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With a fresh geometric approach that incorporates more than 250illustrations, this textbook sets itself apart from all others inadvanced calculus.  Besides the classicalcapstones--the change of variables formula, implicit and inversefunction theorems, the integral theorems of Gauss and Stokes--the texttreats other important topics in differential analysis, such as Morse'slemma and the Poincaré lemma.  The ideas behindmost topics can be understood with just two or three variables.  The bookincorporates modern computational tools to give visualization realpower.  Using 2D and 3D graphics, the book offers newinsights into fundamental elements of the calculus of differentiablemaps.  The geometric themecontinues with an analysis of the physical meaning of the divergence andthe curl at a level of detail not found in other advanced calculusbooks.  This is atextbook for undergraduates and graduate students in mathematics, thephysical sciences, and economics.  Prerequisitesare an introduction to linear algebra and multivariable calculus.  There is enough material for a year-long course onadvanced calculus and for a variety of semester courses--includingtopics in geometry.  The measured pace ofthe book, with its extensive examples and illustrations, make itespecially suitable for independent study. ... Read more

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Designed as a reference work and also as a graduate-leveltextbook, this volume presents an up-to-date and comprehensive accountof the theories and applications of the various methods and techniquesused in dealing with problems involving closed-form evaluations of(and representations of the Riemann Zeta function at positive integerarguments as) numerous families of series associated with the RiemannZeta function, the Hurwitz Zeta function, and their extensions andgeneralizations such as Lerch's transcendent (or theHurwitz-Lerch Zeta function).
Audience: This book is intended for professional mathematiciansand graduate students in mathematical sciences (both pure andapplied). ... Read more

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This is the first treatment in book format of proof-theoretic transformations - known as proof interpretations - that focuses on applications to ordinary mathematics. It covers both the necessary logical machinery behind the proof interpretations that are used in recent applications as well as – via extended case studies – carrying out some of these applications in full detail. This subject has historical roots in the 1950s. This book for the first time tells the whole story.

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This book presents the authors' personal selection of topics in multivariate statistical analysis with emphasis on tools and techniques. Topics included range from definitions of multivariate moments, multivariate distributions, asymptotic distributions of commonly used statistics and density approximations to a modern treatment of multivariate linear models. The theory used is based on matrix algebra and linear spaces and applies lattice theory in a systematic way. Many of the results are obtained by utilizing matrix derivatives which in turn are built up from the Kronecker product and vec-operator. The matrix normal, Wishart and elliptical distributions are studied in detail. In particular, several moment relations are given. Together with the derivatives of density functions, formulae are presented for density approximations, generalizing classical Edgeworth expansions. The asymptotic distributions of many commonly used statistics are also derived. In the final part of the book the Growth Curve model and its various extensions are studied.

The book will be of particular interest to researchers but could also be appropriate as a text-book for graduate courses on multivariate analysis or matrix algebra.

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Topics in Hyperplane Arrangements, Polytopes and Box-Splines brings together many areas of research that focus on methods to compute the number of integral points in suitable families or variable polytopes. The topics introduced expand upon differential and difference equations, approximation theory, cohomology, and module theory.

This book, written by two distinguished authors, engages a broad audience by proving the a strong foudation. This book may be used in the classroom setting as well as a reference for researchers.

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A collection of articles on various aspects of q-series and special functions dedicated to Mizan Rahman. It also includes an article by Askey, Ismail, and Koelink on Rahman’s mathematical contributions and how they influenced the recent upsurge in the subject.

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The main aim of the present work is to present a number of selected results on Ostrowski-type integral inequalities. Results for univariate and multivariate real functions and their natural applications in the error analysis of numerical quadratures for both simple and multiple integrals as well as for the Riemann-Stieltjes integral are given. Topics dealt with include generalisations of the Ostrowski inequality and its applications; integral inequalities for n-times differentiable mappings; three-point quadrature rules; product-branched Peano kernels and numerical integration; Ostrowski-type inequalities for multiple integrals; results for double integrals based on an Ostrowski-type inequality; product inequalities and weighted quadrature; and some inequalities for the Riemann-Stieltjes integral.
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This textbook is an application-oriented introduction to the theory of distributions, a powerful tool used in mathematical analysis. The treatment emphasizes applications that relate distributions to linear partial differential equations and Fourier analysis problems found in mechanics, optics, quantum mechanics, quantum field theory, and signal analysis. The book is motivated by many exercises, hints, and solutions that guide the reader along a path requiring only a minimal mathematical background.

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This textbook introduces sparse and redundant representations with a focus on applications in signal and image processing. The theoretical and numerical foundations are tackled before the applications are discussed. Mathematical modeling for signal sources is discussed along with how to use the proper model for tasks such as denoising, restoration, separation, interpolation and extrapolation, compression, sampling, analysis and synthesis, detection, recognition, and more. The presentation is elegant and engaging.

Sparse and Redundant Representations is intended for graduate students in applied mathematics and electrical engineering, as well as applied mathematicians, engineers, and researchers who are active in the fields of signal and image processing.

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This book presents theoretical results relevant to Edgeworth and saddlepoint expansions to densities and distribution functions. It provides examples of their application in some simple and a few complicated settings, along with numerical, as well as asymptotic assessments of their accuracy. Variants on these expansions, including much of modern likelihood theory, are discussed and applications to lattice distributions are extensively treated. This book is intended primarily for advanced graduate students and researchers in the field needing a collection of core results in a uniform notation, with bibliographical references to further examples and applications. It assumes familiarity with real analysis, vector calculus, and complex analysis. This third edition features an expansion of the material on the Blackwell approximation and an expansion of the discussions on saddlepoint approximation. ... Read more

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Textbook on multivariate integration, with primary emphasis on dimensionally reducing expansion, or DRE, which has become an important mathematical tool in the computerized age. Offers good balance between theory and application, exposition covering the history of the subject, and new results from a number of research fields. DLC: Numerical integration. ... Read more

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The well known transport laws of Navier-Stokes and Fourier fail for the simulation of processes on lengthscales in the order of the mean free path of a particle that is when the Knudsen number is not small enough. Thus, the proper simulation of flows in rarefied gases requires a more detailed description.

This book discusses classical and modern methods to derive macroscopic transport equations for rarefied gases from the Boltzmann equation, for small and moderate Knudsen numbers, i.e. at and above the Navier-Stokes-Fourier level. The main methods discussed are the classical Chapman-Enskog and Grad approaches, as well as the new order of magnitude method, which avoids the short-comings of the classical methods, but retains their benefits. The relations between the various methods are carefully examined, and the resulting equations are compared and tested for a variety of standard problems.

The book develops the topic starting from the basic description of an ideal gas, over the derivation of the Boltzmann equation, towards the various methods for deriving macroscopic transport equations, and the test problems which include stability of the equations, shock waves, and Couette flow.

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From the reviews: "A good introduction to a subject important for its capacity to circumvent theoretical and practical obstacles, and therefore particularly prized in the applications of mathematics. The book presents a balanced view of the methods and their usefulness: integrals on the real line and in the complex plane which arise in different contexts, and solutions of differential equations not expressible as integrals. Murray includes both historical remarks and references to sources or other more complete treatments. More useful as a guide for self-study than as a reference work, it is accessible to any upperclass mathematics undergraduate. Some exercises and a short bibliography included. Even with E.T. Copson's Asymptotic Expansions or N.G. de Bruijn's Asymptotic Methods in Analysis (1958), any academic library would do well to have this excellent introduction." (S. Puckette, University of the South) #Choice Sept. 1984#1 ... Read more

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This unique book provides a concise presentation of many of the fundamental strategies for calculating approximations to the oscillatory solutions of "truly nonlinear" (TNL) oscillator equations. The volume gives a general overview of the author's work on harmonic balance, iteration and combined linearization-averaging methods. However, full discussions are also presented on parameter expansion procedures and a first-order averaging technique for TNL oscillators. The calculational basis of each method is clarified by applying them to a set of standard TNL oscillator equations. This allows a direct comparison to be made among the various methods.

The book is self-contained and therefore suitable for both classroom use and self-study by students and professionals who desire to learn, understand, and apply these technique to the field of nonlinear oscillations. ... Read more

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Intended for a wide range of readers, this book covers the main ideas of convex analysis and approximation theory. The author discusses the sources of these two trends in mathematical analysis, develops the main concepts and results, and mentions some beautiful theorems. The relationship of convex analysis to optimization problems, to the calculus of variations, to optimal control and to geometry is considered, and the evolution of the ideas underlying approximation theory, from its origins to the present day, is discussed. The book is addressed both to students who want to acquaint themselves with these trends and to lecturers in mathematical analysis, optimization and numerical methods, as well as to researchers in these fields who would like to tackle the topic as a whole and seek inspiration for its further development. ... Read more

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This book is written for beginning graduate students in applied mathematics, science, and engineering, and is appropriate as a one-year course in applied mathematical techniques (although I have never been able to cover all of this material in one year). We assume that the students have studied at an introductory undergraduate level material on linear algebra, ordinary and partial differential equations, and complex variables. The emphasis of the book is a working, systematic understanding of classical techniques in a modern context. Along the way, students are exposed to models from a variety of disciplines. It is hoped that this course will prepare students for further study of modern techniques and in-depth modeling in their own specific discipline. ... Read more

Customer Reviews (4)

Broad, shallow and uneven
When an author invites a reader to purchase and read his book, the assumption one makes is that the book provides a new or a different insight into a familiar or difficult subject or alternatively, that it opens new vistas and level of comprehension into the subject matter. First, writing a book on advanced applied mathematics is quite an endeavor and thus, I applaud the author for embarking in such an effort.However, the book overall provides panorama, but very little detail and even less new vistas into critical areas. Thus, there does not seem to be a compelling reason for this book to have been published, because most of the topics have already been adequately covered in other reputable sources. Clearly, the author set out with a lofty and challenging goal; to present clearly and compactly the theory underlying a very broad range of complex mathematical tools. In fairness needed a fuller and more coherent development to accomplish this, that is simply not present in the book.However, there are some bright spots, for which the book deserves commendation and at least two stars. The best written chapters, albeit too short, are Chapters 1-3 and Chapters 6 on Complex variables. But the treatment is neither compelling or original.In chapter 6 the author introduces the reader to orthogonal polynomials and special functions, but leaves the reader with such a limited view of these topics as to make the effort at teasing the reader's curiosity quite frustrating.The book is generous with bibliography at the end of each chapter and a great source for further research. Nevertheless, it would be an improvement if in future editions the author points out to the reader which pages of the suggested readings the reader should be focused on for the chapter in which it is being cited as a reference. The author engages in serial titillation by introducing the reader to exotic topics such as Sobolev spaces but quickly retreats from this topic by stating that many of the operators in Sobolev space lack "self-adjointness" and thus of not much practical use for applied mathematics. (See page 66-67). Similarly, he also introduces the reader to Mellin and Hankel transforms but, does not explain where, why and how they are used.More critically it does not explain when one of these particular transforms arebetter suited than Laplace and Fourier Transforms. So what is the point of bringing the subject to the attention of the reader in such an incomplete manner?The hurried and desultory treatment of Green functions is too compressed beyond belieffor it to be of any significant value to the reader. The same goes for the discusion of Sturm-Liouville systems.The physical examples provided throughout the book are also too condensed and for the most part difficult to follow and relate to the theory the author just discussed that ostensibly equips the reader to understand the example.

The discussion of Laplace and in particular Fourier transforms seems not to logically and seamlessly emerge from the previous discussion of the theory of operators.A few words on the proofs provided throughout the book; some of are so abbreviated or lack explanation that it is a frustrating experience to attempt to follow to their conclusion.

In sum, I commend the author to either expand the length of chapters or consider choosing less topics and develop them more fully and coherently so the book attains depth and eminence.

Every chapter can be a book
This is a book with very broad coverage, which is also its strength. It happens to be my textbook for a graduate course when I worked toward my phd degree in engineering. I had a hard time through it, although I had a high grade in this course. Even though, I don't think this book is well written.

As my title said, every chapter in this book can be written as a book. I am not exaggerating. It's true. Since the author condenses so much material into a small book, there is sacrifice certainly. For example, the coverage is not thorough for a specific topic, the proof is too short or even not given...etc. To overcome this, usually you need to consult other books to get a more clear understanding. In addition, I had a bad experience that some examples have nothing to do with what he has just said above. That drove me crazy. The author mentions in the preface that he intends to seek a balance on application and theory in this book. I don't think he gets the job done.

In spite of so many drawbacks, there are still bright sides. For example, the broad coverage is good for me to get into or acquainted with some topics. When you are not understanding what he said, try to find a reference book. That usually helps. In addition, the motivation part is good. He tells you why you need this, why that way doesn't work...etc.

Simply speaking, this book is kind of opening a door for you and then you need to work out the rest not depending on it but by yourself.

If you need a book for self study, this is not the one. If not for that course, I wouldn't force myself reading through it. I think Logan's "Applied Mathematics" is much better for self study at the expense of narrower coverage than this one. This one is better for course use accompanying instructor's good supplement.

Excellently organized book.
This book presents various mathematical principles in an organization I have not seen before.It starts with the idea of a transformation, then goes on to relate eigenvalues and eigenvectors to general spectral theory,explain how the need for closed function spaces naturally leads to Lebesgueintegration (I know about Lebesgue integration before but I didn't know whyit was needed), and show how the definition of certain inverse operatorsleads to distribution theory.This is a very natural way of organizingthese principles.While other books, such as Strang's Intro to AppliedMathematics and Rudin's Real & Complex Analysis, provide you with onemathematical "toy" after another (Fourier series, Lebesgueintegration, etc.), Keener's book tells you why you need the toy beforegiving it to you.

Am I stupid or this book is too advance???
This book doesn't clearify anything for you. No examples, No further explanation. It only keeps introduce various theories to you. It can compact theories that other books take 2-3 pages to explain it into 5lines! I think you can imagine. Obviously, this book is not suitable to beyour first book (not the second also). Buy it if you are sure that you aresmart enough to understand it! ... Read more

Product Description
This digital document is a journal article from Journal of Econometrics, published by Elsevier in 2006. The article is delivered in HTML format and is available in your Amazon.com Media Library immediately after purchase. You can view it with any web browser.

Description:
The finite sample behavior is analyzed of particular least squares (LS) and a range of (generalized) method of moments (MM) estimators in panel data models with individual effects and both a lagged dependent variable regressor and another explanatory variable. The latter may be affected by lagged feedbacks from the dependent variable too. Asymptotic expansions indicate how the order of magnitude of bias of MM estimators tends to increase with the number of moment conditions exploited. They also provide analytic evidence on how the bias of the various estimators depends on the feedbacks and on other model characteristics such as prominence of individual effects and correlation between observed and unobserved heterogeneity. Simulation results corroborate the theoretical findings and reveal that in small samples of models with dynamic feedbacks none of the techniques examined dominates regarding bias and mean squared error over all parametrizations examined. ... Read more

Product Description
This digital document is a journal article from Journal of Econometrics, published by Elsevier in 2006. The article is delivered in HTML format and is available in your Amazon.com Media Library immediately after purchase. You can view it with any web browser.

Description:
The finite sample behavior is analyzed of particular least squares (LS) and a range of (generalized) method of moments (MM) estimators in panel data models with individual effects and both a lagged dependent variable regressor and another explanatory variable. The latter may be affected by lagged feedbacks from the dependent variable too. Asymptotic expansions indicate how the order of magnitude of bias of MM estimators tends to increase with the number of moment conditions exploited. They also provide analytic evidence on how the bias of the various estimators depends on the feedbacks and on other model characteristics such as prominence of individual effects and correlation between observed and unobserved heterogeneity. Simulation results corroborate the theoretical findings and reveal that in small samples of models with dynamic feedbacks none of the techniques examined dominates regarding bias and mean squared error over all parametrizations examined. ... Read more