in stochastic optimal control allows for previously uncomputable problems to be solved quickly, scaling to such complex systems as the Quadcopter and VTOL aircraft. This technique may be combined with the SOS approach, yielding not only a numerical technique, but also an analytical one that allows for entirely new classes of systems to be studied and for stability properties to be guaranteed.

The analysis of the linear HJB is completed by the study of its implications in application. It is shown that the HJB and a popular technique in robotics, the use of navigation functions, sit on opposite ends of a spectrum of optimization problems, upon which tradeoffs may be made in problem complexity. Analytical solutions to the HJB in these settings are available in simplified domains, yielding guidance towards optimality for approximation schemes. Finally, the use of HJB equations in temporal multi-task planning problems is investigated. It is demonstrated that such problems are reducible to a sequence of SOC problems linked via boundary conditions. The linearity of the PDE allows us to pre-compute control policy primitives and then compose them, at essentially zero cost, to satisfy a complex temporal logic specification.

Value chain analysis for sea cucumber in the Philippines

This study examined the sea cucumber industry in the Philippines through the value chain lens. The intent was to identify effective pathways for the successful introduction of sandfish culture as livelihood support for coastal communities. Value chain analysis is a high-resolution analytical tool that enables industry examination at a detailed level. Previous industry assessments have provided a general picture of the sea cucumber industry in the country. The present study builds on the earlier work and supplies additional details for a better understanding of the industry's status and problems, especially their implications for the Australian Center for International Agricultural Research (ACIAR) funded sandfish project "Culture of sandfish (Holothuria scabra) in Asia- Pacific" (FIS/2003/059). (PDF contains 54 pages)

Value Estimation and Comparison in Multi-Attribute Choice

The following work explores the processes individuals utilize when making multi-attribute choices. With the exception of extremely simple or familiar choices, most decisions we face can be classified as multi-attribute choices. In order to evaluate and make choices in such an environment, we must be able to estimate and weight the particular attributes of an option. Hence, better understanding the mechanisms involved in this process is an important step for economists and psychologists. For example, when choosing between two meals that differ in taste and nutrition, what are the mechanisms that allow us to estimate and then weight attributes when constructing value? Furthermore, how can these mechanisms be influenced by variables such as attention or common physiological states, like hunger?

In order to investigate these and similar questions, we use a combination of choice and attentional data, where the attentional data was collected by recording eye movements as individuals made decisions. Chapter 1 designs and tests a neuroeconomic model of multi-attribute choice that makes predictions about choices, response time, and how these variables are correlated with attention. Chapter 2 applies the ideas in this model to intertemporal decision-making, and finds that attention causally affects discount rates. Chapter 3 explores how hunger, a common physiological state, alters the mechanisms we utilize as we make simple decisions about foods.

Fast numerical methods for mixed, singular Helmholtz boundary value problems and Laplace eigenvalue problems - with applications to antenna design, sloshing, electromagnetic scattering and spectral geometry

This thesis presents a novel class of algorithms for the solution of scattering and eigenvalue problems on general two-dimensional domains under a variety of boundary conditions, including non-smooth domains and certain "Zaremba" boundary conditions - for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the methods for the Zaremba problems on smooth domains concern detailed information, which is put forth for the first time in this thesis, about the singularity structure of solutions of the Laplace operator under boundary conditions of Zaremba type. The new methods, which are based on use of Green functions and integral equations, incorporate a number of algorithmic innovations, including a fast and robust eigenvalue-search algorithm, use of the Fourier Continuation method for regularization of all smooth-domain Zaremba singularities, and newly derived quadrature rules which give rise to high-order convergence even around singular points for the Zaremba problem. The resulting algorithms enjoy high-order convergence, and they can tackle a variety of elliptic problems under general boundary conditions, including, for example, eigenvalue problems, scattering problems, and, in particular, eigenfunction expansion for time-domain problems in non-separable physical domains with mixed boundary conditions.

Stationary random response of multidegree-of-freedom systems

An approximate approach is presented for determining the stationary random response of a general multidegree-of-freedom nonlinear system under stationary Gaussian excitation. This approach relies on defining an equivalent linear system for the nonlinear system. Two particular systems which possess exact solutions have been solved by this approach, and it is concluded that this approach can generate reasonable solutions even for systems with fairly large nonlinearities. The approximate approach has also been applied to two examples for which no exact or approximate solutions were previously available.

Also presented is a matrix algebra approach for determining the stationary random response of a general multidegree-of-freedom linear system. Its derivation involves only matrix algebra and some properties of the instantaneous correlation matricies of a stationary process. It is therefore very direct and straightforward. The application of this matrix algebra approach is in general simpler than that of commonly used approaches.

On the uniqueness of singular solutions to boundary-initial value problems in linear elastodynamics

This investigation deals with certain generalizations of the classical uniqueness theorem for the second boundary-initial value problem in the linearized dynamical theory of not necessarily homogeneous nor isotropic elastic solids. First, the regularity assumptions underlying the foregoing theorem are relaxed by admitting stress fields with suitably restricted finite jump discontinuities. Such singularities are familiar from known solutions to dynamical elasticity problems involving discontinuous surface tractions or non-matching boundary and initial conditions. The proof of the appropriate uniqueness theorem given here rests on a generalization of the usual energy identity to the class of singular elastodynamic fields under consideration.

Following this extension of the conventional uniqueness theorem, we turn to a further relaxation of the customary smoothness hypotheses and allow the displacement field to be differentiable merely in a generalized sense, thereby admitting stress fields with square-integrable unbounded local singularities, such as those encountered in the presence of focusing of elastic waves. A statement of the traction problem applicable in these pathological circumstances necessitates the introduction of "weak solutions'' to the field equations that are accompanied by correspondingly weakened boundary and initial conditions. A uniqueness theorem pertaining to this weak formulation is then proved through an adaptation of an argument used by O. Ladyzhenskaya in connection with the first boundary-initial value problem for a second-order hyperbolic equation in a single dependent variable. Moreover, the second uniqueness theorem thus obtained contains, as a special case, a slight modification of the previously established uniqueness theorem covering solutions that exhibit only finite stress-discontinuities.

Dynamic response of hysteretic systems with application to a system containing limited slip

A general class of single degree of freedom systems possessing rate-independent hysteresis is defined. The hysteretic behavior in a system belonging to this class is depicted as a sequence of single-valued functions; at any given time, the current function is determined by some set of mathematical rules concerning the entire previous response of the system. Existence and uniqueness of solutions are established and boundedness of solutions is examined.

An asymptotic solution procedure is used to derive an approximation to the response of viscously damped systems with a small hysteretic nonlinearity and trigonometric excitation. Two properties of the hysteresis loops associated with any given system completely determine this approximation to the response: the area enclosed by each loop, and the average of the ascending and descending branches of each loop.

The approximation, supplemented by numerical calculations, is applied to investigate the steady-state response of a system with limited slip. Such features as disconnected response curves and jumps in response exist for a certain range of system parameters for any finite amount of slip.

To further understand the response of this system, solutions of the initial-value problem are examined. The boundedness of solutions is investigated first. Then the relationship between initial conditions and resulting steady-state solution is examined when multiple steady-state solutions exist. Using the approximate analysis and numerical calculations, it is found that significant regions of initial conditions in the initial condition plane lead to the different asymptotically stable steady-state solutions.

Práticas religiosas Afro-Brasileiras, marco regulatório e uso do meio ambiente e do espaço urbano da cidade do Rio de Janeiro

A Constituição Federal brasileira relaciona dentre as garantias do cidadão o direito ao meio ambiente sadio e a liberdade religiosa e de liturgia. Também prevê como valor constitucional a ser defendido pelo Estado brasileiro as matrizes culturais africanas. A problemática da presente pesquisa é o conflito entre esses valores e garantias em um Estado democrático de direito, conflito este que indentificamos no caso selecionado para estudo: a proibição de oferendas das religiões afrobrasileiras no Parque Nacional da Tijuca, no Rio de Janeiro, pela administração da entidade gestora do Parque. A partir deste estudo de caso, propomos questionar: 1) como o conflito é construído numa perspectiva multidimensional (da geografia cultural, da teologia, da sociologia etc); 2) se e por que as religiões de matrizes africanas foram excluídas do arcabouço jurídico ambiental brasileiro; 3) se este arcabouço pode ser interpretado de modo a favorecer a prática de oferendas e 4) se há uma consciência e uma ética ambientais emergentes naquelas comunidades religiosas, facilitadoras do argumento defensivo da prática de oferendas em áreas verdes públicas. Assim, o objetivo da presente pesquisa é contribuir para a solução exitosa deste conflito, de modo que esta solução seja válida e exeqüível em qualquer área verde sob administração pública. Desse modo, advogamos a tese de que é possível ponderar as duas garantias constitucionais em conflito, de forma que as oferendas, ao invés de proibidas, sejam aceitas de modo disciplinado, não agressivo ou menos agressivo ao meio ambiente, pela negociação dos atores envolvidos. Através da metodologia qualitativa demonstraremos que há um conflito entre atores que dão distintos significados ao meio ambiente, a partir de racionalidades distintas, sendo a da administração ambiental fortemente ancorada na própria doutrina formatadora dos parques nacionais. Aditaremos que o conflito poderia ter sido evitado ou minorado se as comunidades religiosas urbanas afrobrasileiras tivessem sido reconhecidas como populações tradicionais pelo movimento socioambientalista, fortemente inspirador da legislação brasileira. Demonstraremos ainda que, apesar desta lacuna, a legislação que já está dada pode ser interpretada de modo a chancelar a prática das oferendas, e que a proibição seria um equívoco legal da administração ambiental, tendo em vista que o direito ambiental oferece um sistema principiológico favorável à prática das oferendas, tarefa facilitada por uma emergente ética ambiental naqueles grupos religiosos. Não obstante, uma proposta de inclusão de um artigo na Lei do Sistema Nacional de Unidades de Conservação será elaborada, para evitar que a solução do conflito dependa de interpretações. Por fim, recomendaremos que a interdição no Parque da Tijuca seja exemplarmente substituída por uma negociação entre as partes envolvidas, de modo a que sejam preservados todos os interesses constitucionais envolvidos, proporcionando o avanço da democracia brasileira.

Autonomia do indivíduo com deficiência: estudo de caso no transtorno invasivo do desenvolvimento

A presente dissertação viu no estudo do conceito da autonomia o caminho para valorizar a liberdade de decisão do deficiente cognitivo, especificamente o portador de transtorno invasivo do desenvolvimento, ou autista. Como base teórica, foi utilizada a Bioética, pelo seu papel de mediadora entre as ciências médicas e a filosofia. Primeiramente, realizou-se um relato histórico do conceito de autonomia e a contextualização da sua relação com a Bioética. Como o conceito filosófico de pessoa nem sempre é atribuído ao autista, entender a razão dessa distinção é complementar a compreensão do porque não ser atribuída autonomia para esses indivíduos. Os portadores de deficiências vivem uma situação especial de vulnerabilidade, portanto, o conceito de vulnerabilidade e sua relação com a deficiência também fazem parte do estudo. Para uma melhor compreensão de quem é o autista realizou-se uma breve revisão bibliográfica sobre o autismo, priorizando as informações atuais. Somou-se ao levantamento teórico um estudo de caso, onde foi acompanhado um portador adulto da patologia. Com isso, esperava-se avaliar de maneira qualitativa a autonomia do indivíduo. Ter autonomia é fundamental para garantir uma boa qualidade de vida, por isso a Bioética sugere sua proteção e ampliação. Discutir como isso será realizado é fundamental para evitar as formas de paternalismo que reduzem a pessoalidade. Explorar o conceito de autonomia não visa limitar e sim ampliar as possibilidades para todo e qualquer ser-humano. A presente dissertação viu no estudo do conceito da autonomia o caminho para valorizar a liberdade de decisão do deficiente cognitivo, especificamente o portador de transtorno invasivo do desenvolvimento, ou autista. Como base teórica, foi utilizada a Bioética, pelo seu papel de mediadora entre as ciências médicas e a filosofia. Primeiramente, realizou-se um relato histórico do conceito de autonomia e a contextualização da sua relação com a Bioética. Como o conceito filosófico de pessoa nem sempre é atribuído ao autista, entender a razão dessa distinção é complementar a compreensão do porque não ser atribuída autonomia para esses indivíduos. Os portadores de deficiências vivem uma situação especial de vulnerabilidade, portanto, o conceito de vulnerabilidade e sua relação com a deficiência também fazem parte do estudo. Para uma melhor compreensão de quem é o autista realizou-se uma breve revisão bibliográfica sobre o autismo, priorizando as informações atuais. Somou-se ao levantamento teórico um estudo de caso, onde foi acompanhado um portador adulto da patologia. Com isso, esperava-se avaliar de maneira qualitativa a autonomia do indivíduo. Ter autonomia é fundamental para garantir uma boa qualidade de vida, por isso a Bioética sugere sua proteção e ampliação. Discutir como isso será realizado é fundamental para evitar as formas de paternalismo que reduzem a pessoalidade. Explorar o conceito de autonomia não visa limitar e sim ampliar as possibilidades para todo e qualquer ser-humano.

Value of information in product recovery decisions: A Bayesian approach

There is a widespread recognition of the need for better information sharing and provision to improve the viability of end-of-life (EOL) product recovery operations. The emergence of automated data capture and sharing technologies such as RFID, sensors and networked databases has enhanced the ability to make product information; available to recoverers, which will help them make better decisions regarding the choice of recovery option for EOL products. However, these technologies come with a cost attached to it, and hence the question 'what is its value?' is critical. This paper presents a probabilistic approach to model product recovery decisions and extends the concept of Bayes' factor for quantifying the impact of product information on the effectiveness of these decisions. Further, we provide a quantitative examination of the factors that influence the value of product information, this value depends on three factors: (i) penalties for Type I and Type II errors of judgement regarding product quality; (ii) prevalent uncertainty regarding product quality and (iii) the strength of the information to support/contradict the belief. Furthermore, we show that information is not valuable under all circumstances and derive conditions for achieving a positive value of information. © 2010 Taylor & Francis.