Mathematics

This book covers multivariable and vector calculus. It can be used as a textbook for a one-semester course or self-study. It includes worked-through exercises, with answers provided for many of the basic computational ones and hints for the more complex ones.. This second edition features new exercises, new sections on twist and binormal vectors for curves in space, linear approximations, and the Laplace and Poisson equations.

This book is an introduction to machine learning, with a strong focus on the mathematics behind the standard algorithms and techniques in the field, aimed at senior undergraduates and early graduate students of Mathematics.

There is a focus on well-known supervised machine learning algorithms, detailing the existing theory to provide some theoretical guarantees, featuring intuitive proofs and exposition of the material in a concise and precise manner. A broad set of topics is covered, giving an overview of the field. A summary of the topics covered is: statistical learning theory, approximation theory, linear models, kernel methods, Gaussian processes, deep neural networks, ensemble methods and unsupervised learning techniques, such as clustering and dimensionality reduction.

This book is suited for students who are interested in entering the field, by preparing them to master the standard tools in Machine Learning. The reader will be equipped to understand the main theoretical questions of the current research and to engage with the field.

This graduate-level mathematics textbook provides an in-depth and readable exposition of selected topics in complex analysis. The material spans both the standard theory at a level suitable for a first-graduate class on the subject and several advanced topics delving deeper into the subject and applying the theory in different directions. The focus is on beautiful applications of complex analysis to geometry and number theory. The text is accompanied by beautiful figures illustrating many of the concepts and proofs.

Among the topics covered are asymptotic analysis; conformal mapping and the Riemann mapping theory; the Euler gamma function, the Riemann zeta function, and a proof of the prime number theorem; elliptic functions, and modular forms. The final chapter gives the first detailed account in textbook format of the recent solution to the sphere packing problem in dimension 8, published by Maryna Viazovska in 2016 — a groundbreaking proof for which Viazovska was awarded the Fields Medal in 2022.

The book is suitable for self-study by graduate students or advanced undergraduates with an interest in complex analysis and its applications, or for use as a textbook for graduate mathematics classes, with enough material for 2-3 semester-long classes. Researchers in complex analysis, analytic number theory, modular forms, and the theory of sphere packing, will also find much to enjoy in the text, including new material not found in standard textbooks.

An accessible explanation of Kurt Gödel's groundbreaking work in mathematical logic

In 1931 Kurt Gödel published his fundamental paper, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Gödel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement in the natural sciences—perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as "one of the greatest contributions to the sciences in recent times."

However, few mathematicians of the time were equipped to understand the young scholar's complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Gödel's discovery. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject.

New York University Press is proud to publish this special edition of one of its bestselling books. With a new introduction by Douglas R. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.

The philosophy of the book, which makes it quite distinct from many existing texts on the subject, is based on treating the concepts of measure and integration starting with the most general abstract setting and then introducing and studying the Lebesgue measure and integration on the real line as an important particular case.　

The book consists of nine chapters and appendix, with the material flowing from the basic set classes, through　measures, outer measures and the general procedure of measure extension, through measurable functions　and　various types of convergence of sequences of such based on the idea of measure, to the fundamentals of the abstract Lebesgue integration, the basic limit theorems, and the comparison of the Lebesgue and Riemann integrals. Also, studied are Lp spaces, the basics of normed vector spaces, and signed measures. The novel approach based on the Lebesgue measure and integration theory is applied to develop a better understanding of differentiation and extend the classical total change formula linking differentiation with integration to a substantially wider class of functions.

Being designed as a text to be used in a classroom, the book constantly calls for the student's actively mastering the knowledge of the subject matter. There are problems at the end of each chapter, starting with Chapter 2 and totaling at 125. Many important statements are given as problems and frequently referred to in the main body. There are also 358 Exercises throughout the text, including Chapter 1 and the Appendix, which require of the student to prove or verify a statement or an example, fill in certain details in a proof, or provide an intermediate step or a counterexample. They are also an inherent part of the material. More difficult problems are marked with an asterisk, many problems and exercises are supplied with ``existential'' hints.　

The book is generous on Examples and contains numerous Remarks accompanying definitions, examples, and statements to discuss certain subtleties, raise questions on whether the converse assertions are true, whenever appropriate, or whether the conditions are essential.　 　　

With　plenty of examples, problems, and exercises, this well-designed text is ideal for a one-semester Master's level graduate course on real analysis with emphasis on the measure and integration theory for students majoring in mathematics, physics, computer science, and engineering.　

• A concise but profound and detailed presentation of the basics of real analysis with emphasis on the measure and integration theory.
• Designed　for a one-semester graduate course, with plethora of examples, problems, and exercises.
• Is of interest to students and instructors in mathematics, physics, computer science, and engineering.
• Prepares the students for more advanced courses in functional analysis and operator theory.　　

Contents
Preliminaries
Basic Set Classes
Measures
Extension of Measures
Measurable Functions
Abstract Lebesgue Integral
Lp Spaces
Differentiation and Integration
Signed Measures
The Axiom of Choice and Equivalents

In the two-volume set ‘A Selection of Highlights’ we present basics of mathematics in an exciting and pedagogically sound way. This volume examines fundamental results in Algebra and Number Theory along with their proofs and their history. In the second edition, we include additional material on perfect and triangular numbers. We also added new sections on elementary Group Theory, p-adic numbers, and Galois Theory.

• A true collection of mathematical gems in Algebra and Number Theory, including the integers, the reals, and the complex numbers, along with beautiful results from Galois Theory and associated geometric applications.
• Valuable for lecturers, teachers and students of mathematics as well as for all who are mathematically interested.

The book consists of eight chapters, each focusing on different aspects of multiple integrals and related topics in mathematical analysis.
In Chapter 1, multiple integrals are defined and developed. The Jordan measure in n-dimensional unit balls is introduced, along with the definition and criteria for multiple integrals, as well as their properties.
Chapter 2 delves into advanced techniques for computing multiple integrals. It introduces the Taylor formula, discusses linear maps on measurable sets, and explores the metric properties of differentiable maps.
In Chapter 3, we focus on improper multiple integrals and their properties. The chapter deduces criteria for the integrability of functions of several variables and develops concepts such as improper integrals of nonnegative functions, comparison criteria, and absolute convergence.
Chapter 4 investigates the Stieltjes integral and its properties. Topics covered include the differentiation of monotone functions of finite variation and the Helly principle of choice, as well as continuous functions of finite variation.
Chapter 5 addresses curvilinear integrals, defining line integrals of both the first and second kinds. It also discusses the independence of line integrals from the path of integration.
In Chapter 6, surface integrals of the first and second kinds are introduced. The chapter presents the Gauss-Ostrogradsky theorem and Stokes’ formulas, along with advanced practical problems to practice these concepts.

This book focuses on functional phase space methods in quantum optics and provides a comprehensive introduction to the Wigner functional formalism. The book shows how this formalism can be used for analyzing and designing practical photonic quantum information systems. It also explains in detail the application of the Wigner functional formalism to parametric down-conversion, an important process in quantum optics.

The book contains six chapters. In Chapter 1, the set Rn is defined. n-dimensional ball, sphere, and rectangular neighborhood of a point are defined. Sequences in Rn are defined by their properties. Open, closed and compact sets in Rn are investigated. The Heine-Borel theorem is formulated and proved. In Chapter 2, functions of several variables are introduced. Limits of functions of several variables are defined with their properties are deducted. Continuous functions of several variables are investigated. Uniform continuity is introduced and explored. In Chapter 3, partial derivatives of higher order and differentials for functions of several variables and their properties are introduced. Criteria for the differentiability of functions of several variables are deducted. Gradient of a function explored. Directional derivatives are defined and investigated. In Chapter 4, higher order partial derivatives of functions of several variables are investigated. The minimum and maximum of functions of several variables are introduced and investigated. Implicit functions are defined and explored. The method of Lagrange multipliers is introduced. The book contains proofs, numerous examples, and exercises with solutions in the two last chapters.

Designed for a broad spectrum of mathematics majors, not only those pursuing graduate school, this book also provides a thorough explanation of undergraduate Real Analysis. Through a developmentally appropriate narrative that integrates informal discussion, motivation, and basic proof writing approaches with mathematical rigor and clarity, the aim is to assist all students in learning more about the real number system and calculus theory.

This book provides an introduction to probabilistic methods in finance, based on stochastic models in discrete time. It is aimed primarily at graduate students in mathematics but may also benefit mathematicians in academia and the financial industry.　

In this fifth edition, the entire text has been thoroughly revised to enhance clarity and completeness. This includes new sections on

The aim of the textbook is two-fold: first to serve as an introductory graduate course in Algebraic Topology and then to provide an application-oriented presentation of some fundamental concepts in Algebraic Topology to the fixed point theory.

A simple approach based on point-set Topology is used throughout to introduce many standard constructions of fundamental and homological groups of surfaces and topological spaces. The approach does not rely on Homological Algebra. The constructions of some spaces using the quotient spaces such as the join, the suspension, and the adjunction spaces are developed in the setting of Topology only.

The computations of the fundamental and homological groups of many surfaces and topological spaces occupy large parts of the book (sphere, torus, projective space, Mobius band, Klein bottle, manifolds, adjunctions spaces). Borsuk's theory of retracts which is intimately related to the problem of the extendability of continuous functions is developed in details. This theory together with the homotopy theory, the lifting and covering maps may serve as additional course material for students involved in General Topology.

The book comprises 280 detailed worked examples, 320 exercises (with hints or references), 80 illustrative figures, and more than 80 commutative diagrams to make it more oriented towards applications (maps between spheres, Borsuk-Ulam Theory, Fixed Point Theorems, …) As applications, the book offers some existence results on the solvability of some nonlinear differential equations subject to initial or boundary conditions.

The book is suitable for students primarily enrolled in Algebraic Topology, General Topology, Homological Algebra, Differential Topology, Differential Geometry, and Topological Geometry. It is also useful for advanced undergraduate students who aspire to grasp easily some new concepts in Algebraic Topology and Applications. The textbook is practical both as a teaching and research document for Bachelor, Master students, and first-year PhD students since it is accessible to any reader with a modest understanding of topological spaces.

The book aspires to fill a gap in the existing literature by providing a research and teaching document which investigates both the theory and the applications of Algebraic Topology in an accessible way without missing the main results of the topics covered.

The idea of the book is to present a text that is useful for both students of quantitative sciences and practitioners who work with univariate or multivariate probabilistic models. Since the text should also be suitable for self-study, excessive formalism is avoided though mathematical rigor is retained. A deeper insight into the topics is provided by detailed examples and illustrations. The book covers the standard content of a course in probability and statistics. However, the second edition includes two new chapters about distribution theory and exploratory data analysis. The first-mentioned chapter certainly goes beyond the standard material. It is presented to reflect the growing practical importance of developing new distributions. The second new chapter studies intensively one- and bidimensional concepts like assymetry, kurtosis, correlation and determination coefficients. In particular, examples are intended to enable the reader to take a critical look at the appropriateness of the geometrically motivated concepts.

This book is intended as an introduction to Probability Theory and Mathematical Statistics for students in mathematics, the physical sciences, engineering, and related fields. It is based on the author’s 25 years of experience teaching probability and is squarely aimed at helping students overcome common difficulties in learning the subject. The focus of the book is an explanation of the theory, mainly by the use of many examples. Whenever possible, proofs of stated results are provided. All sections conclude with a short list of problems. The book also includes several optional sections on more advanced topics. This textbook would be ideal for use in a first course in Probability Theory.

Contents:
Probabilities
Conditional Probabilities and Independence
Random Variables and Their Distribution
Operations on Random Variables
Expected Value, Variance, and Covariance
Normally Distributed Random Vectors
Limit Theorems

Introduction to Stochastic Processes
Mathematical Statistics
Appendix
Bibliography
Index

This book offers an introduction to concepts of probability theory, probability distributions relevant in the applied sciences, as well as basics of sampling distributions, estimation and hypothesis testing. As a companion for classes for engineers and scientists, the book also covers applied topics such as model building and experiment design.

Contents
Random phenomena
Probability
Random variables
Expected values
Commonly used discrete distributions
Commonly used density functions
Joint distributions
Some multivariate distributions
Collection of random variables
Sampling distributions
Estimation
Interval estimation
Tests of statistical hypotheses
Model building and regression
Design of experiments and analysis of variance
Questions and answers

Mathematical models cannot be solved using the traditional analytical methods for dynamic equations on time scales. These models must be dealt with using computational methods. This textbook introduces numerical methods for initial value problems for dynamic equations on time scales. Hands-on examples utilizing MATLAB and practical problems illustrate a wide variety of solution techniques.

This book is a concise yet complete calculus textbook covering all essential topics in multi-variable calculus, including geometry in three-dimensional space, partial derivatives, maximum/minimum, multiple integrals and vector calculus as well as a chapter for ODE. All the chapters are constructed in a logical way to outline the essence of each topic and to address potential difficulties arising from learning.

This book is devoted to recent developments of linear and nonlinear integral inequalities on time scales. The book is intended for the use in the field of dynamic calculus on time scales, dynamic equation and integral equations on time scales. It is also suitable for graduate courses in the above fields. The book is designed for those who have mathematical background on time scales calculus.

This book is ideal as an introduction to algebraic topology and applied algebraic topology featuring a streamlined approach including coverage of basic categorical notions, simplicial, cellular, and singular homology, persistent homology, cohomology groups, cup products, Poincare Duality, homotopy theory, and spectral sequences. The focus is on examples and computations, and there are many end of chapter exercises and extensive student projects.

This is the fi rst textbook leading coherently from classical character theory to the theory of lattices over orders and integral representations of fi nite groups. Character theory is developed in a highly pedagogical way including many examples and exercises covering at once the fi rst defi nitions up to Clifford theory, Brauer’s induction theorem and the splitting fi eld theorem, as well as self-dual simple modules allowing bilinear forms. This latter part is done step by step using the approach given by Sin and Willems. Dirichlet characters and Dirichlet’s result on primes in arithmetic progressions are given as an application. Examples of integral representations of fi nite groups are already detailed at a quite early stage where appropriate, so that the more systematic treatment of lattices over orders is natural. After that, the necessary number theory and homological algebra is developed as far as needed. Maximal as well as hereditary orders are introduced and the Auslander- Buchsbaum theorem is proved. The Jordan-Zassenhaus theorem on the fi niteness of lattices in a given vector space is fully proved. Then the development and properties of class groups of orders is a fi rst focus. As a further highlight Swan’s example of a stably free but not free ideal over the integral group ring of the generalised quaternion group of order 32 is developed in great detail.

• A student friendly introduction to ordinary representation theory
• Many examples and exercises, including solutions for some of them, make the book well suited for self-study
• Leads coherently from ordinary character theory to the integral representation theory of lattices over orders
• Several topics appear for the fi rst time in a textbook, such as Sin-Willems’ approach to self-dual simple modules and Swan‘s example of a stably free non free ideal.

This book offers an introduction to differential geometry for the non-specialist. It includes most of the required material from multivariable calculus, linear algebra, and basic analysis. An intuitive approach and a minimum of prerequisites make it a valuable companion for students of mathematics and physics.

The main focus is on manifolds in Euclidean space and the metric properties they inherit from it. Among the topics discussed are curvature and how it affects the shape of space, and the generalization of the fundamental theorem of calculus known as Stokes' theorem.

Derivations on von Neumann algebras are well understood and are always inner, meaning that they act as commutators with a fixed element from the algebra itself. The purpose of this book is to provide a complete description of derivations on algebras of operators affiliated with a von Neumann algebra. The book is designed to serve as an introductory graduate level to various measurable operators affiliated with a von Neumann algebras and their properties. These classes of operators form their respective algebras and the problem of describing derivations on these algebras was raised by Ayupov, and later by Kadison and Liu. A principal aim of the book is to fully resolve the Ayupov-Kadison-Liu problem by proving a necessary and sufficient condition of the existence of non-inner derivation of algebras of measurable operators. It turns out that only for a finite type I von Neumann algebra M may there exist a non-inner derivation on the algebra of operators affiliated with M. In particular, it is established that the classical derivation d/dt of functions of real variables can be extended up to a derivation on the algebra of all measurable functions. This resolves a long-standing problem in classical analysis.

The second edition of this well-received textbook is devoted to Combinatorics and Graph Theory, which are cornerstones of Discrete Mathematics. Every section begins with simple model problems. Following their detailed analysis, the reader is led through the derivation of definitions, concepts, and methods for solving typical problems. Theorems then are formulated, proved, and illustrated by more problems of increasing difficulty.

The idea behind this book is to provide the mathematical foundations for assessing modern developments in the Information Age. It deepens and complements the basic concepts, but it also considers instructive and more advanced topics. The treatise starts with a general chapter on algebraic structures; this part provides all the necessary knowledge for the rest of the book. The next chapter gives a concise overview of cryptography. Chapter 3 on number theoretic algorithms is important for developping cryptosystems, Chapter 4 presents the deterministic primality test of Agrawal, Kayal, and Saxena. The account to elliptic curves again focuses on cryptographic applications and algorithms. With combinatorics on words and automata theory, the reader is introduced to two areas of theoretical computer science where semigroups play a fundamental role.The last chapter is devoted to combinatorial group theory and its connections to automata.

Contents:
Algebraic structures
Cryptography
Number theoretic algorithms
Polynomial time primality test
Elliptic curves
Combinatorics on words
Automata
Discrete infinite groups

This book is intended to be used as a rather informal, and surely not complete, textbook on the subjects indicated in the title. It collects my Lecture Notes held during three academic years at the University of Siena for a one semester course on "Basic Mathematical Physics", and is organized as a short presentation of few important points on the arguments indicated in the title.

It aims at completing the students' basic knowledge on Ordinary Differential Equations (ODE) - dealing in particular with those of higher order - and at providing an elementary presentation of the Partial Differential Equations (PDE) of Mathematical Physics, by means of the classical methods of separation of variables and Fourier series. For a reasonable and consistent discussion of the latter argument, some elementary results on Hilbert spaces and series expansion in othonormal vectors are treated with some detail in Chapter 2.

Prerequisites for a satisfactory reading of the present Notes are not only a course of Calculus for functions of one or several variables, but also a course in Mathematical Analysis where - among others - some basic knowledge of the topology of normed spaces is supposed to be included. For the reader's convenience some notions in this context are explicitly recalled here and there, and in particular as an Appendix in Section 1.4. An excellent reference for this general background material is W. Rudin's classic Principles of Mathematical Analysis. On the other hand, a complete discussion of the results on ODE and PDE that are here just sketched are to be found in other books, specifically and more deeply devoted to these subjects, some of which are listed in the Bibliography.

In conclusion and in brief, my hope is that the present Notes can serve as a second quick reading on the theme of ODE, and as a first introductory reading on Fourier series, Hilbert spaces, and PDE

Analysis and Probability on graphs is an introduction to random graphs, Markov chains on digraphs, entropy of Markov Chains, and discrete Lyapunov exponents and Hausdorff dimension, requiring only minimal background in probability, mathematical analysis, and graphs. This textbook includes constructive discussions about the motivation of basic concepts, and many worked-out problems in each chapter, making it ideal for classroom use or self-study.

This book gives an advanced overview of several topics in infinite group theory. It can also be considered as a rigorous introduction to combinatorial and geometric group theory. The philosophy of the book is to describe the interaction between these two important parts of infinite group theory. In this line of thought, several theorems are proved multiple times with different methods either purely combinatorial or purely geometric while others are shown by a combination of arguments from both perspectives.

The first part of the book deals with Nielsen methods and introduces the reader to results and examples that are helpful to understand the following parts. The second part focuses on covering spaces and fundamental groups, including covering space proofs of group theoretic results. The third part deals with the theory of hyperbolic groups. The subjects are illustrated and described by prominent examples and an outlook on solved and unsolved problems.

New edition now includes the topics on universal free groups, quasiconvex subgroups and hyperbolic groups, and also Stallings foldings and subgroups of free groups. New results on groups of F-types are added.

In this textbook, a concise approach to complex analysis of one and several variables is presented. After an introduction of Cauchy‘s integral theorem general versions of Runge‘s approximation theorem and Mittag-Leffler‘s theorem are discussed. The fi rst part ends with an analytic characterization of simply connected domains. The second part is concerned with functional analytic methods: Fréchet and Hilbert spaces of holomorphic functions, the Bergman kernel, and unbounded operators on Hilbert spaces to tackle the theory of several variables, in particular the inhomogeneous Cauchy-Riemann equations and the d-bar Neumann operator.

Contents
Complex numbers and functions
Cauchy’s Theorem and Cauchy’s formula
Analytic continuation
Construction and approximation of holomorphic functions
Harmonic functions
Several complex variables
Bergman spaces
The canonical solution operator to
Nuclear Fréchet spaces of holomorphic functions
The -complex
The twisted -complex and Schrödinger operators

This textbook offers a high-level introduction to multi-variable differential calculus. Differential forms are introduced incrementally in the narrative, eventually leading to a unified treatment of Green’s, Stokes’ and Gauss’ theorems. Furthermore, the presentation offers a natural route to differential geometry.

Contents:
Calculus of Vector Functions
Tangent Spaces and 1-forms
Line Integrals
Differential Calculus of Mappings
Applications of Differential Calculus
Double and Triple Integrals
Wedge Products and Exterior Derivatives
Integration of Forms
Stokes’ Theorem and Applications

This book is a comprehensive guide to Linear Algebra and covers all the fundamental topics such as vector spaces, linear independence, basis, linear transformations, matrices, determinants, inner products, eigenvectors, bilinear forms, and canonical forms. It also introduces concepts such as fields, rings, group homomorphism, and binary operations early on, which gives students a solid foundation to understand the rest of the material.

Unlike other books on Linear Algebra that are either too theory-oriented with fewer solved examples or too problem-oriented with less good quality theory, this book strikes a balance between the two. It provides easy-to-follow theorem proofs and a considerable number of worked examples with various levels of difficulty. The fundamentals of the subject are explained in a methodical and straightforward way.

This book is aimed at undergraduate and graduate students of Mathematics and Engineering Mathematics who are studying Linear Algebra. It is also a useful resource for students preparing for exams in higher education competitions such as NET, GATE, lectureships, etc. The book includes some of the most recent and challenging questions from these exams.

This book introduces quadratic ideal numbers as objects of study with applications to binary quadratic forms and other topics. The text requires only minimal background in number theory, much of which is reviewed as needed. Computational methods are emphasized throughout, making this subject appropriate for individual study or research at the undergraduate level or above.

• Uses a unique programmed approach that takes readers through the mathematics in a step-by-step fashion with a wealth of worked examples and exercises.
• Contains many Quizzes, Learning Outcomes, and Can You? checklists.
• Ideal as a classroom textbook or a self-learning manual.

The subject of applied complex variables is so fundamental that most of the other topics in advanced engineering mathematics (AEM) depend on it. The present book contains complete coverage of the subject, summarizing the more elementary aspects that you find in most AEM textbooks and delving into the more specialized topics that are less commonplace. The book represents a one-stop reference for complex variables in engineering analysis. The applications of conformal mapping in this book are significantly more extensive than in other AEM textbooks. The treatments of complex integral transforms enable a much larger class of functions that can be transformed, resulting in an expanded use of complex-transform techniques in engineering analysis. The inclusion of the asymptotics of complex integrals enables the analysis of models with irregular singular points. The book, which has more than 300 illustrations, is generous with realistic example problems.

This text offers a unique balance of theory and a variety of standard and new applications along with solved technology-aided problems.

The book includes the fundamental mathematical theory, as well as a wide range of applications, numerical methods, projects, and technology-assisted problems and solutions in Maple, Mathematica, and MATLAB. Some of the applications are new, some are unique, and some are discussed in an essay. There is a variety of exercises which include True/False questions, questions that require proofs, and questions that require computations.

The goal is to provide the student with is a solid foundation of the mathematical theory and an appreciation of some of the important real-life applications. Emphasis is given on geometry, matrix transformations, orthogonality, and least-squares.

Designed for maximum flexibility, it is written for a one-semester/two semester course at the sophomore or junior level for students of mathematics or science.

This is a high level introduction to abstract algebra which is aimed at readers whose interests lie in mathematics and the information and physical sciences. In addition to introducing the main concepts of modern algebra – groups, rings, modules and fields – the book contains numerous applications, which are intended to illustrate the concepts and to show the utility and relevance of algebra today. In particular applications to Polya coloring theory, latin squares, Steiner systems, error correcting codes and economics are described. There is ample material here for a two semester course in abstract algebra. Proofs of almost all results are given. The reader led through the proofs in gentle stages. There are more than 500 problems, of varying degrees of diffi culty. The book should be suitable for advanced undergraduate students in their fi nal year of study and for fi rst or second year graduate students at a university in Europe or North America. In this third edition three new chapters have been added: an introduction to the representation theory of fi nite groups, free groups and presentations of groups, an introduction to category theory.

This text offers a comprehensive introduction to cryptology, based on elementary number theory and essential numerical algorithms. It covers core mathematical topics—number theory, algebraic structures, matrices, determinants, and polynomials—through their use in cryptography. Symmetric-key cryptography and practical techniques are introduced, followed by algorithms and computational complexity. Public-key systems based on integer factorization and discrete logarithms, as well as hash functions and digital signatures, are discussed. Advanced topics include elliptic curve and multivariate polynomial cryptography, primality testing, and integer factorization are also explored. The book is self-contained, with all necessary proofs, theory, and tools to analyze cryptographic security and attacks.

A fascinating journey into the mind-bending world of prime numbers

Cicadas of the genus Magicicada appear once every 7, 13, or 17 years. Is it just a coincidence that these are all prime numbers? How do twin primes differ from cousin primes, and what on earth (or in the mind of a mathematician) could be sexy about prime numbers? What did Albert Wilansky find so fascinating about his brother-in-law's phone number?
Mathematicians have been asking questions about prime numbers for more than twenty-five centuries, and every answer seems to generate a new rash of questions. In Prime Numbers: The Most Mysterious Figures in Math, you'll meet the world's most gifted mathematicians, from Pythagoras and Euclid to Fermat, Gauss, and Erd?o?s, and you'll discover a host of unique insights and inventive conjectures that have both enlarged our understanding and deepened the mystique of prime numbers. This comprehensive, A-to-Z guide covers everything you ever wanted to know--and much more that you never suspected--about prime numbers, including:
* The unproven Riemann hypothesis and the power of the zeta function
* The "Primes is in P" algorithm
* The sieve of Eratosthenes of Cyrene
* Fermat and Fibonacci numbers
* The Great Internet Mersenne Prime Search
* And much, much more

Algebra in the Middle Grades addresses topics that are formalized in the first half of an Algebra 1 course, focusing on linear equations, their graphs and their applications to problem solving. The book is intended to be used by students in the middle school in preparation for the formal study of Algebra 1. In this, it is a truly Pre-Algebra program.

Algebra in the Middle Grades is an approach that brings Algebra to the students. It is written to have the essential elements of Algebra be understood on an inductive level so that when students later address the topic formally, there will be an informal background to create deep understanding.

The approach is extremely visual using both algebra tile-type diagrams along with graphs for equations and grids for ratio. In this way, the text gives students 3 aspects of a topic, the Numerical, the Geometric and the Algebraic. The text places the Numerical and Geometric first having understanding in these contexts leading to understanding in an Algebraic, symbolic context.

This book introduces the theory of multigrid methods for the fast numerical solution of linear and weakly nonlinear elliptic PDE. We use the finite element method to discretize the PDE problems, as this is the most natural choice, and the reader will get a thorough treatment of finite elements. No previous exposure to numerical discretization methods is assumed. All that is required of the reader is some knowledge of matrix theory. Coding the multigrid method is difficult. This book will help the reader build basic multigrid codes using easy-to-read sample Matlab codes. We use a matrix-based approach in the first part of the book, both as a way of presenting the theory in a natural way, and as a means for translating the theory into practical codes. The operators in the text and codes have the same names, which makes reading the sample codes simple, even if the reader has never coded. We deviate from the matrix-based approach only in the presentation of the nonlinear theory in the second part, which represents an area of current research. The book takes the reader from the basics and simple implementation issues all the way to the front lines of research.

A computer algebra system such as Mathematica® is able to do much more than just numerics: This revised text shows how to tackle real mathematical problems from basic analysis. The reader learns how Mathematica® represents domains, qualifiers and limits to implement actual proofs – a requirement to unlock the huge potential of Mathematica® for a variety of applications.

This is a book for an undergraduate number theory course, senior thesis work, graduate level study, or for those wishing to learn about applications of number theory to data encryption and security. With no abstract algebra background required, it covers congruences, the Euclidean algorithm, linear Diophantine equations, the Chinese Remainder Theorem, Mobius inversion formula, Pythagorean triplets, perfect numbers and amicable pairs, Law of Quadratic Reciprocity, theorems on sums of squares, Farey fractions, periodic continued fractions, best rational approximations, and Pell’s equation. Results are applied to factoring and primality testing including those for Mersenne and Fermat primes, probabilistic primality tests, Pollard’s rho and p-1 factorization algorithms, and others. Also an introduction to cryptology with a full discussion of the RSA algorithm, discrete logarithms, and digital signatures.

Chapters on analytic number theory including the Riemann zeta function, average orders of the lattice and divisor functions, Chebyshev’s theorems, and Bertrand’s Postulate. A chapter introduces additive number theory with discussion of Waring’s Problem, the pentagonal number theorem for partitions, and Schnirelmann density.

Also available in a black + white version

AMTE, in the Standards for Preparing Teachers of Mathematics (SPTM), puts forward a national vision of initial preparation for all Pre-K–12 teachers who teach mathematics. SPTM contains critical messages for all who teach mathematics, including elementary school teachers teaching all disciplines, middle and high school mathematics teachers who may teach mathematics exclusively, special education teachers, teachers of emergent multilingual students, and other teaching professionals and administrators who have responsibility for students’ mathematical learning. SPTM has broad implications for teacher preparation programs, in which stakeholders include faculty and administrators in both education and mathematics at the university level; teachers, principals, and district leaders in the schools with which preparation programs partner; and the communities in which preparation programs and their school partners are situated.

SPTM is intended as a national guide that articulates a vision for mathematics teacher preparation and supports the continuous improvement of teacher preparation programs. Such continuous improvement includes changes to preparation program courses and structures, partnerships involving schools and universities and their leaders, the ongoing accreditation of such programs regionally and nationally, and the shaping of state and national mathematics teacher preparation policy. SPTM is also designed to inform assessment practices for mathematics teacher preparation programs, to influence policies related to preparation of teachers of mathematics, and to promote national dialogue around preparing teachers of mathematics. The vision articulated in SPTM is aspirational in that it describes a set of high expectations for developing a well-prepared beginning teacher of mathematics who can support meaningful student learning. The vision is research-based and establishes a set of goals for the continued development and refinement of a mathematics teacher preparation program and a research agenda for the study of the effects of such a program. SPTM contains detailed depictions of what a well-prepared beginning teacher knows and is able to do related to content, pedagogy, and disposition, and what a strong preparation program entails with respect to learning experiences, assessments, and partnerships. Stakeholders in mathematics teacher preparation will find messages related to their roles.

Standards for Preparing Teachers of Mathematics includes standards and indicators for teacher candidates and for the design of teacher preparation programs. SPTM outlines assessment practices related to overall quality, program effectiveness, and candidate performance. SPTM describes specific focal practices by grade band and provides guidance to stakeholders regarding processes for productive change.

The teaching and learning of mathematics in British Columbia has a long and storied history. An integral part of the past 50 years (1962-2012) of this history has been Vector: Journal of the British Columbia Association of Mathematics Teachers. This volume, which presents ten memorable articles from each of the past five decades, that is, 50 articles from the past 50 years of the journal, provides an opportunity to share this rich history with a wide range of individuals interested in the teaching and learning of mathematics and mathematics education. Each decade begins with an introduction, providing a historical context, and concludes with a commentary from a prominent member of the British Columbia mathematics education community. As a result, this monograph provides a historical account as well as a contemporary view of many of the trends and issues in the teaching and learning of mathematics. This volume is meant to serve as a resource for a variety of individuals including: teachers of mathematics, mathematics teacher educators, mathematics education researchers, historians, and undergraduate and graduate students. Most importantly, this volume is a celebratory retrospective on the work of the British Columbia Association of Mathematics Teachers.

Also available in a color version

AMTE, in the Standards for Preparing Teachers of Mathematics (SPTM), puts forward a national vision of initial preparation for all Pre-K–12 teachers who teach mathematics. SPTM contains critical messages for all who teach mathematics, including elementary school teachers teaching all disciplines, middle and high school mathematics teachers who may teach mathematics exclusively, special education teachers, teachers of emergent multilingual students, and other teaching professionals and administrators who have responsibility for students’ mathematical learning. SPTM has broad implications for teacher preparation programs, in which stakeholders include faculty and administrators in both education and mathematics at the university level; teachers, principals, and district leaders in the schools with which preparation programs partner; and the communities in which preparation programs and their school partners are situated.

SPTM is intended as a national guide that articulates a vision for mathematics teacher preparation and supports the continuous improvement of teacher preparation programs. Such continuous improvement includes changes to preparation program courses and structures, partnerships involving schools and universities and their leaders, the ongoing accreditation of such programs regionally and nationally, and the shaping of state and national mathematics teacher preparation policy. SPTM is also designed to inform assessment practices for mathematics teacher preparation programs, to influence policies related to preparation of teachers of mathematics, and to promote national dialogue around preparing teachers of mathematics. The vision articulated in SPTM is aspirational in that it describes a set of high expectations for developing a well-prepared beginning teacher of mathematics who can support meaningful student learning. The vision is research-based and establishes a set of goals for the continued development and refinement of a mathematics teacher preparation program and a research agenda for the study of the effects of such a program. SPTM contains detailed depictions of what a well-prepared beginning teacher knows and is able to do related to content, pedagogy, and disposition, and what a strong preparation program entails with respect to learning experiences, assessments, and partnerships. Stakeholders in mathematics teacher preparation will find messages related to their roles.

Standards for Preparing Teachers of Mathematics includes standards and indicators for teacher candidates and for the design of teacher preparation programs. SPTM outlines assessment practices related to overall quality, program effectiveness, and candidate performance. SPTM describes specific focal practices by grade band and provides guidance to stakeholders regarding processes for productive change.

Reflecting the World: A Guide to Incorporating Equity in Mathematics Teacher Education is a guide for mathematics teacher educators interested in incorporating equity concerns into their teaching. The book draws on the authors’ research and experience integrating issues of equity, diversity, and social justice into their work as mathematics teacher educators of preservice and inservice preK-9 teachers. Reflecting the World includes both a framework for integrating issues of equity into mathematics teacher education courses and professional development and example lessons. The lessons are organized by content area and include guidance for using them effectively.

Elementary and middle grades pre-service teachers are often uncomfortable with mathematics, uncertain about their ability to teach it, and unsure of how it connects to the real world. For many preservice teachers a focus on the real world—and in particular on issues of equity, diversity, and social justice—is more engaging than their past experiences with mathematics and can help lessen their mathematical anxieties.

Reflecting the World will assist teacher educators in designing and teaching mathematics content and methods courses in ways that support future teachers to see the relevance of mathematics to our world and in becoming critical, questioning citizens in an increasingly mathematical world. The book provides a set of tools for helping future teachers connect mathematics to the lives, interests, and political realities of an increasingly diverse student body, and in doing so it provides a meaningful answer to the question, “when will I ever use this?”

The diversity of research in mathematics education has been addressed as both, a problem and a strength. When manifested through adherence to different intellectual roots and theoretical orientations, diversions constitute ‘refractions’ of mathematics education. The collection and analysis of empirical data in a study are by necessity refracted through the specific analytical lens employed, as well as the aim of the study itself. Refractions can also refer to looking at old phenomena through new lenses.

The chapters in this book are refracted through philosophical, political, mathematical and personal lenses by distinguished authors in the field, addressing issues about the elusive experience of doing mathematics, purification of texts, refractions, mathematics and ethnomathematics, political messages in textbook tasks, mathematics education policy debate, the political in mathematics education research, philosophy and mathematics, meanings and representations, identity of mathematical modeling, and dilemmas in the teaching of calculus.

An ancient Sanskrit adage states that Knowledge is something that grows when shared, but shrinks when hoarded. Academics engaged in the generation of new Knowledge are blessed with both the time and the freedom to engage in pursuits that allow for intellectual pleasure. As a phenomenon of the Zeitgeist many have succumbed to the increased corporatization of academic work, engaging in activities for monetary and self advancement purposes. Are there any real intellectuals left in academia, a là Adorno, Bourdieu, Chomsky, Foucault, among others? This Festschrift is dedicated to academics that don't bother with self promotion or aggrandizement of themselves or their ideas in simplistic terms.

Mathematics teacher education has a critical role to play in preparing teachers to put at center stage goals to support equity in mathematics education and to diversify student interest and participation in mathematics. These goals must also resonate with broader public interest goals to improve educational and social conditions both in the U.S. and abroad. The Mathematics Teacher Education in the Public Interest book aims to support mathematics teacher educators to prepare teachers with new knowledge and skills to support all students to learn mathematics and to become informed, engaged, and critical citizens within their community, nation, and world. While internationally there is considerable interest among mathematics educators in issues of equity and social justice, the literature on mathematics teacher education for equity and social justice thus far has been very limited.The book provides theoretical discussions on the need for equity and social justice emphases in mathematics teacher education, as well as practical examples from mathematics teacher educators, documenting their own professional efforts to center practices on equity and social justice. Section emphases include critical perspectives on mathematics teacher education, the use of equity and social justice-themed activities in mathematics teacher preparation courses, and issues of identity and community and cultural contexts in mathematics teacher education. In addition syntheses of major ideas of the book are offered by experienced researchers.

The Association of Mathematics Teacher Educators (AMTE) in its 2015 position paper on Equity in Mathematics Teacher Education provides a list of actions for mathematics teacher educators (MTE’s) to help them develop and implement equitable practices. The position paper states it is critical that mathematics teacher educators: “Model equity-based pedagogy that emphasizes rich and rigorous mathematics; elicit and build on children’s and young adults’ mathematical thinking; connect to P-12 students’ cultural/linguistic knowledge and backgrounds as well as individual interests; facilitate mathematical discourse; minimize status issues by expanding broader participation and engagement where varied mathematical strengths are valued; and promote positive mathematical identity and agency (p. 2)”.

Cases for Mathematics Teacher Educators: Facilitating Conversations about Inequities in Mathematics Classrooms provides an excellent resource to start conversations describing the enactment of these actions. The book is organized into three main sections: (1) Conversations About Inequities in Mathematics Methods Courses, (2) Conversations About Inequities in Mathematics Content Courses, and (3) Conversations about Inequities in Graduate and Professional Development Contexts. Across these sections there are 19 cases and 57 corresponding commentaries focused on dilemmas that arise when mathematics teacher educators foreground equity in their work. This book of cases provides a needed resource for MTEs to engage prospective teachers, practicing teachers, and future teacher educators in discussions about inequities, privilege, and oppression in society, in schools, and in the mathematics classroom. It is the product of the thinking and experiences of 87 authors who are committed to the improvement of mathematics teacher education.

This book is dedicated to metric spaces and their topology. The book starts with ZFC axioms. The real number system is constructed by both the Dedekind cut and the Cauchy sequence approach. The various examples and properties of metric spaces and normed linear spaces are discussed. The different distances between the sets are highlighted. The research work on metric-preserving maps and isometries on different p-norms has been discussed. Homeomorphism and different equivalent metrics have also been discussed. A detailed description of a metric on the product and the quotient set is also provided. The completion of a metric space as a universal property and applications of the Baire Category Theorem are covered. A special focus is on compactness and the relation between a compact metric space, the Hilbert Cube, and the Cantor set. The properties of connected and path-connected metric spaces are provided.

If you’ve spent a fair share of time in math class glancing at the clock, you’re not alone. Glancing at the clouds and wondering how well counting the flower petals on the windowsill can help you run out the clock is a pretty common phenomenon for people who aren’t into numbers. But those connections you make aren’t just in your head. You might just be closer than you think to understanding the universe.

The Voice of Math takes a deep dive into understanding numbers through the lens of humanity. You’ll see the struggles humanity has taken to understand life and the hiccups in understanding. If you look closely, you’ll probably see yourself, too.

If you’ve always been curious about math but don’t know where to start, pick up this book. Inside you’ll find Easter eggs we’ve scattered through its pages. Inside you’ll find mathematical problems broken down to become more understandable. Inside you’ll unlock mysteries history has solved and a few you can unlock on your own.

This book is designed to inspire curiosity. It’s to develop a love for math from an anecdotal point of view. Who you are may just be hidden in these pages, and your journey to become a mathematician might just be right around the corner.

Here are topics that are usually published in briefer form in more advanced texts as part of their supporting mathematics. In this work, vectors and matrices are the main subjects. This text offers an easier-to-understand introduction to the main ideas behind vectors and matrices, stripped of formalism, and leading directly to geometric modeling. It demonstrates the relationships between vectors, matrices, basis vectors and barycentric coordinates, all of which are not usually seen in ordinary texts. This text also discusses how these concepts are applied to produce curves and surfaces, and how they facilitate the analysis of spatial relationships. 

For those readers beginning studies in geometric and 3D modeling, animation, CGI, or CAD/CAM, this work serves as an introduction to vectors and matrices, and provides a good start to understanding how they are applied. For instructors, this book can be a primary text or supplement to more advanced or specialized texts on geometric and 3D modeling.
           
Features

• More than 150 illustrations support visual understanding of the content.
• 100+ exercises and extended solutions enhance the classroom environment.
• A comprehensive range of topics offers an in-depth look at the math underlying 3D modeling and animation courses.
• Linear algebra and calculus are not prerequisites.

Ante la percepción popular de que las matemáticas son inútiles para nuestro día a día, el profesor Ian Stewart nos demuestra que esta disciplina va mucho más allá de los cálculos aburridos que todos recordamos de la escuela y nos propone un curioso recorrido por los usos de las matemáticas que a menudo permanecen ocultos a simple vista, pero contribuyen a nuestras vidas. Desde la trigonometría que mantiene un satélite en órbita hasta los números primos utilizados por los sistemas de seguridad más avanzados del mundo, pasando por los números imaginarios que permiten la realidad aumentada, los métodos más eficientes para los trasplantes de riñón, las aplicaciones en política, la predicción del cambio climático o, incluso, cómo una curva extraña e infinitamente ondulada optimiza las entregas a domicilio; las matemáticas no solo son relevantes para nuestras vidas, sino que sin ellas el mundo moderno como lo conocemos se desmoronaría.

The purpose of this book is to present results on the subject of weak convergence in function spaces to study invariance principles in statistical applications to dependent random variables, U-statistics, censor data analysis. Different techniques, formerly available only in a broad range of literature, are for the first time presented here in a self-contained fashion.

Contents:
Weak convergence of stochastic processes
Weak convergence in metric spaces
Weak convergence on C[0, 1] and D[0,∞)
Central limit theorem for semi-martingales and applications
Central limit theorems for dependent random variables
Empirical process
Bibliography

The aim of the book is to give a broad introduction of topology to undergraduate students. It covers the most important and useful parts of the point-set as well as the combinatorial topology. The development of the material is from simple to complex, concrete to abstract, and appeals to the intuition of readers. Attention is also paid to how topology is actually used in the other fields of mathematics. Over 150 illustrations, 160 examples and 600 exercises will help readers to practice and fully understand the subject.

Contents:
Set and Map
Metric Space
Graph
Topology
Topological Concepts
Complex
Topological Properties
Surface
Topics in Point Set Topology
Index

The book presents 18 games and develops the concepts of game analysis and winning strategies. Students are encouraged to play these mathematical games together, collect data developed through their play, and analyze the data to develop a winning strategy. The book provides the basis for a six- to eight-week unit on mathematical games. Each chapter also functions as a self-contained and independent exploration so that selected chapters may be used as supplementary classroom investigations or as independent projects. The book includes both familiar games (such as Quadrangles and Nim) and many others that will be new and exciting to most readers. Through the exploration of mathematical games, ‘The Play’s the Thing’ introduces teachers and students to the fun of play and to the mathematics behind the fun.

Sponsored by the American Educational Research Association's Special Interest Group for Educational Statisticians

This volume is the second edition of Hancock and Mueller’s highly-successful 2006 volume, with all of the original chapters updated as well as four new chapters. The second edition, like the first, is intended to serve as a didactically-oriented resource for graduate students and research professionals, covering a broad range of advanced topics often not discussed in introductory courses on structural equation modeling (SEM). Such topics are important in furthering the understanding of foundations and assumptions underlying SEM as well as in exploring SEM, as a potential tool to address new types of research questions that might not have arisen during a first course. Chapters focus on the clear explanation and application of topics, rather than on analytical derivations, and contain materials from popular SEM software.

What is latent class analysis? If you asked that question thirty or forty years ago you would have gotten a different answer than you would today. Closer to its time of inception, latent class analysis was viewed primarily as a categorical data analysis technique, often framed as a factor analysis model where both the measured variable indicators and underlying latent variables are categorical. Today, however, it rests within much broader mixture and diagnostic modeling framework, integrating measured and latent variables that may be categorical and/or continuous, and where latent classes serve to define the subpopulations for whom many aspects of the focal measured and latent variable model may differ.

For latent class analysis to take these developmental leaps required contributions that were methodological, certainly, as well as didactic. Among the leaders on both fronts was C. Mitchell “Chan” Dayton, at the University of Maryland, whose work in latent class analysis spanning several decades helped the method to expand and reach its current potential. The current volume in the Center for Integrated Latent Variable Research (CILVR) series reflects the diversity that is latent class analysis today, celebrating work related to, made possible by, and inspired by Chan’s noted contributions, and signaling the even more exciting future yet to come.

Die Wahrscheinlichkeitstheorie gehört zu den Kerndisziplinen der modernen Mathematikausbildung. Sie ist die Grundlage für alle Modelle, die „Risiko" und „Unsicherheit" einbeziehen. Dieses Lehrbuch gibt einen direkten, verlässlichen und modernen Zugang zu den wichtigsten Ergebnissen der mathematischen Wahrscheinlichkeitstheorie. Aufbauend auf dem Band „Maß & Integral" werden zunächst elementare Fragen Wahrscheinlichkeitsverteilungen, Zufallsvariable, Unabhängigkeit, bedingte Wahrscheinlichkeiten und charakteristische Funktionen – bis hin zu einfachen Grenzwertsätzen behandelt. Diese Themen werden dann um das Studium von Summen unabhängiger Zufallsvariablen – Gesetze der Großen Zahlen, Null-Eins-Gesetze, random walks, zentraler Grenzwertsatz von Lindeberg-Feller – ergänzt. Allgemeine bedingte Erwartungen, Anwendungen von charakteristischen Funktionen und eine Einführung in die Theorie unendlich teilbarer Verteilungen und der großen Abweichungen runden die Darstellung ab. In gleicher Ausstattung erscheint der Folgeband „Martingale & Prozesse".

Lösungen zu den im Buch befindlichen Übungsaufgaben unter: http://www.motapa.de/stoch/index.shtml

Allgemeine Maße und das Lebesgue-Integral gehören zu den unverzichtbaren Hilfsmitteln der modernen Analysis, der Funktionalanalysis und der Stochastik. Das vorliegende Lehrbuch bietet eine Einführung in die wesentlichen Aspekte der Theorie – Maße, Integrale, Konvergenzsätze, Parameterintegrale, Satz von Fubini –, die durch weiterführende Themen – allgemeiner Transformationssatz, Satz von Radon-Nikodým, Fouriertransformation von Maßen, topologische Maßtheorie – abgerundet wird. Mehr als 150 Übungsaufgaben (mit vollständigen Lösungen im Internet) vertiefen und erweitern den Stoff.

Die kompakte Darstellung bietet sich als Fortsetzung der Grundvorlesungen "Analysis" oder als Einstieg in die "Stochastik" an. Da nur Grundkenntnisse in Analysis und linearer Algebra vorausgesetzt werden, ist der Text auch für Studierende der Physik und Ingenieurswissenschaften sowie zum Selbststudium geeignet.

In gleicher Ausstattung erscheinen die Folgebände "Wahrscheinlichkeit" und "Martingale & Prozesse".

Lösungen zu den im Buch befindlichen Übungsaufgaben unter: http://www.motapa.de/mint/index.shtml

This textbook, now in its second revised and extended edition, presents the fundamental ideas and results of both probability theory and statistics. It comprises the material of a one-year course, which is addressed to students of mathematics and to scientists with an interest in the mathematical side of stochastics.
The stochastic concepts, models and methods are motivated by examples and then developed and analysed systematically. Some measure theory is included, but this is done at an elementary level that is in accordance with the introductory character of the book. A large number of problems, now in part with solutions, offer applications and supplements to the text.

This volume deals with two complementary topics. On one hand the book deals with the problem of determining the the probability distribution of a positive compound random variable, a problem which appears in the banking and insurance industries, in many areas of operational research and in reliability problems in the engineering sciences.

On the other hand, the methodology proposed to solve such problems, which is based on an application of the maximum entropy method to invert the Laplace transform of the distributions, can be applied to many other problems.

The book contains applications to a large variety of problems, including the problem of dependence of the sample data used to estimate empirically the Laplace transform of the random variable.

This book is an introduction to financial mathematics. It is intended for graduate students in mathematics and for researchers working in academia and industry.
The focus on stochastic models in discrete time has two immediate benefits. First, the probabilistic machinery is simpler, and one can discuss right away some of the key problems in the theory of pricing and hedging of financial derivatives. Second, the paradigm of a complete financial market, where all derivatives admit a perfect hedge, becomes the exception rather than the rule. Thus, the need to confront the intrinsic risks arising from market incomleteness appears at a very early stage.
The first part of the book contains a study of a simple one-period model, which also serves as a building block for later developments. Topics include the characterization of arbitrage-free markets, preferences on asset profiles, an introduction to equilibrium analysis, and monetary measures of financial risk.
In the second part, the idea of dynamic hedging of contingent claims is developed in a multiperiod framework. Topics include martingale measures, pricing formulas for derivatives, American options, superhedging, and hedging strategies with minimal shortfall risk.
This fourth, newly revised edition contains more than one hundred exercises. It also includes material on risk measures and the related issue of model uncertainty, in particular a chapter on dynamic risk measures and sections on robust utility maximization and on efficient hedging with convex risk measures.

Contents:
Part I: Mathematical finance in one period
Arbitrage theory
Preferences
Optimality and equilibrium
Monetary measures of risk
Part II: Dynamic hedging
Dynamic arbitrage theory
American contingent claims
Superhedging
Efficient hedging
Hedging under constraints
Minimizing the hedging error
Dynamic risk measures

Dieses Buch ist für Anwender der Statistik mit Programmierkenntnissen. Rechnerprogramme helfen, die theoretischen Schlussfolgerungen zu verstehen. Die notwendigen und unverzichtbaren Theoreme werden allein durch Experimente illustriert.
Das Buch ist anwendungsorientiert und verzichtet weitgehend auf theoretische Ausführungen.

Inhalt:
Erzeugung von Zufallszahlen undMonte-Carlo-Experimente
Spielexperimente
Wahrscheinlichkeitsfunktionen, Dichten, Verteilungen
Punktschatzungen
Konfidenzschatzungen
Statistische Tests
Funktionstests fur Zufallszahlengeneratoren

This is the third volume of the series "Moderne Stochastik" (Modern Stochastics). As a follow-up to the volume "Wahrscheinlichkeit" (Probability Theory) it gives an intrdouction to dynamical aspects of probability theory using stochastic processes in discrete time. The first part of the book covers discrete martingales - their convergenc behaviour, optional sampling and stopping, uniform integrability and essential martingale inequalities. The power of martingale techniques is illustrated in the chapters on applications of martingales in classical probability and on the Burkholder-Davis-Gundy inequalities. The second half of the book treats random walks on Zd and Rd, their fluctuation behaviour, recurrence and transience. The last two chapters give a brief introduction to probabilistic potential theory and an outlook of further developments: Brownian motion and Donsker's invariance principle

Contents
Fair Play
Conditional Expectation
Martingale
Stopping and Localizing
Martingale Convergence
L²-Martingales
Uniformly Integrable Martingales
Some Classical Results of Probability
Elementary Inequalities for Martingales
The Burkholder–Davis–Gundy Inequalities
Random Walks on ℤd – the first steps
Fluctuations of Simple Random Walks on Z
Recurrence and Transience of General Random Walks
Random Walks and Analysis
Donsker's Invariance Principle and Brownian Motion

Stochastic processes occur everywhere in the sciences, economics and engineering, and they need to be understood by (applied) mathematicians, engineers and scientists alike. This book gives a gentle introduction to Brownian motion and stochastic processes, in general. Brownian motion plays a special role, since it shaped the whole subject, displays most random phenomena while being still easy to treat, and is used in many real-life models. Im this new edition, much material is added, and there are new chapters on ''Wiener Chaos and Iterated Itô Integrals'' and ''Brownian Local Times''.

Im Laufe der Jahrhunderte wurde in der Stochastik und in der in ihr verwendeten Kombinatorik eine Vielzahl von Problemen aufgeworfen, und diese höchst geistreich gelöst. Das vorliegende Buch bietet eine Auswahl der interessantesten Probleme in ihrer historischen Abfolge. Aufgaben und Lösungen werden im historischen Kontext und in moderner Fassung vorgestellt. Dabei überrascht immer wieder, mit welchen Methoden und welcher Rechenfertigkeit in früherer Zeit diese Probleme bewältigt wurden.
Das Werk bietet Anregungen für alle Interessierten vom mathematisch gebildeten Laien bis zum professionellen Mathematiker. Viele der aufgeführten Problemstellungen hatten großen Einfluss auf die Entwicklung der Aufgabenkultur in der Stochastik. Bis heute lassen sich die Spuren in Aufgaben von Schul- und Hochschulbüchern finden.

Inhalt:
Von den Anfängen bis zum Ende des Mittelalters
Beginn der Neuzeit
1. Hälfte des 17. Jahrhunderts
HUYGENS’ Tractatus de Ratiociniis in Aleae Ludo von 1657
2. Hälfte des 17. Jahrhunderts
Beginn des 18. Jahrhunderts
JAKOB BERNOULLIs Ars Conjectandi von 1713
MONTMORTs Essay d’Analyse sur les Jeux de Hazard von 1713
NIKOLAUS BERNOULLIs Petersburger Problem von 1713
Die Jahre nach 1713 bis 1750
2. Hälfte des 18. Jahrhunderts
1. Hälfte des 19. Jahrhunderts
2. Hälfte des 19. Jahrhunderts
Das 20. Jahrhundert
Nachtrag
Lebensdaten
Literatur
Abbildungsverzeichnis
Personenregister
Sachregister

A fast-food chain once tried to compete with McDonald’s quarter-pounder by introducing a third-pound hamburger—only for it to flop when consumers thought a third pound was less than a quarter pound because three is less than four. Separately, a rash of suicides by teenagers who played Dungeons and Dragons caused a panic in parents and the media. They thought D&D was causing teenage suicides—when in fact teenage D&D players died by suicide at a much lower rate than the national average. Errors of this type can be found from antiquity to the present, from the Peloponnesian War to the COVID-19 pandemic. How and why do we keep falling into these traps?

James C. Zimring argues that many of the mistakes that the human mind consistently makes boil down to misperceiving fractions. We see slews of statistics that are essentially fractions, such as percentages, probabilities, frequencies, and rates, and we tend to misinterpret them. Sometimes bad actors manipulate us by cherry-picking data or distorting how information is presented; other times, sloppy communicators inadvertently mislead us. In many cases, we fool ourselves and have only our own minds to blame. Zimring also explores the counterintuitive reason that these flaws might benefit us, demonstrating that individual error can be highly advantageous to problem solving by groups. Blending key scientific research in cognitive psychology with accessible real-life examples, Partial Truths helps readers spot the fallacies lurking in everyday information, from politics to the criminal justice system, from religion to science, from business strategies to New Age culture.

The book is designed to cover all major aspects of applied numerical methods, including numerical computations, solution of algebraic and transcendental equations, finite differences and interpolation, curve fitting, correlation and regression, numerical differentiation and integration, matrices and linear system of equations, numerical solution of ordinary differential equations, and numerical solution of partial differential equations. MATLAB is incorporated throughout the text and most of the problems are executed in MATLAB code. It uses a numerical problem-solving orientation with numerous examples, figures, and end of chapter exercises. Presentations are limited to very basic topics to serve as an introduction to more advanced topics.

Features:

• Integrates MATLAB throughout the text
• Includes over 600 fully-solved problems with step-by-step solutions
• Limits presentations to basic concepts of solving numerical methods

Die Numerische Mathematik ist einer der Grundpfeiler des Mathematik-, Ingenieur-, Physik- und Informatikstudiums. Dieses zweibändige Lehrbuch ist für Einführungsvorlesungen konzipiert und legt eine solide Basis für weiterführende Lerneinheiten. Der Text ist aus Vorlesungsmanuskripten hervorgegangen, die der Verfasser seit etwa 30 Jahren für seine Grundvorlesungen auf dem Gebiet der Numerischen Mathematik und des Wissenschaftlichen Rechnens an der Friedrich-Schiller-Universität Jena verwendet.

Das Buch deckt den gesamten Bereich der Numerischen Mathematik von den klassischen Techniken wie Gaußscher Algorithmus und Newtonsches Verfahren bis hin zu modernen Algorithmen wie kubische Spline-Interpolation, Kleinste-Quadrate-Approximation mittels Householder- und Givens-Transformationen sowie Deflationstechniken ab. Die Verfahren werden mathematisch exakt beschrieben, in MATLAB-Codes implementiert und anhand von Beispielen demonstriert. Die MATLAB-Codes sind auf der Webseite des Verlages zum Download bereitgestellt, so dass der Leser seine eigenen Experimente mit den numerischen Verfahren durchführen kann.

Durch seinen didaktischen Aufbau und die zahlreichen anschaulichen Beispiele und Übungsaufgaben eignet sich dieses Buch hervorragend als vorlesungsbegleitende Lektüre und als Grundlage für ein erfolgreiches Selbststudium. Gleichzeitig kann es von Mathematikern, Naturwissenschaftlern und Ingenieuren als profundes Nachschlagewerk herangezogen werden.

Mit der 4. Auflage wurde das umfangreiche Standardwerk der Numerischen Mathematik so in zwei Bände aufgeteilt, dass diese relativ unabhängig voneinander gelesen werden können. An vielen Stellen wurde der Text überarbeitet und ergänzt. Das betrifft insbesondere diejenigen Abschnitte, die für Lehrerstudenten relevant sind sowie die Implementierung der numerischen Verfahren in der Programmiersprache MATLAB.

Die Numerische Mathematik ist einer der Grundpfeiler des Mathematik-, Ingenieur-, Physik- und Informatikstudiums. Dieses zweibändige Lehrbuch ist für Einführungsvorlesungen konzipiert und legt eine solide Basis für weiterführende Lerneinheiten. Der Text ist aus Vorlesungsmanuskripten hervorgegangen, die der Verfasser seit etwa 30 Jahren für seine Grundvorlesungen auf dem Gebiet der Numerischen Mathematik und des Wissenschaftlichen Rechnens an der Friedrich-Schiller-Universität Jena verwendet.

Das Buch deckt den gesamten Bereich der Numerischen Mathematik von den klassischen Techniken wie Gaußscher Algorithmus und Newtonsches Verfahren bis hin zu modernen Algorithmen wie kubische Spline-Interpolation, Kleinste-Quadrate-Approximation mittels Householder- und Givens-Transformationen sowie Deflationstechniken ab. Die Verfahren werden mathematisch exakt beschrieben, in MATLAB-Codes implementiert und anhand von Beispielen demonstriert. Die MATLAB-Codes sind auf der Webseite des Verlages zum Download bereitgestellt, so dass der Leser seine eigenen Experimente mit den numerischen Verfahren durchführen kann.

Durch seinen didaktischen Aufbau und die zahlreichen anschaulichen Beispiele und Übungsaufgaben eignet sich dieses Buch hervorragend als vorlesungsbegleitende Lektüre und als Grundlage für ein erfolgreiches Selbststudium. Gleichzeitig kann es von Mathematikern, Naturwissenschaftlern und Ingenieuren als profundes Nachschlagewerk herangezogen werden.

Mit der 4. Auflage wurde das umfangreiche Standardwerk der Numerischen Mathematik so in zwei Bände aufgeteilt, dass diese relativ unabhängig voneinander gelesen werden können. An vielen Stellen wurde der Text überarbeitet und ergänzt. Das betrifft insbesondere diejenigen Abschnitte, die für Lehrerstudenten relevant sind sowie die Implementierung der numerischen Verfahren in der Programmiersprache MATLAB.

Die Numerische Mathematik ist einer der Grundpfeiler des Mathematik-, Ingenieur-, Physik- und Informatikstudiums. Dieses zweibändige Lehrbuch ist für Einführungsvorlesungen konzipiert und legt eine solide Basis für weiterführende Lerneinheiten. Der Text ist aus Vorlesungsmanuskripten hervorgegangen, die der Verfasser seit etwa 30 Jahren für seine Grundvorlesungen auf dem Gebiet der Numerischen Mathematik und des Wissenschaftlichen Rechnens an der Friedrich-Schiller-Universität Jena verwendet.

Das Buch deckt den gesamten Bereich der Numerischen Mathematik von den klassischen Techniken wie Gaußscher Algorithmus und Newtonsches Verfahren bis hin zu modernen Algorithmen wie kubische Spline-Interpolation, Kleinste-Quadrate-Approximation mittels Householder- und Givens-Transformationen sowie Deflationstechniken ab. Die Verfahren werden mathematisch exakt beschrieben, in MATLAB-Codes implementiert und anhand von Beispielen demonstriert. Die MATLAB-Codes sind auf der Webseite des Verlages zum Download bereitgestellt, so dass der Leser seine eigenen Experimente mit den numerischen Verfahren durchführen kann.

Durch seinen didaktischen Aufbau und die zahlreichen anschaulichen Beispiele und Übungsaufgaben eignet sich dieses Buch hervorragend als vorlesungsbegleitende Lektüre und als Grundlage für ein erfolgreiches Selbststudium. Gleichzeitig kann es von Mathematikern, Naturwissenschaftlern und Ingenieuren als profundes Nachschlagewerk herangezogen werden.

Mit der 4. Auflage wurde das umfangreiche Standardwerk der Numerischen Mathematik so in zwei Bände aufgeteilt, dass diese relativ unabhängig voneinander gelesen werden können. An vielen Stellen wurde der Text überarbeitet und ergänzt. Das betrifft insbesondere diejenigen Abschnitte, die für Lehrerstudenten relevant sind sowie die Implementierung der numerischen Verfahren in der Programmiersprache MATLAB.

The Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability continues the tradition of this esteemed series, which has been held every five years since 1945. This iteration, spanning multiple sessions from 1970 to 1971, reflects the breadth and evolution of probability theory, statistical analysis, and their applications to emerging scientific challenges. The symposium was structured to cover not only theoretical advances in statistics and probability but also pressing interdisciplinary issues such as molecular evolution, biological health, and environmental pollution, demonstrating the increasingly expansive role of statistical methodologies in scientific inquiry.

Comprising over 3,000 pages of research, these proceedings serve as a vital compendium of contemporary thought in probability and statistics. The volumes include contributions from leading mathematicians and scientists worldwide, fostering collaboration across disciplines and nations. Notably, the symposium acknowledges the loss of two intellectual giants, William Feller and Alfred Rényi, whose influence on probability theory remains profound. With rigorous editorial oversight and contributions from experts across academia and government research agencies, these proceedings exemplify the enduring impact of the Berkeley Symposia in shaping the future of mathematical sciences and their application to real-world problems.

This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1972.

The Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability represents a monumental effort in advancing the discourse on probability theory and statistical applications across a wide range of scientific disciplines. Held every five years, the Berkeley Symposia have established themselves as crucial platforms for stimulating research and fostering collaboration among scholars from around the world. The Fifth Symposium, extending over two sessions in 1965 and organized by renowned statisticians Lucien Le Cam and Jerzy Neyman, brought together leading minds to discuss both theoretical and applied aspects of probability and statistics.

Divided into multiple volumes, this extensive compilation reflects the breadth of topics covered, from foundational statistical theory and probability to specialized applications in astronomy, meteorology, biology, medicine, and even artificial weather modification. The inclusion of interdisciplinary research highlights the growing impact of probabilistic and statistical methodologies beyond pure mathematics, influencing fields as diverse as genetics, epidemiology, and climate science. The symposium's Proceedings provide not only a record of cutting-edge research at the time but also a valuable resource for contemporary statisticians and scientists seeking to understand the evolution of these crucial disciplines.

This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1967.

Dieses Buch hat sich seit der zweiten Auflage zu einem vielbeachteten Klassiker im deutschsprachigen Raum entwickelt. Wesentliche Konzepte der Numerik von Differentialgleichungen werden an einfacheren Problemen wie Drei-Term-Rekursionen oder numerische Quadratur dargestellt. Neu hinzu gekommen in der 5. Auflage sind insbesondere aktuelle Forschungsergebnisse zur effizienten Polynominterpolation, die auf Trefethen et al. zurückgehen.

Numerical analysis deals with the development and analysis of algorithms for scientific computing, and is in itself a very important part of mathematics, which has become more and more prevalent across the mathematical spectrum. This book is an introduction to numerical methods for solving linear and nonlinear systems of equations as well as ordinary and partial differential equations, and for approximating curves, functions, and integrals.

The organization of the material is presented as follows:

This introductory chapter I represents a theoretical analysis of the computational algorithms for a numerical solution of the basic equations in continuum mechanics. In this chapter, the general requirements for computational grids, discretization, and iterative methods for black-box software are examined. Finally, a concept of a two-grid algorithm for (de-)coupled solving multidimensional non-linear (initial-)boundary value problems in continuum mechanics (multiphysics simulation) in complex domains is presented.

Chapter II contains descriptions of the sequential Robust Multigrid Technique which is developed as a general-purpose solver in black-box codes. This chapter presents the main components of the Robust Multigrid Technique (RMT) used in the two-grid algorithm (Chapter I) to compute the auxiliary (structured) grid correction. This includes the generation of multigrid structures, computation of index mapping, and integral evaluation. Finite volume discretization on the multigrid structures will be explained by studying a 1D linear model problem. In addition, the algorithmic complexity of RMT and black-box optimization of the problem-dependent components of RMT are analysed.

Chapter III provides a description of parallel RMT. This chapter introduces parallel RMT-based algorithms for solving the boundary value problems and initial-boundary value problems in unified manner. Section 1 presents a comparative analysis of the parallel RMT and the sequential V-cycle. Sections 2 and 3 present a geometric and an algebraic parallelism of RMT, i.e. parallelization of the smoothing iterations on the coarse and the levels. A parallel multigrid cycle will be considered in Section 4. A parallel RMT for the time-dependent problems is given in Section 5. Finally, the basic properties of parallel RMT will be summarized in Section 6.

Theoretical aspects of the used algorithms for solving multidimensional problems are discussed in Chapters IV. This chapter contains the theoretical aspects of the algorithms used for the numerical solving of the resulting system of linear algebraic equations obtained from discrete multidimensional (initial-)boundary value problems.

Die Mathematik ist in allen Bereichen der modernen Gesellschaft ein wichtiges Hilfsmittel, um bei der Lösung der vielfältigen Probleme mit zu wirken, insbesondere in Technik, Naturwissenschaften und Ökonomie. Die Mitwirkung mathematischer Methoden bei der Lösung praktischer Aufgaben in diesen Bereichen vollzieht sich in folgenden Schritten:

• Präzise Formulierung des Problems: die Aufgabenstellung muss klar erkennbar werden. Es ist kenntlich zu machen, welche Schwerpunkte zu setzen sind.
• Erarbeitung eines mathematischen Modells des Problems: aus der technischen, ökonomischen oder sonstigen Aufgabenstellung ist ein für die mathematische Behandlung geeignetes Modell zu entwickeln und einer mathematischen Behandlung zugänglich zu machen.
• Numerische Berechnung der Lösung: es sind geeignete Algorithmen und die erforderlichen Rechenhilfsmittel auszuwählen. Es sind Anforderungen an den numerischen Algorithmus bezüglich Rechenzeit und Rechengenauigkeit zu formulieren, sowie Abschätzungen der zu erwartenden Rechenfehler durchzuführen.
• Auswertung der erzielten Resultate: Es ist zu entscheiden, ob eine ausreichende Lösung des praktischen Problems erzielt worden ist oder ob eine Verbesserung des mathematischen Modells erfolgen muss.

All relevant implementation aspects of finite element methods are discussed in this book. The focus is on algorithms and data structures as well as on their concrete implementation. Theory is covered only as far as it gives insight into the construction of algorithms. In the exercises, a complete FE-solver for stationary 2D problems is implemented in Matlab/Octave.

Contents:
Finite Element Fundamentals
Grids and Finite Elements
Assembly
Solvers
Error Estimation
Mesh Refinement
Multigrid
Elastomechanics
Fluid Mechanics
Grid Data Structure
Function Reference

Currently, nonstandard analysis is barely considered in university teaching. The author argues that nonstandard analysis is valuable not only for teaching, but also for understanding standard analysis and mathematics itself. An axiomatic approach wich pays attention to different language levels (for example, in the distinction between sums of ones and the natural numbers of the theory) leads naturally to a nonstandard theory. For motivation historical ideas of Leibniz can be taken up. The book contains an elaborated concept that follows this approach and is suitable, for example, as a basis for a lecture-supplementary course.

The monograph part presents all major approaches to nonstandard analysis and discusses logical, model-theoretic, and set-theoretic investigations to reveal possible mathematical reasons that may lead to reservations about nonstandard analysis. Also various foundational positions as well as ontological, epistemological, and application-related issues are addressed. It turns out that the one-sided preference for standard analysis is justified neither from a didactic, mathematical nor philosophical point of view.

Thus, the book is especially valuable for students and instructors of analysis who are also interested in the foundations of their subject.

Bessel functions have the peculiarity of being functions of two independent variables: argument and order. They have been studied extensively because of their countless applications, but the vast majority of available literature is devoted to the case of fixed order, variable argument. This two-volume work explores the opposite case.

This volume collects tabulations of the first, second, and third derivatives with respect to the order.