In this monograph we study the problem of construction of asymptotic solutions of equations for functions whose number of arguments tends to infinity as the small parameter tends to zero. Such equations arise in statistical physics and in quantum theory of a large number of fi elds. We consider the problem of renormalization of quantum field theory in the Hamiltonian formalism, which encounters additional difficulties related to the Stuckelberg divergences and the Haag theorem. Asymptotic methods for solving pseudodifferential equations with small parameter multiplying the derivatives, as well as the asymptotic methods developed in the present monograph for solving problems in statistical physics and quantum field theory, can be considered from a unified viewpoint if one introduces the notion of abstract canonical operator. The book can be of interest for researchers - specialists in asymptotic methods, statistical physics, and quantum fi eld theory as well as for graduate and undergraduate students of these specialities.

This book is about computational methods based on operator splitting. It consists of twenty-three chapters written by recognized splitting method contributors and practitioners, and covers a vast spectrum of topics and application areas, including computational mechanics, computational physics, image processing, wireless communication, nonlinear optics, and finance. Therefore, the book presents very versatile aspects of splitting methods and their applications, motivating the cross-fertilization of ideas.

This thesis describes pioneering research on the extension of plasmonics schemes to the regime of high-intensity lasers. By presenting a rich and balanced mix of experimentation, theory and simulation, it provides a comprehensive overview of the emerging field of high field plasmonics, including open issues and perspectives for future research. Combining specially designed targets and innovative materials with ultrashort, high-contrast laser pulses, the author experimentally demonstrates the effects of plasmon excitation on electron and ion emission. Lastly, the work investigates possible further developments with the help of numerical simulations, revealing the potential of plasmonics effects in the relativistic regime for advances in laser-driven sources of radiation, and for the manipulation of extreme light at the sub-micron scale.

The first part of a two-volume set concerning the field of Clifford (geometric) algebra, this work consists of thematically organized chapters that provide a broad overview of cutting-edge topics in mathematical physics and the physical applications of Clifford algebras. This volume is devoted specifically to the mathematical aspects of Clifford algebras and their applications in physics. Algebraic geometry, cohomology, non-communicative spaces, "q"-deformations and the related quantum groups, and projective geometry provide the basis for algebraic topics covered. Physical applications and extensions of physical theories such as the theory of quaternionic spin, a projective theory of hadron transformation laws, and electron scattering are also presented, showing the broad applicability of Clifford geometric algebras in solving physical problems. Treatment of the structure theory of quantum Clifford algebras, the connection to logic, group representations, and computational techniques including symbolic calculations and theorem proving rounds out the presentation.

Like the first Abel Symposium, held in 2004, the Abel Symposium 2015 focused on operator algebras. It is interesting to see the remarkable advances that have been made in operator algebras over these years, which strikingly illustrate the vitality of the field. A total of 26 talks were given at the symposium on a variety of themes, all highlighting the richness of the subject. The field of operator algebras was created in the 1930s and was motivated by problems of quantum mechanics. It has subsequently developed well beyond its initial intended realm of applications and expanded into such diverse areas of mathematics as representation theory, dynamical systems, differential geometry, number theory and quantum algebra. One branch, known as "noncommutative geometry", has become a powerful tool for studying phenomena that are beyond the reach of classical analysis. This volume includes research papers that present new results, surveys that discuss the development of a specific line of research, and articles that offer a combination of survey and research. These contributions provide a multifaceted portrait of beautiful mathematics that both newcomers to the field of operator algebras and seasoned researchers alike will appreciate.

This book is the first part of a two volume anthology comprising a selection of 49 articles that illustrate the depth, breadth and scope of Nigel Kalton's research. Each article is accompanied by comments from an expert on the respective topic, which serves to situate the article in its proper context, to successfully link past, present and hopefully future developments of the theory, and to help readers grasp the extent of Kalton's accomplishments. Kalton's work represents a bridge to the mathematics of tomorrow, and this book will help readers to cross it. Nigel Kalton (1946-2010) was an extraordinary mathematician who made major contributions to an amazingly diverse range of fields over the course of his career.

Never before was anticipation more relevant to the life and activity of humankind than it is today. "It is no overstatement to suggest that humanity's future will be shaped by its capacity to anticipate...." (Research Agenda for the 21st Century, National Science Foundation). The sciences and the humanities can no longer risk explaining away the complexity and interactivity that lie at the foundation of life and living. The perspective of the world that anticipation opens justifies the descriptor "the post-Cartesian Revolution." If anticipation is a valid research domain, what practical relevance can we await? Indeed, anticipation is more than just the latest catch-word in marketing the apps developed by the digital technology industry. Due to spectacular advances in the study of the living, anticipation can claim a legitimate place in current investigations and applications in the sciences and the humanities. Biology, genetics, medicine, as well as politics and cognitive, behavioral, and social sciences, provide rich evidence of anticipatory processes at work. Readers seeking a foundation for an ticipation will find in these pages recent outcomes pertinent to plant life, political anticipation, cognitive science, architecture, computation. The authors contributing to this volume frame experimental data in language that can be shared among experts from all fields of endeavor. The major characteristic is the inference from the richness of data to principles and practical consequences.

This multi-contributed volume provides a practical, applications-focused introduction to nonlinear acoustical techniques for nondestructive evaluation. Compared to linear techniques, nonlinear acoustical/ultrasonic techniques are much more sensitive to micro-cracks and other types of small distributed damages. Most materials and structures exhibit nonlinear behavior due to the formation of dislocation and micro-cracks from fatigue or other types of repetitive loadings well before detectable macro-cracks are formed. Nondestructive evaluation (NDE) tools that have been developed based on nonlinear acoustical techniques are capable of providing early warnings about the possibility of structural failure before detectable macro-cracks are formed. This book presents the full range of nonlinear acoustical techniques used today for NDE. The expert chapters cover both theoretical and experimental aspects, but always with an eye towards applications. Unlike other titles currently available, which treat nonlinearity as a physics problem and focus on different analytical derivations, the present volume emphasizes NDE applications over detailed analytical derivations. The introductory chapter presents the fundamentals in a manner accessible to anyone with an undergraduate degree in Engineering or Physics and equips the reader with all of the necessary background to understand the remaining chapters. This self-contained volume will be a valuable reference to graduate students through practising researchers in Engineering, Materials Science, and Physics. Represents the first book on nonlinear acoustical techniques for NDE applications Emphasizes applications of nonlinear acoustical techniques Presents the fundamental physics and mathematics behind nonlinear acoustical phenomenon in a simple, easily understood manner Covers a variety of popular NDE techniques based on nonlinear acoustics in a single volume

This book presents a methodology based on inverse problems for use in solutions for fault diagnosis in control systems, combining tools from mathematics, physics, computational and mathematical modeling, optimization and computational intelligence. This methodology, known as fault diagnosis - inverse problem methodology or FD-IPM, unifies the results of several years of work of the authors in the fields of fault detection and isolation (FDI), inverse problems and optimization. The book clearly and systematically presents the main ideas, concepts and results obtained in recent years. By formulating fault diagnosis as an inverse problem, and by solving it using metaheuristics, the authors offer researchers and students a fresh, interdisciplinary perspective for problem solving in these fields. Graduate courses in engineering, applied mathematics and computing also benefit from this work.

This book presents in detail an alternative approach to solving problems involving both linear and nonlinear oscillations of elastic distributed parameter systems. It includes the so-called variational, projection and iterative gradient methods, which, when applied to nonlinear problems, use the procedure of linearization of the original non-linear equations. These methods are not universal and require a different solution for each problem or class of problems.However, in many cases the combination of the methods shown in this book leads to more efficient algorithms for solving important applied problems.To record these algorithms in a unified form, the first part of the book and its appendix devote considerable attention to compiling the general operator equations, which include (as particular cases) equations for vibrations in rods, plates, shells and three-dimensional bodies. They are mainly considered to be periodic or nearly periodic oscillations, which correspond to stat ionary or nearly stationary regimes of machinery operation. In turn, the second part of the book presents a number of solutions for selected applications.

Numerical Linear Algebra with Julia provides in-depth coverage of fundamental topics in numerical linear algebra, including how to solve dense and sparse linear systems, compute QR factorizations, compute the eigendecomposition of a matrix, and solve linear systems using iterative methods such as conjugate gradient. The style is friendly and approachable and cartoon characters guide the way. Inside this book, readers will find detailed descriptions of algorithms, implementations in Julia that illustrate concepts and allow readers to explore methods on their own, and illustrations and graphics that emphasize core concepts and demonstrate algorithms. Numerical Linear Algebra with Julia is a textbook for undergraduate and graduate students. It is appropriate for the following courses: Advanced Numerical Analysis, Special Topics on Numerical Analysis, Topics on Data Science, Topics on Numerical Optimization, and Topics on Approximation Theory. The book may also serve as a reference for researchers in various fields such as computational engineering, statistics, data-science, and machine learning, who depend on numerical solvers in linear algebra.

This work makes major contributions to the thriving area of social, communication, and distributed networks by introducing novel methodologies and tools toward the study of the evolutionary behaviors of these networks, as well as their computational complexity and rates of convergence. By departing from the classical approaches and results in the literature, this work shows that it is possible to handle more complex and realistic nonlinear models where either the traditional approaches fail or lead to weak results. The author also develops several easily implementable algorithms, delivering excellent performance guarantees while running faster than those that exist in the literature. The study undertaken and the approaches adopted enable the analysis of the evolution of several different types of social and distributed networks, with the potential to apply to and resolve several other outstanding issues in such networks.

This book addresses a broad range of problems commonly encountered in the fields of financial analysis, logistics and supply chain management, such as the use of big data analytics in the banking sector. Divided into twenty chapters, some of the contemporary topics discussed in the book are co-operative/non-cooperative supply chain models for imperfect quality items with trade-credit financing; a non-dominated sorting water cycle algorithm for the cardinality constrained portfolio problem; and determining initial, basic and feasible solutions for transportation problems by means of the "supply demand reparation method" and "continuous allocation method." In addition, the book delves into a comparison study on exponential smoothing and the Arima model for fuel prices; optimal policy for Weibull distributed deteriorating items varying with ramp type demand rate and shortages; an inventory model with shortages and deterioration for three different demand rates; outlier labeling methods for medical data; a garbage disposal plant as a validated model of a fault-tolerant system; and the design of a "least cost ration formulation application for cattle"; a preservation technology model for deteriorating items with advertisement dependent demand and trade credit; a time series model for stock price forecasting in India; and asset pricing using capital market curves. The book offers a valuable asset for all researchers and industry practitioners working in these areas, giving them a feel for the latest developments and encouraging them to pursue further research in this direction.

This monograph presents an application of concepts and methods from algebraic topology to models of concurrent processes in computer science and their analysis. Taking well-known discrete models for concurrent processes in resource management as a point of departure, the book goes on to refine combinatorial and topological models. In the process, it develops tools and invariants for the new discipline directed algebraic topology, which is driven by fundamental research interests as well as by applications, primarily in the static analysis of concurrent programs. The state space of a concurrent program is described as a higher-dimensional space, the topology of which encodes the essential properties of the system. In order to analyse all possible executions in the state space, more than "just" the topological properties have to be considered: Execution paths need to respect a partial order given by the time flow. As a result, tools and concepts from topology have to be extended to take privileged directions into account. The target audience for this book consists of graduate students, researchers and practitioners in the field, mathematicians and computer scientists alike.

Simulation has now become an integral part of research and development across many fields of study. Despite the large amounts of literature in the field of simulation and modeling, one recurring problem is the issue of accuracy and confidence level of constructed models. By outlining the new approaches and modern methods of simulation of stochastic processes, this book provides methods and tools in measuring accuracy and reliability in functional spaces. The authors explore analysis of the theory of Sub-Gaussian (including Gaussian one) and Square Gaussian random variables and processes and Cox processes. Methods of simulation of stochastic processes and fields with given accuracy and reliability in some Banach spaces are also considered.

This monograph describes advances in the theory of extremal problems in classes of functions defined by a majorizing modulus of continuity w. In particular, an extensive account is given of structural, limiting, and extremal properties of perfect w-splines generalizing standard polynomial perfect splines in the theory of Sobolev classes. In this context special attention is paid to the qualitative description of Chebyshev w-splines and w-polynomials associated with the Kolmogorov problem of n-widths and sharp additive inequalities between the norms of intermediate derivatives in functional classes with a bounding modulus of continuity. Since, as a rule, the techniques of the theory of Sobolev classes are inapplicable in such classes, novel geometrical methods are developed based on entirely new ideas. The book can be used profitably by pure or applied scientists looking for mathematical approaches to the solution of practical problems for which standard methods do not work. The scope of problems treated in the monograph, ranging from the maximization of integral functionals, characterization of the structure of equimeasurable functions, construction of Chebyshev splines through applications of fixed point theorems to the solution of integral equations related to the classical Euler equation, appeals to mathematicians specializing in approximation theory, functional and convex analysis, optimization, topology, and integral equations

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This book provides the mathematical foundations for Feynman's operator calculus and for the Feynman path integral formulation of quantum mechanics as a natural extension of analysis and functional analysis to the infinite-dimensional setting. In one application, the results are used to prove the last two remaining conjectures of Freeman Dyson for quantum electrodynamics. In another application, the results are used to unify methods and weaken domain requirements for non-autonomous evolution equations. Other applications include a general theory of Lebesgue measure on Banach spaces with a Schauder basis and a new approach to the structure theory of operators on uniformly convex Banach spaces. This book is intended for advanced graduate students and researchers.

This book puts in one place and in accessible form Richard Berk's most recent work on forecasts of re-offending by individuals already in criminal justice custody. Using machine learning statistical procedures trained on very large datasets, an explicit introduction of the relative costs of forecasting errors as the forecasts are constructed, and an emphasis on maximizing forecasting accuracy, the author shows how his decades of research on the topic improves forecasts of risk. Criminal justice risk forecasts anticipate the future behavior of specified individuals, rather than "predictive policing" for locations in time and space, which is a very different enterprise that uses different data different data analysis tools. The audience for this book includes graduate students and researchers in the social sciences, and data analysts in criminal justice agencies. Formal mathematics is used only as necessary or in concert with more intuitive explanations.

The aim of this book is to give a physical treatment of the kinetic theory of gases and magnetoplasmas, covering the standard material in as simple a way as possible, using mean-free-path arguments when possible and identifying problem areas where received theory has either failed or has fallen short of expectations. Examples are provided by strong shock waves, ultrasonic waves (high Knudsen numbers), and transport across strong magnetic fields. Examples of problem areas provided by strong shock waves, ultrasonic waves (high Knudsen numbers), and transport across strong magnetic fields. One of the paradoxes arising in kinetic theory concerns the fluid pressure. Collisions are necessary for a fluid force to result, yet standard kinetic theory does not entail this, being satisfied to bypass Newton's equations by defining pressure as a momentum flux. This omission usually has no adverse consequences, but with increasing Knudsen number, it leads to errors. This text pays particular attention to pressure, explaining the importance of allowing for its collisional nature from the outset in developing kinetic theory.

This book shows how to develop efficient quantitative methods to characterize neural data and extra information that reveals underlying dynamics and neurophysiological mechanisms. Written by active experts in the field, it contains an exchange of innovative ideas among researchers at both computational and experimental ends, as well as those at the interface. Authors discuss research challenges and new directions in emerging areas with two goals in mind: to collect recent advances in statistics, signal processing, modeling, and control methods in neuroscience; and to welcome and foster innovative or cross-disciplinary ideas along this line of research and discuss important research issues in neural data analysis. Making use of both tutorial and review materials, this book is written for neural, electrical, and biomedical engineers; computational neuroscientists; statisticians; computer scientists; and clinical engineers.

This prizewinning PhD thesis presents a general discussion of the orbital motion close to solar system small bodies (SSSBs), which induce non-central asymmetric gravitational fields in their neighborhoods. It introduces the methods of qualitative theory in nonlinear dynamics to the study of local/global behaviors around SSSBs. Detailed mechanical models are employed throughout this dissertation, and specific numeric techniques are developed to compensate for the difficulties of directly analyzing. Applying this method, several target systems, like asteroid 216 Kleopatra, are explored in great detail, and the results prove to be both revealing and pervasive for a large group of SSSBs.

An Introduction to Wavelets is the first volume in a new series, WAVELET ANALYSIS AND ITS APPLICATIONS. This is an introductory treatise on wavelet analysis, with an emphasis on spline wavelets and time-frequency analysis. Among the basic topics covered in this book are time-frequency localization, integral wavelet transforms, dyadic wavelets, frames, spline-wavelets, orthonormal wavelet bases, and wavelet packets. In addition, the author presents a unified treatment of nonorthogonal, semiorthogonal, and orthogonal wavelets. This monograph is self-contained, the only prerequisite being a basic knowledge of function theory and real analysis. It is suitable as a textbook for a beginning course on wavelet analysis and is directed toward both mathematicians and engineers who wish to learn about the subject. Specialists may use this volume as a valuable supplementary reading to the vast literature that has already emerged in this field.
Key Features
* This is an introductory treatise on wavelet analysis, with an emphasis on spline-wavelets and time-frequency analysis
* This monograph is self-contained, the only prerequisite being a basic knowledge of function theory and real analysis
* Suitable as a textbook for a beginning course on wavelet analysis

Markov process theory is basically an extension of ordinary calculus to accommodate functions whos time evolutions are not entirely deterministic. It is a subject that is becoming increasingly important for many fields of science. This book develops the single-variable theory of both continuous and jump Markov processes in a way that should appeal especially to physicists and chemists at the senior and graduate level.
Key Features
* A self-contained, prgamatic exposition of the needed elements of random variable theory
* Logically integrated derviations of the Chapman-Kolmogorov equation, the Kramers-Moyal equations, the Fokker-Planck equations, the Langevin equation, the master equations, and the moment equations
* Detailed exposition of Monte Carlo simulation methods, with plots of many numerical examples
* Clear treatments of first passages, first exits, and stable state fluctuations and transitions
* Carefully drawn applications to Brownian motion, molecular diffusion, and chemical kinetics

This book presents a new approach to modeling carbon structures such as graphene and carbon nanotubes using finite element methods, and addresses the latest advances in numerical studies for these materials. Based on the available findings, the book develops an effective finite element approach for modeling the structure and the deformation of grapheme-based materials. Further, modeling processing for single-walled and multi-walled carbon nanotubes is demonstrated in detail.

This book provides a comprehensive overview of different biomedical data types, including both clinical and genomic data. Thorough explanations enable readers to explore key topics ranging from electrocardiograms to Big Data health mining and EEG analysis techniques. Each chapter offers a summary of the field and a sample analysis. Also covered are telehealth infrastructure, healthcare information association rules, methods for mass spectrometry imaging, environmental biodiversity, and the global nonlinear fitness function for protein structures. Diseases are addressed in chapters on functional annotation of lncRNAs in human disease, metabolomics characterization of human diseases, disease risk factors using SNP data and Bayesian methods, and imaging informatics for diagnostic imaging marker selection. With the exploding accumulation of Electronic Health Records (EHRs), there is an urgent need for computer-aided analysis of heterogeneous biomedical datasets. Biomedical data is notorious for its diversified scales, dimensions, and volumes, and requires interdisciplinary technologies for visual illustration and digital characterization. Various computer programs and servers have been developed for these purposes by both theoreticians and engineers. This book is an essential reference for investigating the tools available for analyzing heterogeneous biomedical data. It is designed for professionals, researchers, and practitioners in biomedical engineering, diagnostics, medical electronics, and related industries.