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Parallelism in Matrix Computations
by Efstratios Gallopoulos Bernard Philippe Ahmed H. SamehThis book is primarily intended as a research monograph that could also be used in graduate courses for the design of parallel algorithms in matrix computations. It assumes general but not extensive knowledge of numerical linear algebra, parallel architectures, and parallel programming paradigms. The book consists of four parts: (I) Basics; (II) Dense and Special Matrix Computations; (III) Sparse Matrix Computations; and (IV) Matrix functions and characteristics. Part I deals with parallel programming paradigms and fundamental kernels, including reordering schemes for sparse matrices. Part II is devoted to dense matrix computations such as parallel algorithms for solving linear systems, linear least squares, the symmetric algebraic eigenvalue problem, and the singular-value decomposition. It also deals with the development of parallel algorithms for special linear systems such as banded ,Vandermonde ,Toeplitz ,and block Toeplitz systems. Part III addresses sparse matrix computations: (a) the development of parallel iterative linear system solvers with emphasis on scalable preconditioners, (b) parallel schemes for obtaining a few of the extreme eigenpairs or those contained in a given interval in the spectrum of a standard or generalized symmetric eigenvalue problem, and (c) parallel methods for computing a few of the extreme singular triplets. Part IV focuses on the development of parallel algorithms for matrix functions and special characteristics such as the matrix pseudospectrum and the determinant. The book also reviews the theoretical and practical background necessary when designing these algorithms and includes an extensive bibliography that will be useful to researchers and students alike. The book brings together many existing algorithms for the fundamental matrix computations that have a proven track record of efficient implementation in terms of data locality and data transfer on state-of-the-art systems, as well as several algorithms that are presented for the first time, focusing on the opportunities for parallelism and algorithm robustness.
Parameter Advising for Multiple Sequence Alignment
by Dan DeBlasio John KececiogluThis book develops a new approach called parameter advising for finding a parameter setting for a sequence aligner that yields a quality alignment of a given set of input sequences. In this framework, a parameter advisor is a procedure that automatically chooses a parameter setting for the input, and has two main ingredients: (a) the set of parameter choices considered by the advisor, and (b) an estimator of alignment accuracy used to rank alignments produced by the aligner. On coupling a parameter advisor with an aligner, once the advisor is trained in a learning phase, the user simply inputs sequences to align, and receives an output alignment from the aligner, where the advisor has automatically selected the parameter setting. The chapters first lay out the foundations of parameter advising, and then cover applications and extensions of advising. The content * examines formulations of parameter advising and their computational complexity, * develops methods for learning good accuracy estimators, * presents approximation algorithms for finding good sets of parameter choices, and * assesses software implementations of advising that perform well on real biological data. Also explored are applications of parameter advising to * adaptive local realignment, where advising is performed on local regions of the sequences to automatically adapt to varying mutation rates, and * ensemble alignment, where advising is applied to an ensemble of aligners to effectively yield a new aligner of higher quality than the individual aligners in the ensemble. The book concludes by offering future directions in advising research.
Parameter Estimation in Reliability and Life Span Models
by A Clifford Cohen Betty Jones WhittenOffers an applications-oriented treatment of parameter estimation from both complete and censored samples; contains notations, simplified formats for estimates, graphical techniques, and numerous tables and charts allowing users to calculate estimates and analyze sample data quickly and easily. Anno
Parameter Redundancy and Identifiability (Chapman & Hall/CRC Interdisciplinary Statistics)
by Diana ColeStatistical and mathematical models are defined by parameters that describe different characteristics of those models. Ideally it would be possible to find parameter estimates for every parameter in that model, but, in some cases, this is not possible. For example, two parameters that only ever appear in the model as a product could not be estimated individually; only the product can be estimated. Such a model is said to be parameter redundant, or the parameters are described as non-identifiable. This book explains why parameter redundancy and non-identifiability is a problem and the different methods that can be used for detection, including in a Bayesian context. Key features of this book: Detailed discussion of the problems caused by parameter redundancy and non-identifiability Explanation of the different general methods for detecting parameter redundancy and non-identifiability, including symbolic algebra and numerical methods Chapter on Bayesian identifiability Throughout illustrative examples are used to clearly demonstrate each problem and method. Maple and R code are available for these examples More in-depth focus on the areas of discrete and continuous state-space models and ecological statistics, including methods that have been specifically developed for each of these areas This book is designed to make parameter redundancy and non-identifiability accessible and understandable to a wide audience from masters and PhD students to researchers, from mathematicians and statisticians to practitioners using mathematical or statistical models.
The Parameterization Method for Invariant Manifolds
by Àlex Haro Marta Canadell Jordi-Lluis Figueras Alejandro Luque Josep Maria MondeloThismonograph presents some theoretical and computational aspects of theparameterization method for invariant manifolds, focusing on the followingcontexts: invariant manifolds associated with fixed points, invariant tori inquasi-periodically forced systems, invariant tori in Hamiltonian systems andnormally hyperbolic invariant manifolds. This book provides algorithms ofcomputation and some practical details of their implementation. The methodologyis illustrated with 12 detailed examples, many of them well known in theliterature of numerical computation in dynamical systems. A public version of the software used for some of the examples is available online. Thebook is aimed at mathematicians, scientists and engineers interested in thetheory and applications of computational dynamical systems.
Parameterized and Exact Computation
by Marek Cygan Pinar HeggernesThis book constitutes the thoroughly refereed post-conference proceedings of the 9th International Symposium on Parameterized and Exact Computation, IPEC 2014, in Wroclaw, Poland, in September 2014. The 27 revised full papers presented together with one invited paper were carefully reviewed and selected from 42 submissions. The topics addressed cover research in all aspects of parameterized/exact algorithms and complexity including but are not limited to new techniques for the design and analysis of parameterized and exact algorithms, fixed-parameter tractability results; parameterized complexity theory, relationship between parameterized complexity and traditional complexity classifications; applications of parameterized and exact exponential-time computation; and implementation issues of parameterized and exact exponential-time algorithms.
Parameterized Complexity in the Polynomial Hierarchy: Extending Parameterized Complexity Theory to Higher Levels of the Hierarchy (Lecture Notes in Computer Science #11880)
by Ronald de HaanParameterized Complexity in the Polynomial Hierarchy was co-recipient of the E.W. Beth Dissertation Prize 2017 for outstanding dissertations in the fields of logic, language, and information. This work extends the theory of parameterized complexity to higher levels of the Polynomial Hierarchy (PH). For problems at higher levels of the PH, a promising solving approach is to develop fixed-parameter tractable reductions to SAT, and to subsequently use a SAT solving algorithm to solve the problem. In this dissertation, a theoretical toolbox is developed that can be used to classify in which cases this is possible. The use of this toolbox is illustrated by applying it to analyze a wide range of problems from various areas of computer science and artificial intelligence.
Parametric and Nonparametric Inference for Statistical Dynamic Shape Analysis with Applications
by Chiara Brombin Luigi Salmaso Lara Fontanella Luigi Ippoliti Caterina FusilliThis book considers specific inferential issues arising from the analysis of dynamic shapes with the attempt to solve the problems at hand using probability models and nonparametric tests. The models are simple to understand and interpret and provide a useful tool to describe the global dynamics of the landmark configurations. However, because of the non-Euclidean nature of shape spaces, distributions in shape spaces are not straightforward to obtain. The book explores the use of the Gaussian distribution in the configuration space, with similarity transformations integrated out. Specifically, it works with the offset-normal shape distribution as a probability model for statistical inference on a sample of a temporal sequence of landmark configurations. This enables inference for Gaussian processes from configurations onto the shape space. The book is divided in two parts, with the first three chapters covering material on the offset-normal shape distribution, and the remaining chapters covering the theory of NonParametric Combination (NPC) tests. The chapters offer a collection of applications which are bound together by the theme of this book. They refer to the analysis of data from the FG-NET (Face and Gesture Recognition Research Network) database with facial expressions. For these data, it may be desirable to provide a description of the dynamics of the expressions, or testing whether there is a difference between the dynamics of two facial expressions or testing which of the landmarks are more informative in explaining the pattern of an expression.
Parametric and Nonparametric Statistics for Sample Surveys and Customer Satisfaction Data (SpringerBriefs in Statistics)
by Rosa Arboretti Arne Bathke Stefano Bonnini Paolo Bordignon Eleonora Carrozzo Livio Corain Luigi SalmasoThis book deals with problems related to the evaluation of customer satisfaction in very different contexts and ways. Often satisfaction about a product or service is investigated through suitable surveys which try to capture the satisfaction about several partial aspects which characterize the perceived quality of that product or service. This book presents a series of statistical techniques adopted to analyze data from real situations where customer satisfaction surveys were performed.The aim is to give a simple guide of the variety of analysis that can be performed when analyzing data from sample surveys: starting from latent variable models to heterogeneity in satisfaction and also introducing some testing methods for comparing different customers. The book also discusses the construction of composite indicators including different benchmarks of satisfaction. Finally, some rank-based procedures for analyzing survey data are also shown.
A Parametric Approach to Nonparametric Statistics (Springer Series in the Data Sciences)
by Philip L. H. Yu Mayer AlvoThis book demonstrates that nonparametric statistics can be taught from a parametric point of view. As a result, one can exploit various parametric tools such as the use of the likelihood function, penalized likelihood and score functions to not only derive well-known tests but to also go beyond and make use of Bayesian methods to analyze ranking data. The book bridges the gap between parametric and nonparametric statistics and presents the best practices of the former while enjoying the robustness properties of the latter. This book can be used in a graduate course in nonparametrics, with parts being accessible to senior undergraduates. In addition, the book will be of wide interest to statisticians and researchers in applied fields.
Parents Without Papers: The Progress and Pitfalls of Mexican American Integration
by Frank D. Bean James D. Bachmeier Susan K. BrownFor several decades, Mexican immigrants in the United States have outnumbered those from any other country. Though the economy increasingly needs their labor, many remain unauthorized. In Parents Without Papers, immigration scholars Frank D. Bean, Susan K. Brown, and James D. Bachmeier document the extent to which the outsider status of these newcomers inflicts multiple hardships on their children and grandchildren. Parents Without Papers provides both a general conceptualization of immigrant integration and an in-depth examination of the Mexican American case. The authors draw upon unique retrospective data to shed light on three generations of integration. They show in particular that the “membership exclusion” experienced by unauthorized Mexican immigrants—that is, their fear of deportation, lack of civil rights, and poor access to good jobs—hinders the education of their children, even those who are U.S.-born. Moreover, they find that children are hampered not by the unauthorized entry of parents itself but rather by the long-term inability of parents, especially mothers, to acquire green cards. When unauthorized parents attain legal status, the disadvantages of the second generation begin to disappear. These second-generation men and women achieve schooling on par with those whose parents come legally. By the third generation, socioeconomic levels for women equal or surpass those of native white women. But men reach parity only through greater labor-force participation and longer working hours, results consistent with the idea that their integration is delayed by working-class imperatives to support their families rather than attend college. An innovative analysis of the transmission of advantage and disadvantage among Mexican Americans, Parents Without Papers presents a powerful case for immigration policy reforms that provide not only realistic levels of legal less-skilled migration but also attainable pathways to legalization. Such measures, combined with affordable access to college, are more important than ever for the integration of vulnerable Mexican immigrants and their descendants.
Paris-Princeton Lectures on Mathematical Finance 2013
by Ronnie Sircar Paolo Guasoni Vicky Henderson Fred Espen Benth Philip Protter Konstantinos Manolarakis Colm Nee Johannes Muhle-Karbe Dan CrisanThe current volume presents four chapters touching on some of the most important and modern areas of research in Mathematical Finance: asset price bubbles (by Philip Protter); energy markets (by Fred Espen Benth); investment under transaction costs (by Paolo Guasoni and Johannes Muhle-Karbe); and numerical methods for solving stochastic equations (by Dan Crisan, K. Manolarakis and C. Nee).The Paris-Princeton Lecture Notes on Mathematical Finance, of which this is the fifth volume, publish cutting-edge research in self-contained, expository articles from renowned specialists. The aim is to produce a series of articles that can serve as an introductory reference source for research in the field.
Parkettierungen der Ebene: Von Escher Über Möbius Zu Penrose
by Ehrhard BehrendsZiel des Buches ist das Studium von Symmetrien und Parkettierungen, die Künstler und Mathematiker schon seit langer Zeit interessieren. Berühmte Beispiele sind die von den Arabern in der Alhambra geschaffenen Werke und die Bilder des holländischen Malers Maurits Escher. Die Mathematiker haben sich erst im 19. Jahrhundert des Themas intensiv angenommen. Dabei führt die Visualisierung der mathematischen Zusammenhänge zu sehr ansprechenden Bildern. Drei Ansätze werden in diesem Buch beschrieben. In Teil I wird dargestellt, dass es 17 prinzipiell verschiedene Möglichkeiten von Parkettierungen der Ebene gibt, die so genannten "Ebenen Kristallgruppen". Ergänzend dazu werden Ideen von Harald Heesch beschrieben, der zeigte, wie diese theoretischen Ergebnisse praktisch umgesetzt werden können: Er gab einen Katalog von 28 Verfahren an, die man selbst - sozusagen auf den Spuren von Escher - kreativ zur Schaffung künstlerisch anspruchsvoller Parkettierungen verwenden kann. Bei den entsprechenden Untersuchungen für die komplexe Ebene in Teil II werden Bewegungen durch bijektive holomorphe Abbildungen ersetzt. Das führt in die Theorie der Gruppen von Möbiustransformationen: Kleinsche Gruppen, Schottkygruppen usw. Dort gibt es auch interessante Verbindungen zur hyperbolischen Geometrie. Schließlich wird in Teil III noch ein dritter Aspekt des Themas behandelt, die Penroseparkettierungen. Dabei geht es um Ergebnisse aus den siebziger Jahren, als erstmals einfach zu beschreibende und beweisbar nichtperiodische Parkettierungen der Ebene angegeben wurden.
Partial Differential Equation Methods for Image Inpainting
by Carola-Bibiane SchönliebThis book is concerned with digital image processing techniques that use partial differential equations (PDE) for the task of image 'inpainting,' an artistic term for virtual image restoration or interpolation, whereby missing or occluded parts in images are completed based on information provided by intact parts. Computer graphic designers, artists and photographers have long used manual inpainting to restore damaged paintings or manipulate photographs. Today, mathematicians apply powerful methods based on PDE to automate this task. This book introduces the mathematical concept of PDE for virtual image restoration. It gives the full picture, from the first modelling steps originating in Gestalt theory and arts restoration to the analysis of resulting PDE models, numerical realisation and real world application. This broad approach also gives insight into functional analysis, variational calculus, optimisation and numerical analysis and will appeal to researchers and graduate students in mathematics with an interest in image processing and mathematical analysis.
Partial Differential Equations: Theory, Control and Approximation
by Philippe G. Ciarlet Tatsien Li Yvon MadayThis book collects papers mainly presented at the "International Conference on Partial Differential Equations: Theory, Control and Approximation" (May 28 to June 1, 2012 in Shanghai) in honor of the scientific legacy of the exceptional mathematician Jacques-Louis Lions. The contributors are leading experts from all over the world, including members of the Academies of Sciences in France, the USA and China etc. , and their papers cover key fields of research, e. g. partial differential equations, control theory and numerical analysis, that Jacques-Louis Lions created or contributed so much to establishing.
Partial Differential Equations: An Introduction (Dover Books on Mathematics #No. 42)
by David ColtonIntended for a college senior or first-year graduate-level course in partial differential equations, this text offers students in mathematics, engineering, and the applied sciences a solid foundation for advanced studies in mathematics. Classical topics presented in a modern context include coverage of integral equations and basic scattering theory. This complete and accessible treatment includes a variety of examples of inverse problems arising from improperly posed applications. Exercises at the ends of chapters, many with answers, offer a clear progression in developing an understanding of this essential area of mathematics. 1988 edition.
Partial Differential Equations: Modeling, Analysis and Numerical Approximation
by Hervé Le Dret Brigitte LucquinThis book is devoted to the study ofpartial differential equation problems both from the theoretical and numericalpoints of view. After presenting modeling aspects, it develops the theoretical analysis of partialdifferential equation problems for the three main classes of partialdifferential equations:elliptic, parabolic and hyperbolic. Several numerical approximation methodsadapted to each of these examples are analyzed: finite difference, finite elementand finite volumes methods, andthey are illustrated using numerical simulation results. Although parts of the book are accessible toBachelor students in mathematics or engineering, it is primarily aimed at Masters students in applied mathematics orcomputational engineering. The emphasis is on mathematical detail and rigor forthe analysis of both continuous and discrete problems.
Partial Differential Equations
by Prof. Avner FriedmanThis three-part treatment of partial differential equations focuses on elliptic and evolution equations. Largely self-contained, it concludes with a series of independent topics directly related to the methods and results of the preceding sections that helps introduce readers to advanced topics for further study. Geared toward graduate and postgraduate students of mathematics, this volume also constitutes a valuable reference for mathematicians and mathematical theorists.Starting with the theory of elliptic equations and the solution of the Dirichlet problem, the text develops the theory of weak derivatives, proves various inequalities and imbedding problems, and derives smoothness theorems. Part Two concerns evolution equations in Banach space and develops the theory of semigroups. It solves the initial-boundary value problem for parabolic equations and covers backward uniqueness, asymptotic behavior, and lower bounds at infinity. The final section includes independent topics directly related to the methods and results of the previous material, including the analyticity of solutions of elliptic and parabolic equations, asymptotic behavior of solutions of elliptic equations near infinity, and problems in the theory of control in Banach space.
Partial Differential Equations: Analytical Methods and Applications (Textbooks in Mathematics)
by Victor Henner Tatyana Belozerova Alexander NepomnyashchyPartial Differential Equations: Analytical Methods and Applications covers all the basic topics of a Partial Differential Equations (PDE) course for undergraduate students or a beginners’ course for graduate students. It provides qualitative physical explanation of mathematical results while maintaining the expected level of it rigor. This text introduces and promotes practice of necessary problem-solving skills. The presentation is concise and friendly to the reader. The "teaching-by-examples" approach provides numerous carefully chosen examples that guide step-by-step learning of concepts and techniques. Fourier series, Sturm-Liouville problem, Fourier transform, and Laplace transform are included. The book’s level of presentation and structure is well suited for use in engineering, physics and applied mathematics courses. Highlights: Offers a complete first course on PDEs The text’s flexible structure promotes varied syllabi for courses Written with a teach-by-example approach which offers numerous examples and applications Includes additional topics such as the Sturm-Liouville problem, Fourier and Laplace transforms, and special functions The text’s graphical material makes excellent use of modern software packages Features numerous examples and applications which are suitable for readers studying the subject remotely or independently
Partial Differential Equations
by T. Hillen I. E. Leonard H. Van RoesselUniquely provides fully solved problems for linear partial differential equations and boundary value problems Partial Differential Equations: Theory and Completely Solved Problems utilizes real-world physical models alongside essential theoretical concepts. With extensive examples, the book guides readers through the use of Partial Differential Equations (PDEs) for successfully solving and modeling phenomena in engineering, biology, and the applied sciences. The book focuses exclusively on linear PDEs and how they can be solved using the separation of variables technique. The authors begin by describing functions and their partial derivatives while also defining the concepts of elliptic, parabolic, and hyperbolic PDEs. Following an introduction to basic theory, subsequent chapters explore key topics including: * Classification of second-order linear PDEs * Derivation of heat, wave, and Laplace's equations * Fourier series * Separation of variables * Sturm-Liouville theory * Fourier transforms Each chapter concludes with summaries that outline key concepts. Readers are provided the opportunity to test their comprehension of the presented material through numerous problems, ranked by their level of complexity, and a related website features supplemental data and resources. Extensively class-tested to ensure an accessible presentation, Partial Differential Equations is an excellent book for engineering, mathematics, and applied science courses on the topic at the upper-undergraduate and graduate levels.
Partial Differential Equations
by Jürgen JostThis book offers an ideal graduate-level introduction to the theory of partial differential equations. The first part of the book describes the basic mathematical problems and structures associated with elliptic, parabolic, and hyperbolic partial differential equations, and explores the connections between these fundamental types. Aspects of Brownian motion or pattern formation processes are also presented. The second part focuses on existence schemes and develops estimates for solutions of elliptic equations, such as Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. In particular, the reader will learn the basic techniques underlying current research in elliptic partial differential equations. This revised and expanded third edition is enhanced with many additional examples that will help motivate the reader. New features include a reorganized and extended chapter on hyperbolic equations, as well as a new chapter on the relations between different types of partial differential equations, including first-order hyperbolic systems, Langevin and Fokker-Planck equations, viscosity solutions for elliptic PDEs, and much more. Also, the new edition contains additional material on systems of elliptic partial differential equations, and it explains in more detail how the Harnack inequality can be used for the regularity of solutions.
Partial Differential Equations
by Rachel Levy Michael ShearerThis textbook provides beginning graduate students and advanced undergraduates with an accessible introduction to the rich subject of partial differential equations (PDEs). It presents a rigorous and clear explanation of the more elementary theoretical aspects of PDEs, while also drawing connections to deeper analysis and applications. The book serves as a needed bridge between basic undergraduate texts and more advanced books that require a significant background in functional analysis.Topics include first order equations and the method of characteristics, second order linear equations, wave and heat equations, Laplace and Poisson equations, and separation of variables. The book also covers fundamental solutions, Green's functions and distributions, beginning functional analysis applied to elliptic PDEs, traveling wave solutions of selected parabolic PDEs, and scalar conservation laws and systems of hyperbolic PDEs.Provides an accessible yet rigorous introduction to partial differential equationsDraws connections to advanced topics in analysisCovers applications to continuum mechanicsAn electronic solutions manual is available only to professorsAn online illustration package is available to professors
Partial Differential Equations: Theory and Numerical Solution (Chapman And Hall/crc Research Notes In Mathematics Ser. #406)
by J. NecasAs a satellite conference of the 1998 International Mathematical Congress and part of the celebration of the 650th anniversary of Charles University, the Partial Differential Equations Theory and Numerical Solution conference was held in Prague in August, 1998. With its rich scientific program, the conference provided an opportunity for almost 200 participants to gather and discuss emerging directions and recent developments in partial differential equations (PDEs).This volume comprises the Proceedings of that conference. In it, leading specialists in partial differential equations, calculus of variations, and numerical analysis present up-to-date results, applications, and advances in numerical methods in their fields. Conference organizers chose the contributors to bring together the scientists best able to present a complex view of problems, starting from the modeling, passing through the mathematical treatment, and ending with numerical realization. The applications discussed include fluid dynamics, semiconductor technology, image analysis, motion analysis, and optimal control. The importance and quantity of research carried out around the world in this field makes it imperative for researchers, applied mathematicians, physicists and engineers to keep up with the latest developments. With its panel of international contributors and survey of the recent ramifications of theory, applications, and numerical methods, Partial Differential Equations: Theory and Numerical Solution provides a convenient means to that end.
Partial Differential Equations: An Introduction (Mathematical Engineering, Manufacturing, and Management Sciences)
by Nita H. Shah Mrudul Y. JaniDifferential equations play a noticeable role in engineering, physics, economics, and other disciplines. They permit us to model changing forms in both mathematical and physical problems. These equations are precisely used when a deterministic relation containing some continuously varying quantities and their rates of change in space and/or time is recognized or postulated. This book is intended to provide a straightforward introduction to the concept of partial differential equations. It provides a diversity of numerical examples framed to nurture the intellectual level of scholars. It includes enough examples to provide students with a clear concept and also offers short questions for comprehension. Construction of real-life problems is considered in the last chapter along with applications. Research scholars and students working in the fields of engineering, physics, and different branches of mathematics need to learn the concepts of partial differential equations to solve their problems. This book will serve their needs instead of having to use more complex books that contain more concepts than needed.
Partial Differential Equations: Topics in Fourier Analysis
by M. W. WongPartial Differential Equations: Topics in Fourier Analysis, Second Edition explains how to use the Fourier transform and heuristic methods to obtain significant insight into the solutions of standard PDE models. It shows how this powerful approach is valuable in getting plausible answers that can then be justified by modern analysis. Using Fourier analysis, the text constructs explicit formulas for solving PDEs governed by canonical operators related to the Laplacian on the Euclidean space. After presenting background material, it focuses on: Second-order equations governed by the Laplacian on Rn; the Hermite operator and corresponding equation; and the sub-Laplacian on the Heisenberg group Designed for a one-semester course, this text provides a bridge between the standard PDE course for undergraduate students in science and engineering and the PDE course for graduate students in mathematics who are pursuing a research career in analysis. Through its coverage of fundamental examples of PDEs, the book prepares students for studying more advanced topics such as pseudo-differential operators. It also helps them appreciate PDEs as beautiful structures in analysis, rather than a bunch of isolated ad-hoc techniques. New to the Second Edition Three brand new chapters covering several topics in analysis not explored in the first edition Complete revision of the text to correct errors, remove redundancies, and update outdated material Expanded references and bibliography New and revised exercises.