Integrable models have a fascinating history with many important discoveries that dates back to the famous Kepler problem of planetary motion. Nowadays it is well recognised that integrable systems play a ubiquitous role in many research areas ranging from quantum field theory, string theory, solvable models of statistical mechanics, black hole physics, quantum chaos and the AdS/CFT correspondence, to pure mathematics, such as representation theory, harmonic analysis, random matrix theory and complex geometry. Starting with the Liouville theorem and finite-dimensional integrable models, this book covers the basic concepts of integrability including elements of the modern geometric approach based on Poisson reduction, classical and quantum factorised scattering and various incarnations of the Bethe Ansatz. Applications of integrability methods are illustrated in vast detail on the concrete examples of the Calogero-Moser-Sutherland and Ruijsenaars-Schneider models, the Heisenberg spin chain and the one-dimensional Bose gas interacting via a delta-function potential. This book has intermediate and advanced topics with details to make them clearly comprehensible.

This book is a course in methods and models rooted in physics and used in modelling economic and social phenomena. It covers the discipline of econophysics, which creates an interface between physics and economics. Besides the main theme, it touches on the theory of complex networks and simulations of social phenomena in general.
After a brief historical introduction, the book starts with a list of basic empirical data and proceeds to thorough investigation of mathematical and computer models. Many of the models are based on hypotheses of the behaviour of simplified agents. These comprise strategic thinking, imitation, herding, and the gem of econophysics, the so-called minority game. At the same time, many other models view the economic processes as interactions of inanimate particles. Here, the methods of physics are especially useful. Examples of systems modelled in such a way include books of stock-market orders, and redistribution of wealth among individuals. Network effects are investigated in the interaction of economic agents. The book also describes how to model phenomena like cooperation and emergence of consensus.
The book will be of benefit to graduate students and researchers in both Physics and Economics.

Homogeneous and, more generally, quasihomogeneous distributions represent an important subclass of L. Schwartz's distributions. In this book, the meromorphic dependence of these distributions on the order of homogeneity and on further parameters is studied. The analytic continuation, the residues and the finite parts of these distribution-valued functions are investigated in some detail. This research was initiated by Marcel Riesz in his seminal article in Acta Mathematica in 1949. It leads to the so-called elliptic and hyperbolic M. Riesz kernels referring to the Laplace and the wave operator. The distributional formulation goes back to J. Dieudonne and J. Horvath. The analytic continuation of these distribution-valued functions yields convolution groups and fundamental solutions of the corresponding linear partial differential operators with constant coefficients. The convolvability and the convolution of distributions and, in particular, of quasihomogeneous distributions are investigated systematically. In contrast to most textbooks on distribution theory, the general concept of convolution of distributions is employed. It was defined by L. Schwartz and further analyzed by R. Shiraishi and J. Horvath. The authors Norbert Ortner (* 1945, Vorarlberg) and Peter Wagner (* 1956, Tirol) are well-known researchers in the fields of Distribution Theory and Partial Differential Equations. The latter is professor for mathematics at the Technical Faculty, the first one was professor for mathematics at the Faculty of Mathematics, Computer Science and Physics of the Innsbruck University.

This book contains around 80 articles on major writings in mathematics published between 1640 and 1940. All aspects of mathematics are covered: pure and applied, probability and statistics, foundations and philosophy. Sometimes two writings from the same period and the same subject are taken together. The biography of the author(s) is recorded, and the circumstances of the preparation of the writing are given. When the writing is of some lengths an analytical table of its contents is supplied. The contents of the writing is reviewed, and its impact described, at least for the immediate decades. Each article ends with a bibliography of primary and secondary items.
.First book of its kind
.Covers the period 1640-1940 of massive development in mathematics
.Describes many of the main writings of mathematics
.Articles written by specialists in their field

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 shows how to decompose high-dimensional microarrays into small subspaces (Small Matryoshkas, SMs), statistically analyze them, and perform cancer gene diagnosis. The information is useful for genetic experts, anyone who analyzes genetic data, and students to use as practical textbooks.Discriminant analysis is the best approach for microarray consisting of normal and cancer classes. Microarrays are linearly separable data (LSD, Fact 3). However, because most linear discriminant function (LDF) cannot discriminate LSD theoretically and error rates are high, no one had discovered Fact 3 until now. Hard-margin SVM (H-SVM) and Revised IP-OLDF (RIP) can find Fact3 easily. LSD has the Matryoshka structure and is easily decomposed into many SMs (Fact 4). Because all SMs are small samples and LSD, statistical methods analyze SMs easily. However, useful results cannot be obtained. On the other hand, H-SVM and RIP can discriminate two classes in SM entirely. RatioSV is the ratio of SV distance and discriminant range. The maximum RatioSVs of six microarrays is over 11.67%. This fact shows that SV separates two classes by window width (11.67%). Such easy discrimination has been unresolved since 1970. The reason is revealed by facts presented here, so this book can be read and enjoyed like a mystery novel. Many studies point out that it is difficult to separate signal and noise in a high-dimensional gene space. However, the definition of the signal is not clear. Convincing evidence is presented that LSD is a signal. Statistical analysis of the genes contained in the SM cannot provide useful information, but it shows that the discriminant score (DS) discriminated by RIP or H-SVM is easily LSD. For example, the Alon microarray has 2,000 genes which can be divided into 66 SMs. If 66 DSs are used as variables, the result is a 66-dimensional data. These signal data can be analyzed to find malignancy indicators by principal component analysis and cluster analysis.

Bursting with brilliant exam practice for A-Level AQA Maths, these CGP Practice Papers are the best way for students to prepare for the tough exams! This pack contains two complete sets of exam-style tests (six papers in total) - plus a formula booklet with all the formulas students will need for their exams. We've also included detailed answers with step-by-step solutions and full mark schemes to make marking easy. For even more practice, don't miss CGP's Exam Practice Workbook for both years of A-Level AQA Maths (9781782947417).

An in-depth look at real analysis and its applications, including an introduction to wavelet
analysis, a popular topic in "applied real analysis." This text makes a very natural connection between the classic pure analysis and the applied topics, including measure theory, Lebesgue Integral,
harmonic analysis and wavelet theory with many associated applications.
*The text is relatively elementary at the start, but the level of difficulty steadily increases
*The book contains many clear, detailed examples, case studies and exercises
*Many real world applications relating to measure theory and pure analysis
*Introduction to wavelet analysis

It has widely been recognized that submodular functions play essential roles in efficiently solvable combinatorial optimization problems. Since the publication of the 1st edition of this book fifteen years ago, submodular functions have been showing further increasing importance in optimization, combinatorics, discrete mathematics, algorithmic computer science, and algorithmic economics, and there have been made remarkable developments of theory and algorithms in submodular functions. The 2nd edition of the book supplements the 1st edition with a lot of remarks and with new two chapters: "Submodular Function Minimization" and "Discrete Convex Analysis." The present 2nd edition is still a unique book on submodular functions, which is essential to students and researchers interested in combinatorial optimization, discrete mathematics, and discrete algorithms in the fields of mathematics, operations research, computer science, and economics.

Key features:

- Self-contained exposition of the theory of submodular functions.
- Selected up-to-date materials substantial to future developments.
- Polyhedral description of Discrete Convex Analysis.
- Full description of submodular function minimization algorithms.
- Effective insertion of figures.
- Useful in applied mathematics, operations research, computer science, and economics.
- Self-contained exposition of the theory of submodular functions.
- Selected up-to-date materials substantial to future developments.
- Polyhedral description of Discrete Convex Analysis.
- Full description of submodular function minimization algorithms.
- Effective insertion of figures.
- Useful in applied mathematics, operations research, computer science, and economics.

Mary Leng offers a defense of mathematical fictionalism, according to which we have no reason to believe that there are any mathematical objects. Perhaps the most pressing challenge to mathematical fictionalism is the indispensability argument for the truth of our mathematical theories (and therefore for the existence of the mathematical objects posited by those theories). According to this argument, if we have reason to believe anything, we have reason to believe that the claims of our best empirical theories are (at least approximately) true. But since claims whose truth would require the existence of mathematical objects are indispensable in formulating our best empirical theories, it follows that we have good reason to believe in the mathematical objects posited by those mathematical theories used in empirical science, and therefore to believe that the mathematical theories utilized in empirical science are true. Previous responses to the indispensability argument have focussed on arguing that mathematical assumptions can be dispensed with in formulating our empirical theories. Leng, by contrast, offers an account of the role of mathematics in empirical science according to which the successful use of mathematics in formulating our empirical theories need not rely on the truth of the mathematics utilized.

This book (hardcover) is part of the TREDITION CLASSICS. It contains classical literature works from over two thousand years. Most of these titles have been out of print and off the bookstore shelves for decades. The book series is intended to preserve the cultural legacy and to promote the timeless works of classical literature. Readers of a TREDITION CLASSICS book support the mission to save many of the amazing works of world literature from oblivion. With this series, tredition intends to make thousands of international literature classics available in printed format again - worldwide.

The Boussinesq equation is the first model of surface waves in shallow water that considers the nonlinearity and the dispersion and their interaction as a reason for wave stability known as the Boussinesq paradigm. This balance bears solitary waves that behave like quasi-particles. At present, there are some Boussinesq-like equations. The prevalent part of the known analytical and numerical solutions, however, relates to the 1d case while for multidimensional cases, almost nothing is known so far. An exclusion is the solutions of the Kadomtsev-Petviashvili equation. The difficulties originate from the lack of known analytic initial conditions and the nonintegrability in the multidimensional case. Another problem is which kind of nonlinearity will keep the temporal stability of localized solutions. The system of coupled nonlinear Schroedinger equations known as well as the vector Schroedinger equation is a soliton supporting dynamical system. It is considered as a model of light propagation in Kerr isotropic media. Along with that, the phenomenology of the equation opens a prospect of investigating the quasi-particle behavior of the interacting solitons. The initial polarization of the vector Schroedinger equation and its evolution evolves from the vector nature of the model. The existence of exact (analytical) solutions usually is rendered to simpler models, while for the vector Schroedinger equation such solutions are not known. This determines the role of the numerical schemes and approaches. The vector Schroedinger equation is a spring-board for combining the reduced integrability and conservation laws in a discrete level. The experimental observation and measurement of ultrashort pulses in waveguides is a hard job and this is the reason and stimulus to create mathematical models for computer simulations, as well as reliable algorithms for treating the governing equations. Along with the nonintegrability, one more problem appears here - the multidimensionality and necessity to split and linearize the operators in the appropriate way.

"Presents a summary of selected mathematics topics from college/university level mathematics courses. Fundamental principles are reviewed and presented by way of examples, figures, tables and diagrams. It condenses and presents under one cover basic concepts from several different applied mathematics topics"--P. [4] of cover.

Educational technologies (e-learning environments or learning management systems for individual and collaborative learning, Internet resources for teaching and learning, academic materials in electronic format, specific subject-related software, groupware and social network software, etc.) are changing the way in which higher education is delivered. Teaching Mathematics Online: Emergent Technologies and Methodologies shares theoretical and applied pedagogical models and systems used in math e-learning including the use of computer supported collaborative learning, which is common to most e-learning practices. The book also forecasts emerging technologies and tendencies regarding mathematical software, learning management systems, and mathematics education online and presents up-to-date research work on how mathematics education is changing in a global and Web-based world.

Providing a practical introduction to state space methods as applied to unobserved components time series models, also known as structural time series models, this book introduces time series analysis using state space methodology to readers who are neither familiar with time series analysis, nor with state space methods. The only background required in order to understand the material presented in the book is a basic knowledge of classical linear regression models, of which brief review is provided to refresh the reader's knowledge. Also, a few sections assume familiarity with matrix algebra, however, these sections may be skipped without losing the flow of the exposition.
The book offers a step by step approach to the analysis of the salient features in time series such as the trend, seasonal, and irregular components. Practical problems such as forecasting and missing values are treated in some detail. This useful book will appeal to practitioners and researchers who use time series on a daily basis in areas such as the social sciences, quantitative history, biology and medicine. It also serves as an accompanying textbook for a basic time series course in econometrics and statistics, typically at an advanced undergraduate level or graduate level.

After teaching junior high school mathematics for 10 years and serving as a high school principal for 14 years, Dr. Clarence Johnson conducted research as a doctoral student on improving the mathematics failure rates of African American students. You can read about his findings in Roll Call: 2012.

This book is specially designed to refresh and elevate the level of understanding of the foundational background in probability and distributional theory required to be successful in a graduate-level statistics program. Advanced undergraduate students and introductory graduate students from a variety of quantitative backgrounds will benefit from the transitional bridge that this volume offers, from a more generalized study of undergraduate mathematics and statistics to the career-focused, applied education at the graduate level. In particular, it focuses on growing fields that will be of potential interest to future M.S. and Ph.D. students, as well as advanced undergraduates heading directly into the workplace: data analytics, statistics and biostatistics, and related areas.

The greatly expanded and updated 3rd edition of this textbook offers the reader a comprehensive introduction to the concepts of logic functions and equations and their applications across computer science and engineering. The authors' approach emphasizes a thorough understanding of the fundamental principles as well as numerical and computer-based solution methods. The book provides insight into applications across propositional logic, binary arithmetic, coding, cryptography, complexity, logic design, and artificial intelligence. Updated throughout, some major additions for the 3rd edition include: a new chapter about the concepts contributing to the power of XBOOLE; a new chapter that introduces into the application of the XBOOLE-Monitor XBM 2; many tasks that support the readers in amplifying the learned content at the end of the chapters; solutions of a large subset of these tasks to confirm learning success; challenging tasks that need the power of the XBOOLE software for their solution. The XBOOLE-monitor XBM 2 software is used to solve the exercises; in this way the time-consuming and error-prone manipulation on the bit level is moved to an ordinary PC, more realistic tasks can be solved, and the challenges of thinking about algorithms leads to a higher level of education.

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.