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The discrepancy method has produced the most fruitful line of attack on a pivotal computer science question: What is the computational power of random bits? It has also played a major role in recent developments in complexity theory. This book tells the story of the discrepancy method in a few succinct independent vignettes. The chapters explore such topics as communication complexity, pseudo-randomness, rapidly mixing Markov chains, points on a sphere, derandomization, convex hulls and Voronoi diagrams, linear programming, geometric sampling and VC-dimension theory, minimum spanning trees, circuit complexity, and multidimensional searching. The mathematical treatment is thorough and self-contained, with minimal prerequisites. More information can be found on the book's home page at http://www.cs.princeton.edu/~chazelle/book.html.

Full length, total contact support with an odor control top fabric that helps reduce heat, friction, and perspiration Firm but flexible contoured neutral arch support that can be customized with snap-on heel posts and snap-on heel lifts for increased comfort, stability, and motion control for those with standard arches Premium dual-layer support cushioning features ShockAbsorb™ Premium Foam that provides bounce back with each step and enhanced comfort from heel to toe while absorbing shock at im

Full length, total contact support with an odor control top fabric that helps reduce heat, friction, and perspiration Firm but flexible contoured neutral arch support that can be customized with snap-on heel posts and snap-on heel lifts for increased comfort, stability, and motion control for those with standard arches Premium dual-layer support cushioning features ShockAbsorb™ Premium Foam that provides bounce back with each step and enhanced comfort from heel to toe while absorbing shock at im

Full length, total contact support with an odor control top fabric that helps reduce heat, friction, and perspiration Firm but flexible contoured neutral arch support that can be customized with snap-on heel posts and snap-on heel lifts for increased comfort, stability, and motion control for those with standard arches Premium dual-layer support cushioning features ShockAbsorb™ Premium Foam that provides bounce back with each step and enhanced comfort from heel to toe while absorbing shock at im

Full length, total contact support with an odor control top fabric that helps reduce heat, friction, and perspiration Firm but flexible contoured neutral arch support that can be customized with snap-on heel posts and snap-on heel lifts for increased comfort, stability, and motion control for those with standard arches Premium dual-layer support cushioning features ShockAbsorb™ Premium Foam that provides bounce back with each step and enhanced comfort from heel to toe while absorbing shock at im

Full length, total contact support with an odor control top fabric that helps reduce heat, friction, and perspiration Firm but flexible contoured neutral arch support that can be customized with snap-on heel posts and snap-on heel lifts for increased comfort, stability, and motion control for those with standard arches Premium dual-layer support cushioning features ShockAbsorb™ Premium Foam that provides bounce back with each step and enhanced comfort from heel to toe while absorbing shock at im

Full length, total contact support with an odor control top fabric that helps reduce heat, friction, and perspiration Firm but flexible contoured neutral arch support that can be customized with snap-on heel posts and snap-on heel lifts for increased comfort, stability, and motion control for those with standard arches Premium dual-layer support cushioning features ShockAbsorb™ Premium Foam that provides bounce back with each step and enhanced comfort from heel to toe while absorbing shock at im

Full length, total contact support with an odor control top fabric that helps reduce heat, friction, and perspiration Firm but flexible contoured neutral arch support that can be customized with snap-on heel posts and snap-on heel lifts for increased comfort, stability, and motion control for those with standard arches Premium dual-layer support cushioning features ShockAbsorb™ Premium Foam that provides bounce back with each step and enhanced comfort from heel to toe while absorbing shock at im

Full length, total contact support with an odor control top fabric that helps reduce heat, friction, and perspiration Firm but flexible contoured neutral arch support that can be customized with snap-on heel posts and snap-on heel lifts for increased comfort, stability, and motion control for those with standard arches Premium dual-layer support cushioning features ShockAbsorb™ Premium Foam that provides bounce back with each step and enhanced comfort from heel to toe while absorbing shock at im

The discrepancy method has produced the most fruitful line of attack on a pivotal computer science question: What is the computational power of random bits? It has also played a major role in recent developments in complexity theory. This book tells the story of the discrepancy method in a few succinct independent vignettes. The chapters explore such topics as communication complexity, pseudo-randomness, rapidly mixing Markov chains, points on a sphere, derandomization, convex hulls and Voronoi diagrams, linear programming, geometric sampling and VC-dimension theory, minimum spanning trees, circuit complexity, and multidimensional searching. The mathematical treatment is thorough and self-contained, with minimal prerequisites. More information can be found on the book's home page at http://www.cs.princeton.edu/~chazelle/book.html.

The discrepancy method has produced the most fruitful line of attack on a pivotal computer science question: What is the computational power of random bits? It has also played a major role in recent developments in complexity theory. This book tells the story of the discrepancy method in a few succinct independent vignettes. The chapters explore such topics as communication complexity, pseudo-randomness, rapidly mixing Markov chains, points on a sphere, derandomization, convex hulls and Voronoi diagrams, linear programming, geometric sampling and VC-dimension theory, minimum spanning trees, circuit complexity, and multidimensional searching. The mathematical treatment is thorough and self-contained, with minimal prerequisites. More information can be found on the book's home page at http://www.cs.princeton.edu/~chazelle/book.html.

The discrepancy method has produced the most fruitful line of attack on a pivotal computer science question: What is the computational power of random bits? It has also played a major role in recent developments in complexity theory. This book tells the story of the discrepancy method in a few succinct independent vignettes. The chapters explore such topics as communication complexity, pseudo-randomness, rapidly mixing Markov chains, points on a sphere, derandomization, convex hulls and Voronoi diagrams, linear programming, geometric sampling and VC-dimension theory, minimum spanning trees, circuit complexity, and multidimensional searching. The mathematical treatment is thorough and self-contained, with minimal prerequisites. More information can be found on the book's home page at http://www.cs.princeton.edu/~chazelle/book.html.

In Alleged Discrepancies of the Bible, John Haley directly addresses questions surrounding common misunderstandings of Scripture. This book is essential for your personal library because it explains questionable passages, helps answer skeptics' questions, and contains valuable resource material. Haley lifts the mystery surrounding the supposed contradictions in the Bible and sheds new light on confusing passages. As you study God's Word and see His purposes unfold in your life, you will experience a new power and love for living!

In Alleged Discrepancies of the Bible, John Haley directly addresses questions surrounding common misunderstandings of Scripture. This book is essential for your personal library because it explains questionable passages, helps answer skeptics' questions, and contains valuable resource material. Haley lifts the mystery surrounding the supposed contradictions in the Bible and sheds new light on confusing passages. As you study God's Word and see His purposes unfold in your life, you will experience a new power and love for living!

The main purpose of this book is to give an overview of the developments during the last 20 years in the theory of uniformly distributed sequences. The authors focus on various aspects such as special sequences, metric theory, geometric concepts of discrepancy, irregularities of distribution, continuous uniform distribution and uniform distribution in discrete spaces. Specific applications are presented in detail: numerical integration, spherical designs, random number generation and mathematical finance. Furthermore over 1000 references are collected and discussed. While written in the style of a research monograph, the book is readable with basic knowledge in analysis, number theory and measure theory.

The main purpose of this book is to give an overview of the developments during the last 20 years in the theory of uniformly distributed sequences. The authors focus on various aspects such as special sequences, metric theory, geometric concepts of discrepancy, irregularities of distribution, continuous uniform distribution and uniform distribution in discrete spaces. Specific applications are presented in detail: numerical integration, spherical designs, random number generation and mathematical finance. Furthermore over 1000 references are collected and discussed. While written in the style of a research monograph, the book is readable with basic knowledge in analysis, number theory and measure theory.

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The study of geometric discrepancy, which provides a framework for quantifying the quality of a distribution of a finite set of points, has experienced significant growth in recent decades. This book provides a self-contained course in number theory, Fourier analysis and geometric discrepancy theory, and the relations between them, at the advanced undergraduate or beginning graduate level. It starts as a traditional course in elementary number theory, and introduces the reader to subsequent material on uniform distribution of infinite sequences, and discrepancy of finite sequences. Both modern and classical aspects of the theory are discussed, such as Weyl's criterion, Benford's law, the Koksma-Hlawka inequality, lattice point problems, and irregularities of distribution for convex bodies. Fourier analysis also features prominently, for which the theory is developed in parallel, including topics such as convergence of Fourier series, one-sided trigonometric approximation, the Poisson summation formula, exponential sums, decay of Fourier transforms, and Bessel functions.

The author writes:“According to Seder Olam Rabbah, the work that forms the basis for almost all rabbinic chronology, the .period from the defeat of the Babylonians by the Medeo-Persians until the beginning of Greek rule, encompassed 52 years and spanned the reigns of three Persian kings. According to the chronology that is universally accepted by historians today (conventional chronology), this period of Persian rule over the land of Israel encompassed 207 years (539 to 332 BCE) and during this period more than ten Persian kings reigned.“This discrepancy between the traditional Jewish chronology and conventional chronology has not gone unnoticed. The purpose of this study is to collect and categorize the variety of Jewish responses to this discrepancy, both by Jewish scholars and rabbinic authorities. Part I provides an introduction to the discrepancy. Part II contains the earliest Jewish responses to the discrepancy. In the major part of the study, Part III, the responses to the discrepancy from the time of Azariah de Rossi (16th century) to the present time are collected and categorized. This unified collection and categorization of the many responses will enable students and scholars to have easy access to what has been written by Jewish scholars and rabbinic authorities about the discrepancy and will facilitate scholarly evaluation of the responses.“Part IV is an evaluation of the responses’ attempts to answer the fundamental question raised by the discrepancy. Part V presents observations on the rabbinic responses. Part VI is a summary and conclusion.”