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Este livro é uma introdução ao Cálculo Científico. O seu objectivo consiste em apresentar vários métodos numéricos para resolver no computador certos problemas matemáticos que não podem ser tratados de maneira mais simples. São abordadas questões clássicas como o cálculo de zeros ou de integrais de funções contínuas, a resolução de sistemas lineares, a aproximação de funções por polinómios e a construção de aproximações precisas de soluções de equações diferenciais. Todos os algoritmos são apresentados nas linguagens de programação MATLAB e Octave, cujos comandos e instruções principais se introduzem de forma gradual, visando em particular a sua compatibilidade nas duas linguagens. O leitor pode assim verificar experimentalmente propriedades teóricas como a estabilidade, a precisão e a complexidade. O livro inclui ainda a resolução de problemas através de numerosos exercícios e exemplos, frequentemente ligados a aplicações concretas. No fim de cada capítulo encontra-se uma secção específica que apresenta assuntos não abordados e as referências bibliográficas que permitem ao leitor aprofundar os conhecimentos adquiridos. Este livro dirige-se a estudantes de cursos universitários ou politécnicos nas áreas das ciências e engenharia, no âmbito de disciplinas de métodos numéricos, cálculo científico e matemática computacional. Serve ainda de apoio a actividades de investigação com forte conteúdo computacional, no meio académico ou empresarial.

Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Applied mathematics. --- Engineering mathematics. --- Computer mathematics. --- Numerical analysis. --- Applications of Mathematics. --- Analysis. --- Computational Mathematics and Numerical Analysis. --- Computational Science and Engineering. --- Numerical Analysis. --- Mathematical analysis --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Engineering --- Engineering analysis --- 517.1 Mathematical analysis --- Math --- Science --- Mathematics --- Global analysis (Mathematics). --- Computer science --- Computer science. --- Informatics --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic

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Mathematical modeling of human physiopathology is a tremendously ambitious task. It encompasses the modeling of most diverse compartments such as the cardiovascular, respiratory, skeletal and nervous systems, as well as the mechanical and biochemical interaction between blood flow and arterial walls, or electrocardiac processes and the electric conduction into biological tissues. Mathematical models can be set up to simulate both vasculogenesis (the aggregation and organisation of endothelial cells dispersed in a given environment) and angiogenesis (the formation of new vessels sprouting from an existing vessel) that are relevant to the formation of vascular networks, and in particular to the description of tumor growth. The integration of models aimed at simulating the cooperation and interrelation of different systems is an even more difficult task. It calls for the set up of, for instance, interaction models for the integrated cardio-vascular system and the interplay between central circulation and peripheral compartments, models for the mid-long range cardiovascular adjustments to pathological conditions (e.g. to account for surgical interventions, congenital malformations, or tumor growth), models for the integration among circulation, tissue perfusion, biochemical and thermal regulation, models for parameter identification and sensitivity analysis to parameter changes or data uncertainty – and many others. The heart is a complex system in itself, where electrical phenomena are functionally related with the wall deformation. In its turn, electrical activity is related with heart physiology. It involves nonlinear reaction-diffusion processes and provides the activation stimulus to the heart dynamics and eventually the blood ventricular flow that drives the haemodynamics of the whole circulatory system. In fact, the influence is reciprocal, since the circulatory system in turns affects the heart dynamics and may induce an overload depending upon the individual physiopathologies ( for instance the presence of a stenotic artery or a vascular prosthesis). Virtually, all the fields of mathematics have a role to play in this context. Geometry and approximation theory provide the tools for handling clinical data acquired by tomography or magnetic resonance, identifying meaningful geometrical patterns and producing three-dimensional geometrical models stemming from the original patients data. Mathematical analysis, flow and solid dynamics, stochastic analysis are used to set up the differential models and predict uncertainty. Numerical analysis and high performance computing are needed to numerically solve the complex differential models. Finally, methods from stochastic and statistical analysis are exploited for the modeling and interpretation of space-time patterns.Indeed, the complexity of the problems at hand often stimulates the use of innovative mathematical techniques that are able, for instance, to accurately catch those processes that occur at multiple scales in time and space (like cellular and systemic effects), and that are governed by heterogeneous physical laws. In this book we have collected the contribution from several Italian research groups that are successfully working on this fascinating and challenging field. Every chapter will deal with a specific subfield, with the aim of providing an overview of the subject and an account of the most recent research results.

Physiology, Pathological --- System theory. --- Mathematical models. --- Systems, Theory of --- Systems science --- Science --- Clinical physiology --- Pathological physiology --- Pathophysiology --- Physiopathology --- Pathology --- Physiology --- Philosophy --- Biology --- Computer science. --- Hydraulic engineering. --- Biomedical engineering. --- Computer Appl. in Life Sciences. --- Mathematical and Computational Biology. --- Computational Science and Engineering. --- Mathematical Modeling and Industrial Mathematics. --- Engineering Fluid Dynamics. --- Biomedical Engineering and Bioengineering. --- Data processing. --- Engineering, Hydraulic --- Engineering --- Fluid mechanics --- Hydraulics --- Shore protection --- Informatics --- Clinical engineering --- Medical engineering --- Bioengineering --- Biophysics --- Medicine --- Bioinformatics . --- Computational biology . --- Biomathematics. --- Computer mathematics. --- Fluid mechanics. --- Hydromechanics --- Continuum mechanics --- Models, Mathematical --- Simulation methods --- Computer mathematics --- Electronic data processing --- Mathematics --- Bioinformatics --- Bio-informatics --- Biological informatics --- Information science --- Computational biology --- Systems biology --- Data processing

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Differential equations, Partial --- Numerical analysis. --- Spectral theory (Mathematics). --- Numerical solutions. --- Numerical analysis --- Spectral theory (Mathematics) --- 517.2 --- 519.63 --- 535.33 --- 681.3*G18 --- 519.63 Numerical methods for solution of partial differential equations --- Numerical methods for solution of partial differential equations --- 535.33 Spectra in general. Emission spectra --- Spectra in general. Emission spectra --- 517.2 Differential calculus. Differentiation --- Differential calculus. Differentiation --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Mathematical analysis --- Numerical solutions --- Partitial differential equations: domain decomposition methods; elliptic equations; finite difference methods; finite element methods; finite volume methods; hyperbolic equations; inverse problems; iterative solution techniques; methods of lines; multigrid and multilevel methods; parabolic equations; special methods

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Fluid dynamics --- Numerical analysis --- Spectral theory (Mathematics) --- Approximation methods

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Numerical analysis --- Mathematical physics --- Classical mechanics. Field theory --- Fluid mechanics --- Gases handling. Fluids handling --- Computer. Automation --- informatica --- wiskunde --- algoritmen --- ingenieurswetenschappen --- fysica --- mechanica --- numerieke analyse --- vloeistoffen