The present volume aims to be a comprehensive survey on the derivation of the equations of motion, both in General Relativity as well as in alternative gravity theories. The topics covered range from the description of test bodies, to self-gravitating (heavy) bodies, to current and future observations. Emphasis is put on the coverage of various approximation methods (e.g., multipolar, post-Newtonian, self-force methods) which are extensively used in the context of the relativistic problem of motion. Applications discussed in this volume range from the motion of binary systems -- and the gravitational waves emitted by such systems -- to observations of the galactic center. In particular the impact of choices at a fundamental theoretical level on the interpretation of experiments is highlighted. This book provides a broad and up-do-date status report, which will not only be of value for the experts working in this field, but also may serve as a guideline for students with background in General Relativity who like to enter this field.

This book provides theories on non-parametric shape optimization problems, systematically keeping in mind readers with an engineering background. Non-parametric shape optimization problems are defined as problems of finding the shapes of domains in which boundary value problems of partial differential equations are defined. In these problems, optimum shapes are obtained from an arbitrary form without any geometrical parameters previously assigned. In particular, problems in which the optimum shape is sought by making a hole in domain are called topology optimization problems. Moreover, a problem in which the optimum shape is obtained based on domain variation is referred to as a shape optimization problem of domain variation type, or a shape optimization problem in a limited sense. Software has been developed to solve these problems, and it is being used to seek practical optimum shapes. However, there are no books explaining such theories beginning with their foundations. The structure of the book is shown in the Preface. The theorems are built up using mathematical results. Therefore, a mathematical style is introduced, consisting of definitions and theorems to summarize the key points. This method of expression is advanced as provable facts are clearly shown. If something to be investigated is contained in the framework of mathematics, setting up a theory using theorems prepared by great mathematicians is thought to be an extremely effective approach. However, mathematics attempts to heighten the level of abstraction in order to understand many things in a unified fashion. This characteristic may baffle readers with an engineering background. Hence in this book, an attempt has been made to provide explanations in engineering terms, with examples from mechanics, after accurately denoting the provable facts using definitions and theorems.

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.

Introduces the two most common numerical methods for heat transfer and fluid dynamics equations, using clear and accessible language. This unique approach covers all necessary mathematical preliminaries at the beginning of the book for the reader to sail smoothly through the chapters. Students will work step-by-step through the most common benchmark heat transfer and fluid dynamics problems, firmly grounding themselves in how the governing equations are discretized, how boundary conditions are imposed, and how the resulting algebraic equations are solved. Providing a detailed discussion of the discretization steps and time approximations, and clearly presenting concepts of explicit and implicit formulations, this graduate textbook has everything an instructor needs to prepare students for their exams and future careers. Each illustrative example shows students how to draw comparisons between the results obtained using the two numerical methods, and at the end of each chapter they can test and extend their understanding by working through the problems provided. A solutions manual is also available for instructors.

Probabilistic Foundations of Statistical Network Analysis presents a fresh and insightful perspective on the fundamental tenets and major challenges of modern network analysis. Its lucid exposition provides necessary background for understanding the essential ideas behind exchangeable and dynamic network models, network sampling, and network statistics such as sparsity and power law, all of which play a central role in contemporary data science and machine learning applications. The book rewards readers with a clear and intuitive understanding of the subtle interplay between basic principles of statistical inference, empirical properties of network data, and technical concepts from probability theory. Its mathematically rigorous, yet non-technical, exposition makes the book accessible to professional data scientists, statisticians, and computer scientists as well as practitioners and researchers in substantive fields. Newcomers and non-quantitative researchers will find its conceptual approach invaluable for developing intuition about technical ideas from statistics and probability, while experts and graduate students will find the book a handy reference for a wide range of new topics, including edge exchangeability, relative exchangeability, graphon and graphex models, and graph-valued Levy process and rewiring models for dynamic networks. The author's incisive commentary supplements these core concepts, challenging the reader to push beyond the current limitations of this emerging discipline. With an approachable exposition and more than 50 open research problems and exercises with solutions, this book is ideal for advanced undergraduate and graduate students interested in modern network analysis, data science, machine learning, and statistics. Harry Crane is Associate Professor and Co-Director of the Graduate Program in Statistics and Biostatistics and an Associate Member of the Graduate Faculty in Philosophy at Rutgers University. Professor Crane's research interests cover a range of mathematical and applied topics in network science, probability theory, statistical inference, and mathematical logic. In addition to his technical work on edge and relational exchangeability, relative exchangeability, and graph-valued Markov processes, Prof. Crane's methods have been applied to domain-specific cybersecurity and counterterrorism problems at the Foreign Policy Research Institute and RAND's Project AIR FORCE.

This work provides a detailed and up-to-the-minute survey of the various stability problems that can affect suspension bridges. In order to deduce some experimental data and rules on the behavior of suspension bridges, a number of historical events are first described, in the course of which several questions concerning their stability naturally arise. The book then surveys conventional mathematical models for suspension bridges and suggests new nonlinear alternatives, which can potentially supply answers to some stability questions. New explanations are also provided, based on the nonlinear structural behavior of bridges. All the models and responses presented in the book employ the theory of differential equations and dynamical systems in the broader sense, demonstrating that methods from nonlinear analysis can allow us to determine the thresholds of instability.

This self-contained textbook discusses all major topics in functional analysis. Combining classical materials with new methods, it supplies numerous relevant solved examples and problems and discusses the applications of functional analysis in diverse fields. The book is unique in its scope, and a variety of applications of functional analysis and operator-theoretic methods are devoted to each area of application. Each chapter includes a set of problems, some of which are routine and elementary, and some of which are more advanced. The book is primarily intended as a textbook for graduate and advanced undergraduate students in applied mathematics and engineering. It offers several attractive features making it ideally suited for courses on functional analysis intended to provide a basic introduction to the subject and the impact of functional analysis on applied and computational mathematics, nonlinear functional analysis and optimization. It introduces emerging topics like wavelets, Gabor system, inverse problems and application to signal and image processing.

Originally published in 1984. Since the logic underlying economic theory can only be grasped fully by a thorough understanding of the mathematics, this book will be invaluable to economists wishing to understand vast areas of important research. It provides a basic introduction to the fundamental mathematical ideas of topology and calculus, and uses these to present modern singularity theory and recent results on the generic existence of isolated price equilibria in exchange economies.

Occupancy Estimation and Modeling: Inferring Patterns and Dynamics of Species Occurrence, Second Edition, provides a synthesis of model-based approaches for analyzing presence-absence data, allowing for imperfect detection. Beginning from the relatively simple case of estimating the proportion of area or sampling units occupied at the time of surveying, the authors describe a wide variety of extensions that have been developed since the early 2000s. This provides an improved insight about species and community ecology, including, detection heterogeneity; correlated detections; spatial autocorrelation; multiple states or classes of occupancy; changes in occupancy over time; species co-occurrence; community-level modeling, and more. Occupancy Estimation and Modeling: Inferring Patterns and Dynamics of Species Occurrence, Second Edition has been greatly expanded and detail is provided regarding the estimation methods and examples of their application are given. Important study design recommendations are also covered to give a well rounded view of modeling.

Broadly speaking, there are two general approaches to teaching mathematical modeling: 1) the case study approach, and 2) the method based approach (that teaches mathematical techniques with applications to relevant mathematical models). This text emphasizes instead the scientific issues for modeling different phenomena. For the natural or harvested growth of a fish population, we may be interested in the evolution of the population, whether it reaches a steady state (equilibrium or cycle), stable or unstable with respect to a small perturbation from equilibrium, or whether a small change in the environment would cause a catastrophic change, etc. Each scientific issue requires an appropriate model and a different set of mathematical tools to extract information from the model. Models examined are chosen to help explain or justify empirical observations such as cocktail drug treatments are more effective and regenerations after injuries or illness are fast-tracked (compared to original developments).Volume I of this three-volume set limits its scope to phenomena and scientific issues that are modeled by ordinary differential equations (ODE). Scientific issues such as signal and wave propagation, diffusion, and shock formation involving spatial dynamics to be modeled by partial differential equations (PDE) will be treated in Vol. II. Scientific issues involving randomness and uncertainty are examined in Vol. III.

Broadly speaking, there are two general approaches to teaching mathematical modeling: 1) the case study approach, and 2) the method based approach (that teaches mathematical techniques with applications to relevant mathematical models). This text emphasizes instead the scientific issues for modeling different phenomena. For the natural or harvested growth of a fish population, we may be interested in the evolution of the population, whether it reaches a steady state (equilibrium or cycle), stable or unstable with respect to a small perturbation from equilibrium, or whether a small change in the environment would cause a catastrophic change, etc. Each scientific issue requires an appropriate model and a different set of mathematical tools to extract information from the model. Models examined are chosen to help explain or justify empirical observations such as cocktail drug treatments are more effective and regenerations after injuries or illness are fast-tracked (compared to original developments).Volume I of this three-volume set limits its scope to phenomena and scientific issues that are modeled by ordinary differential equations (ODE). Scientific issues such as signal and wave propagation, diffusion, and shock formation involving spatial dynamics to be modeled by partial differential equations (PDE) will be treated in Vol. II. Scientific issues involving randomness and uncertainty are examined in Vol. III.

This book comprehensively presents the computational design of rolling bearings dealing with many interdisciplinary difficult working fields. They encompass elastohydrodynamics (EHD), Hertzian contact theory, oil-film thickness in elastohydrodynamic lubrication (EHL), bearing dynamics, tribology of surface textures, fatigue failure mechanisms, fatigue lifetimes of rolling bearings and lubricating greases, Weibull distribution, rotor balancing, and airborne noises (NVH) in the rolling bearings. Furthermore, the readers are provided with hands-on essential formulas based on the up-to-date DIN ISO norms and helpful examples for computational design of rolling bearings. The topics are intended for undergraduate and graduate students in mechanical and material engineering, research scientists, and practicing engineers who want to understand the interactions between these working fields and to know how to design the rolling bearings for automotive industry and many other industries.

* Based on the new idea of gathering state of the art topics in Geometric Modeling together with techniques, applications, systems and tools
* The first book of its kind in that it incorporates several topics providing a class of practical solutions to problems in mathematical, engineering, and physical sciences
* Provides a valuable resource, focusing on interdisciplinary methods and affiliate research in the area

Inductive Learning Algorithms for Complex Systems Modeling is a professional monograph that surveys new types of learning algorithms for modeling complex scientific systems in science and engineering. The book features discussions of algorithm development, structure, and behavior; comprehensive coverage of all types of algorithms useful for this subject; and applications of various modeling activities (e.g., environmental systems, noise immunity, economic systems, clusterization, and neural networks). It presents recent studies on clusterization and recognition problems, and it includes listings of algorithms in FORTRAN that can be run directly on IBM-compatible PCs. Inductive Learning Algorithms for Complex Systems Modeling will be a valuable reference for graduate students, research workers, and scientists in applied mathematics, statistics, computer science, and systems science disciplines. The book will also benefit engineers and scientists from applied fields such as environmental studies, oceanographic modeling, weather forecasting, air and water pollution studies, economics, hydrology, agriculture, fisheries, and time series evaluations.

This is a book on interdisciplinary topics of the Mathematical and Biological Sciences. The treatment is both pedagogical and advanced in order to motivate research students as well as to fulfill the requirements of professional practitioners. There are comprehensive reviews written by senior experts on the important problems of growth and agglomeration in biology, on the algebraic modelling of the genetic code and on multi-step biochemical pathways.There are new results on the state of the art research in the pattern recognition of probability distribution of amino acids, on somitogenesis through reaction-diffusion models, on the mathematical modelling of infectious diseases, on the biophysical modelling of physiological disorders, on the sensitive analysis of parameters of malaria models, on the stability and hopf bifurcation of ecological and epidemiological models, on the viral infection of bee colonies and on the structure and motion of proteins. All these contributions are also strongly recommended to professionals from other scientific areas aiming to work on these interdisciplinary fields.

Mathematics of Planet Earth (MPE) was started and continues to be consolidated as a collaboration of mathematical science organisations around the world. These organisations work together to tackle global environmental, social and economic problems using mathematics.This textbook introduces the fundamental topics of MPE to advanced undergraduate and graduate students in mathematics, physics and engineering while explaining their modern usages and operational connections. In particular, it discusses the links between partial differential equations, data assimilation, dynamical systems, mathematical modelling and numerical simulations and applies them to insightful examples.The text also complements advanced courses in geophysical fluid dynamics (GFD) for meteorology, atmospheric science and oceanography. It links the fundamental scientific topics of GFD with their potential usage in applications of climate change and weather variability. The immediacy of examples provides an excellent introduction for experienced researchers interested in learning the scope and primary concepts of MPE.

This book is aimed primarily towards physicists and mechanical engineers specializing in modeling, analysis, and control of discontinuous systems with friction and impacts. It fills a gap in the existing literature by offering an original contribution to the field of discontinuous mechanical systems based on mathematical and numerical modeling as well as the control of such systems. Each chapter provides the reader with both the theoretical background and results of verified and useful computations, including solutions of the problems of modeling and application of friction laws in numerical computations, results from finding and analyzing impact solutions, the analysis and control of dynamical systems with discontinuities, etc. The contents offer a smooth correspondence between science and engineering and will allow the reader to discover new ideas. Also emphasized is the unity of diverse branches of physics and mathematics towards understanding complex piecewise-smooth dynamical systems. Mathematical models presented will be important in numerical experiments, experimental measurements, and optimization problems found in applied mechanics.

This book covers crucial lacunae of the linear discrete-time time-invariant dynamical systems and introduces the reader to their treatment, while functioning under real, natural conditions, in forced regimes with arbitrary initial conditions. It provides novel theoretical tools necessary for the analysis and design of the systems operating in stated conditions. The text completely covers two well-known systems, IO and ISO, along with a new system, IIO. It discovers the concept of the full transfer function matrix F(z) in the z-complex domain, which incorporates the Z-transform of the system, input and another variable, vectors, all with arbitrary initial conditions. Consequently, it addresses the full system matrix P(z) and the full block diagram technique based on the use of F(z), which incorporates the Z-transform of the system, input and another variable, vectors, all with arbitrary initial conditions. The book explores the direct relationship between the system full transfer function matrix F(z) and the Lyapunov stability concept, definitions, and conditions, as well as with the BI stability concept, definitions, and conditions. The goal of the book is to unify the study and applications of all three classes of the linear discrete-time time-invariant system, for short systems.

This book presents a broad range of deep-learning applications related to vision, natural language processing, gene expression, arbitrary object recognition, driverless cars, semantic image segmentation, deep visual residual abstraction, brain-computer interfaces, big data processing, hierarchical deep learning networks as game-playing artefacts using regret matching, and building GPU-accelerated deep learning frameworks. Deep learning, an advanced level of machine learning technique that combines class of learning algorithms with the use of many layers of nonlinear units, has gained considerable attention in recent times. Unlike other books on the market, this volume addresses the challenges of deep learning implementation, computation time, and the complexity of reasoning and modeling different type of data. As such, it is a valuable and comprehensive resource for engineers, researchers, graduate students and Ph.D. scholars.

This book is directed toward the numerical integration (solution) of a system of partial differential equations (PDEs) that describes a combination of chemical reaction and diffusion, that is, reaction-diffusion PDEs. The particular form of the PDEs corresponds to a system discussed by Alan Turing and is therefore termed a Turing model.Specifically, Turing considered how a reaction-diffusion system can be formulated that does not have the usual smoothing properties of a diffusion (dispersion) system, and can, in fact, develop a spatial variation that might be interpreted as a form of morphogenesis, so he termed the chemicals as morphogens.Turing alluded to the important impact computers would have in the study of a morphogenic PDE system, but at the time (1952), computers were still not readily available. Therefore, his paper is based on analytical methods. Although computers have since been applied to Turing models, computer-based analysis is still not facilitated by a discussion of numerical algorithms and a readily available system of computer routines.The intent of this book is to provide a basic discussion of numerical methods and associated computer routines for reaction-diffusion systems of varying form. The presentation has a minimum of formal mathematics. Rather, the presentation is in terms of detailed examples, presented at an introductory level. This format should assist readers who are interested in developing computer-based analysis for reaction-diffusion PDE systems without having to first study numerical methods and computer programming (coding).The numerical examples are discussed in terms of: (1) numerical integration of the PDEs to demonstrate the spatiotemporal features of the solutions and (2) a numerical eigenvalue analysis that corroborates the observed temporal variation of the solutions. The resulting temporal variation of the 2D and 3D plots demonstrates how the solutions evolve dynamically, including oscillatory long-term behavior.In all of the examples, routines in R are presented and discussed in detail. The routines are available through this link so that the reader can execute the PDE models to reproduce the reported solutions, then experiment with the models, including extensions and application to alternative models.

This book presents solutions to control problems in a number of robotic systems and provides a wealth of worked-out examples with full analytical and numerical details, graphically illustrated to aid in reader comprehension. It also presents relevant studies on and applications of robotic system control approaches, as well as the latest findings from interdisciplinary theoretical studies. Featuring chapters on advanced control (fuzzy, neural, backstepping, sliding mode, adaptive, predictive, diagnosis, and fault-tolerant control), the book will equip readers to easily tailor the techniques to their own applications. Accordingly, it offers a valuable resource for researchers, engineers, and students in the field of robotic systems.

Current mathematical models are notoriously unreliable in describing the time evolution of unexpected social phenomena, from financial crashes to revolution. Can such events be forecast? Can we compute probabilities about them? Can we model them? This book investigates and attempts to answer these questions through Goedel's two incompleteness theorems, and in doing so demonstrates how influential Goedel is in modern logical and mathematical thinking. Many mathematical models are applied to economics and social theory, while Goedel's theorems are able to predict their limitations for more accurate analysis and understanding of national and international events.This unique discussion is written for graduate level mathematicians applying their research to the social sciences, including economics, social studies and philosophy, and also for formal logicians and philosophers of science.

This book primarily focuses on rigorous mathematical formulation and treatment of static problems arising in continuum mechanics of solids at large or small strains, as well as their various evolutionary variants, including thermodynamics. As such, the theory of boundary- or initial-boundary-value problems for linear or quasilinear elliptic, parabolic or hyperbolic partial differential equations is the main underlying mathematical tool, along with the calculus of variations. Modern concepts of these disciplines as weak solutions, polyconvexity, quasiconvexity, nonsimple materials, materials with various rheologies or with internal variables are exploited. This book is accompanied by exercises with solutions, and appendices briefly presenting the basic mathematical concepts and results needed. It serves as an advanced resource and introductory scientific monograph for undergraduate or PhD students in programs such as mathematical modeling, applied mathematics, computational continuum physics and engineering, as well as for professionals working in these fields.

INHALT LANG: Introduction: Introductory Survey; Vector Norm. Matrix Norm. Matrix Measure; Functional Analysis, Function Norms and Control Signals.- Differential Sensitivity. Small-Scale Perturbation: Kronecker Calculus in Control Theory; Analysis Using Matrices and Control Theory; Eigenvalue and Eigenvector Differential Sensitivity; Transition Matrix Differential Sensitivity; Characteristic Polynomial Differential Sensitivity; Optimal Control and Performance Sensitivity; Desensitizing Control.- Robustness in the Time Domain: General Stability Bounds in Perturbed Systems; Robust Dynamic Interval Systems; Lyapunov-Based Methods for Perturbed Continuous-Time Systems; Lyapunov-Based Methods for Perturbed Discrete-Time Systems; Robust Pole Assignment; Models for Optimal and Interconnected Systems; Robust State Feedback Using Ellipsoid Sets; Robustness of Observers and Kalman-Bucy Filters; Initial Condition Perturbation, Overshoot and Robustness; Lnp-Stability and Robust Nonlinear Control.- Robustness in the Frequency Domain: Uncertain Polynomials. Interval Polynomials; Eigenvalues and Singular Values of Complex Matrices; Resolvent Matrix and Stability Radius; Robustness Via Singular-Value Analysis; Generalized Nyquist Stability of Perturbed Systems; Block-Structured Uncertainty and Structured Singular Value; Performance Robustness; Robust Controllers Via Spectral Radius Technique.- Coprime Factorization and Minimax Frequency Optimization: Robustness Based on the Internal Model Principle; Parametrization and Factorization of Systems; Hardy Space Robust Design.- Robustness Via Approximative Models: Robust Hyperplane Design in Variable Structure Control; SIngular Perturbaitons. Unmodelled High-Frequendy Dynamics; Control Using Aggregation Models; Optimum Control of Approximate and Nonlinear Systems; System Analysis via Orthogonal Functions; System Analysis Via Pulse Functions and Piecewise Linear Functions; Orthogonal Decomposition Applications.

Geochemical modeling is an important tool in environmental studies, and in the areas of subsurface and surface hydrology, pedology, water resources management, mining geology, geothermal resources, hydrocarbon geology, and related areas dealing with the exploration and extraction of natural resources. The book fills a gap in the literature through its discussion of geochemical modeling, which simulates the chemical and physical processes affecting the distribution of chemical species in liquid, gas, and solid phases. Geochemical modeling applies to a diversity of subsurface environments, from the vadose zone close to the Earth's surface, down to deep-seated geothermal reservoirs. This book provides the fundamental thermodynamic concepts of liquid-gas-solid phase systems. It introduces the principal types of geochemical models, such as speciation, reaction-path or forward, inverse- and reactive-transport models, together with examples of the most common codes and the best-practices for constructing geochemical models. The physical laws describing homogeneous and heterogeneous chemical reactions, their kinetics, and the transport of reactive solutes are presented. The partial differential or algebraic equations representing these laws, and the principal numerical methods that allow approximate solutions of these equations that can provide useful solutions to model different geochemical processes, are discussed in detail. Case studies applying geochemical models in different scientific areas and environmental settings, conclude the book. The book is addressed to students, teachers, other professionals, and to the institutions involved in water, geothermal and hydrocarbon resources, mining, and environmental management. The book should prove useful to undergraduate and graduate students, postgraduates, professional geologists and geophysicists, engineers, environmental scientists, soil scientists, hydrochemists, and others interested in water and geochemistry.