Using Bishop's work on constructive analysis as a framework, this monograph gives a systematic, detailed and general constructive theory of probability theory and stochastic processes. It is the first extended account of this theory: almost all of the constructive existence and continuity theorems that permeate the book are original. It also contains results and methods hitherto unknown in the constructive and nonconstructive settings. The text features logic only in the common sense and, beyond a certain mathematical maturity, requires no prior training in either constructive mathematics or probability theory. It will thus be accessible and of interest, both to probabilists interested in the foundations of their speciality and to constructive mathematicians who wish to see Bishop's theory applied to a particular field.

This authoritative book presents recent research results on nonlinear problems with lack of compactness. The topics covered include several nonlinear problems in the Euclidean setting as well as variational problems on manifolds. The combination of deep techniques in nonlinear analysis with applications to a variety of problems make this work an essential source of information for researchers and graduate students working in analysis and PDE's.

Nonlinear Approaches in Engineering Applications 2 focuses on the application of nonlinear approaches to different engineering and science problems. The selection of the topics for this book is based on the best papers presented in the ASME 2010 and 2011 in the tracks of Dynamic Systems and Control, Optimal Approaches in Nonlinear Dynamics and Acoustics, both of which were organized by the editors. For each selected topic, detailed concept development, derivations and relevant knowledge are provided for the convenience of the readers. The topics that have been selected are of great interest in the fields of engineering and physics and this book is designed to appeal to engineers and researchers working in a broad range of practical topics and approaches.

The theory of Lyapunov exponents originated over a century ago in the study of the stability of solutions of differential equations. Written by one of the subject's leading authorities, this book is both an account of the classical theory, from a modern view, and an introduction to the significant developments relating the subject to dynamical systems, ergodic theory, mathematical physics and probability. It is based on the author's own graduate course and is reasonably self-contained with an extensive set of exercises provided at the end of each chapter. This book makes a welcome addition to the literature, serving as a graduate text and a valuable reference for researchers in the field.

This contemporary first course focuses on concepts and ideas of Measure Theory, highlighting the theoretical side of the subject. Its primary intention is to introduce Measure Theory to a new generation of students, whether in mathematics or in one of the sciences, by offering them on the one hand a text with complete, rigorous and detailed proofs--sketchy proofs have been a perpetual complaint, as demonstrated in the many Amazon reader reviews critical of authors who "omit 'trivial' steps" and "make not-so-obvious 'it is obvious' remarks." On the other hand, Kubrusly offers a unique collection of fully hinted problems. On the other hand, Kubrusly offers a unique collection of fully hinted problems. The author invites the readers to take an active part in the theory construction, thereby offering them a real chance to acquire a firmer grasp on the theory they helped to build. These problems, at the end of each chapter, comprise complements and extensions of the theory, further examples and counterexamples, or auxiliary results. They are an integral part of the main text, which sets them apart from the traditional classroom or homework exercises.
JARGON BUSTER:
measure theory
Measure theory investigates the conditions under which integration can take place.
It considers various ways in which the "size" of a set can be estimated.
This topic is studied in pure mathematics programs but the theory is also foundational for students of statistics and probability, engineering, and financial engineering.
Key Features
* Designed with a minimum of prerequisites (intro analysis, and for Ch 5, linear algebra)
* Includes 140 classical measure-theory problems
* Carefully crafted to present essential elements of the theory in compact form

This book is a unique blend of difference equations theory and its exciting applications to economics. It deals with not only theory of linear (and linearized) difference equations, but also nonlinear dynamical systems which have been widely applied to economic analysis in recent years. It studies most important concepts and theorems in difference equations theory in a way that can be understood by anyone who has basic knowledge of calculus and linear algebra. It contains well-known applications and many recent developments in different fields of economics. The book also simulates many models to illustrate paths of economic dynamics.

Key Features:

.A unique book concentrated on theory of discrete dynamical systems and its traditional as well as advanced applications to economics.
.Mathematical definitions and theorems are introduced in a systematic and easily accessible way.
.Examples are from almost all fields of economics; technically proceeding from basic to advanced topics.
.Lively illustrations with numerous figures.
.Numerous simulation to see paths of economic dynamics.
.Comprehensive treatment of the subject with a comprehensive and easily accessible approach.
Key Features:

.A unique book concentrated on theory of discrete dynamical systems and its traditional as well as advanced applications to economics.
.Mathematical definitions and theorems are introduced in a systematic and easily accessible way.
.Examples are from almost all fields of economics; technically proceeding from basic to advanced topics.
.Lively illustrations with numerous figures.
.Numerous simulation to see paths of economic dynamics.
.Comprehensive treatment of the subject with a comprehensive and easily accessible approach."

The field of fluid mechanics is vast and has numerous and diverse applications. Presented papers from the 11th International Conference on Advances in Fluid Dynamics with emphasis on Multiphase and Complex Flow are contained in this book and cover a wide range of topics, including basic formulations and their computer modelling as well as the relationship between experimental and analytical results. Innovation in fluid-structure approaches including emerging applications as energy harvesting systems, studies of turbulent flows at high Reynold number, or subsonic and hypersonic flows are also among the topics covered. The emphasis placed on multiphase flow in the included research works is due to the fact that fluid dynamics processes in nature are predominantly multi-phased, i.e. involving more than one phase of a component such as liquid, gas or plasma. The range of related problems of interest is vast: astrophysics, biology, geophysics, atmospheric processes, and a large variety of engineering applications. Multiphase fluid dynamics are generating a great deal interest, leading to many notable advances in experimental, analytical, and numerical studies in this area. While progress is continuing in all three categories, advances in numerical solutions are likely the most conspicuous, owing to the continuing improvements in computer power and the software tools available to researchers. Progress in numerical methods has not only allowed for the solution of many practical problems but also helped to improve our understanding of the physics involved. Many unresolved issues are inherent in the very definition of multiphase flow, where it is necessary to consider coupled processes on multiple scales, as well as the interplay of a wide variety of relevant physical phenomena.

The book presents the recent achievements on bifurcation studies of nonlinear dynamical systems. The contributing authors of the book are all distinguished researchers in this interesting subject area. The first two chapters deal with the fundamental theoretical issues of bifurcation analysis in smooth and non-smooth dynamical systems. The cell mapping methods are presented for global bifurcations in stochastic and deterministic, nonlinear dynamical systems in the third chapter. The fourth chapter studies bifurcations and chaos in time-varying, parametrically excited nonlinear dynamical systems. The fifth chapter presents bifurcation analyses of modal interactions in distributed, nonlinear, dynamical systems of circular thin von Karman plates. The theories, methods and results presented in this book are of great interest to scientists and engineers in a wide range of disciplines. This book can be adopted as references for mathematicians, scientists, engineers and graduate students conducting research in nonlinear dynamical systems.
- New Views for Difficult Problems
- Novel Ideas and Concepts
- Hilbert's 16th Problem
- Normal Forms in Polynomial Hamiltonian Systems
- Grazing Flow in Non-smooth Dynamical Systems
- Stochastic and Fuzzy Nonlinear Dynamical Systems
- Fuzzy Bifurcation
- Parametrical, Nonlinear Systems
- Mode Interactions in nonlinear dynamical systems

This book is devoted to an important branch of the dynamical systems theory: the study of the fine (fractal) structure of Poincare recurrences -instants of time when the system almost repeats its initial state. The authors were able to write an entirely self-contained text including many insights and examples, as well as providing complete details of proofs. The only prerequisites are a basic knowledge of analysis and topology. Thus this book can serve as a graduate text or self-study guide for courses in applied mathematics or nonlinear dynamics (in the natural sciences). Moreover, the book can be used by specialists in applied nonlinear dynamics following the way in the book. The authors applied the mathematical theory developed in the book to two important problems: distribution of Poincare recurrences for nonpurely chaotic Hamiltonian systems and indication of synchronization regimes in coupled chaotic individual systems.

* Portions of the book were published in an article that won the title "month's new hot paper in the field of Mathematics" in May 2004
* Rigorous mathematical theory is combined with important physical applications
* Presents rules for immediate action to study mathematical models of real systems
* Contains standard theorems of dynamical systems theory

The book contains a detailed treatment of thermodynamic formalism on general compact metrizable spaces. Topological pressure, topological entropy, variational principle, and equilibrium states are presented in detail. Abstract ergodic theory is also given a significant attention. Ergodic theorems, ergodicity, and Kolmogorov-Sinai metric entropy are fully explored. Furthermore, the book gives the reader an opportunity to find rigorous presentation of thermodynamic formalism for distance expanding maps and, in particular, subshifts of finite type over a finite alphabet. It also provides a fairly complete treatment of subshifts of finite type over a countable alphabet. Transfer operators, Gibbs states and equilibrium states are, in this context, introduced and dealt with. Their relations are explored. All of this is applied to fractal geometry centered around various versions of Bowen's formula in the context of expanding conformal repellors, limit sets of conformal iterated function systems and conformal graph directed Markov systems. A unique introduction to iteration of rational functions is given with emphasize on various phenomena caused by rationally indifferent periodic points. Also, a fairly full account of the classicaltheory of Shub's expanding endomorphisms is given; it does not have a book presentation in English language mathematical literature.

The book addresses many important new developments in the field. All the topics covered are of great interest to the readers because such inequalities have become a major tool in the analysis of various branches of mathematics.
* It contains a variety of inequalities which find numerous applications in various branches of mathematics.
* It contains many inequalities which have only recently appeared in the literature and cannot yet be found in other books.
* It will be a valuable reference for someone requiring a result about inequalities for use in some applications in various other branches of mathematics.
* Each chapter ends with some miscellaneous inequalities for futher study.
* The work will be of interest to researchers working both in pure and applied mathematics, and it could also be used as the text for an advanced graduate course.

This book presents a novel approach to umbral calculus, which uses only elementary linear algebra (matrix calculus) based on the observation that there is an isomorphism between Sheffer polynomials and Riordan matrices, and that Sheffer polynomials can be expressed in terms of determinants. Additionally, applications to linear interpolation and operator approximation theory are presented in many settings related to various families of polynomials.

The Keller-Segel model for chemotaxis is a prototype of nonlocal systems describing concentration phenomena in physics and biology. While the two-dimensional theory is by now quite complete, the questions of global-in-time solvability and blowup characterization are largely open in higher dimensions. In this book, global-in-time solutions are constructed under (nearly) optimal assumptions on initial data and rigorous blowup criteria are derived.

The relaxation method has enjoyed an intensive development during many decades and this new edition of this comprehensive text reflects in particular the main achievements in the past 20 years. Moreover, many further improvements and extensions are included, both in the direction of optimal control and optimal design as well as in numerics and applications in materials science, along with an updated treatment of the abstract parts of the theory.

Originating from the 42nd conference on Boundary Elements and other Mesh Reduction Methods (BEM/MRM), the research presented in this book consist of high quality papers that report on advances in techniques that reduce or eliminate the type of meshes associated with such methods as finite elements or finite differences. The maturity of BEM since 1978 has resulted in a substantial number of industrial applications which demonstrate the accuracy, robustness and easy use of the technique. Their range still needs to be widened, taking into account the potentialities of the Mesh Reduction techniques in general. As design, analysis and manufacture become more integrated the chances are that the users will be less aware of the capabilities of the analytical techniques that are at the core of the process. This reinforces the need to retain expertise in certain specialised areas of numerical methods, such as BEM/MRM, to ensure that all new tools perform satisfactorily in the integrated process. The papers in this volume help to expand the range of applications as well as the type of materials in response to industrial and professional requirements. Some of the topics include: Hybrid foundations; Meshless and mesh reduction methods; Structural mechanics; Solid mechanics; Heat and mass transfer; Electrical engineering and electromagnetics; Fluid flow modelling; Damage mechanics and fracture; Dynamics and vibrations analysis.

The maturity of BEM over the last few decades has resulted in a substantial number of industrial applications of the method; this demonstrates its accuracy, robustness and ease of use. The range of applications still needs to be widened, taking into account the potentialities of the Mesh Reduction techniques in general. Theoretical developments and new formulations have been reported over the last few decades, helping to expand the range of boundary elements and other mesh reduction methods (BEM/MRM) applications as well as the type of modelled materials in response to the requirements of contemporary industrial and professional environments. As design, analysis and manufacture become more integrated, the chances are that software users will be less aware of the capabilities of the analytical techniques that are at the core of the process. This reinforces the need to retain expertise in certain specialised areas of numerical methods, such as BEM/MRM, to ensure that all new tools perform satisfactorily within the aforementioned integrated process. The papers included were presented at the 44th International Conference on Boundary Elements and other Mesh Reduction Methods and report advances in techniques that reduce or eliminate the type of meshes associated with finite elements or finite differences.

The book provides the reader with the different types of functional equations that s/he can find in practice, showing, step by step, how they can be solved.
A general methodology for solving functional equations is provided in Chapter 2. The different types of functional equations are described and solved in Chapters 3 to 8. Many examples, coming from different fields, as geometry, science, engineering, economics, probability, statistics, etc, help the reader to change his/her mind in order to state problems as functional equations as an alternative to differential equations, and to state new problems in terms of functional equations or systems.
An interesting feature of the book is that it deals with functional networks, a powerful generalization of neural networks that allows solving many practical problems. The second part of the book, Chapters 9 to 13, is devoted to the applications of this important paradigm.
The book contains many examples and end of chapter exercises, that facilitates the understanding of the concepts and applications.
- A general methodology for solving functional equations is provided in Chapter 2.
- It deals with functional networks, a powerful generalization of neural networks.
- Many examples, coming from different fields, as geometry, science, engineering, economics, probability, statistics, etc, illustrate the concept of functional equation.
- Functional equations are presented as a powerful alternative to differential equations.
- The book contains end of chapter exercises.

Many fundamental theories of modern physics can be considered as descriptions of dynamical systems subjected to constraints. The study of these constrained dynamical systems, and in particular the problems of formulating them as quantum systems, has many profound links with geometry. These links were explored in the Symposium on Geometry and Gravity held at the Newton Institute in 1994. This book, which arose from a conference held during that symposium, is a collection of papers devoted to problems such as Chern-Simons theory, sigma-models, gauge invariance and loop quantization, general relativity and the notion of time, and quantum gravity. They present a lively, varied and topical perspective on this important branch of theoretical physics from some of the leading authorities in the subject, and will be of value to theoretical physicists and mathematicians interested in the latest advances.

Brunello Terreni (1953-2000) was a researcher and teacher with vision and dedication. The present volume is dedicated to the memory of Brunello Terreni. His mathematical interests are reflected in 20 expository articles written by distinguished mathematicians. The unifying theme of the articles is "evolution equations and functional analysis," which is presented in various and diverse forms: parabolic equations, semigroups, stochastic evolution, optimal control, existence, uniqueness and regularity of solutions, inverse problems as well as applications. Contributors: P. Acquistapace, V. Barbu, A. Briani, L. Boccardo, P. Colli Franzone, G. Da Prato, D. Donatelli, A. Favini, M. Fuhrmann, M. Grasselli, R. Illner, H. Koch, R. Labbas, H. Lange, I. Lasiecka, A. Lorenzi, A. Lunardi, P. Marcati, R. Nagel, G. Nickel, V. Pata, M. M. Porzio, B. Ruf, G. Savare, R. Schnaubelt, E. Sinestrari, H. Tanabe, H. Teismann, E. Terraneo, R. Triggiani, A. Yagi

The focus of this book is on the Schatten-von Neumann properties and the product formulas of localization operators defined in terms of infinite-dimensional and square-integrable representations of locally compact and Hausdorff groups. Wavelet transforms, which are the building blocks of localization operators, are also studied in their own right. Daubechies operators on the Weyl-Heisenberg group, localization operators on the affine group, and wavelet multipliers on the Euclidean space are investigated in detail. The study is carried out in the perspective of pseudo-differential operators, quantization and signal analysis. Although the emphasis is put on locally compact and Hausdorff groups, results in the context of homogeneous spaces are given in order to unify the various localization operators into a single theory. Several new spectral results on pseudo-differential operators in the setting of localization operators are presented for the first time. The book is accessible to graduate students and mathematicians who have a basic knowledge of measure theory and functional analysis and wish to have a fast track to the frontier of research at the interface of pseudo-differential operators, quantization and signal analysis.

This book is an introduction to topological dynamics and ergodic theory. It is divided into a number of relatively short chapters with the intention that each may be used as a component of a lecture course tailored to the particular audience. The authors provide a number of applications, principally to number theory and arithmetic progressions (through Van der Waerden's theorem and Szemerdi's theorem). This text is suitable for advanced undergraduate and beginning graduate students.

I The fixed point theorems of Brouwer and Schauder.- 1 The fixed point theorem of Brouwer and applications.- 2 The fixed point theorem of Schauder and applications.- II Measures of noncompactness.- 1 The general notion of a measure of noncompactness.- 2 The Kuratowski and Hausdorff measures of noncompactness.- 3 The separation measure of noncompactness.- 4 Measures of noncompactness in Banach sequences spaces.- 5 Theorem of Darbo and Sadovskii and applications.- III Minimal sets for a measure of noncompactness.- 1 o-minimal sets.- 2 Minimalizable measures of noncompactness.- IV Convexity and smoothness.- 1 Strict convexity and smoothness.- 2 k-uniform convexity.- 3 k-uniform smoothness.- V Nearly uniform convexity and nearly uniform smoothness.- 1 Nearly uniformly convex Banach spaces.- 2 Nearly uniformly smooth Banach spaces.- 3 Uniform Opial condition.- VI Fixed points for nonexpansive mappings and normal structure.- 1 Existence of fixed points for nonexpansive mappings: Kirk's theorem.- 2 The coefficient N(X) and its connection with uniform convexity.- 3 The weakly convergent sequence coefficient.- 4 Uniform smoothness, near uniform convexity and normal structure.- 5 Normal structure in direct sum spaces.- 6 Computation of the normal structure coefficients in Lp-spaces.- VII Fixed point theorems in the absence of normal structure.- 1 Goebel-Karlovitz's lemma and Lin's lemma.- 2 The coefficient M(X) and the fixed point property.- VIII Uniformly Lipschitzian mappings.- 1 Lifshitz characteristic and fixed points.- 2 Connections between the Lifshitz characteristic and certain geometric coefficients.- 3 The normal structure coefficient and fixed points.- IX Asymptotically regular mappings.- 1 A fixed point theorem for asymptotically regular mappings.- 2 Connections between the ?-characteristic and some other geometric coefficients.- 3 The weakly convergent sequence coefficient and fixed points.- X Packing rates and o-contractiveness constants.- 1 Comparable measures of noncompactness.- 2 Packing rates of a metric space.- 3 Connections between the packing rates and the normal structure coefficients.- 4 Packing rates in lp-spaces.- 5 Packing rates in Lpspaces.- 6 Packing rates in direct sum spaces.- References.- List of Symbols and Notations.

Although the problem of stability and bifurcation is well understood in Mechanics, very few treatises have been devoted to stability and bifurcation analysis in dissipative media, in particular with regard to present and fundamental problems in Solid Mechanics such as plasticity, fracture and contact mechanics. Stability and Nonlinear Solid Mechanics addresses this lack of material, and proposes to the reader not only a unified presentation of nonlinear problems in Solid Mechanics, but also a complete and unitary analysis on stability and bifurcation problems arising within this framework. Main themes include:
* elasticity and plasticity problems in small and finite deformation
* general concepts of stability and bifurcation and basic results
* elastic buckling
* plastic buckling of structures
* standard dissipative systems obeying maximum dissipation.
These themes are developed in 20 chapters and illustrated by various analytical and numerical results. The coverage given here extends beyond the limited boundaries of previous works, resulting in a text of lasting interest and value to postgraduate students, researchers and practitioners working in mechanical, civil and aerospace engineering, as well as materials science.

The fascinating world of canonical moments--a unique look at this practical, powerful statistical and probability tool
Unusual in its emphasis, this landmark monograph on canonical moments describes the theory and application of canonical moments of probability measures on intervals of the real line and measures on the circle. Stemming from the discovery that canonical moments appear to be more intrinsically related to the measure than ordinary moments, the book's main focus is the broad application of canonical moments in many areas of statistics, probability, and analysis, including problems in the design of experiments, simple random walks or birth and death chains, and in approximation theory.
The book begins with an explanation of the development of the theory of canonical moments for measures on intervals [a, b] and then describes the various practical applications of canonical moments. The book's topical range includes:
* Definition of canonical moments both geometrically and as ratios of Hankel determinants
* Orthogonal polynomials viewed geometrically as hyperplanes to moment spaces
* Continued fractions and their link between ordinary moments and canonical moments
* The determination of optimal designs for polynomial regression
* The relationships between canonical moments, random walks, and orthogonal polynomials
* Canonical moments for the circle or trigonometric functions
Finally, this volume clearly illustrates the powerful mathematical role of canonical moments in a chapter arrangement that is as logical and interdependent as is the relationship of canonical moments to statistics, probability, and analysis.