Linear algebra: selected problems

Linear Algebra—Selected Problems is a unique book for senior undergraduate and graduate students to fast review basic materials in Linear Algebra. Vector spaces are presented first, and linear transformations are reviewed secondly. Matrices and Linear systems are presented. Determinants and Basic geometry are presented in the last two chapters. The solutions for proposed excises are listed for readers to references.

Time domain design of fractional differintegrators using least-squares

In this paper we propose the use of the least-squares based methods for obtaining digital rational approximations (IIR filters) to fractional-order integrators and differentiators of type sα, α∈R. Adoption of the Padé, Prony and Shanks techniques is suggested. These techniques are usually applied in the signal modeling of deterministic signals. These methods yield suboptimal solutions to the problem which only requires finding the solution of a set of linear equations. The results reveal that the least-squares approach gives similar or superior approximations in comparison with other widely used methods. Their effectiveness is illustrated, both in the time and frequency domains, as well in the fractional differintegration of some standard time domain functions.

Optimal fuzzy PID control tuned with genetic algorithms

Fuzzy logic controllers (FLC) are intelligent systems, based on heuristic knowledge, that have been largely applied in numerous areas of everyday life. They can be used to describe a linear or nonlinear system and are suitable when a real system is not known or too difficult to find their model. FLC provide a formal methodology for representing, manipulating and implementing a human heuristic knowledge on how to control a system. These controllers can be seen as artificial decision makers that operate in a closed-loop system, in real time. The main aim of this work was to develop a single optimal fuzzy controller, easily adaptable to a wide range of systems – simple to complex, linear to nonlinear – and able to control all these systems. Due to their efficiency in searching and finding optimal solution for high complexity problems, GAs were used to perform the FLC tuning by finding the best parameters to obtain the best responses. The work was performed using the MATLAB/SIMULINK software. This is a very useful tool that provides an easy way to test and analyse the FLC, the PID and the GAs in the same environment. Therefore, it was proposed a Fuzzy PID controller (FL-PID) type namely, the Fuzzy PD+I. For that, the controller was compared with the classical PID controller tuned with, the heuristic Ziegler-Nichols tuning method, the optimal Zhuang-Atherton tuning method and the GA method itself. The IAE, ISE, ITAE and ITSE criteria, used as the GA fitness functions, were applied to compare the controllers performance used in this work. Overall, and for most systems, the FL-PID results tuned with GAs were very satisfactory. Moreover, in some cases the results were substantially better than for the other PID controllers. The best system responses were obtained with the IAE and ITAE criteria used to tune the FL-PID and PID controllers.

Numerical solution of a PDE system with non-linear steady state conditions that translates the air stripping pollutants removal

This work deals with the numerical simulation of air stripping process for the pre-treatment of groundwater used in human consumption. The model established in steady state presents an exponential solution that is used, together with the Tau Method, to get a spectral approach of the solution of the system of partial differential equations associated to the model in transient state.

Spectral solution for the air stripping pollutants removal dynamic model with non linear steady state conditions

This work deals with the numerical simulation of air stripping process for the pre-treatment of groundwater used in human consumption. The model established in steady state presents an exponential solution that is used, together with the Tau Method, to get a spectral approach of the solution of the system of partial differential equations associated to the model in transient state.

Multiple linear and principal component regressions for modelling ecotoxicity bioassay response

The ecotoxicological response of the living organisms in an aquatic system depends on the physical, chemical and bacteriological variables, as well as the interactions between them. An important challenge to scientists is to understand the interaction and behaviour of factors involved in a multidimensional process such as the ecotoxicological response.With this aim, multiple linear regression (MLR) and principal component regression were applied to the ecotoxicity bioassay response of Chlorella vulgaris and Vibrio fischeri in water collected at seven sites of Leça river during five monitoring campaigns (February, May, June, August and September of 2006). The river water characterization included the analysis of 22 physicochemical and 3 microbiological parameters. The model that best fitted the data was MLR, which shows: (i) a negative correlation with dissolved organic carbon, zinc and manganese, and a positive one with turbidity and arsenic, regarding C. vulgaris toxic response; (ii) a negative correlation with conductivity and turbidity and a positive one with phosphorus, hardness, iron, mercury, arsenic and faecal coliforms, concerning V. fischeri toxic response. This integrated assessment may allow the evaluation of the effect of future pollution abatement measures over the water quality of Leça River.

Multiple Linear Regression and Artificial Neural Networks to Predict Time and Efficiency of Soil Vapor Extraction

The prediction of the time and the efficiency of the remediation of contaminated soils using soil vapor extraction remain a difficult challenge to the scientific community and consultants. This work reports the development of multiple linear regression and artificial neural network models to predict the remediation time and efficiency of soil vapor extractions performed in soils contaminated separately with benzene, toluene, ethylbenzene, xylene, trichloroethylene, and perchloroethylene. The results demonstrated that the artificial neural network approach presents better performances when compared with multiple linear regression models. The artificial neural network model allowed an accurate prediction of remediation time and efficiency based on only soil and pollutants characteristics, and consequently allowing a simple and quick previous evaluation of the process viability.

Multiple Roots of Systems of Equations by Repulsion Merit Functions

In this paper we address the problem of computing multiple roots of a system of nonlinear equations through the global optimization of an appropriate merit function. The search procedure for a global minimizer of the merit function is carried out by a metaheuristic, known as harmony search, which does not require any derivative information. The multiple roots of the system are sequentially determined along several iterations of a single run, where the merit function is accordingly modified by penalty terms that aim to create repulsion areas around previously computed minimizers. A repulsion algorithm based on a multiplicative kind penalty function is proposed. Preliminary numerical experiments with a benchmark set of problems show the effectiveness of the proposed method.

Aplicação de um modelo de programação linear inteira mista (PLIM) na expansão da rede de distribuição de energia elétrica em Angola

Em Angola, apenas cerca de 30% da população tem acesso à energia elétrica, nível que decresce para valores inferiores a 10% em zonas rurais mais remotas. Este problema é agravado pelo facto de, na maioria dos casos, as infraestruturas existentes se encontrarem danificadas ou não acompanharem o desenvolvimento da região. Em particular na capital angolana, Luanda que, sendo a menor província de Angola, é a que regista atualmente a maior densidade populacional. Com uma população de cerca de 5 milhões de habitantes, não só há frequentemente problemas relacionados com a falha do fornecimento de energia elétrica como há ainda uma percentagem considerável de municípios onde a rede elétrica ainda nem sequer chegou. O governo de Angola, no seu esforço de crescimento e aproveitamento das suas enormes potencialidades, definiu o setor energético como um dos fatores críticos para o desenvolvimento sustentável do país, tendo assumido que este é um dos eixos prioritários até 2016. Existem objetivos claros quanto à reabilitação e expansão das infraestruturas do setor elétrico, aumentando a capacidade instalada do país e criando uma rede nacional adequada, com o intuito não só de melhorar a qualidade e fiabilidade da rede já existente como de a aumentar. Este trabalho de dissertação consistiu no levantamento de dados reais relativamente à rede de distribuição de energia elétrica de Luanda, na análise e planeamento do que é mais premente fazer relativamente à sua expansão, na escolha dos locais onde é viável localizar novas subestações, na modelação adequada do problema real e na proposta de uma solução ótima para a expansão da rede existente. Depois de analisados diferentes modelos matemáticos aplicados ao problema de expansão de redes de distribuição de energia elétrica encontrados na literatura, optou-se por um modelo de programação linear inteira mista (PLIM) que se mostrou adequado. Desenvolvido o modelo do problema, o mesmo foi resolvido por recurso a software de otimização Analytic Solver e CPLEX. Como forma de validação dos resultados obtidos, foi implementada a solução de rede no simulador PowerWorld 8.0 OPF, software este que permite a simulação da operação do sistema de trânsito de potências.

Gestão de Recursos Energéticos nas SmartGrids

A sustentabilidade do sistema energético é crucial para o desenvolvimento económico e social das sociedades presentes e futuras. Para garantir o bom funcionamento dos sistemas de energia actua-se, tipicamente, sobre a produção e sobre as redes de transporte e de distribuição. No entanto, a integração crescente de produção distribuída, principalmente nas redes de distribuição de média e de baixa tensão, a liberalização dos mercados energéticos, o desenvolvimento de mecanismos de armazenamento de energia, o desenvolvimento de sistemas automatizados de controlo de cargas e os avanços tecnológicos das infra-estruturas de comunicação impõem o desenvolvimento de novos métodos de gestão e controlo dos sistemas de energia. O contributo deste trabalho é o desenvolvimento de uma metodologia de gestão de recursos energéticos num contexto de SmartGrids, considerando uma entidade designada por VPP que gere um conjunto de instalações (unidades produtoras, consumidores e unidades de armazenamento) e, em alguns casos, tem ao seu cuidado a gestão de uma parte da rede eléctrica. Os métodos desenvolvidos contemplam a penetração intensiva de produção distribuída, o aparecimento de programas de Demand Response e o desenvolvimento de novos sistemas de armazenamento. São ainda propostos níveis de controlo e de tomada de decisão hierarquizados e geridos por entidades que actuem num ambiente de cooperação mas também de concorrência entre si. A metodologia proposta foi desenvolvida recorrendo a técnicas determinísticas, nomeadamente, à programação não linear inteira mista, tendo sido consideradas três funções objectivo distintas (custos mínimos, emissões mínimas e cortes de carga mínimos), originando, posteriormente, uma função objectivo global, o que permitiu determinar os óptimos de Pareto. São ainda determinados os valores dos custos marginais locais em cada barramento e consideradas as incertezas dos dados de entrada, nomeadamente, produção e consumo. Assim, o VPP tem ao seu dispor um conjunto de soluções que lhe permitirão tomar decisões mais fundamentadas e de acordo com o seu perfil de actuação. São apresentados dois casos de estudo. O primeiro utiliza uma rede de distribuição de 32 barramentos publicada por Baran & Wu. O segundo caso de estudo utiliza uma rede de distribuição de 114 barramentos adaptada da rede de 123 barramentos do IEEE.

Dimensão do sector público e crescimento económico: uma relação não linear na União Europeia dos 15?

Os Estados-Membros da União Europeia têm tido a preocupação de reduzirem a dimensão da Administração Pública na economia, a par de a tornar muito mais eficiente de forma a promover o crescimento económico. Neste artigo analisam-se as relações entre a despesa pública e o crescimento económico em 14 Estados-Membros da União Europeia dos 15, com o objectivo de determinar a dimensão óptima das Administrações Públicas, tendo por base teórica a Curva de Armey. Os resultados, para o período 1965-2007, sugerem uma dimensão do sector público maximizadora do crescimento económico de 47,37% e 22,17% do PIB, quando avaliada pelas despesas públicas totais e o consumo público, respectivamente.

Disposable immunosensor using a simple method for oriented antibody immobilization for label-free real-time detection of an oxidative stress biomarker implicated in cancer diseases

This work proposes a novel approach for a suitable orientation of antibodies (Ab) on an immunosensing platform, applied here to the determination of 8-hydroxy-2′-deoxyguanosine (8OHdG), a biomarker of oxidative stress that has been associated to chronic diseases, such as cancer. The anti-8OHdG was bound to an amine modified gold support through its Fc region after activation of its carboxylic functions. Non-oriented approaches of Ab binding to the platform were tested in parallel, in order to show that the presented methodology favored Ab/Ag affinity and immunodetection of the antigen. The immunosensor design was evaluated by quartz-crystal microbalance with dissipation, atomic force microscopy, electrochemical impedance spectroscopy (EIS) and square-wave voltammetry. EIS was also a suitable technique to follow the analytical behavior of the device against 8OHdG. The affinity binding between 8OHdG and the antibody immobilized in the gold modified platform increased the charge transfer resistance across the electrochemical set-up. The observed behavior was linear from 0.02 to 7.0 ng/mL of 8OHdG concentrations. The interference from glucose, urea and creatinine was found negligible. An attempt of application to synthetic samples was also successfully conducted. Overall, the presented approach enabled the production of suitably oriented Abs over a gold platform by means of a much simpler process than other oriented-Ab binding approaches described in the literature, as far as we know, and was successful in terms of analytical features and sample application.

Distributed Linear Precoding and User Selection in Coordinated Multicell Systems

In this manuscript we tackle the problem of semidistributed user selection with distributed linear precoding for sum rate maximization in multiuser multicell systems. A set of adjacent base stations (BS) form a cluster in order to perform coordinated transmission to cell-edge users, and coordination is carried out through a central processing unit (CU). However, the message exchange between BSs and the CU is limited to scheduling control signaling and no user data or channel state information (CSI) exchange is allowed. In the considered multicell coordinated approach, each BS has its own set of cell-edge users and transmits only to one intended user while interference to non-intended users at other BSs is suppressed by signal steering (precoding). We use two distributed linear precoding schemes, Distributed Zero Forcing (DZF) and Distributed Virtual Signalto-Interference-plus-Noise Ratio (DVSINR). Considering multiple users per cell and the backhaul limitations, the BSs rely on local CSI to solve the user selection problem. First we investigate how the signal-to-noise-ratio (SNR) regime and the number of antennas at the BSs impact the effective channel gain (the magnitude of the channels after precoding) and its relationship with multiuser diversity. Considering that user selection must be based on the type of implemented precoding, we develop metrics of compatibility (estimations of the effective channel gains) that can be computed from local CSI at each BS and reported to the CU for scheduling decisions. Based on such metrics, we design user selection algorithms that can find a set of users that potentially maximizes the sum rate. Numerical results show the effectiveness of the proposed metrics and algorithms for different configurations of users and antennas at the base stations.

Disposable immunosensor with simple antibody orientation for label-free real-time detection of a cancer biomarker

This work proposes a novel approach for a suitable orientation of antibodies (Ab) on an immunosensing platform, applied here to the determination of 8-hydroxy-2’-deoxyguanosine (8OHdG), a biomarker of oxidative stress that has been associated to chronic diseases, such as cancer. The Anti-8OHdG was bound to an amine modified gold support through its Fc region after activation of its carboxylic functions. Non-oriented approaches of Ab binding to the platform were tested in parallel, in order to show that the presented proposal favored Ab/Ag affinity. The immunosensor design was evaluated by Quartz-Crystal microbalance with Dissipation, Atomic Force Microscopy, Electrochemical Impedance Spectroscopy (EIS) and Square-Wave Voltammetry. EIS was also a suitable technique to follow the analytical behavior of the device against 8OHdG. The affinity binding between 8OHdG and the antibody immobilized in the gold modified platform increased the charged transfer resistance across the electrochemical sep-up. The observed behavior was linear from 0.02 to 7.0 ng/mL of 8OHdG concentrations. The interference from Glucose, Urea and Creatinine was found negligible. An attempt of application to synthetic samples was also successfully conducted. Overall, the presented approach enabled the production of suitably oriented Abs over a gold platform by means of a much simpler process than other oriented-Ab binding approaches described in the literature, as far as we know, and was successful in terms of analytical features and sample application.