Modeling and Simulation in Science, Engineering and Technology – serie

In recent years kinetic theory has developed in many areas of the physical sciences and engineering, and has extended the borders of its traditional fields of application. New applications in traffic flow engineering, granular media modeling, and polymer and phase transition physics have resulted in new numerical algorithms which depart from traditional stochastic Monte--Carlo methods. This monograph is a self-contained presentation of such recently developed aspects of kinetic theory, as well as a comprehensive account of the fundamentals of the theory. Emphasizing modeling techniques and numerical methods, the book provides a unified treatment of kinetic equations not found in more focused theoretical or applied works. The book is divided into two parts. Part I is devoted to the most fundamental kinetic model: the Boltzmann equation of rarefied gas dynamics. Additionally, widely used numerical methods for the discretization of the Boltzmann equation are reviewed: the Monte--Carlo method, spectral methods, and finite-difference methods.Part II considers specific applications: plasma kinetic modeling using the Landau--Fokker--Planck equations, traffic flow modeling, granular media modeling, quantum kinetic modeling, and coagulation-fragmentation problems. "Modeling and Computational Methods of Kinetic Equations" will be accessible to readers working in different communities where kinetic theory is important: graduate students, researchers and practitioners in mathematical physics, applied mathematics, and various branches of engineering. The work may be used for self-study, as a reference text, or in graduate-level courses in kinetic theory and its applications.

This book has evolved from lectures and graduate courses given in Brescia (Italy), Bordeaux and Toulouse (France};' It is intended to serve as an intro­ duction to the stability analysis of noncharacteristic multidimensional small viscosity boundary layers developed in (MZl]. We consider parabolic singular perturbations of hyperbolic systems L(u) - £P(u) = 0, where L is a nonlinear hyperbolic first order system and P a nonlinear spatially elliptic term. The parameter e measures the strength of the diffusive effects. With obvious reference to fluid mechanics, it is referred to as a "viscosity." The equation holds on a domain n and is supplemented by boundary conditions on an.The main goal of this book is to studythe behavior of solutions as etends to O. In the interior of the domain, the diffusive effects are negligible and the nondiffusive or inviscid equations (s = 0) are good approximations. However, the diffusive effects remain important in a small vicinity of the boundary where they induce rapid fluctuations of the solution, called layers. Boundary layers occur in many problems in physics and mechanics. They also occur in free boundary value problems, and in particular in the analysis of shock waves. Indeed, our study of noncharacteristic boundary layers is strongly motivated by the analysis of multidimensional shock waves. At the least, it is a necessary preliminary and important step. We also recall the importance of the viscous approach in the theoretical analysis ofconservation laws (see, e.g., [Lax], (Kru], (Bi-Br]).

Mathematical Modeling and Immunology An enormous amount of human effort and economic resources has been directed in this century to the fight against cancer. The purpose, of course, has been to find strategies to overcome this hard, challenging and seemingly endless struggle. We can readily imagine that even greater efforts will be required in the next century. The hope is that ultimately humanity will be successful; success will have been achieved when it is possible to activate and control the immune system in its competition against neoplastic cells. Dealing with the above-mentioned problem requires the fullest pos­ sible cooperation among scientists working in different fields: biology, im­ munology, medicine, physics and, we believe, mathematics. Certainly, bi­ ologists and immunologists will make the greatest contribution to the re­ search. However, it is now increasingly recognized that mathematics and computer science may well able to make major contributions to such prob­ lems. We cannot expect mathematicians alone to solve fundamental prob­ lems in immunology and (in particular) cancer research, but valuable sup­ port, however modest, can be provided by mathematicians to the research aspirations of biologists and immunologists working in this field.

The present monograph is dedicated to a new branch of the theory of climate, which is titled by the authors, "Mathematical Theory of Climate. " The foundation of this branch is the investigation of climate models by the methods of the qUalitative theory of differential equa­ tions. In the Russian edition the book was named "Fundamentals of the Mathematical Theory of Climate. " Respecting the recommenda­ tions of Wayne Yuhasz (we are truly grateful to him for this advice), we named the English edition of the book "Mathematics of Climate Modelling. " This title appears to be more appropriate, since the con­ structive results of the theory are at present preliminary and have not been fully tested with experiments in climate modelling. This branch of science is yet developing and its practical results will be obtained only in the near future. Nevertheless, we want to keep the terminology which we have used in the introduction to the Russian edition of the book, since the authors hope that this term will be accepted by the scientific community for identification of a given branch of climate theory. On preparing the English edition, new ideas were established con­ necting some significant new research results obtained by the author. We are deeply grateful to G. Marchuk for continual encourage­ ment of this scientific enterprise and fruitful discussions, to our young colleagues A. Gorelov, E. Kazantsev, A. Gritsun, and A.

A reference for the field of particle modelling - the study of dynamical behaviour of solids and fluids in response to external forces, with the solids and fluids modelled as systems of atoms and molecules.

In the modern theory of boundary value problems the following ap­ proach to investigation is agreed upon (we call it the functional approach): some functional spaces are chosen; the statements of boundary value prob­ the basis of these spaces; and the solvability of lems are formulated on the problems, properties of solutions, and their dependence on the original data of the problems are analyzed. These stages are put on the basis of the correct statement of different problems of mathematical physics (or of the definition of ill-posed problems). For example, if the solvability of a prob­ lem in the functional spaces chosen cannot be established then, probably, the reason is in their unsatisfactory choice. Then the analysis should be repeated employing other functional spaces. Elliptical problems can serve as an example of classical problems which are analyzed by this approach. Their investigations brought a number of new notions and results in the theory of Sobolev spaces W;(D) which, in turn, enabled us to create a sufficiently complete theory of solvability of elliptical equations. Nowadays the mathematical theory of radiative transfer problems and kinetic equations is an extensive area of modern mathematical physics. It has various applications in astrophysics, the theory of nuclear reactors, geophysics, the theory of chemical processes, semiconductor theory, fluid mechanics, etc. [25,29,31,39,40, 47, 52, 78, 83, 94, 98, 120, 124, 125, 135, 146].

Fluid mechanics (FM) is a branch of science dealing with the investi­ gation of flows of continua under the action of external forces. The fundamentals of FM were laid in the works of the famous scientists, such as L. Euler, M. V. Lomonosov, D. Bernoulli, J. L. Lagrange, A. Cauchy, L. Navier, S. D. Poisson, and other classics of science. Fluid mechanics underwent a rapid development during the past two centuries, and it now includes, along with the above branches, aerodynamics, hydrodynamics, rarefied gas dynamics, mechanics of multi phase and reactive media, etc. The FM application domains were expanded, and new investigation methods were developed. Certain concepts introduced by the classics of science, however, are still of primary importance and will apparently be of importance in the future. The Lagrangian and Eulerian descriptions of a continuum, tensors of strains and stresses, conservation laws for mass, momentum, moment of momentum, and energy are the examples of such concepts and results. This list should be augmented by the first and second laws of thermodynamics, which determine the character and direction of processes at a given point of a continuum. The availability of the conservation laws is conditioned by the homogeneity and isotrop­ icity properties of the Euclidean space, and the form of these laws is related to the Newton's laws. The laws of thermodynamics have their foundation in the statistical physics.

Modeling and Applied Mathematics Modeling the behavior of real physical systems by suitable evolution equa­ tions is a relevant, maybe the fundamental, aspect of the interactions be­ tween mathematics and applied sciences. Modeling is, however, only the first step toward the mathematical description and simulation of systems belonging to real world. Indeed, once the evolution equation is proposed, one has to deal with mathematical problems and develop suitable simula­ tions to provide the description of the real system according to the model. Within this framework, one has an evolution equation and the re­ lated mathematical problems obtained by adding all necessary conditions for their solution. Then, a qualitative analysis should be developed: this means proof of existence of solutions and analysis of their qualitative be­ havior. Asymptotic analysis may include a detailed description of stability properties. Quantitative analysis, based upon the application ofsuitable methods and algorithms for the solution of problems, ends up with the simulation that is the representation of the dependent variable versus the independent one. The information obtained by the model has to be compared with those deriving from the experimental observation of the real system. This comparison may finally lead to the validation of the model followed by its application and, maybe, further generalization.

Micropolar fluids are fluids with microstructure. They belong to a class of fluids with nonsymmetric stress tensor that we shall call polar fluids, and include, as a special case, the well-established Navier-Stokes model of classical fluids that we shall call ordinary fluids. Physically, micropolar fluids may represent fluids consisting of rigid, randomly oriented (or spherical) particles suspended in a viscous medium, where the deformation of fluid particles is ignored. The model of micropolar fluids introduced in [65] by C. A. Eringen is worth studying as a very well balanced one. First, it is a well-founded and significant generalization of the classical Navier-Stokes model, covering, both in theory and applications, many more phenomena than the classical one. Moreover, it is elegant and not too complicated, in other words, man­ ageable to both mathematicians who study its theory and physicists and engineers who apply it. The main aim of this book is to present the theory of micropolar fluids, in particular its mathematical theory, to a wide range of readers. The book also presents two applications of micropolar fluids, one in the theory of lubrication and the other in the theory of porous media, as well as several exact solutions of particular problems and a numerical method. We took pains to make the presentation both clear and uniform.

“The authors have backgrounds which are ideally suited for writing this book. ...They have produced a compact, moderately general book which encompasses many fluid models of current interest…The book is written very clearly and contains a large number of exercises and their solutions. The level of mathematics is that commonly taught to undergraduates in mathematics departments. This is an excellent book which is highly recommended to students and researchers in fluid mechanics.” (Mathematical Reviews)“The writing style is quintessential Truesdellania: purely mathematical, breathtaking, irrepressible, irreverent, uncompromising, taking no prisoners...The book is filled with historical nuggets…Its pure, exact mathematics will baptize, enlighten and exhilarate.” (Applied Mechanics Review)“The most positive aspect of this book is its brevity; a large number of topics are covered within the space of a little more than 250 pages.” (Current Science)“This advanced monograph presents one of the best new views on the subject for those who like relative simplicity and certain abstractness combined with mathematical rigor and elegance…The book should be useful for graduates and researchers not only in applied mathematics and mechanical engineering but also in advanced materials science and technology…Each public scientific library as well as hydrodynamics hand libraries should own this timeless book…Everyone who decides to buy this book can be sure to have bought a classic of science and the heritage of an outstanding scientist.” (Silikáty)“All applied mathematicians, mechanical engineers, aerospace engineers, and engineering mechanics graduates and researchers will find the book an essential reading resource for fluids.” (Simulation News Europe)

The subject of mathematical modeling has expanded considerably in the past twenty years. This is in part due to the appearance of the text by Kemeny and Snell, "Mathematical Models in the Social Sciences," as well as the one by Maki and Thompson, "Mathematical Models and Applica­ tions. " Courses in the subject became a widespread if not standard part of the undergraduate mathematics curriculum. These courses included var­ ious mathematical topics such as Markov chains, differential equations, linear programming, optimization, and probability. However, if our own experience is any guide, they failed to teach mathematical modeling; that is, few students who completed the course were able to carry out the mod­ eling paradigm in all but the simplest cases. They could be taught to solve differential equations or find the equilibrium distribution of a regular Markov chain, but could not, in general, make the transition from "real world" statements to their mathematical formulation. The reason is that this process is very difficult, much more difficult than doing the mathemat­ ical analysis. After all, that is exactly what engineers spend a great deal of time learning to do. But they concentrate on very specific problems and rely on previous formulations of similar problems. It is unreasonable to expect students to learn to convert a large variety of real-world problems to mathematical statements, but this is what these courses require.

The goal of this is book to give a detailed presentation of multicomponent flow models and to investigate the mathematical structure and properties of the resulting system of partial differential equations. These developments are also illustrated by simulating numerically a typical laminar flame. Our aim in the chapters is to treat the general situation of multicomponent flows, taking into account complex chemistry and detailed transport phe­ nomena. In this book, we have adopted an interdisciplinary approach that en­ compasses a physical, mathematical, and numerical point of view. In par­ ticular, the links between molecular models, macroscopic models, mathe­ matical structure, and mathematical properties are emphasized. We also often mention flame models since combustion is an excellent prototype of multicomponent flow. This book still does not pretend to be a complete survey of existing models and related mathematical results. In particular, many subjects like multi phase-flows , turbulence modeling, specific applications, porous me­ dia, biological models, or magneto-hydrodynamics are not covered. We rather emphasize the fundamental modeling of multicomponent gaseous flows and the qualitative properties of the resulting systems of partial dif­ ferential equations. Part of this book was taught at the post-graduate level at the Uni­ versity of Paris, the University of Versailles, and at Ecole Poly technique in 1998-1999 to students of applied mathematics.

Most materials used in contemporary life and industry are heterogeneous (composites) and multicomponent, possessing a rich and complex internal structure. This internal structure, or microstructure, plays a key role in understanding and controlling the continuum behavior, or macroscopic, of a wide variety of materials. The modeling process is a critical tool for scientists and engineers studying the analysis and experimentation for the micromechanics and behavior of these materials. "Heterogeneous Media" is a critical, in-depth edited survey of the major topics surrounding the modeling and analysis of problems in micromechanics of multicomponent systems, including conceptual and practical aspects. The goal of this extensive and comprehensive survey is to provide both specialists and nonspecialists with an authoritative and interdisciplinary perspective of current ideas and methods used for modeling heterogeneous materials behavior and their applications. Topics and Features: * all chapters use interdisciplinary modeling perspective for investigating heterogeneous media*Five chapters provide self-contained discussions, with background provided*Focuses only upon most important techniques and models, fully exploring micro-macro interconnections*extensive introductory survey chapter on micromechanics of heterogeneous media*microstructure characterization via statistical correlation functions*micro-scale deformation of pore space*wave fields and effective dynamical properties*modeling of the complex production technologies for composite materials The book is ideal for a general scientific and engineering audience needing an in-depth view and guide to current ideas, methods and

Despite the fact that fluid dynamics and filtration through porous media and mathematics, there are classical research areas in engineering, physics, are still many industrial processes that require the study of new mathemat­ ical models for flows of particular complexity, due to the peculiar properties of the systems involved. The aim of this book is to provide a number of examples showing how frequently such situations arise in various branches of industrial technology. The selection of the subjects was motivated not only by their industrial rel­ evance and mathematical interest. What I had in mind was a collection of problems having a really distinctive character, thus bringing some fresh air into one of the oldest and most revered domains of applied mathematics. The incredible richness of nonstandard flow problems in industrial appli­ cations has always been, and still is, a constant surprise to me. Therefore I tried to offer a very large spectrum of subjects, with special attention devoted to those problems in which the modeling phase is far from being obvious, and the mathematical content is absolutely nontrivial. With such a view to diversity, topics have been selected from a variety of sources (such as glass industry, polymers science, coffee brewing, fuels pipelining), and contributors from different backgrounds (mathematics, physics, chemical engineering) have been included. Consequently, the mathematical nature of the problems formulated spans over a large range, so that their theoret­ ical investigation and numerical computation require a variety of different techniques.

Modeling complex biological, chemical, and physical systems, in the context of spatially heterogeneous mediums, is a challenging task for scientists and engineers using traditional methods of analysis. Modeling in Applied Sciences is a comprehensive survey of modeling large systems using kinetic equations, and in particular the Boltzmann equation and its generalizations. An interdisciplinary group of leading authorities carefully develop the foundations of kinetic models and discuss the connections and interactions between model theories, qualitative and computational analysis and real-world applications. This book provides a thoroughly accessible and lucid overview of the different aspects, models, computations, and methodology for the kinetic-theory modeling process. Topics and Features: * Integrated modeling perspective utilized in all chapters * Fluid dynamics of reacting gases * Self-contained introduction to kinetic models * Becker–Doring equations * Nonlinear kinetic models with chemical reactions * Kinetic traffic-flow models * Models of granular media * Large communication networks * Thorough discussion of numerical simulations of Boltzmann equation This new book is an essential resource for all scientists and engineers who use large-scale computations for studying the dynamics of complex systems of fluids and particles. Professionals, researchers, and postgraduates will find the book a modern and authoritative guide to the topic.

Many interesting behaviors of real physical, biological, economical, and chemical systems can be described by ordinary differential equations (ODEs). Scientific Computing with Mathematica for Ordinary Differential Equations provides a general framework useful for the applications, on the conceptual aspects of the theory of ODEs, as well as a sophisticated use of Mathematica software for the solutions of problems related to ODEs. In particular, a chapter is devoted to the use ODEs and Mathematica in the Dynamics of rigid bodies. Mathematical methods and scientific computation are dealt with jointly to supply a unified presentation. The main problems of ordinary differential equations such as, phase portrait, approximate solutions, periodic orbits, stability, bifurcation, and boundary problems are covered in an integrated fashion with numerous worked examples and computer program demonstrations using Mathematica. Topics and Features:*Explains how to use the Mathematica package ODE.m to support qualitative and quantitative problem solving *End-of- chapter exercise sets incorporating the use of Mathematica programs *Detailed description and explanation of the mathematical procedures underlying the programs written in Mathematica *Appendix describing the use of ten notebooks to guide the reader through all the exercises. This book is an essential text/reference for students, graduates and practitioners in applied mathematics and engineering interested in ODE's problems in both the qualitative and quantitative description of solutions with the Mathematica program. It is also suitable as a self-

This book presents the language of random field theory and the principles of stochastic geometry in order to give the systematic and concise knowledge necessary for modeling real random heterogeneous media. This book is ideal for a general scientific audience needing an in-depth guide to current ideas, methods and models for the stochastic modeling of microstructures. It is an essential reference for practitioners, researchers and professionals in material modeling, mechanical engineering, materials science, chemical, civil, environmental engineering and applied mathematics.

Modelling of granular and porous materials in science and engineering touches all kinds of problems arising from the necessity to successfully describe the behaviour of materials naturally existing in our surroundings. The book brings together scientists in engineering and mathematicians interested in solid behaviour and experts in particulate material modelling. The modelling process is a critical tool for studying the analysis and experimentation of micromechanics and behavior of granular and porous material. The text focuses upon the most important techniques and models currently in use, looking at: mechanics of porous media; modelling of soil behaviour; flow and transport phenomena in particulate materials; numerical simulation methods; and numerical model for finite elastoplastic deformations. The text should appeal to research workers, graduate students, and scientists in applied mathematics, continuum mechanics, finite element methods, solid mechanics and hydraulics engineering.

This monograph is intended to provide a comprehensive description of the rela­ tion between kinetic theory and fluid dynamics for a time-independent behavior of a gas in a general domain. A gas in a steady (or time-independent) state in a general domain is considered, and its asymptotic behavior for small Knudsen numbers is studied on the basis of kinetic theory. Fluid-dynamic-type equations and their associated boundary conditions, together with their Knudsen-layer corrections, describing the asymptotic behavior of the gas for small Knudsen numbers are presented. In addition, various interesting physical phenomena derived from the asymptotic theory are explained. The background of the asymptotic studies is explained in Chapter 1, accord­ ing to which the fluid-dynamic-type equations that describe the behavior of a gas in the continuum limit are to be studied carefully. Their detailed studies depending on physical situations are treated in the following chapters. What is striking is that the classical gas dynamic system is incomplete to describe the behavior of a gas in the continuum limit (or in the limit that the mean free path of the gas molecules vanishes). Thanks to the asymptotic theory, problems for a slightly rarefied gas can be treated with the same ease as the corresponding classical fluid-dynamic problems. In a rarefied gas, a temperature field is di­ rectly related to a gas flow, and there are various interesting phenomena which cannot be found in a gas in the continuum limit.

Sensitivity analysis and optimal shape design are key issues in engineering that have been affected by advances in numerical tools currently available. This book, and its supplementary online files, presents basic optimization techniques that can be used to compute the sensitivity of a given design to local change, or to improve its performance by local optimization of these data. The relevance and scope of these techniques have improved dramatically in recent years because of progress in discretization strategies, optimization algorithms, automatic differentiation, software availability, and the power of personal computers. Numerical Methods in Sensitivity Analysis and Shape Optimization will be of interest to graduate students involved in mathematical modeling and simulation, as well as engineers and researchers in applied mathematics looking for an up-to-date introduction to optimization techniques, sensitivity analysis, and optimal design.

The current use of complex materials in nanotechnology and industrial engineering has led to a number of intricate problems in mechanics, and the macroscopic behavior of such materials often depends critically on their substructures. This work, written by leading mathematicians and engineers, features a broad range of topics that offer both experimental results and clear, detailed answers to fundamental questions about the general formulation of multifield theories within the context of modeling material substructures. Applied mathematicians, mechanical and structural engineers, material scientists, graduate students, and researchers in the above areas will benefit from the work and may find suggestions for further research in its rich collection of open problems.

Molecular Gas Dynamics originates from lectures and seminars delivered by the author at various universities and institutions worldwide. These materials are supplemented and arranged in a form appropriate to a graduate textbook on molecular gas dynamics, or gas dynamics on the basis of kinetic theory. The book provides an up-to-date description of the basic theory of molecular gas dynamics and its various applications giving interesting and important gas - namic phenomena. The progress of molecular gas dynamics in the last forty years has greatly enhanced the contents of the basic theory and provided inf- mation on various interesting and important gas dynamic problems. This has made it possible to compile a new graduate textbook on molecular gas dyn- ics. The present book re?ects these developments providing working knowledge: theory,techniques,andtypicalphenomenainarare?edgas(low-densityand- cro ?ows), for future theoretical development and applications. The book begins with a brief presentation of the fundamental properties of the Boltzmann equation and a summary of notation used globally in subsequent chapters of the book. A full explanation of the fundamental properties is given in Appendix A. The author hopes that readers of various backgrounds can proceed quickly to the main subject, with reference to Appendix A if necessary.

Increasingly, neural networks are used and implemented in a wide range of fields and have become useful tools in probabilistic analysis and prediction theory. This book—unique in the literature—studies the application of neural networks to the analysis of time series of sea data, namely significant wave heights and sea levels. The particular problem examined as a starting point is the reconstruction of missing data, a general problem that appears in many cases of data analysis.Specific topics covered include:* Presentation of general information on the phenomenology of waves and tides, as well as related technical details of various measuring processes used in the study.* Description of the model of wind waves (WAM) used to determine the spectral function of waves and predict the behavior of SWH (significant wave heights); a comparison is made of the reconstruction of SWH time series obtained by means of neural networks algorithms versus SWH computed by WAM.* Principles of artificial neural networks, approximation theory, and extreme-value theory necessary to understand the main applications of the book.* Application of artificial neural networks (ANN) to reconstruct SWH and sea levels (SL).* Comparison of the ANN approach and the approximation operator approach, displaying the advantages of ANN.* Examination of extreme-event analysis applied to the time series of sea data in specific locations.* Generalizations of ANN to treat analogous problems for other types of phenomena and data.This book, a careful blend of theory and applications, is an excellent introduction to the use of ANN, which may encourage readers to try analogous approaches in other important application areas. Researchers, practitioners, and advanced graduate students in neural networks, hydraulic and marine engineering, prediction theory, and data analysis will benefit from the resultsand novel ideas presented in this useful resource.

Modern aerospace vehicles, such as the space shuttle, other launch vehicles, and long-range ballistic missiles, do not discriminate between atmospheric and space flight. Most texts on flight dynamics, however, make this artificial distinction and therefore do not simultaneously cover aircraft and spacecraft. Bridging this gap in the literature, Atmospheric and Space Flight Dynamics is a unified presentation, demonstrating that the two disciplines have actually evolved from the same set of physical principles. Key features inclue an introduction to a broad range of modern topics in an accessible, yet mathematically rigorous presentation; many numerical examples and simulations utilizing MATLAB® and Simulink®; software used as an instructional, hands-on tool, moving away from the "cookbook" approach found in other works; and numerous illustrations and end-of-chapter exercises. Primarily useful as a textbook for advanced undergraduate and beginning graduate-level students, the work is also an excellent reference or self-study guide for researchers and practitioners in aerospace engineering, aviation, mechanical engineering, dynamics, astrodynamics, aeronautics, and astronautics.

Shapes have been among man’s fascinations from thestoneage to thespace age. The scienti?c study of shapes may indeed be traced back to D’Arcy Thompson in his pioneering book On Growth and Form where shape was shown to be dependent on functionality [6]. Numerous de?nitions of a notion of a shape have been proposed in the past, each and every one highlighting aspects relevant to a particular application of interest. The advent of digital imagery, together with the ubiquitous exploitation of its characteristics in a variety of applications, have triggered a renewed and keen interest in further re?ning and possibly unifying the notion ofshape. The present contributed book is, to a large extent, motivated by this upsurge in activity and by the need for an update on recent accomplishments and trends. Theresearchactivityinshapeanalysisisdistinguishedbytwomainschools of thought: — The?rstapproximatesshapesbya?nite-dimensionalrepresentation(a set of landmarks), which is then subjected to various transformations to account for variability and to subsequently derive models. — The second, on the other hand, interprets shapes as closed contours in an in?nite-dimensional space, which, when subjected to transformations, morph into other shapes, thereby yielding a notion of similarity in the space of shapes. 1 Landmark-BasedShapeRepresentation Shapeisaboutscale,orientation,andrelationshipamongtheso-calledchar- teristic points/landmarks of an object-delineating contour. Such information about a data set better de?nes a shape. Equivalently, when such information is taken out of two data sets, the resulting shapes may be compared.

ContentsandScienti?cAims The scienti?c community is aware that the great scienti?c revolution of this century will be the mathematical formalization, by methods of applied mathematics, of complex biological systems. A fascinating prospect is that biological sciences will ?nally be supported by rigorous investigation me- ods and tools, similar to what happened in the past two centuries in the case of mechanical and physical sciences. It is not an easy task, considering that new mathematical methods maybeneededtodealwiththeinnercomplexityofbiologicalsystemswhich exhibit features and behaviors very di?erent from those of inert matter. Microscopic entities in biology, say cells in a multicellular system, are characterized by biological functions and the ability to organize their dynamics and interactions with other cells. Indeed, cells organize their dynamics according to the above functions, while classical particles follow deterministic laws of Newtonian mechanics. Cells have a life according to a cell cycle which ends up with a programmed death. The dialogue among cells can modify their behavior. The activity of cells includes proliferation and/or destructive events which may, in some cases, result in dangerously reproductive events. Finally, a cellular system may move far from eq- librium in physical situations where classical particles generally show a tendency toward equilibrium. An additional source of complexity is that biological systems always need a multiscale approach. Speci?cally, the dynamics of a cell, including its life, are ruled by sub-cellular entities, while most of the phenomena can be e?ectively observed only at the macroscopic scale.

One of the primary purposes and obligations of science, in addition to - derstandingnatureingeneralandlifeinparticular, istoassistinenhancing the quality and longevity of life, indeed a most daunting challenge. To be able to meaningfully meet the last of the above expectations, it is nec- sary to provide the practitioner of medicine with diagnostic and predictive capabilities that science will accord when its seemingly disparate parts are melded together and brought to bear on the problems that they face. The development of interdisciplinary activities involving the various basic sciences—biology, physics, chemistry, and mathematics, and their applied counterparts, engineering and technology—is a necessary key to unlocking the mysteries of medicine, which at the moment is a curious admixture of art, craft, and science. Signi?cant strides have been taken during the past decades for putting intoplaceamethodologythattakesintoaccounttheinterplayofthevarious basic sciences. Considerable progress has been made in understanding the role that mechanics has to play in the development of medical procedures. Thiscollectionofsurveyarticlesaddressestheroleofmechanicswithregard to advances in the medical sciences. In particular, these survey articles bring to one’s attention the central role played by mathematical modeling in general and the modeling of mechanical issues in particular that have a bearing on the biology, chemistry, and physics of living matter.

Modern aerospace vehicles, such as the space shuttle, other launch vehicles, and long-range ballistic missiles, do not discriminate between atmospheric and space flight. Most texts on flight dynamics, however, make this artificial distinction and therefore do not simultaneously cover aircraft and spacecraft. Bridging this gap in the literature, Atmospheric and Space Flight Dynamics is a unified presentation, demonstrating that the two disciplines have actually evolved from the same set of physical principles. Key features inclue an introduction to a broad range of modern topics in an accessible, yet mathematically rigorous presentation; many numerical examples and simulations utilizing MATLAB® and Simulink®; software used as an instructional, hands-on tool, moving away from the "cookbook" approach found in other works; and numerous illustrations and end-of-chapter exercises. Primarily useful as a textbook for advanced undergraduate and beginning graduate-level students, the work is also an excellent reference or self-study guide for researchers and practitioners in aerospace engineering, aviation, mechanical engineering, dynamics, astrodynamics, aeronautics, and astronautics.

This volume aims to provide an overview of some recent developments of mathematical kinetic theory focused on its application in modelling complex systems in various ?elds of applied sciences. Mathematical kinetic theory is essentially based on the Boltzmann eq- tion, which describes the evolution, possibly far from equilibrium, of a class of particles modelled as point masses. The equation de?nes the evolution in time and space of the distribution function over the possible microscopic states of the test particle, classically position and velocity. The test particle is subject to pair collisions with the ?eld particles. The interested reader can ?nd in the book, Theory and Application of the Boltzmann Equation, by C. Cercignani, R. Illner, and M. Pulvirenti, Springer, Heidelberg, 1993, all necessary knowledge of the physics and mathematical topics related to this celebrated model of non-equilibrium statistical mech- ics. Another important model of mathematical kinetic theory is the Vlasov equation, where interactions between particles are not speci?cally collisions, but mean ?eld actions of the ?eld particles over the test particle. The model de?nes again an evolution equation for the one-particle distribution function over the microscopic state of the test particle. The two models brie?y mentioned above can be regarded as the fun- mental models of mathematical kinetic theory and the essential background o?ered from the kinetic theory for classical particles towards the modelling of large systems of several particles undergoing non classical interactions.