Two central problems in the pure theory of economic growth are analysed in this monograph: 1) the dynamic laws governing the economic growth processes, 2) the kinematic and geometric properties of the set of solutions to the dynamic systems. With allegiance to rigor and the emphasis on the theoretical fundamentals of prototype mathematical growth models, the treatise is written in the theorem-proof style. To keep the exposition orderly and as smooth as possible, the economic analysis has been separated from the purely mathematical issues, and hence the monograph is organized in two books. Regarding the scope and content of the two books, an "Introduction and Over view" has been prepared to offer both motivation and a brief account. The introduc tion is especially designed to give a recapitulation of the mathematical theory and results presented in Book II, which are used as the unifying mathematical framework in the analysis and exposition of the different economic growth models in Book I. Economists would probably prefer to go directly to Book I and proceed by consult ing the mathematical theorems of Book II in confirming the economic theorems in Book I. Thereby, both the independence and interdependence of the economic and mathematical argumentations are respected."

This book, intended for researchers and graduate students in physics, applied mathematics and engineering, presents a detailed comparison of the important methods of solution for linear differential and difference equations - variation of constants, reduction of order, Laplace transforms and generating functions - bringing out the similarities as well as the significant differences in the respective analyses. Equations of arbitrary order are studied, followed by a detailed analysis for equations of first and second order. Equations with polynomial coefficients are considered and explicit solutions for equations with linear coefficients are given, showing significant differences in the functional form of solutions of differential equations from those of difference equations. An alternative method of solution involving transformation of both the dependent and independent variables is given for both differential and difference equations. A comprehensive, detailed treatment of Green's functions and the associated initial and boundary conditions is presented for differential and difference equations of both arbitrary and second order. A dictionary of difference equations with polynomial coefficients provides a unique compilation of second order difference equations obeyed by the special functions of mathematical physics. Appendices augmenting the text include, in particular, a proof of Cramer's rule, a detailed consideration of the role of the superposition principal in the Green's function, and a derivation of the inverse of Laplace transforms and generating functions of particular use in the solution of second order linear differential and difference equations with linear coefficients.

This book covers pattern recognition techniques applied to various areas of biomedicine, including disease diagnosis and prognosis, and several problems of classification, with a special focus on-but not limited to-pattern recognition modeling of biomedical signals and images. Multidisciplinary by definition, the book's topic blends computing, mathematics and other technical sciences towards the development of computational tools and methodologies that can be applied to pattern recognition processes. In this work, the efficacy of such methods and techniques for processing medical information is analyzed and compared, and auxiliary criteria for determining the correct diagnosis and treatment strategies are recommended and applied. Researchers in applied mathematics, the computer sciences, engineering and related fields with a focus on medical applications will benefit from this book, as well as professionals with a special interest in state-of-the-art pattern recognition techniques as applied to biomedicine.

For more than 250 years partial di?erential equations have been clearly the most important tool available to mankind in order to understand a large variety of phenomena, natural at ?rst and then those originating from - man activity and technological development. Mechanics, physics and their engineering applications were the ?rst to bene't from the impact of partial di?erential equations on modeling and design, but a little less than a century ago the Schr] odinger equation was the key opening the door to the application of partial di?erential equations to quantum chemistry, for small atomic and molecular systems at ?rst, but then for systems of fast growing complexity. The place of partial di?erential equations in mathematics is a very particular one: initially, the partial di?erential equations modeling natural phenomena were derived by combining calculus with physical reasoning in order to - press conservation laws and principles in partial di?erential equation form, leading to the wave equation, the heat equation, the equations of elasticity, the Euler and Navier-Stokes equations for ?uids, the Maxwell equations of electro-magnetics, etc. It is in order to solve 'constructively' the heat equation that Fourier developed the series bearing his name in the early 19th century; Fourier series (and later integrals) have played (and still play) a fundamental roleinbothpureandappliedmathematics, includingmanyareasquiteremote from partial di?erential equations. On the other hand, several areas of mathematics such as di?erential ge- etry have bene?ted from their interactions with partial di?erential equations."

This book represents the HELS (Helmholtz equation least squares) theory and its applications for visualizing acoustic radiation from an arbitrarily shaped vibrating structure in free or confined space. It culminates the most updated research work of the author and his graduate students since 1997. The book contains six chapters. The first serves as a review of the fundamentals in acoustics and the rest cover five specific topics on the HELS theory.

This book is a collection of articles by leading researchers working at the cutting edge of neuro-computational modelling of neurological and psychiatric disorders. Each article contains model validation techniques used in the context of the specific problem being studied. Validation is essential for neuro-inspired computational models to become useful tools in the understanding and treatment of disease conditions. Currently, the immense diversity in neuro-computational modelling approaches for investigating brain diseases has created the need for a structured and coordinated approach to benchmark and standardise validation methods and techniques in this field of research. This book serves as a step towards a systematic approach to validation of neuro-computational models used for studying brain diseases and should be useful for all neuro-computational modellers.

This text discusses mathematical modelling, analysis and control of the immune system and disease dynamics. The purpose of the book is the practical application of mathematics to immunology and medicine in order to establish a basis for more effective treatment, to provide a tutorial systematic description of how the immune system controls diseases and to present several significant examples such as malignant tumour dynamics and control, and viral hepatitis. The book is multidisciplinary in content, with the intended readers including biomathematicians, biologists and physicists. It combines immunological principles, mathematical models, computer simulations and methods of analysis.

The purpose of this monograph is to show how a compliant offshore structure in an ocean environment can be modeled in two and three di mensions. The monograph is divided into five parts. Chapter 1 provides the engineering motivation for this work, that is, offshore structures. These are very complex structures used for a variety of applications. It is possible to use beam models to initially study their dynamics. Chapter 2 is a review of variational methods, and thus includes the topics: princi ple of virtual work, D'Alembert's principle, Lagrange's equation, Hamil ton's principle, and the extended Hamilton's principle. These methods are used to derive the equations of motion throughout this monograph. Chapter 3 is a review of existing transverse beam models. They are the Euler-Bernoulli, Rayleigh, shear and Timoshenko models. The equa tions of motion are derived and solved analytically using the extended Hamilton's principle, as outlined in Chapter 2. For engineering purposes, the natural frequencies of the beam models are presented graphically as functions of normalized wave number and geometrical and physical pa rameters. Beam models are useful as representations of complex struc tures. In Chapter 4, a fluid force that is representative of those that act on offshore structures is formulated. The environmental load due to ocean current and random waves is obtained using Morison's equa tion. The random waves are formulated using the Pierson-Moskowitz spectrum with the Airy linear wave theory."

Recent interest in nanotechnology is challenging the community to analyse, develop and design nanometer to micrometer-sized devices for applications in new generations of computer, electronics, photonics and drug delivery systems. To successfully design and fabricate novel nanomaterials and nanosystems, we must necessarily bridge the gap in our understanding of mechanical properties and processes at length scales ranging from 100 nanometers (where atomistic simulations are currently possible) to a micron (where continuum mechanics is experimentally validated). For this purpose the difficulties and complexity originate in the substantial differences in philosophy and viewpoints between conventional continuum mechanics and quantum theories. The challenge lies in how to establish the relationship between a continuum mechanical system and its atomistic counterpart in order to define continuum variables that are calculable within an atomic system.

This book maximizes reader insights into the field of mathematical models and methods for the processing of two-dimensional remote sensing images. It presents a broad analysis of the field, encompassing passive and active sensors, hyperspectral images, synthetic aperture radar (SAR), interferometric SAR, and polarimetric SAR data. At the same time, it addresses highly topical subjects involving remote sensing data types (e.g., very high-resolution images, multiangular or multiresolution data, and satellite image time series) and analysis methodologies (e.g., probabilistic graphical models, hierarchical image representations, kernel machines, data fusion, and compressive sensing) that currently have primary importance in the field of mathematical modelling for remote sensing and image processing. Each chapter focuses on a particular type of remote sensing data and/or on a specific methodological area, presenting both a thorough analysis of the previous literature and a methodological and experimental discussion of at least two advanced mathematical methods for information extraction from remote sensing data. This organization ensures that both tutorial information and advanced subjects are covered. With each chapter being written by research scientists from (at least) two different institutions, it offers multiple professional experiences and perspectives on each subject. The book also provides expert analysis and commentary from leading remote sensing and image processing researchers, many of whom serve on the editorial boards of prestigious international journals in these fields, and are actively involved in international scientific societies. Providing the reader with a comprehensive picture of the overall advances and the current cutting-edge developments in the field of mathematical models for remote sensing image analysis, this book is ideal as both a reference resource and a textbook for graduate and doctoral students as well as for remote sensing scientists and practitioners.

This monograph presents the newly developed method of rigged Hilbert spaces as a modern approach in singular perturbation theory. A key notion of this approach is the Lax-Berezansky triple of Hilbert spaces embedded one into another, which specifies the well-known Gelfand topological triple. All kinds of singular interactions described by potentials supported on small sets (like the Dirac -potentials, fractals, singular measures, high degree super-singular expressions) admit a rigorous treatment only in terms of the equipped spaces and their scales. The main idea of the method is to use singular perturbations to change inner products in the starting rigged space, and the construction of the perturbed operator by the Berezansky canonical isomorphism (which connects the positive and negative spaces from a new rigged triplet). The approach combines three powerful tools of functional analysis based on the Birman-Krein-Vishik theory of self-adjoint extensions of symmetric operators, the theory of singular quadratic forms, and the theory of rigged Hilbert spaces. The book will appeal to researchers in mathematics and mathematical physics studying the scales of densely embedded Hilbert spaces, the singular perturbations phenomenon, and singular interaction problems.

Expositions of quantitative methods and algorithms for biological data tend to be scattered through the technical literature, often across different fields, and are thus awkward to assimilate. This book documents one example of this: the relationship between the cell biology idea of metabolic networks and the mathematical idea of polyhedral cones. Such cones can be used to describe the set of steady-state admissible fluxes through metabolic networks, and consequently have become important constructs in the field of microbiology. Via convex cone concepts, fundamental objects called elementary flux modes (EFMs) can be described mathematically. The fundamental algorithm of this relationship is the double description method, which has an extended history in the field of computational geometry. This monograph addresses its relatively recent use in the context of cellular metabolism. Metabolic Networks, Elementary Flux Modes, and Polyhedral Cones: Addresses important topics in the mathematical description of metabolic activity that have not previously appeared in unified form. Introduces a central topic of mathematical systems biology in a manner accessible to nonmathematicians with some mathematical and computational experience. Presents a careful study of the double description method, a fundamental algorithm of computational geometry, in the context of metabolic analysis. The core audience for this book includes mathematicians, engineers, and biologists interested in cell metabolism. Computational geometers will also find it of interest.

This book contributes to the developing dialogue between cognitive science and social sciences. It focuses on a central issue in both fields, i.e. the nature and the limitations of the rationality of beliefs and action. The development of cognitive science is one of the most important and fascinating intellectual advances of recent decades, and social scientists are paying increasing attention to the findings of this new branch of science that forces us to consider many classical issues related to epistemology and philosophy of action in a new light. Analysis of the concept of rationality is a leitmotiv in the history of the social sciences and has involved endless disputes. Since it is difficult to give a precise definition of this concept, and there is a lack of agreement about its meaning, it is possible to say that there is a 'mystery of rationality'. What is it to be rational? Is rationality merely instrumental or does it also involve the endorsement of values, i.e. the choice of goals? Should we consider rationality to be a normative principle or a descriptive one? Can rationality be only Cartesian or can it also be argumentative? Is rationality a conscious skill or a partly tacit one? This book, which has been written by an outstanding collection of authors, including both philosophers and social scientists, tries to make a useful contribution to the debates on these problems and shed some light on the mystery of rationality. The target audience primarily comprises researchers and experts in the field.

This authored monograph provides a detailed discussion of the boundary layer flow due to a moving plate. The topical focus lies on the 2- and 3-dimensional case, considering axially symmetric and unsteady flows. The author derives a criterion for the self-similar and non-similar flow, and the turbulent flow due to a stretching or shrinking sheet is also discussed. The target audience primarily comprises research experts in the field of boundary layer flow, but the book will also be beneficial for graduate students.

This volume contains the proceedings of the 19th International Conference on Difference Equations and Applications, held at Sultan Qaboos University, Muscat, Oman in May 2013. The conference brought together experts and novices in the theory and applications of difference equations and discrete dynamical systems. The volume features papers in difference equations and discrete time dynamical systems with applications to mathematical sciences and, in particular, mathematical biology, ecology, and epidemiology. It includes four invited papers and eight contributed papers. Topics covered include: competitive exclusion through discrete time models, Benford solutions of linear difference equations, chaos and wild chaos in Lorenz-type systems, advances in periodic difference equations, the periodic decomposition problem, dynamic selection systems and replicator equations, and asymptotic equivalence of difference equations in Banach Space. This book will appeal to researchers, scientists, and educators who work in the fields of difference equations, discrete time dynamical systems and their applications.

This book gathers the peer-reviewed proceedings of the 13th Annual Meeting of the Bulgarian Section of the Society for Industrial and Applied Mathematics, BGSIAM'18, held in Sofia, Bulgaria. The general theme of BGSIAM'18 was industrial and applied mathematics with particular focus on: mathematical physics, numerical analysis, high performance computing, optimization and control, mathematical biology, stochastic modeling, machine learning, digitization and imaging, advanced computing in environmental, biomedical and engineering applications.

This book is an intellectually stimulating excursion into mathematical machines and structures capable for a universal computation. World top experts in computer science and mathematics overview exciting and intriguing topics of logical theory of monoids, geometry of Gauss word, philosophy of mathematics in computer science, asynchronous and parallel P-systems, decidability in cellular automata, splicing systems, reversible Turing machines, information flows in two-way finite automata, prime generators in automaton arrays, Grossone and Turing machines, automaton models of atomic lattices. The book is full of visually attractive examples of mathematical machines, open problems and challenges for future research. Those interested in the advancement of a theory of computation, philosophy of mathematics, future and emergent computing paradigms, architectures and implementations will find the book vital for their research and development.

This three-part book provides a comprehensive and systematic introduction to these challenging topics such as model calibration, parameter estimation, reliability assessment, and data collection design. Part 1 covers the classical inverse problem for parameter estimation in both deterministic and statistical frameworks, Part 2 is dedicated to system identification, hyperparameter estimation, and model dimension reduction, and Part 3 considers how to collect data and construct reliable models for prediction and decision-making. For the first time, topics such as multiscale inversion, stochastic field parameterization, level set method, machine learning, global sensitivity analysis, data assimilation, model uncertainty quantification, robust design, and goal-oriented modeling, are systematically described and summarized in a single book from the perspective of model inversion, and elucidated with numerical examples from environmental and water resources modeling. Readers of this book will not only learn basic concepts and methods for simple parameter estimation, but also get familiar with advanced methods for modeling complex systems. Algorithms for mathematical tools used in this book, such as numerical optimization, automatic differentiation, adaptive parameterization, hierarchical Bayesian, metamodeling, Markov chain Monte Carlo, are covered in details. This book can be used as a reference for graduate and upper level undergraduate students majoring in environmental engineering, hydrology, and geosciences. It also serves as an essential reference book for professionals such as petroleum engineers, mining engineers, chemists, mechanical engineers, biologists, biology and medical engineering, applied mathematicians, and others who perform mathematical modeling.

This monograph provides a definitive overview of recent advances in the stability and oscillation of autonomous delay differential equations. Topics include linear and nonlinear delay and integrodifferential equations, which have potential applications to both biological and physical dynamic processes. Chapter 1 deals with an analysis of the dynamical characteristics of the delay logistic equation, and a number of techniques and results relating to stability, oscillation and comparison of scalar delay and integrodifferential equations are presented. Chapter 2 provides a tutorial-style introduction to the study of delay-induced Hopf bifurcation to periodicity and the related computations for the analysis of the stability of bifurcating periodic solutions. Chapter 3 is devoted to local analyses of nonlinear model systems and discusses many methods applicable to linear equations and their perturbations. Chapter 4 considers global convergence to equilibrium states of nonlinear systems, and includes oscillations of nonlinear systems about their equilibria. Qualitative analyses of both competitive and cooperative systems with time delays feature in both Chapters 3 and 4. Finally, Chapter 5 deals with recent developments in models of neutral differential equations and their applications to population dynamics. Each chapter concludes with a number of exercises and the overall exposition recommends this volume as a good supplementary text for graduate courses. For mathematicians whose work involves functional differential equations, and whose interest extends beyond the boundaries of linear stability analysis.

Providing an introduction to major topics in macroeconomic theory, this book offers the reader three keys for comparing different models. The first key is a mathematical reformulation of Say's Law. The second key is the use of the income velocity of the circulation of money as a behavioural function in accordance with the Friedman tradition. The third is the use of the Phillips curve to represent the labour market. The book also deals with rationing equilibria and some extensions. It provides a neo-Walrasian synthesis of Patinkin's suggestions and new perspectives on Keynesian criticism of neoclassical theory.

Agent-based modeling/simulation is an emergent approach to the analysis of social and economic systems. It provides a bottom-up experimental method to be applied to social sciences such as economics, management, sociology and politics as well as some engineering fields dealing with social activities. This book includes selected papers presented at the Eighth International Workshop on Agent-Based Approaches in Economic and Social Complex Systems held in Tokyo, Japan, in 2013. At the workshop, 23 reviewed full papers were presented and of those, 13 were selected to be included in this volume.

Understanding and predicting the performance of electromechanical systems is crucially important in the design of many modern products, and today s engineers and researchers are constantly seeking methods for optimizing these complex systems.

This important text/reference highlights a unique combination of numerical tools and strategies for handling the challenges of multiphysics simulation. As multiphysics simulation is a broad and rapidly growing field, requiring an array of technical skills in different intersecting disciplines, this book presents a specific focus on electromechanical systems as the target application.

Topics and features: introduces the concept of design via simulation, along with the role of multiphysics simulation in today s engineering environment; discusses the importance of structural optimization techniques in the design and development of electromechanical systems; provides an overview of the physics commonly involved with electromechanical systems for applications such as electronics, magnetic components, RF components, actuators, and motors; reviews the governing equations for the simulation of related multiphysics problems; outlines relevant (topology and parametric size) optimization methods for electromechanical systems; describes in detail several multiphysics simulation and optimization example studies in both two and three dimensions, with sample numerical code.

Researchers and engineers in industry and academia will find this work to be an invaluable reference on advanced electromechanical system design. The book is also suitable for students at undergraduate and graduate level, and many of the design examples will be of interest to anyone curious about the unique design solutions that arise from the coupling of optimization methods with multiphysics simulation techniques."

This thesis is devoted to the study of the Bohman-Frieze-Wormald percolation model, which exhibits a discontinuous transition at the critical threshold, while the phase transitions in random networks are originally considered to be robust continuous phase transitions. The underlying mechanism that leads to the discontinuous transition in this model is carefully analyzed and many interesting critical behaviors, including multiple giant components, multiple phase transitions, and unstable giant components are revealed. These findings should also be valuable with regard to applications in other disciplines such as physics, chemistry and biology.

This book-unique in the literature-provides readers with the mathematical background needed to design many of the optical combinations that are used in astronomical telescopes and cameras. The results presented in the work were obtained by using a different approach to third-order aberration theory as well as the extensive use of the software package Mathematica (R). Replete with workout examples and exercises, Geometric Optics is an excellent reference for advanced graduate students, researchers, and practitioners in applied mathematics, engineering, astronomy, and astronomical optics. The work may be used as a supplementary textbook for graduate-level courses in astronomical optics, optical design, optical engineering, programming with Mathematica, or geometric optics.

Recent groundbreaking discoveries in physics, including the discovery of the Higgs Boson and gravitational waves, have relied on chi-squared analysis and model testing, a data analysis method. This is the first book to make chi-squared model testing accessible to students in introductory physics lab courses and others who need to learn this method, such as beginning researchers in astrophysics and particle physics, beginners in data science, and lab students in other experimental sciences. For over a decade, Harvard University's introductory physics lab sequence has made chi-squared model testing its central theme. Written by two faculty members, the book is based on years of experience teaching students learn how to think like scientists by testing their models using chi-squared analysis. By including uncertainties in the curve fitting technique, chi-squared data analysis improves on the centuries old ordinary least squares and linear regression methods and combines best fit parameter estimation and model testing in one method. A toolkit of essential statistical and experimental concepts is developed from the ground up with novel features to interest even those familiar with the material. The presentation of one and two parameter chi-squared model testing, requiring only elementary probability and algebra, is followed by case studies that apply the methods to simple introductory physics lab experiments. More challenging topics requiring calculus are addressed in an advanced topic chapter. This self-contained and student-friendly introduction includes a glossary, end of chapter problems with complete solutions, and software scripts available in several popular programming languages that the reader can use for chi-squared model testing.