New books by subject
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Mathematics  Concordia University Libraries Recent Acquisitions
Titles in the call number range QA 1  QA 73, QA 90  QA 699 (Mathematics) that were added to the Concordia University Libraries collection in the last 60 days.

Ergodic theory and negative curvature : CIRM JeanMorlet Chair, Fall 2013 / Boris Hasselblatt, editorQA313
These lectures center on ergodicity of the (WeilPetersson) geodesic flow on a nonpositively curved space whose points are negatively curved metrics on surfaces. The subject matter is anchored by a selfcontained introduction to hyperbolic dynamics and ergodic theory and complemented by lectures that show the deep connections of geodesic flows in negative curvature with on one hand Diophantine approximation and on the other hand with the ergodic theory of horocycle flows. 
Waves in Continuous Media / by Place of publication not identified Gavrilyuk, N.I. Makarenko, S.V. SukhininQA380Starting with the basic notions and facts of the mathematical theory of waves illustrated by numerous examples, exercises, and methods of solving typical problems Chapters 1 & 2 show e.g. how to recognize the hyperbolicity property, find characteristics, Riemann invariants and conservation laws for quasilinear systems of equations, construct and analyze solutions with weak or strong discontinuities, and how to investigate equations with dispersion and to construct travelling wave solutions for models reducible to nonlinear evolution equations.
Chapter 3 deals with surface and internal waves in an incompressible fluid. The efficiency of mathematical methods is demonstrated on a hierarchy of approximate submodels generated from the Euler equations of homogeneous and nonhomogeneous fluids.
The selfcontained presentations of the material is complemented by 200+ problems of different level of difficulty, numerous illustrations, and bibliographical recommendations. 
Spatial interaction models : facility location using game theory / Lina Mallozzi, Egidio D'Amato, Panos M. Pardalos, editorsQA269Facility location theory develops the idea of locating one or more facilities by optimizing suitable criteria such as minimizing transportation cost, or capturing the largest market share. The contributions in this book focus an approach to facility location theory through game theoretical tools highlighting situations where a location decision is faced by several decision makers and leading to a game theoretical framework in noncooperative and cooperative methods. Models and methods regarding the facility location via game theory are explored and applications are illustrated through economics, engineering, and physics. Mathematicians, engineers, economists and computer scientists working in theory, applications and computational aspects of facility location problems using game theory will find this book useful.

Geometry over nonclosed fields / Fedor Bogomolov, Brendan Hassett, Yuri Tschinkel, editorsQA564
Based on the Simons Symposia held in 2015, the proceedings in this volume focus on rational curves on higherdimensional algebraic varieties and applications of the theory of curves to arithmetic problems. There has been significant progress in this field with major new results, which have given new impetus to the study of rational curves and spaces of rational curves on K3 surfaces and their higherdimensional generalizations. One main recent insight the book covers is the idea that the geometry of rational curves is tightly coupled to properties of derived categories of sheaves on K3 surfaces. The implementation of this idea led to proofs of longstanding conjectures concerning birational properties of holomorphic symplectic varieties, which in turn should yield new theorems in arithmetic. This proceedings volume covers these new insights in detail. 
Mathematical practitioners and the transformation of natural knowledge in early modern Europe / Lesley B. Cormack, Steven A. Walton, John A. Schuster, editorsQA27.E85This book argues that we can only understand transformations of nature studies in the Scientific Revolution if we take seriously the interaction between practitioners (those who know by doing) and scholars (those who know by thinking). These are not in opposition, however. Theory and practice are end points on a continuum, with some participants interested only in the practical, others only in the theoretical, and most in the murky intellectual world in between. It is this borderland where influence, appropriation, and collaboration have the potential to lead to new methods, new subjects of enquiry, and new social structures of natural philosophy and science.
The case for connection between theory and practice can be most persuasively drawn in the area of mathematics, which is the focus of this book. Practical mathematics was a growing field in early modern Europe and these essays are organised into three parts which contribute to the debate about the role of mathematical practice in the Scientific Revolution. First, they demonstrate the variability of the identity of practical mathematicians, and of the practices involved in their activities in early modern Europe. Second, readers are invited to consider what practical mathematics looked like and that although practical mathematical knowledge was transmitted and circulated in a wide variety of ways, participants were able to recognize them all as practical mathematics. Third, the authors show how differences and nuances in practical mathematics typically depended on the different contexts in which it was practiced: social, cultural, political, and economic particularities matter. Historians of science, especially those interested in the Scientific Revolution period and the history of mathematics will find this book and its groundbreaking approach of particular interest.

Study guide for Single variable calculus : early transcendentals, eighth edition / Richard St. AndreQA 303.2 S772 2016 Guide
For each section of the text, the Study Guide provides students with a brief introduction, a short list of concepts to master, as well as summary and focus questions with explained answers. The Study Guide also contains "Technology Plus" questions, and multiplechoice "On Your Own" examstyle questions. 
An introduction to partial differential equations / Daniel J. ArrigoQA374 .A773 2018eb
This book is an introduction to methods for solving partial differential equations (PDEs). After the introduction of the main four PDEs that could be considered the cornerstone of Applied Mathematics, the reader is introduced to a variety of PDEs that come from a variety of fields in the Natural Sciences and Engineering and is a springboard into this wonderful subject. The chapters include the following topics: Firstorder PDEs, Secondorder PDEs, Fourier Series, Separation of Variables, and the Fourier Transform. The reader is guided through these chapters where techniques for solving first and secondorder PDEs are introduced. Each chapter ends with a series of exercises illustrating the material presented in each chapter. The book can be used as a textbook for any introductory course in PDEs typically found in both science and engineering programs and has been used at the University of Central Arkansas for over ten years. 
Numerical integration of space fractional partial differential equations. Younes Salehi, William E. SchiesserQA372 .S2662 2018eb
Partial differential equations (PDEs) are one of the most used widely forms of mathematics in science and engineering. PDEs can have partial derivatives with respect to (1) an initial value variable, typically time, and (2) boundary value variables, typically spatial variables. Therefore, two fractional PDEs can be considered, (1) fractional in time (TFPDEs), and (2) fractional in space (SFPDEs). The two volumes are directed to the development and use of SFPDEs, with the discussion divided as: *Vol 1: Introduction to Algorithms and Computer Coding in R *Vol 2: Applications from Classical Integer PDEs. Various definitions of space fractional derivatives have been proposed. We focus on the Caputo derivative, with occasional reference to the RiemannLiouville derivative. In the second volume, the emphasis is on applications of SFPDEs developed mainly through the extension of classical integer PDEs to SFPDEs. The example applications are: *Fractional diffusion equation with Dirichlet, Neumann and Robin boundary conditions *FisherKolmogorov SFPDE *Burgers SFPDE *FokkerPlanck SFPDE *BurgersHuxley SFPDE *FitzhughNagumo SFPDE. These SFPDEs were selected because they are integer first order in time and integer second order in space. The variation in the spatial derivative from order two (parabolic) to order one (first order hyperbolic) demonstrates the effect of the spatial fractional order ?? with 1 ≤ ?? ≤ 2. All of the example SFPDEs are one dimensional in Cartesian coordinates. Extensions to higher dimensions and other coordinate systems, in principle, follow from the examples in this second volume. The examples start with a statement of the integer PDEs that are then extended to SFPDEs. The format of each chapter is the same as in the first volume. The R routines can be downloaded and executed on a modest computer (R is readily available from the Internet). 
Numerical integration of space fractional partial differential equations. Younes Salehi, William E. SchiesserQA372 .S266 2018eb
Partial differential equations (PDEs) are one of the most used widely forms of mathematics in science and engineering. PDEs can have partial derivatives with respect to (1) an initial value variable, typically time, and (2) boundary value variables, typically spatial variables. Therefore, two fractional PDEs can be considered, (1) fractional in time (TFPDEs), and (2) fractional in space (SFPDEs). The two volumes are directed to the development and use of SFPDEs, with the discussion divided as: Vol 1: Introduction to Algorithms and Computer Coding in R Vol 2: Applications from Classical Integer PDEs. Various definitions of space fractional derivatives have been proposed. We focus on the Caputo derivative, with occasional reference to the RiemannLiouville derivative. Partial differential equations (PDEs) are one of the most used widely forms of mathematics in science and engineering. PDEs can have partial derivatives with respect to (1) an initial value variable, typically time, and (2) boundary value variables, typically spatial variables. Therefore, two fractional PDEs can be considered, (1) fractional in time (TFPDEs), and (2) fractional in space (SFPDEs). The two volumes are directed to the development and use of SFPDEs, with the discussion divided as: Vol 1: Introduction to Algorithms and Computer Coding in R Vol 2: Applications from Classical Integer PDEs. Various definitions of space fractional derivatives have been proposed. We focus on the Caputo derivative, with occasional reference to the RiemannLiouville derivative. The Caputo derivative is defined as a convolution integral. Thus, rather than being local (with a value at a particular point in space), the Caputo derivative is nonlocal (it is based on an integration in space), which is one of the reasons that it has properties not shared by integer derivatives. A principal objective of the two volumes is to provide the reader with a set of documented R routines that are discussed in detail, and can be downloaded and executed without having to first study the details of the relevant numerical analysis and then code a set of routines. In the first volume, the emphasis is on basic concepts of SFPDEs and the associated numerical algorithms. The presentation is not as formal mathematics, e.g., theorems and proofs. Rather, the presentation is by examples of SFPDEs, including a detailed discussion of the algorithms for computing numerical solutions to SFPDEs and a detailed explanation of the associated source code. 
Diffusiondriven wavelet design for shape analysis / Tingbo Hou, Hong QinQA403.3 .H68 2015From Design Methods and Generation Schemes to StateoftheArt Applications
Wavelets are powerful tools for functional analysis and geometry processing, enabling researchers to determine the structure of data and analyze 3D shapes. Suitable for researchers in computer graphics, computer vision, visualization, medical imaging, and geometric modeling as well as graduate and senior undergraduate students in computer science, DiffusionDriven Wavelet Design for Shape Analysispresents recent research results in wavelet designs on 3D shapes and their applications in shape analysis. It explains how to apply the design methods to various types of 3D data, such as polygonal meshes, point clouds, manifolds, and volumetric images.Extensions of Wavelet Generation on Volumetric and Manifold Data
The first part of the book introduces design methods of wavelets on manifold data, incorporating interdisciplinary knowledge from differential geometry, functional analysis, Fourier transform, spectral graph theory, and stochastic processes. The authors show how wavelets are purely determined by the shape geometry and how wavelet transforms are computed as inner products of wavelet kernels and input functions.Wavelets for Solving Computer Graphics Problems
The second part presents applications in shape analysis/representation. The book looks at wavelets as spectral tools for geometry processing with filters in a joint spacefrequency domain and examines wavelets as detail extractors for shape feature definition and detection. Going beyond these fundamental applications, the book also covers middle and highlevel applications, including shape matching, shape registration, and shape retrieval.EasytoUnderstand Implementations and Algorithms
Unlike many other wavelet books, this one does not involve complicated mathematics. Instead, the book uses simplified formulations and illustrative examples to explain deep theories. Code and other materials are available on a supplementary website. 
Modeling shallow water flows using the discontinuous Galerkin method / Abdul A. Khan, Wencong LaiQA372 .K43 2014Replacing the Traditional Physical Model Approach
Computational models offer promise in improving the modeling of shallow water flows. As new techniques are considered, the process continues to change and evolve. Modeling Shallow Water Flows Using the Discontinuous Galerkin Methodexamines a technique that focuses on hyperbolic conservation laws and includes onedimensional and twodimensional shallow water flows and pollutant transports.
Combines the Advantages of Finite Volume and Finite Element Methods
This book explores the discontinuous Galerkin (DG) method, also known as the discontinuous finite element method, in depth. It introduces the DG method and its application to shallow water flows, as well as background information for implementing and applying this method for natural rivers. It considers dambreak problems, shock wave problems, and flows in different regimes (subcritical, supercritical, and transcritical).
Readily Adaptable to the Real World
While the DG method has been widely used in the fields of science and engineering, its use for hydraulics has so far been limited to simple cases. The book compares numerical results with laboratory experiments and field data, and includes a set of tests that can be used for a wide range of applications.
Provides stepbystep implementation details Presents the different forms in which the shallow water flow equations can be written Places emphasis on the details and modifications required to apply the scheme to realworld flow problemsThis text enables readers to readily understand and develop an efficient computer simulation model that can be used to model flow, contaminant transport, and other aspects in rivers and coastal environments. It is an ideal resource for practicing environmental engineers and researchers in the area of computational hydraulics and fluid dynamics, and graduate students in computational hydraulics.

Monte Carlo methods for particle transport / Alireza HaghighatQA298 .H34 2015The Monte Carlo method has become the de facto standard in radiation transport. Although powerful, if not understood and used appropriately, the method can give misleading results.
Monte Carlo Methods for Particle Transport teaches appropriate use of the Monte Carlo method, explaining the method's fundamental concepts as well as its limitations. Concise yet comprehensive, this wellorganized text:
Introduces the particle importance equation and its use for variance reduction Describes general and particletransportspecific variance reduction techniques Presents particle transport eigenvalue issues and methodologies to address these issues Explores advanced formulations based on the author's research activities Discusses parallel processing concepts and factors affecting parallel performanceFeaturing illustrative examples, mathematical derivations, computer algorithms, and homework problems, Monte Carlo Methods for Particle Transport provides nuclear engineers and scientists with a practical guide to the application of the Monte Carlo method.

Regularization, optimization, kernels, and support vector machines / edited by Johan A.K. Suykens, KU Leuven, Belgium, Marco Signoretto, KU Leuven, Belgium, Andreas Argyriou, Ecole Centrale Paris, FranceQA401 .R56 2013Regularization, Optimization, Kernels, and Support Vector Machinesoffers a snapshot of the current state of the art of largescale machine learning, providing a single multidisciplinary source for the latest research and advances in regularization, sparsity, compressed sensing, convex and largescale optimization, kernel methods, and support vector machines. Consisting of 21 chapters authored by leading researchers in machine learning, this comprehensive reference:
Covers the relationship between support vector machines (SVMs) and the Lasso Discusses multilayer SVMs Explores nonparametric feature selection, basis pursuit methods, and robust compressive sensing Describes graphbased regularization methods for single and multitask learning Considers regularized methods for dictionary learning and portfolio selection Addresses nonnegative matrix factorization Examines lowrank matrix and tensorbased models Presents advanced kernel methods for batch and online machine learning, system identification, domain adaptation, and image processing Tackles largescale algorithms including conditional gradient methods, (nonconvex) proximal techniques, and stochastic gradient descentRegularization, Optimization, Kernels, and Support Vector Machines is ideal for researchers in machine learning, pattern recognition, data mining, signal processing, statistical learning, and related areas.

Finite element mesh generation / Daniel S.H. LoQA377 .L574 2015Highlights the Progression of Meshing Technologies and Their Applications
Finite Element Mesh Generationprovides a concise and comprehensive guide to the application of finite element mesh generation over 2D domains, curved surfaces, and 3D space. Organised according to the geometry and dimension of the problem domains, it develops from the basic meshing algorithms to the most advanced schemes to deal with problems with specific requirements such as boundary conformity, adaptive and anisotropic elements, shape qualities, and mesh optimization.
It sets out the fundamentals of popular techniques, including:
Delaunay triangulation Advancingfront (ADF) approach Quadtree/Octree techniques Refinement and optimizationbased strategiesFrom the geometrical and the topological aspects and their associated operations and interrelationships, each approach is vividly described and illustrated with examples. Beyond the algorithms, the book also explores the practice of using metric tensor and surface curvatures for generating anisotropic meshes on parametric space. It presents results from research including 3D anisotropic meshing, mesh generation over unbounded domains, meshing by means of intersection, remeshing by DelaunayADF approach, mesh refinement and optimization, generation of hexahedral meshes, and large scale and parallel meshing, along with innovative unpublished meshing methods. The author provides illustrations of major meshing algorithms, pseudo codes, and programming codes in C++ or FORTRAN.
Geared toward research centers, universities, and engineering companies, Finite Element Mesh Generationdescribes mesh generation methods and fundamental techniques, and also serves as a valuable reference for laymen and experts alike.

Iterative optimization in inverse problems / Charles L. Byrne, University of Massachusetts, Lowell, USAQA297.8 .B974 2014Iterative Optimization in Inverse Problemsbrings together a number of important iterative algorithms for medical imaging, optimization, and statistical estimation. It incorporates recent work that has not appeared in other books and draws on the author's considerable research in the field, including his recently developed class of SUMMA algorithms. Related to sequential unconstrained minimization methods, the SUMMA class includes a wide range of iterative algorithms well known to researchers in various areas, such as statistics and image processing.
Organizing the topics from general to more specific, the book first gives an overview of sequential optimization, the subclasses of auxiliaryfunction methods, and the SUMMA algorithms. The next three chapters present particular examples in more detail, including barrier and penaltyfunction methods, proximal minimization, and forwardbackward splitting. The author also focuses on fixedpoint algorithms for operators on Euclidean space and then extends the discussion to include distance measures other than the usual Euclidean distance. In the final chapters, specific problems illustrate the use of iterative methods previously discussed. Most chapters contain exercises that introduce new ideas and make the book suitable for selfstudy.
Unifying a variety of seemingly disparate algorithms, the book shows how to derive new properties of algorithms by comparing known properties of other algorithms. This unifying approach also helps researchersfrom statisticians working on parameter estimation to image scientists processing scanning data to mathematicians involved in theoretical and applied optimizationdiscover useful related algorithms in areas outside of their expertise.

Sinusoids : theory and technological applications / Prem K. Kythe, University of New Orleans, Louisiana, USAQA403 .K98 2015A Complete Treatment of Current Research Topics in Fourier Transforms and Sinusoids
Sinusoids: Theory and Technological Applicationsexplains how sinusoids and Fourier transforms are used in a variety of application areas, including signal processing, GPS, optics, xray crystallography, radioastronomy, poetry and music as sound waves, and the medical sciences. With more than 200 illustrations, the book discusses electromagnetic force and sychrotron radiation comprising all kinds of waves, including gamma rays, xrays, UV rays, visible light rays, infrared, microwaves, and radio waves. It also covers topics of common interest, such as quasars, pulsars, the Big Bang theory, Olbers' paradox, black holes, Mars mission, and SETI.
The book begins by describing sinusoidswhich are periodic sine or cosine functionsusing wellknown examples from wave theory, including traveling and standing waves, continuous musical rhythms, and the human liver. It next discusses the Fourier series and transform in both continuous and discrete cases and analyzes the Dirichlet kernel and Gibbs phenomenon. The author shows how invertibility and periodicity of Fourier transforms are used in the development of signals and filters, addresses the general concept of communication systems, and explains the functioning of a GPS receiver. The author then covers the theory of Fourier optics, synchrotron light and xray diffraction, the mathematics of radioastronomy, and mathematical structures in poetry and music. The book concludes with a focus on tomography, exploring different types of procedures and modern advances. The appendices make the book as selfcontained as possible.

Robust response surfaces, regression, and positive data analyses / Rabindra Nath DasQA279 .D355 2014Although widely used in science and technology for experimental data generating, modeling, and optimization, the response surface methodology (RSM) has many limitations. Showing how robust response surface methodology (RRSM) can overcome these limitations, Robust Response Surfaces, Regression, and Positive Data Analysespresents RRS designs, along with the relevant regression and positive data analysis techniques. It explains how to use RRSM in experimental designs and regression analysis.
The book addresses problems of RRS designs, such as rotatability, sloperotatability, weak rotatability, and optimality. It describes methods for estimating model parameters as well as positive data analysis techniques. The author illustrates the concepts and methods with real examples of lifetime responses, resistivity, replicated measures, and more.
The range of topics and applications gives the book broad appeal both to theoreticians and practicing professionals. The book helps quality engineers, scientists in any area, medical practitioners, demographers, economists, and statisticians understand the theory and applications of RRSM. It can also be used in a second course on the design of experiments.

Analysis with ultrasmall numbers / Karel Hrbacek, The City College of New York, USA, Olivier Lessmann, Coll?ge Rousseau, Geneva, Switzerland, Richard O'Donovan, CEC Andr?Chavanne, Geneva, SwitzerlandQA303.2 .H73 2015Analysis with Ultrasmall Numberspresents an intuitive treatment of mathematics using ultrasmall numbers. With this modern approach to infinitesimals, proofs become simpler and more focused on the combinatorial heart of arguments, unlike traditional treatments that use epsilondelta methods. Students can fully prove fundamental results, such as the Extreme Value Theorem, from the axioms immediately, without needing to master notions of supremum or compactness.
The book is suitable for a calculus course at the undergraduate or high school level or for selfstudy with an emphasis on nonstandard methods. The first part of the text offers material for an elementary calculus course while the second part covers more advanced calculus topics.
The text provides straightforward definitions of basic concepts, enabling students to form good intuition and actually prove things by themselves. It does not require any additional "black boxes" once the initial axioms have been presented. The text also includes numerous exercises throughout and at the end of each chapter.

Pseudolinear functions and optimization / Shashi Kant Mishra, Balendu Bhooshan UpadhyayQA353.P74 M57 2015Pseudolinear Functions and Optimizationis the first book to focus exclusively on pseudolinear functions, a class of generalized convex functions. It discusses the properties, characterizations, and applications of pseudolinear functions in nonlinear optimization problems.
The book describes the characterizations of solution sets of various optimization problems. It examines multiobjective pseudolinear, multiobjective fractional pseudolinear, static minmax pseudolinear, and static minmax fractional pseudolinear optimization problems and their results. The authors extend these results to locally Lipschitz functions using Clarke subdifferentials. They also present optimality and duality results for h pseudolinear and semiinfinite pseudolinear optimization problems.
The authors go on to explore the relationships between vector variational inequalities and vector optimization problems involving pseudolinear functions. They present characterizations of solution sets of pseudolinear optimization problems on Riemannian manifolds as well as results on pseudolinearity of quadratic fractional functions. The book also extends npseudolinear functions to pseudolinear and npseudolinear fuzzy mappings and characterizations of solution sets of pseudolinear fuzzy optimization problems and npseudolinear fuzzy optimization problems. The text concludes with some applications of pseudolinear optimization problems to hospital management and economics.
This book encompasses nearly all the published literature on the subject along with new results on semiinfinite nonlinear programming problems. It will be useful to readers from mathematical programming, industrial engineering, and operations management.

Significant figures : the lives and work of great mathematicians / Ian StewartQA 28 S74 2017
A celebrated mathematician traces the history of math through the lives and work of twentyfive pioneering mathematicians
In Significant Figures , acclaimed mathematician Ian Stewart introduces the visionaries of mathematics throughout history. Delving into the lives of twentyfive great mathematicians, Stewart examines the roles they played in creating, inventing, and discovering the mathematics we use today. Through these short biographies, we get acquainted with the history of mathematics from Archimedes to Benoit Mandelbrot, and learn about those too often left out of the cannon, such as Muhammad ibn Musa alKhwarizmi (c. 780850), the creator of algebra, and Augusta Ada King (18151852), Countess of Lovelace, the world's first computer programmer.
Tracing the evolution of mathematics over the course of two millennia, Significant Figures will educate and delight aspiring mathematicians and experts alike.

Probability and computing : randomization and probabilistic techniques in algorithms and data analysis / Michael Mitzenmacher, Eli UpfalQA 274 M574 2017
Greatly expanded, this new edition requires only an elementary background in discrete mathematics and offers a comprehensive introduction to the role of randomization and probabilistic techniques in modern computer science. Newly added chapters and sections cover topics including normal distributions, sample complexity, VC dimension, Rademacher complexity, power laws and related distributions, cuckoo hashing, and the Lovasz Local Lemma. Material relevant to machine learning and big data analysis enables students to learn modern techniques and applications. Among the many new exercises and examples are programmingrelated exercises that provide students with excellent training in solving relevant problems. This book provides an indispensable teaching tool to accompany a one or twosemester course for advanced undergraduate students in computer science and applied mathematics. 
Numerical linear algebra : an introduction / Holger WendlandQA 184.2 W46 2018
This selfcontained introduction to numerical linear algebra provides a comprehensive, yet concise, overview of the subject. It includes standard material such as direct methods for solving linear systems and leastsquares problems, error, stability and conditioning, basic iterative methods and the calculation of eigenvalues. Later chapters cover more advanced material, such as Krylov subspace methods, multigrid methods, domain decomposition methods, multipole expansions, hierarchical matrices and compressed sensing. The book provides rigorous mathematical proofs throughout, and gives algorithms in generalpurpose languageindependent form. Requiring only a solid knowledge in linear algebra and basic analysis, this book will be useful for applied mathematicians, engineers, computer scientists, and all those interested in efficiently solving linear problems. 
Reverse mathematics : proofs from the inside out / John StillwellQA 9.25 S75 2018This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics , John Stillwell gives a representative view of this field, emphasizing basic analysisfinding the "right axioms" to prove fundamental theoremsand giving a novel approach to logic.
Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenthcentury project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentiethcentury arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it.
By using a minimum of mathematical logic in a wellmotivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

Enlargement of filtration with finance in view Anna Aksamit, Monique JeanblancQA274This volume presents classical results of the theory of enlargement of filtration. The focus is on the behavior of martingales with respect to the enlarged filtration and related objects. The study is conducted in various contexts including immersion, progressive enlargement with a random time and initial enlargement with a random variable.
The aim of this book is to collect the main mathematical results (with proofs) previously spread among numerous papers, great part of which is only available in French. Many examples and applications to finance, in particular to credit risk modelling and the study of asymmetric information, are provided to illustrate the theory. A detailed summary of further connections and applications is given in bibliographic notes which enables to deepen study of the topic.
This book fills a gap in the literature and serves as a guide for graduate students and researchers interested in the role of information in financial mathematics and in econometric science. A basic knowledge of the general theory of stochastic processes is assumed as a prerequisite.