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Selected Works of Eberhard Hopf with Commentaries
Edited by: Cathleen S. Morawetz, New York University-Courant Institute, NY, James B. Serrin, University of Minnesota, Minneapolis, MN, and Yakov G. Sinai, Princeton University, NJ
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Collected Works
2002; 396 pp; hardcover
Volume: 17
ISBN-10: 0-8218-2077-X
ISBN-13: 978-0-8218-2077-3
List Price: US$114
Member Price: US$91.20
Order Code: CWORKS/17
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This work celebrates the work of Eberhard Hopf, a founding father of ergodic theory, a mathematician who produced many beautiful, elegantly written, and now classical, results in integral equations and partial differential equations. Hopf's results remain at the core of these fields, and the title includes Hopf's original mathematical papers, still notable for their elegance and clarity of the writing, with accompanying summaries and commentary by well-known mathematicians.

Today, ergodic theory and P.D.E. continue to be active, important areas of mathematics. In this volume the reader will find the roots of many ergodic theory concepts and theorems. Hopf authored fundamental results for P.D.E., such as the maximum principle of elliptic equations and the complete solution of Burger's equation. The familiar properties of elliptic equations were proved for the first time in his earliest work and are included here. His bifurcation theorem, still used over and over again, is a particular gem. The proof of the Wiener-Hopf Theorem is a stunning application of deep analysis.

The volume is presented in two main parts. The first section is dedicated to classical papers in analysis and fluid dynamics, and the second to ergodic theory.

These works and all the others in the Selected Works carry commentaries by a stellar group of mathematicians who write of the origin of the problems, the important results that followed. Many a mathematical researcher and graduate student will find these collected works to be an excellent resource.

Readership

Graduate students and research mathematicians interested in analysis.

Reviews

"The value and interest of the book is also highly increased by the commentaries on Hopf's main achievements, written by experts ... Most of them are real gems. The book also contains a short curriculum vitae, a complete list of publications and a portrait of Eberhard Hopf. It should be in every mathematical library."

-- Zentralblatt MATH

Table of Contents

Part I
  • E. Hopf -- Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus
  • J. B. Serrin -- Commentary
  • E. Hopf -- A remark on linear elliptic differential equations of second order
  • J. B. Serrin -- Commentary
  • E. Hopf -- Zum analytischen Charakter der Lösungen regulärer zweidimensionaler Variationsprobleme
  • H. Weinberger -- Commentary
  • N. Wiener and E. Hopf -- Über eine Klasse singulärer Integralgleichungen
  • H. Widom -- Commentary
  • E. Hopf -- Über den funktionalen, insbesondere den analytischen Charakter der Lösungen elliptscher Differentialgleichungen zweiter Ordnung
  • H. Weinberger -- Commentary
  • E. Hopf -- Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems
  • M. Golubitsky and P. H. Rabinowitz -- Commentary
  • E. Hopf -- Repeated branching through loss of stability. An example
  • E. Hopf -- A mathematical example displaying features of turbulence
  • R. Temam -- Commentary
  • E. Hopf -- On S. Bernstein's theorem on surfaces \(z(x,y)\) of nonpositive curvature
  • L. Nirenberg -- Commentary
  • E. Hopf -- The partial differential equation \(u_t + uu_x = \mu_{xx}\)
  • P. D. Lax -- Commentary
  • E. Hopf -- Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen
  • J. B. Serrin -- Commentary
  • E. D. Conway and E. Hopf -- Hamilton's theory and generalized solutions of the Hamilton-Jacobi equation
  • C. S. Morawetz -- Commentary
Part II
  • E. Hopf -- Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung
  • E. Hopf -- Statistik der Lösungen geodätischer Probleme vom unstabilen Typus. II
  • Ya. G. Sinai -- Commentary
  • E. Hopf -- Closed surfaces without conjugate points
  • Ya. G. Sinai -- Commentary
  • E. Hopf -- Statistical hydromechanics and functional calculus
  • Ya. G. Sinai -- Commentary
  • E. Hopf -- On the ergodic theorem for positive linear operators
  • D. Ornstein -- Commentary
  • Acknowledgments
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