Description : The aim of this monograph is to give a detailed exposition of the summation method that Ramanujan uses in Chapter VI of his second Notebook. This method, presented by Ramanujan as an application of the Euler-MacLaurin formula, is here extended using a difference equation in a space of analytic functions. This provides simple proofs of theorems on the summation of some divergent series. Several examples and applications are given. For numerical evaluation, a formula in terms of convergent series is provided by the use of Newton interpolation. The relation with other summation processes such as those of Borel and Euler is also studied. Finally, in the last chapter, a purely algebraic theory is developed that unifies all these summation processes. This monograph is aimed at graduate students and researchers who have a basic knowledge of analytic function theory.
Description : The author demonstrates that the Dirichlet series representation of the Riemann zeta function converges geometrically at the roots in the critical strip. The Dirichlet series parts of the Riemann zeta function diverge everywhere in the critical strip. It has therefore been assumed for at least 150 years that the Dirichlet series representation of the zeta function is useless for characterization of the non-trivial roots. The author shows that this assumption is completely wrong. Reduced, or simplified, asymptotic expansions for the terms of the zeta function series parts are equated algebraically with reduced asymptotic expansions for the terms of the zeta function series parts with reflected argument, constraining the real parts of the roots of both functions to the critical line. Hence, the Riemann hypothesis is correct. Formulae are derived and solved numerically, yielding highly accurate values of the imaginary parts of the roots of the zeta function.
Description : #1 New York Times bestselling author Veronica Roth retells a pivotal Divergent scene (chapter 13) from Tobias's point of view. This thirteen-page scene reveals unknown facts and fascinating details about Four's character, his past, his own initiation, and his thoughts about new Dauntless initiate Tris Prior.
Description : Providing an elementary introduction to analytic continuation and monodromy, the first part of this volume applies these notions to the local and global study of complex linear differential equations, their formal solutions at singular points, their monodromy and their differential Galois groups. The Riemann-Hilbert problem is discussed from Bolibrukh’s point of view. The second part expounds 1-summability and Ecalle’s theory of resurgence under fairly general conditions. It contains numerous examples and presents an analysis of the singularities in the Borel plane via “alien calculus”, which provides a full description of the Stokes phenomenon for linear or non-linear differential or difference equations. The first of a series of three, entitled Divergent Series, Summability and Resurgence, this volume is aimed at graduate students, mathematicians and theoretical physicists interested in geometric, algebraic or local analytic properties of dynamical systems. It includes useful exercises with solutions. The prerequisites are a working knowledge of elementary complex analysis and differential algebra.
Description : The first companion guide to the blockbuster bestselling Divergent trilogy—soon to be a major motion picture Written by the New York Times bestselling author of The Twilight Companion and The Hunger Games Companion, the book takes fans deeper into the post-apocalyptic world created by Veronica Roth—a dystopian Chicago in which humanity has organized itself into five factions, each with its own core value to uphold. At the age of sixteen, Beatrice Prior must choose to which one she will devote her life. The Divergent Companion includes fascinating background facts about the action in all three books—the third book, Allegiant, publishes in October 2013—a revealing biography of the author, and amazing insights into the trilogy's major themes and features. It's everything fans have been hungering for since the very first book! This book is not authorized by Veronica Roth, Katherine Tegen Books, or anyone involved in the Divergent movie. The Divergent Companion is a must-read and a terrific gift for the millions of fans both young and old—especially with the Summit Entertainment film version of Divergent, the first book in the trilogy, hitting theaters in March 2014.
Description : The explosive conclusion to Veronica Roth's #1 New York Times bestselling Divergent trilogy reveals the secrets of the dystopian world that captivated millions of readers and film fans in Divergent and Insurgent. One choice will define you. What if your whole world was a lie? What if a single revelation—like a single choice—changed everything? What if love and loyalty made you do things you never expected? Told from a riveting dual perspective, this third installment in the series follows Tris and Tobias as they battle to comprehend the complexities of human nature—and their selves—while facing impossible choices of courage, allegiance, sacrifice, and love. And don't miss The Fates Divide, Veronica Roth's powerful sequel to the bestselling Carve the Mark!
Description : Complete your Divergent library with Four! Fans of the Divergent trilogy by #1 New York Times bestselling author Veronica Roth will be thrilled by Four: A Divergent Collection, a companion volume that includes four pre-Divergent stories plus three additional scenes from Divergent, all told from Tobias's point of view. This collection also makes a great pick for fans of the blockbuster movies who want to delve deeper into the character played by Theo James. Readers first encountered Tobias as "Four" in Divergent. His voice is an integral part of Allegiant. Readers will find more of this charismatic character's backstory told from his own perspective in Four: A Divergent Collection. When read together, these long narrative pieces illuminate the defining moments in Tobias's life. The first three pieces in this volume—"The Transfer," "The Initiate," and "The Son"—follow Tobias's transfer from Abnegation to Dauntless, his Dauntless initiation, and the first clues that a foul plan is brewing in the leadership of two factions. The fourth story, "The Traitor," runs parallel with the events of Divergent, giving readers a glimpse into the decisions of loyalty—and love—that Tobias makes in the weeks after he meets Tris Prior. Also includes three additional scenes from Divergent, told from Tobias's point of view!
Description : Society is divided into five factions, each dedicated to the cultivation of a particular virtue, in the attempt to form a 'perfect society'. On her Choosing Day, Beatrice Prior renames herself Tris, rejects her family's Abnegation group, and chooses another faction.