This second edition of Elements of Operator Theory is a concept-driven textbook that includes a significant expansion of the problems and solutions used to illustrate the principles of operator theory. Written in a user-friendly, motivating style intended to avoid the formula-computational approach, fundamental topics are presented in a systematic fashion, i.e., set theory, algebraic structures, topological structures, Banach spaces, and Hilbert spaces, culminating with the Spectral Theorem.

Included in this edition: more than 150 examples, with several interesting counterexamples that demonstrate the frontiers of important theorems, as many as 300 fully rigorous proofs, specially tailored to the presentation, 300 problems, many with hints, and an additional 20 pages of problems for the second edition.

*This self-contained work is an excellent text for the classroom as well as a self-study resource for researchers.

This volume contains the proceedings of the OTAMP 2008 (Operator Theory, Analysis and Mathematical Physics) conference held at the Mathematical Research and Conference Center in Bedlewo near Poznan. It is composed of original research articles describing important results presented at the conference, some with extended review sections, as well as presentations by young researchers. Special sessions were devoted to random and quasi-periodic differential operators, orthogonal polynomials, Jacobi and CMV matrices, and quantum graphs. This volume also reflects new trends in spectral theory, where much emphasis is given to operators with magnetic fields and non-self-adjoint problems.

The book is geared towards scientists from advanced undergraduate students to researchers interested in the recent development on the borderline between operator theory and mathematical physics, especially spectral theory for Schr dinger operators and Jacobi matrices.

This book examines in detail the planning and modelling of local infrastructure like energy systems, including the complexities resulting from various uncertainties. Readers will discover the individual steps involved in infrastructure planning in cities and territories, as well as the primary requirements and supporting quality factors. Further topics covered concern the field of uncertainty and its synergies with infrastructure planning. Theories, methodological backgrounds and concrete case studies will not only help readers to understand the proposed methodologies for modelling and uncertainty analysis, but will also show them how these approaches are implemented in practice.

This textbook presents in a concise and self-contained way the advanced fundamental mathematical structures in quantum theory. It is based on lectures prepared for a 6 months course for MSc students. The reader is introduced to the beautiful interconnection between logic, lattice theory, general probability theory, and general spectral theory including the basic theory of von Neumann algebras and of the algebraic formulation, naturally arising in the study of the mathematical machinery of quantum theories. Some general results concerning hidden-variable interpretations of QM such as Gleason's and the Kochen-Specker theorems and the related notions of realism and non-contextuality are carefully discussed. This is done also in relation with the famous Bell (BCHSH) inequality concerning local causality. Written in a didactic style, this book includes many examples and solved exercises. The work is organized as follows. Chapter 1 reviews some elementary facts and properties of quantum systems. Chapter 2 and 3 present the main results of spectral analysis in complex Hilbert spaces. Chapter 4 introduces the point of view of the orthomodular lattices' theory. Quantum theory form this perspective turns out to the probability measure theory on the non-Boolean lattice of elementary observables and Gleason's theorem characterizes all these measures. Chapter 5 deals with some philosophical and interpretative aspects of quantum theory like hidden-variable formulations of QM. The Kochen-Specker theorem and its implications are analyzed also in relation BCHSH inequality, entanglement, realism, locality, and non-contextuality. Chapter 6 focuses on the algebra of observables also in the presence of superselection rules introducing the notion of von Neumann algebra. Chapter 7 offers the idea of (groups of) quantum symmetry, in particular, illustrated in terms of Wigner and Kadison theorems. Chapter 8 deals with the elementary ideas and results of the so called algebraic formulation of quantum theories in terms of both *-algebras and C*-algebras. This book should appeal to a dual readership: on one hand mathematicians that wish to acquire the tools that unlock the physical aspects of quantum theories; on the other physicists eager to solidify their understanding of the mathematical scaffolding of quantum theories.

Vortices comprising swirling motion of matter are observable in classical systems at all scales ranging from atomic size to the scale of galaxies. In quantum mechanical systems, such vortices are robust entities whose behaviours are governed by the strict rules of topology. The physics of quantum vortices is pivotal to basic science of quantum turbulence and high temperature superconductors, and underpins emerging quantum technologies including topological quantum computation. This handbook is aimed at providing a dictionary style portal to the fascinating quantum world of vortices.

Unifying two decades of research, this book is the first to establish a comprehensive foundation for a systematic analysis and design of linear systems with general state and input constraints. For such systems, which can be used as models for most nonlinear systems, the issues of stability, controller design, additonal constraints, and satisfactory performance are addressed. The book is an excellent reference for practicing engineers, graduate students, and researchers in control systems theory and design. It may also serve as an advanced graduate text for a course or a seminar in nonlinear control systems theory and design in applied mathematics or engineering departments. Minimal prerequisites include a first graduate course in state-space methods as well as a first course in control systems design.

This book employs homogeneous coordinate notation to compute the first- and second-order derivative matrices of various optical quantities. It will be one of the important mathematical tools for automatic optical design. The traditional geometrical optics is based on raytracing only. It is very difficult, if possible, to compute the first- and second-order derivatives of a ray and optical path length with respect to system variables, since they are recursive functions. Consequently, current commercial software packages use a finite difference approximation methodology to estimate these derivatives for use in optical design and analysis. Furthermore, previous publications of geometrical optics use vector notation, which is comparatively awkward for computations for non-axially symmetrical systems.

Recent advances in the fields of knowledge representation, reasoning and human-computer interaction have paved the way for a novel approach to treating and handling context. The field of research presented in this book addresses the problem of contextual computing in artificial intelligence based on the state of the art in knowledge representation and human-computer interaction. The author puts forward a knowledge-based approach for employing high-level context in order to solve some persistent and challenging problems in the chosen showcase domain of natural language understanding. Specifically, the problems addressed concern the handling of noise due to speech recognition errors, semantic ambiguities, and the notorious problem of underspecification. Consequently the book examines the individual contributions of contextual composing for different types of context. Therefore, contextual information stemming from the domain at hand, prior discourse, and the specific user and real world situation are considered and integrated in a formal model that is applied and evaluated employing different multimodal mobile dialog systems. This book is intended to meet the needs of readers from at least three fields - AI and computer science; computational linguistics; and natural language processing - as well as some computationally oriented linguists, making it a valuable resource for scientists, researchers, lecturers, language processing practitioners and professionals as well as postgraduates and some undergraduates in the aforementioned fields. "The book addresses a problem of great and increasing technical and practical importance - the role of context in natural language processing (NLP). It considers the role of context in three important tasks: Automatic Speech Recognition, Semantic Interpretation, and Pragmatic Interpretation. Overall, the book represents a novel and insightful investigation into the potential of contextual information processing in NLP." Jerome A Feldman, Professor of Electrical Engineering and Computer Science, UC Berkeley, USA http://dm.tzi.de/research/contextual-computing/

This monograph tackles three challenges. First, show a mathematics-based meta-model that matches known elementary particles. Second, apply models, based on the meta-model, to match other known physics data. Third, predict future physics data. The math features solutions to isotropic pairs of isotropic quantum harmonic oscillators. This monograph matches some solutions to known elementary particles. Matched properties include spin, types of interactions in which the particles partake, and (for elementary bosons) approximate masses. Other solutions point to possible elementary particles. This monograph applies the models and the extended particle list. Results narrow gaps between physics data and theory. Results pertain to elementary particles, astrophysics, and cosmology. For example, this monograph predicts properties for beyond-the-Standard-Model elementary particles, proposes descriptions of dark matter and dark energy, provides new relationships between known physics constants (including masses of some elementary particles), includes theory that dovetails with the ratio of dark matter to ordinary matter, includes math that dovetails with the number of elementary-fermion generations, suggests forces that govern the rate of expansion of the universe, and suggests additions to and details for the cosmology timeline.

The non-Gaussianity in the primordial density fluctuations is a key feature to clarify the early Universe and it has been probed with the Cosmic Microwave Background (CMB) bispectrum. In recent years, we have treated the novel-type CMB bispectra, which originate from the vector- and tensor-mode perturbations and include the violation of the rotational or parity invariance. On the basis of our current works, this thesis provides the general formalism for the CMB bispectrum sourced by the non-Gaussianity in the scalar, vector and tensor-mode perturbations. Applying this formalism, we calculate the CMB bispectra from the two scalars and a graviton correlation and primordial magnetic fields, and then outline new constraints on these magnitudes. Furthermore, this formalism can be easily extended to the cases where the rotational or parity invariance is broken. We also compute the CMB bispectra from the scalar-mode non-Gaussianities with a preferred direction and the tensor-mode non-Gaussianities induced by the parity-violating Weyl cubic terms. Here, we show that these bispectra include unique signals, which any symmetry-invariant models can never produce.

This book sheds new light on the development and use of quantitative models to describe the process of skin permeation. It critically reviews the development of quantitative predictive models of skin absorption and discusses key recommendations for model development. Topics presented include an introduction to skin physiology; the underlying theories of skin absorption; the physical laboratory-based processes used to generate skin absorption data, which is in turn used to construct mathematical models describing the skin permeation process; algorithms of skin permeability including quantitative structure-activity (or permeability) relationships (QSARs or QSPRs); relationships between permeability and molecular properties; the development of formulation-focused approaches to models of skin permeability prediction; the use of artificial membranes, e.g. polydimethylsiloxane as alternatives to mammalian skin; and lastly, the use of novel Machine Learning methods in developing the next generation of predictive skin permeability models. The book will be of interest to all researchers in academia and industry working in pharmaceutical discovery and development, as well as readers from the field of occupational exposure and risk assessment, especially those whose work involves agrochemicals, bulk chemicals and cosmetics.

This work concerns the computational modelling of the dynamics of partially ionized gases, with emphasis on electrodischarge processes. Understanding gas discharges is fundamental for many processes in mechanics, manufacturing, materials science, and aerospace engineering. This second edition has been expanded to include the latest developments in the field, especially regarding the drift-diffusion model and rarefied hypersonic flow.

Most networks and databases that humans have to deal with contain large, albeit finite number of units. Their structure, for maintaining functional consistency of the components, is essentially not random and calls for a precise quantitative description of relations between nodes (or data units) and all network components. This book is an introduction, for both graduate students and newcomers to the field, to the theory of graphs and random walks on such graphs. The methods based on random walks and diffusions for exploring the structure of finite connected graphs and databases are reviewed (Markov chain analysis). This provides the necessary basis for consistently discussing a number of applications such diverse as electric resistance networks, estimation of land prices, urban planning, linguistic databases, music, and gene expression regulatory networks.

This unified volume is a collection of invited chapters presenting recent developments in the field of data analysis, with applications to reliability and inference, data mining, bioinformatics, lifetime data, and neural networks. The book is a useful reference for graduate students, researchers, and practitioners in statistics, mathematics, engineering, economics, social science, bioengineering, and bioscience.

This volume is dedicated to Harold Widom, a distinguished mathematician and renowned expert in the area of Toeplitz, Wiener-Hopf and pseudodifferential operators, on the occasion of his sixtieth birthday. The book opens with biographical material and a list of the mathematician's publications, this being followed by two papers based on Toeplitz lectures which he delivered at Tel Aviv University in March, 1993. The rest of the book consists of a selection of papers containing some recent achievements in the following areas: SzegA-Widom asymptotic formulas for determinants of finite sections of Toeplitz matrices and their generalizations, the Fisher-Hartwig conjecture, random matrices, analysis of kernels of Toeplitz matrices, projectional methods and eigenvalue distribution for Toeplitz matrices, the Fredholm theory for convolution type operators, the Nehari interpolation problem with generalizations and applications, and Toeplitz-Hausdorff type theorems. The book will appeal to a wide audience of pure and applied mathematicians.

This book describes a promising approach to problems in the foundations of quantum mechanics, including the measurement problem. The dynamics of ensembles on configuration space is shown here to be a valuable tool for unifying the formalisms of classical and quantum mechanics, for deriving and extending the latter in various ways, and for addressing the quantum measurement problem. A description of physical systems by means of ensembles on configuration space can be introduced at a very fundamental level: the basic building blocks are a configuration space, probabilities, and Hamiltonian equations of motion for the probabilities. The formalism can describe both classical and quantum systems, and their thermodynamics, with the main difference being the choice of ensemble Hamiltonian. Furthermore, there is a natural way of introducing ensemble Hamiltonians that describe the evolution of hybrid systems; i.e., interacting systems that have distinct classical and quantum sectors, allowing for consistent descriptions of quantum systems interacting with classical measurement devices and quantum matter fields interacting gravitationally with a classical spacetime.

This book presents several new findings in the field of turbulent duct flows, which are important for a range of industrial applications. It presents both high-quality experiments and cutting-edge numerical simulations, providing a level of insight and rigour rarely found in PhD theses. The scientific advancements concern the effect of the Earth's rotation on large duct flows, the experimental confirmation of marginal turbulence in a pressure-driven square duct flow (previously only predicted in simulations), the identification of similar marginal turbulence in wall-driven flows using simulations (for the first time by any means) and, on a separate but related topic, a comprehensive experimental study on the phenomenon of drag reduction via polymer additives in turbulent duct flows. In turn, the work on drag reduction resulted in a correlation that provides a quantitative prediction of drag reduction based on a single, measurable material property of the polymer solution, regardless of the flow geometry or concentration. The first correlation of its kind, it represents an important advancement from both a scientific and practical perspective.

The behavior of materials at the nanoscale is a key aspect of modern nanoscience and nanotechnology. This book presents rigorous mathematical techniques showing that some very useful phenomenological properties which can be observed at the nanoscale in many nonlinear reaction-diffusion processes can be simulated and justified mathematically by means of homogenization processes when a certain critical scale is used in the corresponding framework.

This book surveys significant modern contributions to the mathematical theories of generalized heat wave equations. The first three chapters form a comprehensive survey of most modern contributions also describing in detail the mathematical properties of each model. Acceleration waves and shock waves are the focus in the next two chapters. Numerical techniques, continuous data dependence, and spatial stability of the solution in a cylinder, feature prominently among other topics treated in the following two chapters. The final two chapters are devoted to a description of selected applications and the corresponding formation of mathematical models. Illustrations are taken from a broad range that includes nanofluids, porous media, thin films, nuclear reactors, traffic flow, biology, and medicine, all of contemporary active technological importance and interest. This book will be of value to applied mathematicians, theoretical engineers and other practitioners who wish to know both the theory and its relevance to diverse applications.

Network Science is the emerging field concerned with the study of large, realistic networks. This interdisciplinary endeavor, focusing on the patterns of interactions that arise between individual components of natural and engineered systems, has been applied to data sets from activities as diverse as high-throughput biological experiments, online trading information, smart-meter utility supplies, and pervasive telecommunications and surveillance technologies. This unique text/reference provides a fascinating insight into the state of the art in network science, highlighting the commonality across very different areas of application and the ways in which each area can be advanced by injecting ideas and techniques from another. The book includes contributions from an international selection of experts, providing viewpoints from a broad range of disciplines. It emphasizes networks that arise in nature-such as food webs, protein interactions, gene expression, and neural connections-and in technology-such as finance, airline transport, urban development and global trade. Topics and Features: begins with a clear overview chapter to introduce this interdisciplinary field; discusses the classic network science of fixed connectivity structures, including empirical studies, mathematical models and computational algorithms; examines time-dependent processes that take place over networks, covering topics such as synchronisation, and message passing algorithms; investigates time-evolving networks, such as the World Wide Web and shifts in topological properties (connectivity, spectrum, percolation); explores applications of complex networks in the physical and engineering sciences, looking ahead to new developments in the field. Researchers and professionals from disciplines as varied as computer science, mathematics, engineering, physics, chemistry, biology, ecology, neuroscience, epidemiology, and the social sciences will all benefit from this topical and broad overview of current activities and grand challenges in the unfolding field of network science.

This is the revised and enlarged 2nd edition of the authors' original text, which was intended to be a modest complement to Grenander's fundamental memoir on stochastic processes and related inference theory. The present volume gives a substantial account of regression analysis, both for stochastic processes and measures, and includes recent material on Ridge regression with some unexpected applications, for example in econometrics. The first three chapters can be used for a quarter or semester graduate course on inference on stochastic processes. The remaining chapters provide more advanced material on stochastic analysis suitable for graduate seminars and discussions, leading to dissertation or research work. In general, the book will be of interest to researchers in probability theory, mathematical statistics and electrical and information theory.

All the existing books in Infinite Dimensional Complex Analysis focus on the problems of locally convex spaces. However, the theory without convexity condition is covered for the first time in this book. This shows that we are really working with a new, important and interesting field.

Theory of functions and nonlinear analysis problems are widespread in the mathematical modeling of real world systems in a very broad range of applications. During the past three decades many new results from the author have helped to solve multiextreme problems arising from important situations, non-convex and non linear cases, in function theory.

Foundations of Complex Analysis in Non Locally Convex Spaces is a comprehensive book that covers the fundamental theorems in Complex and Functional Analysis and presents much new material.

The book includes generalized new forms of: Hahn-Banach Theorem, Multilinear maps, theory of polynomials, Fixed Point Theorems, p-extreme points and applications in Operations Research, Krein-Milman Theorem, Quasi-differential Calculus, Lagrange Mean-Value Theorems, Taylor series, Quasi-holomorphic and Quasi-analytic maps, Quasi-Analytic continuations, Fundamental Theorem of Calculus, Bolzano's Theorem, Mean-Value Theorem for Definite Integral, Bounding and weakly-bounding (limited) sets, Holomorphic Completions, and Levi problem.

Each chapter contains illustrative examples to help the student and researcher to enhance his knowledge of theory of functions.

The new concept of Quasi-differentiability introduced by the author represents the backbone of the theory of Holomorphy for non-locally convex spaces. In fact it is different but much stronger than the Frechet one.

The book is intended not only for Post-Graduate (M.Sc.& Ph.D.) students and researchers in Complex and Functional Analysis, but for all Scientists in various disciplines whom need nonlinear or non-convex analysis and holomorphy methods without convexity conditions to model and solve problems.

bull; The book contains new generalized versions of:
i) Fundamental Theorem of Calculus, Lagrange Mean-Value Theorem in real and complex cases, Hahn-Banach Theorems, Bolzano Theorem, Krein-Milman Theorem, Mean value Theorem for Definite Integral, and many others.
ii) Fixed Point Theorems of Bruower, Schauder and Kakutani's.

bull; The book contains some applications in Operations research and non convex analysis as a consequence of the new concept p-Extreme points given by the author.

bull; The book contains a complete theory for Taylor Series representations of the different types of holomorphic maps in F-spaces without convexity conditions.

bull; The book contains a general new concept of differentiability stronger than the Frechet one. This implies a new Differentiable Calculus called Quasi-differential (or Bayoumi differential) Calculus. It is due to the author's discovery in 1995.

bull; The book contains the theory of polynomials and Banach Stienhaus theorem in non convex spaces.

This work presents a study of methods useful for modeling and understanding dynamical systems in the Galaxy. A natural coordinate system for the study of dynamical systems is the angle-action coordinate system. New methods for the approximation of the action-angle variables in general potentials are presented and discussed. These new tools are applied to the construction of dynamical models for two of the Galaxy's components: tidal streams and the Galactic disc. Tidal streams are remnants of tidally stripped satellites in the Milky Way that experience the effects of the large scale structure of the Galactic gravitational potential, while the Galactic disc provides insights into the nature of the Galaxy near the Sun. Appropriate action-based models are presented and discussed for these components, and extended to include further information such as the metallicity of stars.