This book explores the numerical algorithms underpinning modern finite element based computational mechanics software. It covers all the major numerical methods that are used in computational mechanics. It reviews the basic concepts in linear algebra and advanced matrix theory, before covering solution of systems of equations, symmetric eigenvalue solution methods, and direct integration of discrete dynamic equations of motion, illustrated with numerical examples. This book suits a graduate course in mechanics based disciplines, and will help software developers in computational mechanics. Increased understanding of the underlying numerical methods will also help practicing engineers to use the computational mechanics software more effectively.

This book is devoted to research in the actual field of mathematical modeling in modern problems of plasma physics associated with vibrations and wake waves excited by a short high-power laser pulse. The author explores the hydrodynamic model of the wake wave in detail and from different points of view, within the framework of its regular propagation, a development suitable for accelerating electrons, and the final tipping effect resulting in unregulated energy transfer to plasma particles. Key selling features: Presents research directly related to the propagation of super-power short laser pulses (subject of the 2018 Nobel Prize in Physics). Presents mathematical modeling of plasma physics associated with vibrations and wake waves excited by a short high-power laser pulse. Includes studies of large-amplitude plasma oscillations. Most of the presented results are of original nature and have not appeared in the domestic and foreign scientific literature Written at a level accessible for researchers, academia, and engineers.

Presenting excellent material for a first course on functional analysis , Functional Analysis in Applied Mathematics and Engineering concentrates on material that will be useful to control engineers from the disciplines of electrical, mechanical, and aerospace engineering. This text/reference discusses: rudimentary topology Banach's fixed point theorem with applications L^p-spaces density theorems for testfunctions infinite dimensional spaces bounded linear operators Fourier series open mapping and closed graph theorems compact and differential operators Hilbert-Schmidt operators Volterra equations Sobolev spaces control theory and variational analysis Hilbert Uniqueness Method boundary element methods Functional Analysis in Applied Mathematics and Engineering begins with an introduction to the important, abstract basic function spaces and operators with mathematical rigor, then studies problems in the Hilbert space setting. The author proves the spectral theorem for unbounded operators with compact inverses and goes on to present the abstract evolution semigroup theory for time dependent linear partial differential operators. This structure establishes a firm foundation for the more advanced topics discussed later in the text.

Perturbation Methods for Engineers and Scientists examines the main techniques of perturbation expansions applied to both differential equations and integral expressions. It describes several fluid dynamics applications, including aerofoils, boundary layers in momentum heat, and mass transfer. In addition, it applies the multiple scale technique to the description of surface roughness effects in lubrication. The book's intuitive, rather than formal, approach enables these advanced techniques to be used by scientists and engineers as well as by students.

"Integrates two classical approaches to computability. Offers detailed coverage of recent research at the interface of logic, computability theory, nd theoretical computer science. Presents new, never-before-published results and provides informtion not easily accessible in the literature."

Stochastic Finance provides an introduction to mathematical finance that is unparalleled in its accessibility. Through classroom testing, the authors have identified common pain points for students, and their approach takes great care to help the reader to overcome these difficulties and to foster understanding where comparable texts often do not. Written for advanced undergraduate students, and making use of numerous detailed examples to illustrate key concepts, this text provides all the mathematical foundations necessary to model transactions in the world of finance. A first course in probability is the only necessary background. The book begins with the discrete binomial model and the finite market model, followed by the continuous Black-Scholes model. It studies the pricing of European options by combining financial concepts such as arbitrage and self-financing trading strategies with probabilistic tools such as sigma algebras, martingales and stochastic integration. All these concepts are introduced in a relaxed and user-friendly fashion.

Hormones as Tokens of Selection addresses deep questions in biology: How are biological systems controlled? How can one formulate general theories of homeostasis and control and instantiate such theories in mathematical models? How can one use evolutionary arguments to guide our answers to these questions, recognising that the control mechanisms themselves are a product of evolution? Biological systems are exceptionally varied and extremely difficult to understand, because they are complex and experimentation remains limited relative to the challenges at hand. Moreover, biological phenomena occur at a wide range of temporal and spatial scales. Such a deeply convoluted subject calls for a unifying and coherent theoretical foundation - one which recognises and departs from the primary importance of mathematical modelling and key physicochemical principles to theory formation in the life sciences. This Focus monograph proposes and outlines such a foundation, departing from the deceptively simple proposition that hormones are tokens of evolutionary pressures. Features Provides a coherent and unified approach to a multifaceted problem Pays close attention to both the biological and mathematical modelling aspects of the subject matter, exploring the philosophical background where appropriate Written in a concise and innovative style

Just as the circle number or the Euler constant e determines mathematics, fundamental constants of nature define the scales of the natural sciences. This book presents a new perspective by means of a few axioms and compares the resulting validity with experimental data. By the axiomatic approach Sommerfeld's mysterious fine-structure constant and Dirac's cosmic number are fixed as pure number constants. Thanks to these number constants, it is possible to calculate the value for the anomalous magnetic-moment of the electron in a simple way compared to QED calculations. With the same number constants it is also possible to calculate masses, partial lifetimes, magnetic-moments or charge radii of fundamental particles. The expressions used for the calculations, with few exceptions, yield values within the experimental error limits of the Particle Data Group. The author shows that the introduced number constants give even better predictions than the complicated QED calculations of today's doctrine. In the first part only experimental data from the literature for checking the postulates are used. In the second part the author explains electrical transport measurements with emergent behaviour, which were carried out in a professional environment.

We lift a veil of obscurity from a branch of mathematical physics in a straightforward manner that can be understood by motivated and prepared undergraduate students as well as graduate students specializing in relativity. Our book on 'Einstein Fields' clarifies Einstein's very first principle of equivalence (1907) that is the basis of his theory of gravitation. This requires the exploration of homogeneous Riemannian manifolds, a program that was suggested by Elie Cartan in 'Riemannian Geometry in an Orthogonal Frame,' a 2001 World Scientific publication.Einstein's first principle of equivalence, the key to his General Relativity, interprets homogeneous fields of acceleration as gravitational fields. The general theory of these 'Einstein Fields' is given for the first time in our monograph and has never been treated in such exhaustive detail. This study has yielded significant new insights to Einstein's theory. The volume is heavily illustrated and is accessible to well-prepared undergraduate and graduate students as well as the professional physics community.

The unique compendium is an introductory reference to learn the most popular numerical methods cohesively. The text focuses on practical applications rather than on abstract and heavy analytical concepts. The key elements of the numerical methods are Taylor series and linear algebra. Based on the authors' years of experience, most materials on the text are tied to those elements in a unified manner.The useful reference manual benefits professionals, researchers, academics, senior undergraduate and graduate students in chemical engineering, civil engineering, mechanical engineering and aerospace engineering.

This book is designed to be a vital companion to math textbooks covering the topics of precalculus, calculus, linear algebra, differential equations, and probability and statistics. While these existing textbooks focus mainly on solving mathematic problems using the old paper-and-pencil method, this book teaches how to solve these problems using Maxima open-source software. Maxima is a system for the manipulation of symbolic and numerical expressions, including differentiation, integration, Taylor series, Laplace transforms, ordinary differential equations, systems of linear equations, polynomials, sets, lists, vectors, and matrices. One of the benefits of using Maxima to solve mathematics problems is the immediacy with which it produces answers. Investing in learning Maxima now will pay off in the future, particularly for students and beginning professionals in mathematics, science, and engineering. The volume will help readers to apply nearly all of the Maxima skills discussed here to future courses and research.

Pathwise estimation and inference for diffusion market models discusses contemporary techniques for inferring, from options and bond prices, the market participants' aggregate view on important financial parameters such as implied volatility, discount rate, future interest rate, and their uncertainty thereof. The focus is on the pathwise inference methods that are applicable to a sole path of the observed prices and do not require the observation of an ensemble of such paths. This book is pitched at the level of senior undergraduate students undertaking research at honors year, and postgraduate candidates undertaking Master's or PhD degree by research. From a research perspective, this book reaches out to academic researchers from backgrounds as diverse as mathematics and probability, econometrics and statistics, and computational mathematics and optimization whose interest lie in analysis and modelling of financial market data from a multi-disciplinary approach. Additionally, this book is also aimed at financial market practitioners participating in capital market facing businesses who seek to keep abreast with and draw inspiration from novel approaches in market data analysis. The first two chapters of the book contains introductory material on stochastic analysis and the classical diffusion stock market models. The remaining chapters discuss more special stock and bond market models and special methods of pathwise inference for market parameter for different models. The final chapter describes applications of numerical methods of inference of bond market parameters to forecasting of short rate. Nikolai Dokuchaev is an associate professor in Mathematics and Statistics at Curtin University. His research interests include mathematical and statistical finance, stochastic analysis, PDEs, control, and signal processing. Lin Yee Hin is a practitioner in the capital market facing industry. His research interests include econometrics, non-parametric regression, and scientific computing.

New Edition: The Nonlinear Workbook (6th Edition)The Nonlinear Workbook provides a comprehensive treatment of all the techniques in nonlinear dynamics together with C++, Java and SymbolicC++ implementations. The book not only covers the theoretical aspects of the topics but also provides the practical tools. To understand the material, more than 100 worked out examples and 150 ready to run programs are included. New topics added to the fifth edition are Langton's ant, chaotic data communication, self-controlling feedback, differential forms and optimization, T-norms and T-conorms with applications.

Variational Techniques for Elliptic Partial Differential Equations, intended for graduate students studying applied math, analysis, and/or numerical analysis, provides the necessary tools to understand the structure and solvability of elliptic partial differential equations. Beginning with the necessary definitions and theorems from distribution theory, the book gradually builds the functional analytic framework for studying elliptic PDE using variational formulations. Rather than introducing all of the prerequisites in the first chapters, it is the introduction of new problems which motivates the development of the associated analytical tools. In this way the student who is encountering this material for the first time will be aware of exactly what theory is needed, and for which problems. Features A detailed and rigorous development of the theory of Sobolev spaces on Lipschitz domains, including the trace operator and the normal component of vector fields An integration of functional analysis concepts involving Hilbert spaces and the problems which can be solved with these concepts, rather than separating the two Introduction to the analytical tools needed for physical problems of interest like time-harmonic waves, Stokes and Darcy flow, surface differential equations, Maxwell cavity problems, etc. A variety of problems which serve to reinforce and expand upon the material in each chapter, including applications in fluid and solid mechanics

The aim of this book is to make the subject easier to understand. This book provides clear concepts, tools, and techniques to master the subject -tensor, and can be used in many fields of research. Special applications are discussed in the book, to remove any confusion, and for absolute understanding of the subject. In most books, they emphasize only the theoretical development, but not the methods of presentation, to develop concepts. Without knowing how to change the dummy indices, or the real indices, the concept cannot be understood. This book takes it down a notch and simplifies the topic for easy comprehension. Features Provides a clear indication and understanding of the subject on how to change indices Describes the original evolution of symbols necessary for tensors Offers a pictorial representation of referential systems required for different kinds of tensors for physical problems Presents the correlation between critical concepts Covers general operations and concepts

The book is an introduction, for both graduate students and newcomers to the field of the modern theory of mesoscopic complex systems, time series, hypergraphs and graphs, scaled random walks, and modern information theory. As these are applied for the exploration and characterization of complex systems. Our self-consistent review provides the necessary basis for consistency. We discuss a number of applications such diverse as urban structures and musical compositions.

See also GEOMETRIC MECHANICS - Part II: Rotating, Translating and Rolling (2nd Edition) This textbook introduces the tools and language of modern geometric mechanics to advanced undergraduates and beginning graduate students in mathematics, physics and engineering. It treats the fundamental problems of dynamical systems from the viewpoint of Lie group symmetry in variational principles. The only prerequisites are linear algebra, calculus and some familiarity with Hamilton's principle and canonical Poisson brackets in classical mechanics at the beginning undergraduate level.The ideas and concepts of geometric mechanics are explained in the context of explicit examples. Through these examples, the student develops skills in performing computational manipulations, starting from Fermat's principle, working through the theory of differential forms on manifolds and transferring these ideas to the applications of reduction by symmetry to reveal Lie-Poisson Hamiltonian formulations and momentum maps in physical applications.The many Exercises and Worked Answers in the text enable the student to grasp the essential aspects of the subject. In addition, the modern language and application of differential forms is explained in the context of geometric mechanics, so that the importance of Lie derivatives and their flows is clear. All theorems are stated and proved explicitly.The organisation of the first edition has been preserved in the second edition. However, the substance of the text has been rewritten throughout to improve the flow and to enrich the development of the material. In particular, the role of Noether's theorem about the implications of Lie group symmetries for conservation laws of dynamical systems has been emphasised throughout, with many applications.

'This book offers one of the few places where a collection of results from the literature can be found ... The book has an extensive bibliography ... It is very nice to have the compendium of results that is presented here.'zbMATHA mathematical billiard is a mechanical system consisting of a billiard ball on a table of any form (which can be planar or even a multidimensional domain) but without billiard pockets. The ball moves and its trajectory is defined by the ball's initial position and its initial speed vector. The ball's reflections from the boundary of the table are assumed to have the property that the reflection and incidence angles are the same. This book comprehensively presents known results on the behavior of a trajectory of a billiard ball on a planar table (having one of the following forms: circle, ellipse, triangle, rectangle, polygon and some general convex domains). It provides a systematic review of the theory of dynamical systems, with a concise presentation of billiards in elementary mathematics and simple billiards related to geometry and physics.The description of these trajectories leads to the solution of various questions in mathematics and mechanics: problems related to liquid transfusion, lighting of mirror rooms, crushing of stones in a kidney, collisions of gas particles, etc. The analysis of billiard trajectories can involve methods of geometry, dynamical systems, and ergodic theory, as well as methods of theoretical physics and mechanics, which has applications in the fields of biology, mathematics, medicine, and physics.

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.

Some people fear and mistrust numbers. Others want to use them for everything. After a long career as a statistician, Paul Goodwin has learned the hard way that the ones who want to use them for everything are a very good reason for the rest of us to fear and mistrust them. Something Doesn't Add Up is a fieldguide to the numbers that rule our world, even though they don't make sense. Wry, witty and humane, Goodwin explains mathematical subtleties so painlessly that you hardly need to think about numbers at all. He demonstrates how statistics that are meant to make life simpler often make it simpler than it actually is, but also reveals some of the ways we really can use maths to make better decisions. Enter the world of fitness tracking, the history of IQ testing, China's social credit system, Effective Altruism, and learn how someone should have noticed that Harold Shipman was killing his patients years before they actually did. In the right hands, maths is a useful tool. It's just a pity there are so many of the wrong hands about.

This book explores the remarkable information correspondences and probability structures of proteins. Correspondences are pervasive in biochemistry and bioinformatics: proteins share homologies, folding patterns, and mechanisms. Probability structures are just as paramount: folded state graphics reflect Angstrom-scale maps of electron density. The author explores protein sequences (primary structures), both individually and in sets (systems) with the help of probability and information tools. This perspective will enhance the reader's knowledge of how an important class of molecules is designed and put to task in natural systems, and how we can approach class members in hands-on ways.

The widespread use of Geographical Information Systems (GIS) has significantly increased the demand for knowledge about spatial analytical techniques across a range of disciplines. As growing numbers of researchers realise they are dealing with spatial data, the demand for specialised statistical and mathematical methods designed to deal with spatial data is undergoing a rapid increase. Responding to this demand, The Handbook of Spatial Analysis is a comprehensive and authoritative discussion of issues and techniques in the field of Spatial Data Analysis. Its principal focus is on: * why the analysis of spatial data needs separate treatment * the main areas of spatial analysis * the key debates within spatial analysis * examples of the application of various spatial analytical techniques * problems in spatial analysis * areas for future research Aimed at an international audience of academics, The Handbook of Spatial Analysis will also prove essential to graduate level students and researchers in government agencies and the private sector.

The application of fractals and fractal geometry in soil science has become increasingly important over the last few years. This self-contained and timely book was designed to provide detailed and comprehensive information on the current status of the application of fractal geometry in soil science, and on prospects for its future use. With a detailed and specific introductory chapter, particular attention is paid to comparing and contrasting "fractal" and "fragmentation" concepts. Some uses of fractals, such as to quantify the retention and transport properties of soils, to describe the intricate geometry of pore surfaces and macropore networks, or to elucidate the rooting patterns of various plants, are discussed. Applications of fractals in soil science are both relatively recent and in constant evolution. This book reflects accurately existing trends, by allowing sharp differences among the viewpoints expressed in contributed chapters to be presented to the reader in one self-contained volume.

This authoritative text offers a complete overview on the statistical mechanics and electrodynamics of physical processes in dense plasma systems. The author emphasizes laboratory-based experiments and astrophysical observations of plasma phenomena, elucidated through the fundamentals. The coverage encompasses relevant condensed matter physics, atomic physics, nuclear physics, and astrophysics, including such key topics as phase transitions, transport, optical and nuclear processes. This essential resource also addresses exciting, cutting edge topics in the field, including metallic hydrogen, stellar and planetary magnetisms, pycnonuclear reactions, and gravitational waves. Scientists, researchers, and students in plasma physics, condensed matter physics, materials science, atomic physics, nuclear physics, and astrophysics will benefit from this work. Setsuo Ichimaru is a distinguished professor at the University of Tokyo, and has been a visiting member at The Institute for Advanced Study in Princeton, New Jersey, at the University of California, San Diego (UCSD), the Institute for Theoretical Physics at Johannes Kepler University, and the Max Planck Institute for Quantum Optics. He is a recipient of the Subramanyan Chandrasekhar Prize of Plasma Physics from the Association of Asia-Pacific Physical Societies and the Humboldt Research Award from the Alexander von Humboldt Foundation.

This book presents a new, efficient numerical-analytical method for solving the Laplace equation on an arbitrary polygon. This method, called the approximate block method, overcomes indicated difficulties and has qualitatively more rapid convergence than well-known difference and variational-difference methods. The block method also solves the complicated problem of approximate conformal mapping of multiply-connected polygons onto canonical domains with no preliminary information required. The high-precision results of calculations carried out on the computer are presented in an abundance of tables substantiating the exponential convergence of the block method and its strong stability concerning the rounding-off of errors.