The new 6th edition of Applied Combinatorics builds on the previous editions with more in depth analysis of computer systems in order to help develop proficiency in basic discrete math problem solving.

As one of the most widely used book in combinatorial problems, this edition explains how to reason and model combinatorically while stressing the systematic analysis of different possibilities, exploration of the logical structure of a problem, and ingenuity. Although important uses of combinatorics in computer science, operations research, and finite probability are mentioned, these applications are often used solely for motivation. Numerical examples involving the same concepts use more interesting settings such as poker probabilities or logical games.

This book is designed for use by students with a wide range of ability and maturity (sophomores through beginning graduate students). The stronger the students, the harder the exercises that can be assigned. The book can be used for one-quarter, two-quarter, or one-semester course depending on how much material is used.

Economic theories can be expressed in words, numbers, graphs and symbols. The existing traditional economics textbooks cover all four methods, but the general focus is often more on writing about the theory and methods, with few practical examples. With an increasing number of universities having introduced mathematical economics at undergraduate level, Basic mathematics for economics students aims to fill this gap in the field. Basic mathematics for economics students begins with a comprehensive chapter on basic mathematical concepts and methods (suitable for self-study, revision or tutorial purposes) to ensure that students have the necessary foundation. The book is written in an accessible style and is extremely practical. Numerous mathematical economics examples and exercises are provided as well as fully worked solutions using numbers, graphs and symbols. Basic mathematics for economics students is aimed at all economics students. It focuses on quantitative aspects and especially complements the three highly popular theoretical economics textbooks, Understanding microeconomics, Understanding macroeconomics and Economics for South African students, all written by Philip Mohr and published by Van Schaik Publishers.

In their bestselling MATHEMATICAL STATISTICS WITH APPLICATIONS, premiere authors Dennis Wackerly, William Mendenhall, and Richard L. Scheaffer present a solid foundation in statistical theory while conveying the relevance and importance of the theory in solving practical problems in the real world.

The authors' use of practical applications and excellent exercises helps you discover the nature of statistics and understand its essential role in scientific research.

Bundles, connections, metrics and curvature are the 'lingua franca' of modern differential geometry and theoretical physics. This book will supply a graduate student in mathematics or theoretical physics with the fundamentals of these objects.
Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the definition of characteristic classes, and also an introduction to complex and Kahler geometry.
Differential Geometry uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life. Helpfully, proofs are offered for almost all assertions throughout. All of the introductory material is presented in full and this is the only such source with the classical examples presented in detail.

This book provides a rigorous, physics-focused introduction to set theory that is geared towards natural science majors. The science major is presented with a robust introduction to set theory, which concentrates on the specific knowledge and skills that will be needed in calculus topics and natural science topics in general.

This is the first book in a four-part series designed to give a comprehensive and coherent description of Fluid Dynamics, starting with chapters on classical theory suitable for an introductory undergraduate lecture course, and then progressing through more advanced material up to the level of modern research in the field. The present Part 1 consists of four chapters. Chapter 1 begins with a discussion of Continuum Hypothesis, which is followed by an introduction to macroscopic functions, the velocity vector, pressure, density, and enthalpy. We then analyse the forces acting inside a fluid, and deduce the Navier-Stokes equations for incompressible and compressible fluids in Cartesian and curvilinear coordinates. In Chapter 2 we study the properties of a number of flows that are presented by the so-called exact solutions of the Navier-Stokes equations, including the Couette flow between two parallel plates, Hagen-Poiseuille flow through a pipe, and Karman flow above an infinite rotating disk. Chapter 3 is devoted to the inviscid incompressible flow theory, with particular focus on two-dimensional potential flows. These can be described in terms of the "complex potential", allowing the full power of the theory of functions of complex variables to be used. We discuss in detail the method of conformal mapping, which is then used to study various flows of interest, including the flows past Joukovskii aerofoils. The final Chapter 4 is concerned with compressible flows of perfect gas, including supersonic flows. Particular attention is given to the theory of characteristics, which is used, for example, to analyse the Prandtl-Meyer flow over a body surface bend and a corner. Significant attention is also devoted to the shock waves. The chapter concludes with analysis of unsteady flows, including the theory of blast waves.

A series of seminal technological revolutions has led to a new generation of electronic devices miniaturized to such tiny scales where the strange laws of quantum physics come into play. There is no doubt that, unlike scientists and engineers of the past, technology leaders of the future will have to rely on quantum mechanics in their everyday work. This makes teaching and learning the subject of paramount importance for further progress. Mastering quantum physics is a very non-trivial task and its deep understanding can only be achieved through working out real-life problems and examples. It is notoriously difficult to come up with new quantum-mechanical problems that would be solvable with a pencil and paper, and within a finite amount of time. This book remarkably presents some 700+ original problems in quantum mechanics together with detailed solutions covering nearly 1000 pages on all aspects of quantum science. The material is largely new to the English-speaking audience. The problems have been collected over about 60 years, first by the lead author, the late Prof. Victor Galitski, Sr. Over the years, new problems were added and the material polished by Prof. Boris Karnakov. Finally, Prof. Victor Galitski, Jr., has extended the material with new problems particularly relevant to modern science.

In the high energy gas flows, associating high velocities and high temperatures, physical and chemical processes such as molecular vibrational excitation, dissociation, ionisation or various reactions take palce and deeply influence the structure of the flows. The characteristic times of these processes have the same order of magnitude as aerodynamic characteristic times so that these reactive media are generally in thermodynamic and chemical non-equilibrium. This book presents a general introductory study of these media. In the first part their fundamental statistical aspects are described, starting from their discrete structure and taking into account the interactions between elementary particles: the transport phenomena, relaxation and kinetics as well as their coupling are thus analysed and illustrated by many examples. The second part deals with the macroscopic re-entry bodies. Finally the experimental aspects of these flows, their simulations in shock tube and shock tunnel are described as well as their application, particularly in the aero- spatial domain.
This book is intended for researchers and students that have acquired basic knowledge in thermodynamics, statistical physics and fluid mechanics. It must also interest the engineers engaged in research and industry related to the applications of the reactive flows, in particular in the aerospace field and, more generally, all the researchers trying to simulate and calculate complex reactive flows.

General Relativity is a beautiful geometric theory, simple in its mathematical formulation but leading to numerous consequences with striking physical interpretations: gravitational waves, black holes, cosmological models, and so on. This introductory textbook is written for mathematics students interested in physics and physics students interested in exact mathematical formulations (or for anyone with a scientific mind who is curious to know more of the world we live in), recent remarkable experimental and observational results which confirm the theory are clearly described and no specialised physics knowledge is required. The mathematical level of Part A is aimed at undergraduate students and could be the basis for a course on General Relativity. Part B is more advanced, but still does not require sophisticated mathematics. Based on Yvonne Choquet-Bruhat's more advanced text, General Relativity and the Einstein Equations, the aim of this book is to give with precision, but as simply as possible, the foundations and main consequences of General Relativity. The first five chapters from General Relativity and the Einstein Equations have been updated with new sections and chapters on black holes, gravitational waves, singularities, and the Reissner-Nordstroem and interior Schwarzchild solutions. The rigour behind this book will provide readers with the perfect preparation to follow the great mathematical progress in the actual development, as well as the ability to model, the latest astrophysical and cosmological observations. The book presents basic General Relativity and provides a basis for understanding and using the fundamental theory.

This book takes a fresh look at understanding how financial markets behave. Using recent ideas from the highly-topical science of complexity and complex systems, the book provides the basis for a unified theoretical description of how today's markets really work. Since financial markets are an excellent example of a complex system, the book also doubles as a science textbook.

This book discusses the study and analysis of the physical aspects of social systems and models, inspired by the analogy with familiar models of physical systems and possible applications of statistical physics tools. Unlike the traditional analysis of the physics of macroscopic many-body or condensed matter systems, which is now an established and mature subject, the upsurge in the physical analysis and modelling of social systems, which are clearly many-body dynamical systems, is a recent phenomenon. Though the major developments in sociophysics have taken place only recently, the earliest attempts of proposing "Social Physics" as a discipline are more than one and a half centuries old. Various developments in the mainstream physics of condensed matter systems have inspired and induced the recent growth of sociophysical analysis and models. In spite of the tremendous efforts of many scientists in recent years, the subject is still in its infancy and major challenges are yet to be taken up. An introduction to these challenges is the main motivation for this book.

This hands-on guide is primarily intended to be used in undergraduate laboratories in the physical sciences and engineering. It assumes no prior knowledge of statistics. It introduces the necessary concepts where needed, with key points illustrated with worked examples and graphic illustrations. In contrast to traditional mathematical treatments it uses a combination of spreadsheet and calculus-based approaches, suitable as a quick and easy on-the-spot reference. The emphasis throughout is on practical strategies to be adopted in the laboratory.
Error analysis is introduced at a level accessible to school leavers, and carried through to research level. Error calculation and propagation is presented though a series of rules-of-thumb, look-up tables and approaches amenable to computer analysis. The general approach uses the chi-square statistic extensively. Particular attention is given to hypothesis testing and extraction of parameters and their uncertainties by fitting mathematical models to experimental data. Routines implemented by most contemporary data analysis packages are analysed and explained. The book finishes with a discussion of advanced fitting strategies and an introduction to Bayesian analysis.

This book gives a rigorous yet physics focused introduction to mathematical logic that is geared towards natural science majors. We present the science major with a robust introduction to logic, focusing on the specific knowledge and skills that will unavoidably be needed in calculus topics and natural science topics in general rather than taking a philosophical-math-fundamental oriented approach that is commonly found in mathematical logic textbooks.

We experience elasticity everywhere in daily life: in the straightening or curling of hairs, the irreversible deformations of car bodies after a crash, or the bouncing of elastic balls in ping-pong or soccer. The theory of elasticity is essential to the recent developments of applied and fundamental science, such as the bio-mechanics of DNA filaments and other macro-molecules, and the animation of virtual characters in computer graphics and materials science. In this book, the emphasis is on the elasticity of thin bodies (plates, shells, rods) in connection with geometry. It covers such topics as the mechanics of hairs (curled and straight), the buckling instabilities of stressed plates, including folds and conical points appearing at larger stresses, the geometric rigidity of elastic shells, and the delamination of thin compressed films. It applies general methods of classical analysis, including advanced nonlinear aspects (bifurcation theory, boundary layer analysis), to derive detailed, fully explicit solutions to specific problems. These theoretical concepts are discussed in connection with experiments. The book is self-contained. Mathematical prerequisites are vector analysis and differential equations. The book can serve as a concrete introduction to nonlinear methods in analysis.

As an introductory account of the theory of phase transitions and critical phenomena, Elements of Phase Transitions and Critical Phenomena reflects lectures given by the authors to graduate students at their departments and is thus classroom-tested to help beginners enter the field. Most parts are written as self-contained units and every new concept or calculation is explained in detail without assuming prior knowledge of the subject. The book significantly enhances and revises a Japanese version which is a bestseller in the Japanese market and is considered a standard textbook in the field. It contains new pedagogical presentations of field theory methods, including a chapter on conformal field theory, and various modern developments hard to find in a single textbook on phase transitions. Exercises are presented as the topics develop, with solutions found at the end of the book, making the text useful for self-teaching, as well as for classroom learning.

In the field of economic analysis, computability in the formation of economic hypotheses is seen as the way forward. In this book, Professor Velupillai implements a theoretical research program along these lines. Choice theory, learning rational expectations equlibria, the persistence of adaptive behaviour, arithmetical games, aspects of production theory, and economic dynamics are given recursion theoretic (i.e. computable) interpretations. These interpretations lead to new kinds of questions being posed by the economic theorist. In particular, recurison theoretic decision problems replace standard optimisation paradigms in economic analysis. Economic theoretic questions, posed recursion-theoretically, lead to answers that are ambiguous: undecidable choices, uncomputable learning processes, and algorithmically unplayable games become standard answers. Professor Velupillai argues that a recursion theoretic formalisation of economic analysisComputable Economicsmakes the subject intrinsically inductive and computational.

This is a book written primarily for graduate students and early researchers in the fields of Analysis and Partial Differential Equations (PDEs). Coverage of the material is essentially self-contained, extensive and novel with great attention to details and rigour. The strength of the book primarily lies in its clear and detailed explanations, scope and coverage, highlighting and presenting deep and profound inter-connections between different related and seemingly unrelated disciplines within classical and modern mathematics and above all the extensive collection of examples, worked-out and hinted exercises. There are well over 700 exercises of varying level leading the reader from the basics to the most advanced levels and frontiers of research. The book can be used either for independent study or for a year-long graduate level course. In fact it has its origin in a year-long graduate course taught by the author in Oxford in 2004-5 and various parts of it in other institutions later on. A good number of distinguished researchers and faculty in mathematics worldwide have started their research career from the course that formed the basis for this book.

This is a book written primarily for graduate students and early researchers in the fields of Analysis and Partial Differential Equations (PDEs). Coverage of the material is essentially self-contained, extensive and novel with great attention to details and rigour. The strength of the book primarily lies in its clear and detailed explanations, scope and coverage, highlighting and presenting deep and profound inter-connections between different related and seemingly unrelated disciplines within classical and modern mathematics and above all the extensive collection of examples, worked-out and hinted exercises. There are well over 700 exercises of varying level leading the reader from the basics to the most advanced levels and frontiers of research. The book can be used either for independent study or for a year-long graduate level course. In fact it has its origin in a year-long graduate course taught by the author in Oxford in 2004-5 and various parts of it in other institutions later on. A good number of distinguished researchers and faculty in mathematics worldwide have started their research career from the course that formed the basis for this book.

Quadratic equations, Pythagoras' theorem, imaginary numbers, and pi - you may remember studying these at school, but did anyone ever explain why? Never fear - bestselling science writer, and your new favourite maths teacher, Michael Brooks, is here to help. In The Maths That Made Us, Brooks reminds us of the wonders of numbers: how they enabled explorers to travel far across the seas and astronomers to map the heavens; how they won wars and halted the HIV epidemic; how they are responsible for the design of your home and almost everything in it, down to the smartphone in your pocket. His clear explanations of the maths that built our world, along with stories about where it came from and how it shaped human history, will engage and delight. From ancient Egyptian priests to the Apollo astronauts, and Babylonian tax collectors to juggling robots, join Brooks and his extraordinarily eccentric cast of characters in discovering how maths made us who we are today.

Theoretical advances and new foundations have been reported at the Conference for more than 40 years which has helped expand the range of applications as well as the type of materials in response to industrial and professional requirements. Since the conference started it has attracted high quality papers that report further advances in techniques that reduce or eliminate the type of meshes associated with finite elements or finite differences, for instance. 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 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. The included papers originate from the 46th conference on Boundary Elements and other Mesh Reduction Methods (BEM/MRM) which acts as a forum to discuss new ideas and critically compare results before the solution and tools are released to the end users.

This second book on Unity Root Matrix Theory extends its original three-dimensional formulation, as given in the first book, to an arbitrary number of higher dimensions. Unity Root Matrix Theory is formulated with strong adherence to concepts in mathematical physics and it is thought it may provide a discrete formulation of physical phenomena at the Planck level and upward. Consequently, it is essential that the theory incorporates the geometric dimensionality present in established physical theories. In particular, it must naturally embody the four-dimensional spacetime of Special Relativity, the five dimensions of Kaluza-Klein theory, and the eleven or more dimensions of Grand Unified Theories such as String Theory. Not only has an n-dimensional extension of Unity Root Matrix Theory successfully been achieved, whilst retaining all the three-dimensional mathematical and physical properties detailed in the first book, but a complete n-dimensional solution has been obtained which exhibits the geometric property of compactification, or dimensional reduction. This solution shows that dimensional shrinkage of higher dimensions may occur over long evolutionary timescales. The emergence of compactification and other physical phenomena gives further confidence that n-dimensional Unity Root Matrix Theory may, indeed, offer a discrete formulation of Physics starting at its most elemental level.

Developed on surprisingly simple but fundamental concepts, it provides a rich mathematical and physical structure, justifying it as a subject to be studied in its own right by physicists and mathematicians alike. Ultimately, it is thought that unity root matrix theory may provide an alternative reformulation of some fundamental concepts in physics and an integer-based escape from the current, unification impasse.