Study Of Linear And Nonlinear Models With Mathematica
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Study Of Linear And Nonlinear Models With "Mathematica" by Czeslaw Maczka,Sergii Skurativskyi,Vsevolod Vladimirov Pdf
The book is devoted to the problems of modeling physical systems and fields using the tools and capabilities of the 'Mathematica' software package. In the process of teaching classical courses in mechanics and mathematical physics, one often has to overcome significant difficulties associated with the cumbersomeness of the mathematical apparatus, which more than once distracts from the essence of the problems under consideration. The use of the 'Mathematica' package, which has a rich set of analytical and graphic tools, makes the presentation of classic issues related to modeling and interpretation of physical processes much more transparent. This package enables the visualization of both analytical solutions of nonlinear differential equations and solutions obtained in the form of infinite series or special functions.The textbook consists of two parts that can be studied independently of each other. The first part deals with the issues of nonlinear mechanics and the theory of oscillations. The second part covers linear problems of classical mathematical physics and nonlinear evolution models describing, inter alia, transport phenomena and propagation of waves. The book contains the codes of programs written in the 'Mathematica' package environment. Supplementary materials of programs illustrating and often complementing the presented material are available on the publisher's website.
Applications of Linear and Nonlinear Models by Erik W. Grafarend,Silvelyn Zwanzig,Joseph L. Awange Pdf
This book provides numerous examples of linear and nonlinear model applications. Here, we present a nearly complete treatment of the Grand Universe of linear and weakly nonlinear regression models within the first 8 chapters. Our point of view is both an algebraic view and a stochastic one. For example, there is an equivalent lemma between a best, linear uniformly unbiased estimation (BLUUE) in a Gauss–Markov model and a least squares solution (LESS) in a system of linear equations. While BLUUE is a stochastic regression model, LESS is an algebraic solution. In the first six chapters, we concentrate on underdetermined and overdetermined linear systems as well as systems with a datum defect. We review estimators/algebraic solutions of type MINOLESS, BLIMBE, BLUMBE, BLUUE, BIQUE, BLE, BIQUE, and total least squares. The highlight is the simultaneous determination of the first moment and the second central moment of a probability distribution in an inhomogeneous multilinear estimation by the so-called E-D correspondence as well as its Bayes design. In addition, we discuss continuous networks versus discrete networks, use of Grassmann–Plucker coordinates, criterion matrices of type Taylor–Karman as well as FUZZY sets. Chapter seven is a speciality in the treatment of an overjet. This second edition adds three new chapters: (1) Chapter on integer least squares that covers (i) model for positioning as a mixed integer linear model which includes integer parameters. (ii) The general integer least squares problem is formulated, and the optimality of the least squares solution is shown. (iii) The relation to the closest vector problem is considered, and the notion of reduced lattice basis is introduced. (iv) The famous LLL algorithm for generating a Lovasz reduced basis is explained. (2) Bayes methods that covers (i) general principle of Bayesian modeling. Explain the notion of prior distribution and posterior distribution. Choose the pragmatic approach for exploring the advantages of iterative Bayesian calculations and hierarchical modeling. (ii) Present the Bayes methods for linear models with normal distributed errors, including noninformative priors, conjugate priors, normal gamma distributions and (iii) short outview to modern application of Bayesian modeling. Useful in case of nonlinear models or linear models with no normal distribution: Monte Carlo (MC), Markov chain Monte Carlo (MCMC), approximative Bayesian computation (ABC) methods. (3) Error-in-variables models, which cover: (i) Introduce the error-in-variables (EIV) model, discuss the difference to least squares estimators (LSE), (ii) calculate the total least squares (TLS) estimator. Summarize the properties of TLS, (iii) explain the idea of simulation extrapolation (SIMEX) estimators, (iv) introduce the symmetrized SIMEX (SYMEX) estimator and its relation to TLS, and (v) short outview to nonlinear EIV models. The chapter on algebraic solution of nonlinear system of equations has also been updated in line with the new emerging field of hybrid numeric-symbolic solutions to systems of nonlinear equations, ermined system of nonlinear equations on curved manifolds. The von Mises–Fisher distribution is characteristic for circular or (hyper) spherical data. Our last chapter is devoted to probabilistic regression, the special Gauss–Markov model with random effects leading to estimators of type BLIP and VIP including Bayesian estimation. A great part of the work is presented in four appendices. Appendix A is a treatment, of tensor algebra, namely linear algebra, matrix algebra, and multilinear algebra. Appendix B is devoted to sampling distributions and their use in terms of confidence intervals and confidence regions. Appendix C reviews the elementary notions of statistics, namely random events and stochastic processes. Appendix D introduces the basics of Groebner basis algebra, its careful definition, the Buchberger algorithm, especially the C. F. Gauss combinatorial algorithm.
Applications of Linear and Nonlinear Models by Erik Grafarend,Joseph L. Awange Pdf
Here we present a nearly complete treatment of the Grand Universe of linear and weakly nonlinear regression models within the first 8 chapters. Our point of view is both an algebraic view as well as a stochastic one. For example, there is an equivalent lemma between a best, linear uniformly unbiased estimation (BLUUE) in a Gauss-Markov model and a least squares solution (LESS) in a system of linear equations. While BLUUE is a stochastic regression model, LESS is an algebraic solution. In the first six chapters we concentrate on underdetermined and overdeterimined linear systems as well as systems with a datum defect. We review estimators/algebraic solutions of type MINOLESS, BLIMBE, BLUMBE, BLUUE, BIQUE, BLE, BIQUE and Total Least Squares. The highlight is the simultaneous determination of the first moment and the second central moment of a probability distribution in an inhomogeneous multilinear estimation by the so called E-D correspondence as well as its Bayes design. In addition, we discuss continuous networks versus discrete networks, use of Grassmann-Pluecker coordinates, criterion matrices of type Taylor-Karman as well as FUZZY sets. Chapter seven is a speciality in the treatment of an overdetermined system of nonlinear equations on curved manifolds. The von Mises-Fisher distribution is characteristic for circular or (hyper) spherical data. Our last chapter eight is devoted to probabilistic regression, the special Gauss-Markov model with random effects leading to estimators of type BLIP and VIP including Bayesian estimation. A great part of the work is presented in four Appendices. Appendix A is a treatment, of tensor algebra, namely linear algebra, matrix algebra and multilinear algebra. Appendix B is devoted to sampling distributions and their use in terms of confidence intervals and confidence regions. Appendix C reviews the elementary notions of statistics, namely random events and stochastic processes. Appendix D introduces the basics of Groebner basis algebra, its careful definition, the Buchberger Algorithm, especially the C. F. Gauss combinatorial algorithm.
Linear and Nonlinear Models by Erik W. Grafarend Pdf
This monograph contains a thorough treatment of methods for solving over- and underdetermined systems of equations, e.g. the minimum norm solution method with respect to weighted norms. The considered equations can be nonlinear or linear, and deterministic models as well as probabilistic ones are considered. An extensive appendix provides all necessary prerequisites like matrix algebra, matrix analysis and Lagrange multipliers, and a long list of references is also included.
A First Course in Linear Model Theory by Nalini Ravishanker,Zhiyi Chi,Dipak K. Dey Pdf
Thoroughly updated throughout, A First Course in Linear Model Theory, Second Edition is an intermediate-level statistics text that fills an important gap by presenting the theory of linear statistical models at a level appropriate for senior undergraduate or first-year graduate students. With an innovative approach, the authors introduce to students the mathematical and statistical concepts and tools that form a foundation for studying the theory and applications of both univariate and multivariate linear models. In addition to adding R functionality, this second edition features three new chapters and several sections on new topics that are extremely relevant to the current research in statistical methodology. Revised or expanded topics include linear fixed, random and mixed effects models, generalized linear models, Bayesian and hierarchical linear models, model selection, multiple comparisons, and regularized and robust regression. New to the Second Edition: Coverage of inference for linear models has been expanded into two chapters. Expanded coverage of multiple comparisons, random and mixed effects models, model selection, and missing data. A new chapter on generalized linear models (Chapter 12). A new section on multivariate linear models in Chapter 13, and expanded coverage of the Bayesian linear models and longitudinal models. A new section on regularized regression in Chapter 14. Detailed data illustrations using R. The authors' fresh approach, methodical presentation, wealth of examples, use of R, and introduction to topics beyond the classical theory set this book apart from other texts on linear models. It forms a refreshing and invaluable first step in students' study of advanced linear models, generalized linear models, nonlinear models, and dynamic models.
Numerical Methods for Nonlinear Engineering Models by John R. Hauser Pdf
There are many books on the use of numerical methods for solving engineering problems and for modeling of engineering artifacts. In addition there are many styles of such presentations ranging from books with a major emphasis on theory to books with an emphasis on applications. The purpose of this book is hopefully to present a somewhat different approach to the use of numerical methods for - gineering applications. Engineering models are in general nonlinear models where the response of some appropriate engineering variable depends in a nonlinear manner on the - plication of some independent parameter. It is certainly true that for many types of engineering models it is sufficient to approximate the real physical world by some linear model. However, when engineering environments are pushed to - treme conditions, nonlinear effects are always encountered. It is also such - treme conditions that are of major importance in determining the reliability or failure limits of engineering systems. Hence it is essential than engineers have a toolbox of modeling techniques that can be used to model nonlinear engineering systems. Such a set of basic numerical methods is the topic of this book. For each subject area treated, nonlinear models are incorporated into the discussion from the very beginning and linear models are simply treated as special cases of more general nonlinear models. This is a basic and fundamental difference in this book from most books on numerical methods.
This book focuses on several key aspects of nonlinear systems including dynamic modeling, state estimation, and stability analysis. It is intended to provide a wide range of readers in applied mathematics and various engineering disciplines an excellent survey of recent studies of nonlinear systems. With its thirteen chapters, the book brings together important contributions from renowned international researchers to provide an excellent survey of recent studies of nonlinear systems. The first section consists of eight chapters that focus on nonlinear dynamic modeling and analysis techniques, while the next section is composed of five chapters that center on state estimation methods and stability analysis for nonlinear systems.
A First Course in Linear Model Theory by Nalini Ravishanker,Zhiyi Chi,Dipak K. Dey Pdf
Thoroughly updated throughout, A First Course in Linear Model Theory, Second Edition is an intermediate-level statistics text that fills an important gap by presenting the theory of linear statistical models at a level appropriate for senior undergraduate or first-year graduate students. With an innovative approach, the authors introduce to students the mathematical and statistical concepts and tools that form a foundation for studying the theory and applications of both univariate and multivariate linear models. In addition to adding R functionality, this second edition features three new chapters and several sections on new topics that are extremely relevant to the current research in statistical methodology. Revised or expanded topics include linear fixed, random and mixed effects models, generalized linear models, Bayesian and hierarchical linear models, model selection, multiple comparisons, and regularized and robust regression. New to the Second Edition: Coverage of inference for linear models has been expanded into two chapters. Expanded coverage of multiple comparisons, random and mixed effects models, model selection, and missing data. A new chapter on generalized linear models (Chapter 12). A new section on multivariate linear models in Chapter 13, and expanded coverage of the Bayesian linear models and longitudinal models. A new section on regularized regression in Chapter 14. Detailed data illustrations using R. The authors' fresh approach, methodical presentation, wealth of examples, use of R, and introduction to topics beyond the classical theory set this book apart from other texts on linear models. It forms a refreshing and invaluable first step in students' study of advanced linear models, generalized linear models, nonlinear models, and dynamic models.
Nonlinear Differential Equation Models by Ansgar Jüngel,Raul Manasevich,Peter A. Markowich,Henrik Shahgholian Pdf
The papers in this book originate from lectures which were held at the "Vienna Workshop on Nonlinear Models and Analysis" – May 20–24, 2002. They represent a cross-section of the research field Applied Nonlinear Analysis with emphasis on free boundaries, fully nonlinear partial differential equations, variational methods, quasilinear partial differential equations and nonlinear kinetic models.
Non-Linear Time Series by Kamil Feridun Turkman,Manuel González Scotto,Patrícia de Zea Bermudez Pdf
This book offers a useful combination of probabilistic and statistical tools for analyzing nonlinear time series. Key features of the book include a study of the extremal behavior of nonlinear time series and a comprehensive list of nonlinear models that address different aspects of nonlinearity. Several inferential methods, including quasi likelihood methods, sequential Markov Chain Monte Carlo Methods and particle filters, are also included so as to provide an overall view of the available tools for parameter estimation for nonlinear models. A chapter on integer time series models based on several thinning operations, which brings together all recent advances made in this area, is also included. Readers should have attended a prior course on linear time series, and a good grasp of simulation-based inferential methods is recommended. This book offers a valuable resource for second-year graduate students and researchers in statistics and other scientific areas who need a basic understanding of nonlinear time series.
Linear and Nonlinear Optimization by Richard W. Cottle,Mukund N. Thapa Pdf
This textbook on Linear and Nonlinear Optimization is intended for graduate and advanced undergraduate students in operations research and related fields. It is both literate and mathematically strong, yet requires no prior course in optimization. As suggested by its title, the book is divided into two parts covering in their individual chapters LP Models and Applications; Linear Equations and Inequalities; The Simplex Algorithm; Simplex Algorithm Continued; Duality and the Dual Simplex Algorithm; Postoptimality Analyses; Computational Considerations; Nonlinear (NLP) Models and Applications; Unconstrained Optimization; Descent Methods; Optimality Conditions; Problems with Linear Constraints; Problems with Nonlinear Constraints; Interior-Point Methods; and an Appendix covering Mathematical Concepts. Each chapter ends with a set of exercises. The book is based on lecture notes the authors have used in numerous optimization courses the authors have taught at Stanford University. It emphasizes modeling and numerical algorithms for optimization with continuous (not integer) variables. The discussion presents the underlying theory without always focusing on formal mathematical proofs (which can be found in cited references). Another feature of this book is its inclusion of cultural and historical matters, most often appearing among the footnotes. "This book is a real gem. The authors do a masterful job of rigorously presenting all of the relevant theory clearly and concisely while managing to avoid unnecessary tedious mathematical details. This is an ideal book for teaching a one or two semester masters-level course in optimization – it broadly covers linear and nonlinear programming effectively balancing modeling, algorithmic theory, computation, implementation, illuminating historical facts, and numerous interesting examples and exercises. Due to the clarity of the exposition, this book also serves as a valuable reference for self-study." Professor Ilan Adler, IEOR Department, UC Berkeley "A carefully crafted introduction to the main elements and applications of mathematical optimization. This volume presents the essential concepts of linear and nonlinear programming in an accessible format filled with anecdotes, examples, and exercises that bring the topic to life. The authors plumb their decades of experience in optimization to provide an enriching layer of historical context. Suitable for advanced undergraduates and masters students in management science, operations research, and related fields." Michael P. Friedlander, IBM Professor of Computer Science, Professor of Mathematics, University of British Columbia
Linear and Nonlinear Aspects of Vortices by Frank Pacard,Tristan Riviere Pdf
Aimed at mathematicians, physicists, engineers, and grad students, this monograph will be useful for the nonlinear analysis of problems arising in geometry or mathematical physics. The material presented covers recent and original results by the authors, and serves as an excellent classroom text or a valuable self-study resource.
Geographical Models with Mathematica by Andre Dauphine Pdf
Geographical Models with Mathematica provides a fairly comprehensive overview of the types of models necessary for the development of new geographical knowledge, including stochastic models, models for data analysis, for geostatistics, for networks, for dynamic systems, for cellular automata and for multi-agent systems, all discussed in their theoretical context. The author then provides over 65 programs, written in the Mathematica language, that formalize these models. Case studies are provided to help the reader apply these programs to their own studies. Provides theoretical, stochastic and dynamic system models Covers data science, both in a spatial and spatio-temporal analysis Presents a microstructural understanding of the mechanical behavior of granular materials
Nonlinear Dynamical Systems And Carleman Linearization by Krzysztof Kowalski,Willi-hans Steeb Pdf
The Carleman linearization has become a new powerful tool in the study of nonlinear dynamical systems. Nevertheless, there is the general lack of familiarity with the Carleman embedding technique among those working in the field of nonlinear models. This book provides a systematic presentation of the Carleman linearization, its generalizations and applications. It also includes a review of existing alternative methods for linearization of nonlinear dynamical systems. There are probably no books covering such a wide spectrum of linearization algorithms. This book also gives a comprehensive introduction to the Kronecker product of matrices, whereas most books deal with it only superficially. The Kronecker product of matrices plays an important role in mathematics and in applications found in theoretical physics.
Nonlinear Equations for Beams and Degenerate Plates with Piers by Maurizio Garrione,Filippo Gazzola Pdf
This book develops a full theory for hinged beams and degenerate plates with multiple intermediate piers with the final purpose of understanding the stability of suspension bridges. New models are proposed and new tools are provided for the stability analysis. The book opens by deriving the PDE’s based on the physical models and by introducing the basic framework for the linear stationary problem. The linear analysis, in particular the behavior of the eigenvalues as the position of the piers varies, enables the authors to tackle the stability issue for some nonlinear evolution beam equations, with the aim of determining the “best position” of the piers within the beam in order to maximize its stability. The study continues with the analysis of a class of degenerate plate models. The torsional instability of the structure is investigated, and again, the optimal position of the piers in terms of stability is discussed. The stability analysis is carried out by means of both analytical tools and numerical experiments. Several open problems and possible future developments are presented. The qualitative analysis provided in the book should be seen as the starting point for a precise quantitative study of more complete models, taking into account the action of aerodynamic forces. This book is intended for a two-fold audience. It is addressed both to mathematicians working in the field of Differential Equations, Nonlinear Analysis and Mathematical Physics, due to the rich number of challenging mathematical questions which are discussed and left as open problems, and to Engineers interested in mechanical structures, since it provides the theoretical basis to deal with models for the dynamics of suspension bridges with intermediate piers. More generally, it may be enjoyable for readers who are interested in the application of Mathematics to real life problems.