Inequalities


Author: G. H. Hardy,J. E. Littlewood,G. Pólya
Publisher: Cambridge University Press
ISBN: 9780521358804
Category: Mathematics
Page: 324
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This classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics. First published in 1934, it presents clearly and lucidly both the statement and proof of all the standard inequalities of analysis. The authors were well-known for their powers of exposition and made this subject accessible to a wide audience of mathematicians.

Inequalities: A Journey into Linear Analysis


Author: D. J. H. Garling
Publisher: Cambridge University Press
ISBN: 1139465147
Category: Mathematics
Page: N.A
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This book contains a wealth of inequalities used in linear analysis, and explains in detail how they are used. The book begins with Cauchy's inequality and ends with Grothendieck's inequality, in between one finds the Loomis-Whitney inequality, maximal inequalities, inequalities of Hardy and of Hilbert, hypercontractive and logarithmic Sobolev inequalities, Beckner's inequality, and many, many more. The inequalities are used to obtain properties of function spaces, linear operators between them, and of special classes of operators such as absolutely summing operators. This textbook complements and fills out standard treatments, providing many diverse applications: for example, the Lebesgue decomposition theorem and the Lebesgue density theorem, the Hilbert transform and other singular integral operators, the martingale convergence theorem, eigenvalue distributions, Lidskii's trace formula, Mercer's theorem and Littlewood's 4/3 theorem. It will broaden the knowledge of postgraduate and research students, and should also appeal to their teachers, and all who work in linear analysis.

Isoperimetric Inequalities

Differential Geometric and Analytic Perspectives
Author: Isaac Chavel
Publisher: Cambridge University Press
ISBN: 9780521802673
Category: Mathematics
Page: 268
View: 3546

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This advanced introduction emphasizes the variety of ideas, techniques, and applications of the subject.

Inequalities

Theorems, Techniques and Selected Problems
Author: Zdravko Cvetkovski
Publisher: Springer Science & Business Media
ISBN: 3642237924
Category: Mathematics
Page: 444
View: 8241

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This work is about inequalities which play an important role in mathematical Olympiads. It contains 175 solved problems in the form of exercises and, in addition, 310 solved problems. The book also covers the theoretical background of the most important theorems and techniques required for solving inequalities. It is written for all middle and high-school students, as well as for graduate and undergraduate students. School teachers and trainers for mathematical competitions will also gain benefit from this book.

An Introduction to Harmonic Analysis


Author: Yitzhak Katznelson
Publisher: Cambridge University Press
ISBN: 9780521543590
Category: Mathematics
Page: 314
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First published in 1968, An Introduction to Harmonic Analysis has firmly established itself as a classic text and a favorite for students and experts alike. Professor Katznelson starts the book with an exposition of classical Fourier series. The aim is to demonstrate the central ideas of harmonic analysis in a concrete setting, and to provide a stock of examples to foster a clear understanding of the theory. Once these ideas are established, the author goes on to show that the scope of harmonic analysis extends far beyond the setting of the circle group, and he opens the door to other contexts by considering Fourier transforms on the real line as well as a brief look at Fourier analysis on locally compact abelian groups. This new edition has been revised by the author, to include several new sections and a new appendix.

Analytic Inequalities


Author: Nicholas D. Kazarinoff
Publisher: Courier Corporation
ISBN: 0486798178
Category: Mathematics
Page: 96
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Mathematical analysis is largely a systematic study and exploration of inequalities — but for students the study of inequalities often remains a foreign country, difficult of access. This book is a passport to that country, offering a background on inequalities that will prepare undergraduates (and even high school students) to cope with the concepts of continuity, derivative, and integral. Beginning with explanations of the algebra of inequalities and conditional inequalities, the text introduces a pair of ancient theorems and their applications. Explorations of inequalities and calculus cover the number e, examples from the calculus, and approximations by polynomials. The final sections present modern theorems, including Bernstein's proof of the Weierstrass approximation theorem and the Cauchy, Bunyakovskii, Hölder, and Minkowski inequalities. Numerous figures, problems, and examples appear throughout the book, offering students an excellent foundation for further studies of calculus.

The Cauchy-Schwarz Master Class

An Introduction to the Art of Mathematical Inequalities
Author: J. Michael Steele
Publisher: Cambridge University Press
ISBN: 9780521546775
Category: Mathematics
Page: 306
View: 8667

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This lively, problem-oriented text, first published in 2004, is designed to coach readers toward mastery of the most fundamental mathematical inequalities. With the Cauchy-Schwarz inequality as the initial guide, the reader is led through a sequence of fascinating problems whose solutions are presented as they might have been discovered - either by one of history's famous mathematicians or by the reader. The problems emphasize beauty and surprise, but along the way readers will find systematic coverage of the geometry of squares, convexity, the ladder of power means, majorization, Schur convexity, exponential sums, and the inequalities of Hölder, Hilbert, and Hardy. The text is accessible to anyone who knows calculus and who cares about solving problems. It is well suited to self-study, directed study, or as a supplement to courses in analysis, probability, and combinatorics.

Advanced Problems in Mathematics

Preparing for University
Author: Stephen Siklos
Publisher: Open Book Publishers
ISBN: 1783741449
Category: Mathematics
Page: 186
View: 2212

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This book is intended to help students prepare for entrance examinations in mathematics and scientific subjects, including STEP (Sixth Term Examination Papers). STEP examinations are used by Cambridge colleges as the basis for conditional offers in mathematics and sometimes in other mathematics-related subjects. They are also used by Warwick University, and many other mathematics departments recommend that their applicants practice on past papers to become accustomed to university-style mathematics. Advanced Problems in Mathematics is recommended as preparation for any undergraduate mathematics course, even for students who do not plan to take the Sixth Term Examination Paper. The questions analysed in this book are all based on recent STEP questions selected to address the syllabus for Papers I and II, which is the A-level core (i.e. C1 to C4) with a few additions. Each question is followed by a comment and a full solution. The comments direct the reader’s attention to key points and put the question in its true mathematical context. The solutions point students to the methodology required to address advanced mathematical problems critically and independently. This book is a must read for any student wishing to apply to scientific subjects at university level and for anybody interested in advanced mathematics.

Probability with Martingales


Author: David Williams
Publisher: Cambridge University Press
ISBN: 1139642987
Category: Mathematics
Page: N.A
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Probability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. This book is a modern, lively and rigorous account which has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's Strong Law of Large Numbers and the Three-Series Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions. A distinguishing feature is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals; and it assumes certain key results from measure theory in the main text. These measure-theoretic results are proved in full in appendices, so that the book is completely self-contained. The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital rôle. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction.

A Course of Modern Analysis


Author: E. T. Whittaker,G. N. Watson
Publisher: Cambridge University Press
ISBN: 1107268583
Category: Mathematics
Page: N.A
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This classic text is known to and used by thousands of mathematicians and students of mathematics throughout the world. It gives an introduction to the general theory of infinite processes and of analytic functions together with an account of the principal transcendental functions.

Analytic Methods for Diophantine Equations and Diophantine Inequalities


Author: H. Davenport
Publisher: Cambridge University Press
ISBN: 9781139441230
Category: Mathematics
Page: N.A
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Harold Davenport was one of the truly great mathematicians of the twentieth century. Based on lectures he gave at the University of Michigan in the early 1960s, this book is concerned with the use of analytic methods in the study of integer solutions to Diophantine equations and Diophantine inequalities. It provides an excellent introduction to a timeless area of number theory that is still as widely researched today as it was when the book originally appeared. The three main themes of the book are Waring's problem and the representation of integers by diagonal forms, the solubility in integers of systems of forms in many variables, and the solubility in integers of diagonal inequalities. For the second edition of the book a comprehensive foreword has been added in which three prominent authorities describe the modern context and recent developments. A thorough bibliography has also been added.

Divergent Series


Author: Godfrey Harold Hardy
Publisher: American Mathematical Soc.
ISBN: 0821826492
Category: Mathematics
Page: 396
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Review of the original edition: This is an inspiring textbook for students who know the theory of functions of real and complex variables and wish further knowledge of mathematical analysis. There are no problems displayed and labelled ``problems,'' but one who follows all of the arguments and calculations of the text will find use for his ingenuity and pencil. The book deals with interesting and important problems and topics in many fields of mathematical analysis, to an extent very much greater than that indicated by the titles of the chapters. It is, of course, an indispensable handbook for those interested in divergent series. It assembles a considerable part of the theory of divergent series, which has previously existed only in periodical literature. Hardy has greatly simplified and improved many theories, theorems and proofs. In addition, numerous acknowledgements show that the book incorporates many previously unpublished results and improvements of old results, communicated to Hardy by his colleagues and by others interested in the book. --Mathematical Reviews

Mathematical Miniatures


Author: Svetoslav Savchev,Titu Andreescu
Publisher: MAA
ISBN: 9780883856451
Category: Mathematics
Page: 223
View: 6495

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Problems illustrating important mathematical techniques with solutions and accompanying essays.

Ramsey Theory on the Integers

Second Edition
Author: Bruce M. Landman, Aaron Robertson
Publisher: American Mathematical Soc.
ISBN: 0821898671
Category: Mathematics
Page: 384
View: 2207

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Ramsey theory is the study of the structure of mathematical objects that is preserved under partitions. In its full generality, Ramsey theory is quite powerful, but can quickly become complicated. By limiting the focus of this book to Ramsey theory applied to the set of integers, the authors have produced a gentle, but meaningful, introduction to an important and enticing branch of modern mathematics. Ramsey Theory on the Integers offers students a glimpse into the world of mathematical research and the opportunity for them to begin pondering unsolved problems. For this new edition, several sections have been added and others have been significantly updated. Among the newly introduced topics are: rainbow Ramsey theory, an "inequality" version of Schur's theorem, monochromatic solutions of recurrence relations, Ramsey results involving both sums and products, monochromatic sets avoiding certain differences, Ramsey properties for polynomial progressions, generalizations of the Erdős-Ginzberg-Ziv theorem, and the number of arithmetic progressions under arbitrary colorings. Many new results and proofs have been added, most of which were not known when the first edition was published. Furthermore, the book's tables, exercises, lists of open research problems, and bibliography have all been significantly updated. This innovative book also provides the first cohesive study of Ramsey theory on the integers. It contains perhaps the most substantial account of solved and unsolved problems in this blossoming subject. This breakthrough book will engage students, teachers, and researchers alike.

A User's Guide to Measure Theoretic Probability


Author: David Pollard
Publisher: Cambridge University Press
ISBN: 9780521002899
Category: Mathematics
Page: 351
View: 6090

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This book grew from a one-semester course offered for many years to a mixed audience of graduate and undergraduate students who have not had the luxury of taking a course in measure theory. The core of the book covers the basic topics of independence, conditioning, martingales, convergence in distribution, and Fourier transforms. In addition there are numerous sections treating topics traditionally thought of as more advanced, such as coupling and the KMT strong approximation, option pricing via the equivalent martingale measure, and the isoperimetric inequality for Gaussian processes. The book is not just a presentation of mathematical theory, but is also a discussion of why that theory takes its current form. It will be a secure starting point for anyone who needs to invoke rigorous probabilistic arguments and understand what they mean.

Corruption, Inequality, and the Rule of Law

The Bulging Pocket Makes the Easy Life
Author: Eric M. Uslaner
Publisher: Cambridge University Press
ISBN: 1139472372
Category: Political Science
Page: 345
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Corruption flouts rules of fairness and gives some people advantages that others don't have. Corruption is persistent; there is little evidence that countries can escape the curse of corruption easily - or at all. Instead of focusing on institutional reform, in this book Eric M. Uslaner suggests that the roots of corruption lie in economic and legal inequality, low levels of generalized trust (which are not readily changed), and poor policy choices (which may be more likely to change). Economic inequality provides a fertile breeding ground for corruption, which, in turn, leads to further inequalities. Just as corruption is persistent, inequality and trust do not change much over time, according to Uslaner's cross-national aggregate analyses. He argues that high inequality leads to low trust and high corruption, and then to more inequality - an inequality trap - and identifies direct linkages between inequality and trust in surveys of the mass public and elites in transition countries.

The New Economics of Inequality and Redistribution


Author: Samuel Bowles
Publisher: Cambridge University Press
ISBN: 1107014034
Category: Business & Economics
Page: 188
View: 6213

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Incorporating the latest results from behavioral economics and microeconomic theory, Samuel Bowles argues that conventional economics has mistakenly presented inequality as the price of progress. In place of this view, he offers a novel and optimistic account of the possibility of a more just economy.

Weapons of Math Destruction

How Big Data Increases Inequality and Threatens Democracy
Author: Cathy O'Neil
Publisher: Crown Books
ISBN: 0553418815
Category: Big data
Page: 259
View: 8354

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"A former Wall Street quantitative analyst sounds an alarm on mathematical modeling, a pervasive new force in society that threatens to undermine democracy and widen inequality, "--NoveList.

Aspects of Sobolev-Type Inequalities


Author: L. Saloff-Coste
Publisher: Cambridge University Press
ISBN: 9780521006071
Category: Mathematics
Page: 190
View: 1920

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Focusing on Poincaré, Nash and other Sobolev-type inequalities and their applications to the Laplace and heat diffusion equations on Riemannian manifolds, this text is an advanced graduate book that will also suit researchers.