BARBARA GELLAI Th e Intrinsic Nature of Th ings

Th e Life and Science of Cornelius Lanczos The Intrinsic Nature of Things The Life and Science of Cornelius Lanczos

http://dx.doi.org/10.1090/mbk/076

The Intrinsic Nature of Things The Life and Science of Cornelius Lanczos

BARBARA GELLAI

Providence, Rhode Island Cover photograph courtesy of the late Ida Rhodes: Cornelius Lanczos, visiting researcher at Boeing Aircraft Company, in 1944.

2010 Mathematics Subject Classification. Primary 01A60.

For additional information and updates on this book, visit www.ams.org/bookpages/mbk-76

Library of Congress Cataloging-in-Publication Data Gellai, Barbara, 1933– The intrinsic nature of things : the life and science of Cornelius Lanczos / Barbara Gellai. p. cm. Includes bibliographical references and index. ISBN 978-0-8218-5166-1 (alk. paper) 1. Lanczos, Cornelius, 1893–1974. 2. Mathematicians—Hungary—Biography. 3. Physicists—Hungary—Biography. I. Title. QA29.L285G45 2010 510.92—dc22 [B] 2010017680

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Contents

Preface xi

Chapter 1. Background 1

Chapter 2. Family and Basic Studies 5 §2.1. The Lanczos Family 5 §2.2. From L¨owy to L´anczos 6 §2.3. Basic Studies 7

Chapter 3. A Change in Our World View: The Theory of Relativity 11 §3.1. Unsolved Problems 12 §3.2. New Problem—Again 13 §3.3. The Theory of Relativity 14 §3.4. A Fundamentally New Feature of the Theory 15 §3.5. The Need for a New Theory 16

Chapter 4. Higher Studies 17 §4.1. The First Position 18 §4.2. Contacting Einstein 19 §4.3. Getting His Ph.D. 20

vii viii Contents

Chapter 5. Lanczos’s Early Research in the Theory of Relativity 23 §5.1. The Beautiful Years 23 §5.2. Probing the Theory of Relativity in Cosmology 25 §5.3. The Problem of Motion 26 §5.4. Einstein’s Invitation 27

Chapter 6. Contribution to Quantum Mechanics 31 §6.1. The Frankfurt Mathematics Seminars 32 §6.2. The Quantum Mystery 33 §6.3. Matrix Mechanics—Quantum Mechanics 35 §6.4. Field-like Representation of Quantum Mechanics 36 §6.5. Wave Mechanics 38 §6.6. Farewell to Europe 41

Chapter 7. Purdue Beginnings 45 §7.1. Purdue 45 §7.2. “Workable Mathematics” 47 §7.3. “Telescoping” Power Series 48 §7.4. The Tau Method 49 §7.5. Trigonometric Truncations 50 §7.6. Having a Hard Time 52

Chapter 8. The Educator 55 §8.1. Experimental Curriculum for Women 55 §8.2. The Educational Value of Mathematics 57 §8.3. Some Improvement in Practical Fourier Analysis 59 §8.4. Fast Fourier Transform (FFT) 61 §8.5. Mathematical Tables Project 63 §8.6. Sad News 65

Chapter 9. “Why Mathematics?” 67 §9.1. Divine Occupation or Bread and Butter Question? 69 Contents ix

§9.2. The Language of Nature 70 §9.3. Relation of Mathematics to Its History 71

Chapter 10. Ripples on the Old Pond’s Surface 73 §10.1. Early Efforts 74 §10.2. Lanczos’s Own Way 75 §10.3. Matter Waves and Electricity 76

Chapter 11. The Lanczos Method 81 §11.1. An Applied Mathematician in Industry 83 §11.2. Walker Aimes, Lecturer 84 §11.3. Learning How to Use Computers 85 §11.4. Exceptional History of an Exceptional Method 87 §11.5. Accused of Disloyalty 89 §11.6. Back to Europe 90

Chapter 12. Full-Time Research 93 §12.1. The Dublin Institute for Advanced Studies 93 §12.2. Electricity and General Relativity 95 §12.3. Doing Mathematics 98 §12.4. Well-Posed Problems 101 §12.5. Participating in Dublin’s Social Life 102

Chapter 13. Nature’s Pythagorean Theorem 105 §13.1. Synge the Pragmatic and Lanczos the Mystic 105 §13.2. What is Nature’s Pythagorean Theorem? 106 §13.3. Lanczos’s Philosophy of Science 107 §13.4. Continuing His Research 109 §13.5. Undulatory Riemannian Spaces 110

Chapter 14. Probing Riemannian Space 113 §14.1. The Lecturer 113 §14.2. Quadratic Action and Vector Potential 116 §14.3. Fourier Analysis of Random Sequences 118 x Contents

§14.4. Gravitation and Riemannian Space 120

Epilogue 125

A Brief Professional Chronology of Cornelius Lanczos 129

Published Papers and Books of Cornelius Lanczos 133

Bibliography 149

Index 163 Preface

Whether in the world of truth [science] or in the world of beauty [art], the human mind, on its way towards the ultimate cognition, endeavors to understand the intrinsic nature of things; not to understand things as they are but as they must be—this is the intrinsic necessity.

Lanczos: “Science as a Kind of Art”, 1973

It is widely thought that science and art have hardly anything in com- mon, that they in fact exclude each other. We mention here several of their different characteristics on the basis of Cornelius Lanczos’s paper “Science as a Kind of Art”. Science is concerned with facts established by careful experi- ments. The artistic fantasy can create a world that is not subject to the unchangeable laws of the physical universe. The method of science is logic. Starting from a few universal statements, the sci- entist can obtain new results by logical reasoning. Art need not be logical. In art the logic of pure reasoning is replaced by an emotional and intuitive way of understanding “Whatever the laws of the universe may be, they will not be changed by personal sympathy or antipathy; the scientist makes a recording of the physical events only, while his personality remains in the background. In the world of art, on the contrary, human emotions

xi xii Preface and human reactions to certain situations in life are of interest” [1, p. 4–103]. The sharp distinction, however, that science is factual and art is visionary cannot be maintained in every case. We would hardly make a big mistake by claiming that almost every great discovery in science could be considered as the achievement of an ingenious and inspired mind rather than one of mere factual inference. We are all, scientists or artists, each one in his own way, seekers of the truth. The one who takes this road, however, should not forget that the result is not guaranteed. The history of science teaches us that each time we proudly think that we have it all figured out, we will be surprised at having even more difficult problems than we had before. This is the time to reflect back and marvel at the journey we have taken so far. While going further, we experience, as Einstein did, that “Lessing’s comforting words stay with us: The aspiration to truth is more precious than its assured possession”1 [2]. We invite the reader to discover, in learning about the life and science of Cornelius Lanczos, the intrinsic nature of things, to under- stand things not as they are but as they must be. We believe you will enjoy the journey. Cornelius Lanczos was a great admirer of Greek philosophy and cultural tradition. For him, dealing with science, probing the secrets of nature, was that “pathos philosophikon” (the passion for knowl- edge) with which Aristotle so beautifully describes the essence of the scientist. Having a very deep awareness of beauty in science, Lanczos regarded “science as a kind of art”. For him a beautiful theory was in some sense truer than an unbeautiful theory. We will learn how his philosophical disposition motivated his research. A posthumous appreciation summarized his achievement briefly: He was a great scientist and a great man. These are two great testi- monials, and the second one is no less important than the first. In this book we follow this exceptional career from his homeland, Hungary, through Germany and the United States, then back to Europe into

1Lessing, Gotthold Ephraim (1729–1781), German dramatist and critic, was a vigorous and prolific writer on literary, philosophical, and theological subjects and the outstanding figure of the German Enlightenment. Academic American Encyclopedia, Grolier Incorporated, Danbury, Connecticut, Volume 12, 1991, pp. 298–299. Preface xiii

Ireland. We will analyze how the social and historical circumstances of these countries affected his career. In addition to his life story, his mathematical methods and the novelty of his ideas in physics will be described. We attempt to use simple terminology as far as pos- sible or to explain scientific notions briefly. The goal is not to give the mathematical details of his theorems or to prove their exactness. We could not do it better than he did. Rather, it is to make them understandable not only for professionals but for educated laymen as well. Choosing the subjects to discuss was not easy. Lanczos’s pub- lications cover two huge fields: mathematics and physics. In both fields his extremely flexible mind touched important and interesting topics, producing such unique algorithms and ideas that one gets the overwhelming feeling that everything was important. To write a book of moderate size, however, one must make hard choices. As for mathematics, we received help from the man himself: In the last interview of his career, he named the Tau method, the ma- trix eigenvalue approximation method (now known as the Lanczos Method), and the linear systems in self-adjoint form as his three most important contributions to mathematics. In physics, we investigated how his results contributed to other subjects, determined the course of his research, and influenced his ca- reer. We came up with the following: The geometric model Lanczos applied in his early paper “Surface Distribution of Matter in Ein- stein’s Theory of Gravitation” later proved to be very important in the dynamics of bubbles in inflationary cosmology. His contribution to quantum mechanics was the first continuum representation of quantum mechanics. His publication on the topic preceded Schr¨odinger’s epoch-making paper by two weeks. This as- signed him a special position in the history of science. The results Lanczos achieved in the field of “Matter Waves and Electricity” motivated Schr¨odinger to invite him to the Dublin Insti- tute for Advanced Studies and earned him the senior professorship in theoretical physics, the post he filled until the end of his life. xiv Preface

His investigation of undulatory Riemannian spaces determined the course Lanczos followed in his quest for a unified field theory during the Dublin period. His research in the field culminated in his very last paper in physics on the topic of gravitation and Riemannian space. To readers interested in the mathematical details, we recommend Lanczos’s papers, a complete list of which can be found at the end of the book.

Acknowledgements Thanks are due to Jerry L. Whitten, Richard R. Patty, and William R. Davis for making the author’s participation possible in the Lanczos Centenary Project of the College of the Physical and Mathematical Sciences at the University of North Carolina, Raleigh, NC, in the 1990s. The cooperation of Christopher R. Gould and of Wesley O. Doggett for bringing the publication of Lanczos’s Collected Papers to a conclusion is appreciated. The author thanks the American Mathematical Society for pub- lishing this book. Senior editor Edward G. Dunne’s advice and rec- ommendations were of decisive importance in the publication process. At his suggestion, Cornelius Lanczos’s favorite phrase “the intrinsic nature of things” became the title of the book. The persistent assis- tance of Cristin Zannella, editorial assistant, in the arduous process of getting permissions helped to speed up the acquisition process. Grateful thanks go to the production department for careful edito- rial work. It was a pleasure to work with production editor Arlene O’Sean and to see the book approach its conclusion. The artistic cover designed by Peter Sykes heralds the message of the book. Grateful thanks go to J´ozsef Illy for reading and commenting on the manuscript. Werner Israel’s thoughtful interview which covered Lanczos’s life both as a scientist and as a man is gratefully acknowledged. As a contact person between North Carolina State University and the publisher, Wesley O. Doggett helped the publication process. Preface xv

For reviewing the quotations from the Einstein-Lanczos corre- spondence thanks are due to Barbara Wolff at the Albert Einstein Archive of Jerusalem. The painstaking work of Monika Csuka and Peter Kallai in pro- ducing the LATEX form of the manuscript is acknowledged. Mrs. Eleanor Boyk’s permission to quote from Lanczos’s letters is appreciated.

Bibliography

[1] L´anczos Korn´el, “A tudom´any mint a m¨uv´eszet egyik form´aja” [Sci- ence as a Kind of Art]. Fizikai Szemle, 23, 1973, pp. 225–231. For the English translation see [207], Vol. VI, p. 4–103. [2] Abraham Pais, “Subtle is the Lord”, The Science and the Life of Albert Einstein, Oxford: Oxford University Press, 1982, p. 468. [3] John Lukacs, Budapest 1900, A Historical Portrait of a City & Its Culture, Grove Weidenfeld, New York, 1988, pp. 29-66. [4] Ibid. p. 10. [5] Reuben Hersh and Vera John-Steiner, “A Visit to Hungarian Math- ematics”, The Mathematical Intelligencer, 15, No. 2, 1993, Springer- Verlag New York, p. 18. [6] [3], p. 28. [7] George Lanczos (Lanczos’s nephew, a.k.a. Ravid Benjamin), “A L´anczos csal´ad t¨ort´enete” lecture at the Lanczos Centenary Cele- bration,Sz´ekesfeh´erv´ar, Hungary, 1993. [8] Sur´anyi Istv´an, “A sz´ekesfeh´erv´ari Goldziher Ign´ac Izraelita Elemi N´episkola t¨ort´enete” (“The History of the Jewish Elementary School of Ignac Goldziher, 1842–1943”), Proceedings of the Conference The Jews in County Fejer 1688–1867,Sz´ekesfeh´erv´ar, September 23, 1986, Publications of County Fejer,FarkasG´abor, ed., No. 4, 1989 (in Hungarian). [9] Lakatos D´enes, Asz´ekesfeh´erv´ari cisztercita rendi katholikus f˝ogimn´azium sz´az ´eves t¨ort´enete [The History of the One- Hundred-Year-Old Catholic Gymnasium of the Cistercian Order in Sz´ekesfeh´erv´ar]; Debreceny nyomda, Sz´ekesfeh´erv´ar, 1914.

149 150 Bibliography

[10] Cornelius Lanczos, “Why Mathematics”, Irish Mathematics Teachers Association Newsletter, 9 (1967), p. 3. [11] Peter Coveney and Roger Highfield, The Arrow of Time, Ballantine Books, 1990, pp. 58-59. [12] Isaac Newton, Sir Isaac Newton’s Mathematical Principles of Natural Philosophy and His System of the World, translated by A. Motte and Florian Cajory, University of California Press, Berkeley, 1962, Vol. I, p. 634. [13] Cornelius Lanczos, Albert Einstein and the Cosmic World Order, John Wiley & Sons, 1965, p. 37. [14] Albert Einstein, “Zur Elektrodynamik bewegter K¨orper” [“On the Electrodynamics of Moving Bodies”], Annalen der Physik, 17, 1905, pp. 891-921. [15] Brian Greene, The Fabric of the Cosmos, Alfred A. Knopf, 2004, p. 45. [16] L´anczos Korn´el, “A tudom´any mint a m˝uv´eszet egyik form´aja” [Sci- ence as a Kind of Art], Fizikai Szemle, XXIII, No. 8, August 1973, p. 228. For the English translation see [207], Volume VI, pp. 4-103 – 4-115. [17] Cornelius Lanczos, Albert Einstein—His Life and His Work, Annual Statutory Lecture in the Dublin Institute for Advanced Studies for the academic year 1955–1956. Unpublished lecture notes, p. 18. [18] George Marx, “The Roots of Cornelius Lanczos”, Foreword in Cor- nelius Lanczos Collected Published Papers with Commentaries,W.R. Davis (general ed.), College of Physical and Mathematical Sciences, North Carolina State University, 1998, p. xl. [19] See [17], p. 18. [20] Program courses of the years 1918–1919 of the Hungarian Technical University Kir´aly J´ozsef, Lloyd Press, 1914, pp. 42-43. (Archive of the Budapest Institute of Technology.) [21] Lanczos wrote his Ph.D. dissertation in German with the ti- tle Die functionentheoretischen Beziehungen der Maxwellschen Aethergleihungen—Ein Beitrag zur Relativit¨ats und Elektronentheorie [The Function-theoretical Relationships of the Maxwellian Aether- Equations—A Contribution to the Theory of Relativity and Elec- trons.] [22] Lanczos gave the mathematical details of the most important physical methods and results of his doctoral dissertation in his paper “Die ten- soranalytischen Beziehungen der Diracschen Gleichung” [“The Tensor Analytical Aspects of Dirac’s Equation”]. Zeitschrift f¨ur Physik,57 Bibliography 151

(1929), pp. 449-454. For the English translation see [207], Volume III, pp. 2-1133 – 2-1185. [23] Ibid. [24] Lanczos’s letter to Einstein, December 3, 1919, Einstein Archives (EA),1 16-206. Courtesy of the Albert Einstein Archives, the Hebrew University of Jerusalem, Israel. [25] Einstein’s letter to Lanczos, January 20, 1920, EA 15-207. The Col- lected Papers of Albert Einstein. Vol. 9. English translation, Prince- ton: Princeton University Press, 2004, Doc. 276. [26] Peter F. Sugar, Peter Hanak, and Tibor Frank, eds., AHistoryof Hungary, Indiana University Press, Bloomington and Indianapolis, 1990, pp. 295-318. [27] Ibid. [28] Ortvay’s notes, undated, probably from 1928–1929. MTAK K´eziratt´ar (Archive of the Hungarian Academy of Sciences), Ms 5108/286. [29] Robert Jungk, Brighter Than a Thousands Suns, A Personal History of the Atomic Scientists, Penguin Books, 1960, p. 18. [30] Ibid. p. 29. [31] Ibid. pp. 32-33. [32] Cornelius Lanczos, “Fl¨achenhafte Verteilung der Materie in der Ein- steinschen Gravitationstheorie” [Surface Distribution of Matter in Einstein’s Theory of Gravitation], Annalen der Physik, 74, 1924, p. 519. For the English translation see [207], Volume I, pp. 2-219 – 2-263. [33] Werner Israel, “Singular Hypersurfaces and Thin Shells in General Relativity”, Nuovo Cimento, 44B, 1966, pp. 1-14. [34] Alan H. Guth, The Inflationary Universe: The Quest for a New The- ory of Cosmic Origins, Perseus, Cambridge, Massachusetts, 1997, pp. 329-330. The book written by the discoverer of the inflationary model provides a clear explanation of the theory and a readable story of its discovery. [35] The author’s interview with Werner Israel, University of Alberta, Ed- monton, Alberta, Canada, April 1991. The interview was audiotaped. Later the author transcribed it into hard copy. Further quotations from Werner Israel in this book are from this interview. [36] For further references, see C. W. Misner, K. S. Thorne and J. A. Wheeler,Gravitation, W. H. Freeman and Company, San Francisco, 1973, pp. 553-554. For updated references see, for example, S. K.

1Throughout the Bibliography, references to EA are to the Einstein Archives. 152 Bibliography

Blaue, E. I. Guendelman, and A. H. Guth, “Dynamics of False- Vacuum Bubbles”, Physical Review, D35, 1987, pp. 1747-1766; P. Laguna-Castillo and R. A. Matzner, “Inflation and Bubbles in Gen- eral Relativity”, Physical Review, D34, 1986, pp. 2913-2925; J. Ipser and P. Sikive, “Gravitationally Repulsive Domain Wall”, Physical Review, D30, 1984, pp. 712-719. [37] Cornelius Lanczos, “Ein vereinfachendes Koordinatensystem f¨ur die Einsteinschen Gravitationsgleichungen” [A Simplifying Coodi- nate System for Einstein’s Gravitational Equations]. Physikalische Zeitschrift, 23, 1922, 537-539. For the English translation see [207], Volume I, pp. 2-25 – 2-29. [38] Cornelius Lanczos, “Zum Bewegungsprinzip der allgemeinen Rela- tivit¨atstheorie” [On the Principle of Motion in the Theory of General Relativity]. Physikalische Zeitschrift, 28, 1927, pp. 723-726. For the English translation see [207], Volume II, pp. 2-531 – 2-537. [39] Lanczos to Seelig, September 9, 1955, Seelig Archive, ETH-Bibliothek Zurich. The quotation is taken from the paper: Peter Havas, “The Early History of the ‘Problem of Motion’ in General Relativity”, Einstein and the History of General Relativity,D.HowardandJ. Stachel, eds., Proceedings of the 1986 Osgood Hill Conference, North Andover, Massachusetts, May 8-11, 1986, p. 250. [40] Ibid. [41] Cornelius Lanczos, “Die neue Feldtheorie Einsteins” [The New Field Theory of Einstein], Ergebnisse der exacten Naturwissenschaften, 10, 1931, p. 126. For the English translation see [207], Volume IV, pp. 2-1413 – 2-1483. [42] Wilhelm Magnus, “Vignette of a Cultural Episode”, Studies in Nu- merical Analysis.B.K.P.Scaife,ed.NewYork,NY:RoyalIrish Academy and Academic Press, 1974. Festschrift presented to Cor- nelius Lanczos on the occasion of his eightieth birthday, p. 7. [43] Ibid. p. 10. [44] Carl Ludwig Siegel, “On the History of the Frankfurt Mathemat- ics Seminar. Address Given on June 13, 1964, at the Mathematics Seminar of the University of Frankfurt on the Occasion of the 50th Anniversary of the Johann-Wolfgang-Goethe-University Frankfurt”, The Mathematical Intelligencer, 1, 1978/79, pp. 223-230. [45] Ibid. p. 226. [46] See [42], p. 12. [47] Werner Heisenberg, Physics and Philosophy, The Revolution in Mod- ern Science, Ruskin House, George Allen & Unwin Ltd., London, 1958, p. 33. Bibliography 153

[48] Werner Heisenberg, “Uber¨ die quantentheoretische Umdeutung kine- matischer und mechanischer Beziehungen” [“On a Quantum The- oretical Reinterpretation of Kinematic and Mechanical Relations”], Zeitschrift f¨ur Physik, 33, 1925, pp. 879-893 (received July 29, 1925; published in issue No. 12, September 18, 1925). [49] Max Born and Pascal Jordan, “Zur Quantenmechanik” [“On Quan- tum Mechanics”], Zeitschrift f¨ur Physik, 34, 1925, pp. 858-888 (re- ceived September 27, 1925; published in issue No. 11/12, November 28, 1925). English translations of the Heisenberg-Born-Jordan papers can be found in B. L. van der Waerden, ed. and transl., Sources of Quantum Mechanics (New York: Dover, 1967). [50] Kornel Lanczos2,“Uber¨ eine feldm¨aßige Darstellung der neuen Quan- tenmechanik” [“On a Field-Theoretical Representation of the New Quantum Mechanics”], Zeitschrift f¨ur Physik, 35, 1926, p. 812 (re- ceived by the journal on December 22, 1925; published in issue No. 11/12, February 26, 1926). For the English translation see [207], Volume III, pp. 2-859 – 2-895. A detailed discussion of Lanczos’s quantum mechanical research can be found in J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory,Vol. 3, Springer-Verlag, New York, 1982, pp. 204-220. [51] Kornel Lanczos, “Zum Problem der Atherstrahlung¨ in einer r¨aum- lich geschlossenen Welt” [“On the Problem of Aether Radiation in a Spacially Closed Universe”], Zeitschrift f¨ur Physik, 32, 1925, p. 149. For the English translation see [207], Volume I, pp. 2-383 – 2-411. [52] See the paper in [50]. [53] See [50], pp. 820-821. [54] Ibid. p. 820. [55] Ibid. p. 823. [56] Erwin Schr¨odinger, “Quantisierung als Eigenwertproblem. (Erste Mitteilung)” [“Quantization as an Eigenvalue Problem (Part I)”], An- nalen der Physik, 79, 1926, pp. 361-376 (received January 27, 1926; published in issue No. 4, March 13, 1926); “Quantisierung als Eigen- wertproblem. (Zweite Mitteilung)” [“Quantization as an Eigenvalue Problem (Part II)”], Annalen der Physik, 79 (1926) pp. 489-527 (re- ceived February 23, 1926; published in issue No. 6, April 6, 1926). [57] Erwin Schr¨odinger, “Uber¨ das Verh¨altnis der Heisenberg-Born- Jordanschen Quantenmechanik zu der meinen” [“On the Relation of the Heisenberg-Born-Jordan Quantum Mechanics to Mine”], An- nalen der Physik, (4), 79, 1926, pp. 734-756 (received March 18, 1926;

2From 1922, the beginning of Lanczos’s career in Germany, until July 29, 1927, his papers were published under the name Kornel Lanczos. 154 Bibliography

published in issue No. 8, May 4, 1926). An English translation of Schr¨odinger’s wave mechanics papers can be found in, among other places, Schr¨odinger, Collected papers on wave mechanics,2nded., New York: Chelsea Publishing Co., 1978. [58] Jagdish Mehra, The Physicist’s Conception of Nature,Dordrecht:D. Reidel Publishing Company, 1973, pp. 276-293. [59] See [2], p. 463. [60] Karl Lark-Horowitz’s letter to Einstein, April 28, 1931, EA 15-241. [61] Lanczos to Einstein, May 8, 1931, EA 15-242. [62] See [44], p. 227. [63] Jules Janick: “A Jewish History of Purdue 1920–1940”, Purdue Uni- versity Newsletter, No. 46, January/February 1991, p. 7. [64] Ibid. p. 6. [65] Ibid. p. 6. [66] Ibid. pp. 4, 5. [67] Ibid. p. 5. [68] For Karl Lark-Horovitz’s life see V. A. Johnson, Men of Physics, Pergamon Press, 1969. [69] Undated letter of Lanczos to Rudolf Ortvay (probably from 1931 or 1932). MTAK K´eziratt´ar (Archive of the Hungarian Academy of Sciences), K785/425-437. [70] Lanczos to Einstein, Oct. 20 (no year, probably from 1931), EA 15- 243. [71] Paul Hoffman: The Man Who Loved Only Numbers, Hyperion, New York, New York, 1998, p. 157. [72] Cornelius Lanczos, Applied Analysis, Dover Publication Inc., New York, 1988, p. 440. [73] Cornelius Lanczos, “Trigonometric Interpolation of Empirical and Analytical Functions”, Journal of Mathematics and Physics 17, 1938, p. 198. [74] See Lanczos Collection [207], Volume V, pp. 3-298 and 3-309. [75] “Professor Lanczos’s Contribution to Mathematics,” videotape recording courtesy of the University of Manchester Audio Visual Ser- vice, Manchester, England, 1972. [76] Jacques Hadamard, Lectures on the Cauchy Problem in Linear Partial Differential Equations, New Haven, Yale University Press; reprinted by Dover Publications, Inc., New York, 1954. [77] Cornelius Lanczos, “Evaluation of Noisy Data”, Journal of the Soci- ety for Industrial and Applied Mathematics, B, 1, 1964, p. 85. Bibliography 155

[78] Lanczos’s letter to Einstein, February 15, 1936, EA 15-256. [79] Lanczos’s letter to Einstein, March 12, 1936, EA 15-257. [80] Lanczos’s letter to the Boyks (Sol and Eleanor), July 1, 1939. [81] Lanczos’s letter to the Boyks (Sol and Eleanor), August 15, 1939. [82] Ibid. [83] Personal communication from Alice Lanczos, Lanczos’s daughter-in- law, to the author. [84] Einstein’s letter to Lanczos, December 28, 1929, EA 15-240. [85] See [43], p. 10. [86] Cornelius Lanczos, The History of Mathematics, unpublished lecture notes, p. VII. [87] Ibid. pp. 67-76. [88] Ibid. p. 73. [89] Ibid. p. 74. [90] Ibid. p. 76. [91] Ibid. p. III. [92] C. Runge and H. Koenig, “Vorlesungenuber ¨ numerisches Rech- nen”, Die Grundlehren der Mathematischen Wissenschaften,Band XI, Springer, Berlin, 1924, p. 211. [93] Gordon C. Danielson and Cornelius Lanczos, “Some Improvements in Practical Fourier Analysis and Their Applications to X-Ray Scat- tering from Liquids”, Journal of the Franklin Institute, 233, 1942, pp. 365-380 and 435-452. [94] James W. Cooley and John Tukey, “An Algorithm for the Machine Calculation of Complex Fourier Series”, Mathematics of Computa- tion, 19, 1965, p. 297. [95] Philip Rudnick, “Note on the Calculation of Fourier Series”, Mathe- matics of Computation, 20, 1966, pp. 429-430. [96] James W. Cooley, “The Re-Discovery of the Fast Fourier Transform Algorithm”, Microchimica Acta 3, 1987, pp. 33-45. [97] James Cooley, Electrical Engineer, an oral history, conducted in 1997 by Andrew Goldstein, IEEE History Center, Rutgers University, New Brunswick, NJ. [98] Gertrude Blanch and Ida Rhodes, “Table-making at the National Bureau of Standards”, Studies in Numerical Analysis.B.K.P.Scaife, editor, New York, NY: Royal Irish Academy and Academic Press, 1974. Festschrift, presented to Cornelius Lanczos on the occasion of his eightieth birthday, pp. 1-6. 156 Bibliography

[99] Ibid. pp. 4-5. [100] Ibid. p. 3. [101] Edited from material provided by Israel Rotkin. [102] Cornelius Lanczos, History of Mathematics, Unpublished lecture notes, Prologue, p. 1. [103] Cornelius Lanczos, “Why Mathematics?” Lecture given at the An- nual Meeting of the Irish Mathematics Association, October 31, 1966, Belfield, Dublin. Irish Mathematics Teachers Association Newsletter, 9, 1967, p. 1. [104] Ibid. [105] Cornelius Lanczos, Linear Differential Operators, Dover Publications, New York, 1997, Preface, p. vii. [106] Cornelius Lanczos, “The Inspired Guess in the History of Physics”, Studies, 53, 1964, p. 408. [107] See [103]. [108] Ibid. p. 4. [109] Ibid. [110] See [103], p. 2. [111] Einstein’s letter to Lanczos, January 24, 1938, EA 15-269. [112] Albert Einstein, “Physics and Reality”, Journal of the Franklin In- stitute, 221, 1936, pp. 349-382. [113] See [103], p. 5. [114] Einstein’s letter to Lanczos, September 11, 1935, EA 15-246. [115] Lanczos’s letter to Einstein, September 14, 1935, EA 15-248. [116] Lanczos’s letter to Einstein, October 20, no year (probably from 1931), EA 15-243. [117] Cornelius Lanczos, “Electricity as a Natural Property of Riemannian Geometry”, Physical Review, 39, 1932, pp. 716-736. [118] Cornelius Lanczos, “Matter Waves and Electricity”, Physical Review, 61, 1942, pp. 713-720. [119] Lanczos’s letter to Einstein, March 17, 1942, EA 15-293. [120] Einstein’s letter to Lanczos, March 21, 1942, EA 15-294. [121] Lanczos’s letter to Einstein, December 22, 1945, EA 15-309. [122] “News Bureau” report, September 13, 1944, pp. 1-2. Boeing Airline Company, Seattle, Washington. [123] Cornelius Lanczos, Discourse on Fourier Series, Oliver & Boyd, Ed- inburgh and London, 1966. Bibliography 157

[124] For details of the history of the INA see John Todd, “The Prehis- tory and Early History of Computation at the U.S. National Bureau of Standards,” A History of Scientific Computing, Stephen G. Nash, ed., Addison-Wesley, 1990, pp. 251-263. See also Magnus Hestenes and John Todd, NBS-INA—The Institute for Numerical Analysis— UCLA 1947–1954, National Institute of Standards and Technology (NIST), Special Publication 730, U. S. Government Printing Office, Washington, D.C., August 1991. It has been published commercially by the Mathematical Association of America under the title Mathe- maticians Learning to Use Computers. [125] Cornelius Lanczos, “Iteration Method for the Solution of the Eigen- value Problem of Linear Differential and Integral Operators”, Journal of Research of the National Bureau of Standards, Vol. 45, No. 4, Oc- tober 1950, Research Paper 2133. [126] Ibid. p. 255, footnote 4. [127] For detailed discussion of the Krylov method in English see D. K. Faddeev and V. N. Faddeeva, Computational Methods of Linear Al- gebra, W. H. Freeman and Company, San Francisco and London, 1963, pp. 231-241. [128] Alston S. Householder, The Theory of Matrices in Numerical Analy- sis, Dover Publications Inc., New York, 1964, pp. 149-163. [129] R. W. Humming, “The Frequency Approach to Numerical Analysis”, Studies in Numerical Analysis, Festschrift for Cornelius Lanczos’s 80- th birthday, Royal Society, 1973, p. 152. [130] Istv´an Kardos, “L´anczos Korn´el”, Otszemk¨¨ ozt tud´osokkal [Television Interviews with Scientists] (in Hungarian), Budapest, MRT-Minerva, 1974, p. 442. [131] For a list of references see, for example, G. H. Golub and C. F. van Loan, Matrix Computations, John Hopkins University Press, Balti- more, 1996, 475-504. [132] Jack Dongarra and Francis Sullivan, “Top 10 Algorithms”, Comput- ing in Science & Engineering, 2, 1, (2000), pp. 22-23. [133] Henk A. van der Vorst, “Krylov Subspace Iteration”, Computing in Science & Engineering, 2, 1, (2000), pp. 32-37. [134] Lanczos’s letter to Einstein, March 29, 1954, EA 15-326. The letter suggests that an FBI file on Lanczos should exist. In August of 1990 in an interview with John Todd at Caltech in Pasadena, California, the author was told that Lanczos’s disloyalty charge was completely separate from the security investigation of the INA. Todd, however, had no information whatsoever about the nature of Lanczos’s dis- loyalty accusation. FOIA (Freedom of Information Act) requests to 158 Bibliography

track down the location of Lanczos’s FBI file were sent to the legal successor of the NBS-INA. They have not succeeded up to this time. [135] Lanczos’s letter to Einstein, August 18, 1953, EA 15-323. [136] Abraham Flexner, I Remember: The Autobiography of Abraham Flexner. New York: Simon and Schuster, 1940. Quotation from Wal- ter Moore, Schr¨odinger, Life and Thought, Cambridge University Press, Cambridge, 1989, p. 353. [137] Walter Moore, Schr¨odinger, Life and Thought, Cambridge University Press, Cambridge, 1989, p. 368. [138] John J. O’Connor and Edmund F. Robertson, “William Rowan Hamilton”, in Physicists of Ireland, Mark McCartney and Andrew Whitaker, eds., Institute of Physics Publishing, Bristol and Philadel- phia, 2003, pp. 61-68. [139] Cornelius Lanczos, “William Rowan Hamilton—an Appreciation”, American Scientists, 55, 2, 1967, pp. 129-143. [140] Cornelius Lanczos, “Lagrange Multiplier and Riemannian Spaces”, Reviews of Modern Physics, 21, 1949, pp. 497-502. [141] Cornelius Lanczos, “Electricity and General Relativity”, Reviews of Modern Physics, 29, 1957, pp. 337-350. [142] Cornelius Lanczos, “Electricity as a Natural Property of Riemannian Geometry”, Physical Review, 39, 1932, pp. 716-736. [143] See [141], p. 345. [144] Cornelius Lanczos, “Matter Waves and Electricity”, Physical Review, 61, 1942, pp. 713-721. [145] See [141], p. 350. [146] Cornelius Lanczos, “Electricit´e et Relativit´eG´en´erale” [Electricity and General Relativity], Cahiers de physique, 95, 1958, pp. 247- 255. For the English translation see [207], Volume IV, pp. 2-1733 – 2-1749. [147] See [151], p. 672. [148] See [105], p. 132. [149] E. H. Moore, “On the Reciprocal of the General Algebraic Matrix,” Bulletin of the American Mathematical Society, 26, 1920, pp. 394-395. [150] R. Penrose, “A Generalized Inverse for Matrices”, Proc. Cambridge Philos. Soc., 51, 1955, pp. 406-413. [151] Cornelius Lanczos, “Linear Systems in Self-Adjoint Form”, The American Mathematical Monthly, 65, 1958, pp. 665-679. [152] “Preface” and “William Chauvenet” in “The Chauvenet Papers, A Collection of Prize-Winning Expository Papers in Mathematics”, J. C. Abbott, ed., The Mathematical Association of America, 1978, Vol. 1. Bibliography 159

[153] G. W. Stewart, “On the Early History of the Singular Value Decom- position”, SIAM Review, 35, 1993, pp. 551-566. [154] See for example, G. H. Golub and W. Kahan, “Calculating the Singu- lar Values and Pseudo Inverse of a Matrix”, SIAM J. Numer. Anal- ysis, 2, 1965, pp. 205-224. [155] Cornelius Lanczos, “Extended Boundary Value Problems”, Proceed- ings of the International Congress of Mathematicians (Edinburgh, August 14–21, 1958). Cambridge: Cambridge University Press, 1960, pp. 154-181. [156] See [76]. [157] See [151], p. 677. [158] See [105], pp. 432-435. [159] Ibid, pp. 508-511. [160] Interview with Werner Israel. See [35]. [161] Ibid. [162] Ronald Butler, “The Lanczos Legend”, Computers & Mathematics with Applications, 1, 1975, pp. 258-259. Pergamon Press, 1975. [163] James R. McConnell, “Cornelius Lanczos in Dublin (1953–1974)”, Computers and Mathematics with Applications, 1, 1975, pp. 263-264. [164] Einstein’s letter to Lanczos, January 10, 1946, EA 15-310. [165] For Synge’s biography see Petros S. Florides, “John Synge 1897– 1995”, in Physicists of Ireland, Mark McCartney and Andrew Whitaker, eds., Institute of Physics Publishing, Bristol and Philadel- phia, 2003, pp. 208-219. [166] Eduardo L. Ortiz, “Lanczos and the Institute for Advanced Studies in the Early Sixties”, Computers and Mathematics with Applications, 1, 1975, pp. 264-265. [167] Interview with Werner Israel. See [35]. [168] Cornelius Lanczos, The Einstein Decade, Elek Science, London, 1974, p. 32. [169] Einstein’s letter to Lanczos, September 11, 1953, EA 15-324. [170] Interview with Werner Israel. See [35]. [171] Ibid. [172] Ibid. [173] Ibid. [174] Cornelius Lanczos, “Splitting of the Riemann Tensor”, Reviews of Modern Physics, 34, 1962, pp. 379-89. 160 Bibliography

[175] Proceedings of the Cornelius Lanczos International Centenary Con- ference, J. D. Brown, M. T. Chu, D. C. Ellison, and R. J. Plemmons, eds., Philadelphia, PA: SIAM, 1994. (Conference held at North Car- olina State University, Raleigh, NC, December12–17, 1993.) [176] Cornelius Lanczos, “Undulatory Riemannian Spaces”, Journal of Mathematical Physics, 4, 1963, pp. 951-959. [177] Ibid., p. 959. [178] Ronald Butler, “The Lanczos Legend”, Computers & Mathematics with Applications, 1, 1975, p. 258. [179] Charles W. Clenshaw, “Commentary on Lanczos’s ‘Solution of Ordinary Differential Equations by Trigonometric Interpolation’ ”, William R. Davis, general ed., Vol. 6, Lanczos Collection, p. 3-665, North Carolina State University, Raleigh, North Carolina, 1999. [180] Ibid. [181] Cornelius Lanczos, Space Through the Ages, Academic Press, London, 1970, Preface, p. 2. [182] These videotape recordings include the following: Autobiographi- cal Discourse. His Education, Life and Career; Discussion Interview: “Professor Lanczos’s Contribution to Mathematics”; Discussion In- terview: “Albert Einstein: the Man and Relativity”; Lectures I and II of “The Life and Work of Albert Einstein: Cosmic World Order”. Copyright holder: Department of Mathematics, University of Manch- ester Institute of Science and Technology, Manchester, England. [183] Cornelius Lanczos, “Electricity as a Natural Property of Riemannian Geometry”, Physical Review, 39, 1932, pp. 716-36. [184] Cornelius Lanczos, “Quadratic Action Principle of Relativity”, Jour- nal of Mathematical Physics, 10, 1969, pp. 1057-1065. [185] Ibid, p. 1065. [186] Cornelius Lanczos, “Vector Potential and Quadratic Action”, Foun- dations of Physics, 2, 1972, p. 284. [187] See [168]. [188] Albert Einstein, “Antwort auf eine Abhandlung M. v. Laue ‘Ein Satz der Wahrscheinlikeitsrechnung und seine Anwendung auf die Strahlungstheorie’ ”, (“Answer to M. v. Laue’s essay ‘A Theorem of Probability Calculus and Its Application to the Theory of Ra- diation’ ”), Annalen der Physik, 47, 1915, pp. 879-885. For a brief summary in English see [168], p. 208. [189] G. Marsaglia and T. A. Bray, “A Convenient Method for Generating Normal Variables”, SIAM Rev. 6, 1964, pp. 260-264. [190] Lanczos’s letter to Barbara Gellai, April 18, 1974. Bibliography 161

[191] Lanczos’s letter to Barbara Gellai, May 10, 1974. [192] Ibid. [193] Lanczos’s letter to Barbara Gellai, May 17, 1974. [194] Werner Israel, “Dark Stars, the Development of an Idea”, in Three Hundred Years of Gravitation, Stephen Hawking and Werner Israel, eds., Cambridge University Press, Cambridge, 1987, pp. 231-239. [195] Cornelius Lanczos, “Gravitation and Riemannian Space”, Founda- tions of Physics, 5, 1975 p. 17. [196] Lanczos’s letter to Sol Boyk, April 23, 1974. [197] Lanczos’s letter to Barbara Gellai, March 13, 1974. [198] L´anczos Korn´el, “Einstein ´es a j¨ov˝o”, [“Einstein and the Future”], Fizikai Szemle, 24, 1974, p. 166. For the English translation see [207], Volume IV, p. 2-2079. [199] See [198], pp. 2-2069 – 2-2079. [200] “In Memory of Cornelius Lanczos”, Computers and Mathematics with Applications, 1, 1975, pp. 257-268. [201] C. Lanczos and B. Gellai, “Fourier Analysis of Random Sequences”, Computers and Mathematics with Applications, 1, 1975, pp. 269-276. [202] Cornelius Lanczos, “Gravitation and Riemannian Space”, Founda- tions of Physics, 5, 1975, pp. 9-18. [203] Wolfgang Yourgrau, “Cornelius Lanczos (1893–1974)”, Foundations of Physics, 5, 1975, pp. 19-20. [204] Lanczos’s letter to Einstein, March 9, 1947, EA 15-313. [205] Cornelius Lanczos, Preface, in History of Mathematics, p. 4 (unpub- lished lecture notes). [206] See [146], p. 255. For the English translation see [207], Volume IV, p. 2-1749. [207] Cornelius Lanczos Collected Published Papers with Commentaries (briefly referred to as Lanczos Collection ), William R. Davis, Moody T. Chu, Patrick Dolan, James R. McConnell, Larry K. Norris, Ed- uardo L. Ortiz, Robert J. Plemmons, Don Ridgeway, B. K. P. Scaife, William J. Stewart, James W. York, Jr., eds., Wesley O. Doggett, Barbara M. Gellai, Andr´e Gsponer, Carmine A. Prioli, associate eds., College of Physical and Mathematical Sciences, North Carolina State University, Raleigh, North Carolina, 1998.

Index

Abbott, J. C., 158 Cauchy problem, 154 Abramowitz, Milton, 50, 63 characteristic polynomial, 83, 84, absolute motion, 14 87 absolute time, 14 charged particles, 77 action principle, 97, 118 Chauvenet Prize, 100 action-at-a-distance, 12, 14, 15, 18 Chauvenet, William (1820–1870) , Alexander, Bernath, 17 100 Chebyshev polynomials, 48 B´ek´esy, Georg von, 3 Chu, Moody T., 161 Beltrami, Eugenio (1835–1899), 100 Clark, R. C., 144 Berlin University, 19 Clenshaw, Charles William big-bang model, 25 (1926–2004), 114 Blanch, Gertrude, 63, 64 Cohen, R. S., 143 Boeing Aircraft Company, 81, 83, Condon, Edward U. (1902–1974), 86, 95, 113, 130, 156 89, 141 Bohr, Niels (1885–1962), 24, 34, 35 conservation law, 76 Born, Max (1882–1970), 24, 35, 40 continuum formalism, 39 boundary value problem, 31, 101, correspondence principle, 35 102, 141, 143, 144 Cosmic Background Explorer Boyk, Sol, 120 (COBE), 26 Boyks (Sol and Eleanor), 53 cosmological constant, 97 Broglie, Louis de (1892–1987), 34, Coulomb, Charles (1736–1806), 12 38, 96 Coveney, Peter, 150 Brown, J. D., 160 Curtiss, J. H., 86 Budapest Institute of Technology, curvature tensor, 75, 109 18, 21, 129 Budapest University, 18 Danielson, Gordon Charles Butler, Ronald, 113, 116 (1912–1983), 59, 61, 62, 138, 139, 155 Cardelli, Francesco, 143 dark stars, 120

163 164 Index

Davis, William R., 114, 117, 150, Ford Motor Company, 130 160, 161 Fourier amplitudes, 119 de Hevesy, George (1885–1966), 3 Fourier analysis, 59–61, 118, 119, de Sitter, Willem, 25 122 de Valera, Eamon, 90, 93, 94 Fourier coefficients, 51, 59, 60, 118, Dehn, Max, 32 119 density of probability, 40 Fourier functions, 48 Derrick,G.H.,144 Fourier phase angles, 119 digital computers, 65, 88 Frank, Tibor, 151 Dirac delta function, 40 Frankfurt Mathematics Seminars, Dirac, Paul Adrien Maurice 32 (1902–1984), 39 Fredricksson, Einar, 145 displacement current, 13 distant parallelism, 28, 29, 75 G´abor, Dennis, 3 Doggett, Wesley O., 161 Gamma function, 142 Dongarra, Jack, 157 Gauss, Karl Friedrich (1777–1855), doubling algorithm, 61 69 Dublin Institute for Advanced general relativity, 2, 25, 27, 71, 74, Studies (DIAS), 89, 94, 103, 75, 95–97, 109, 135, 136, 138, 113 140, 141, 145, 151, 152, 158 dynamic world picture, 77 generalized inverse, 100, 158 geodesic line, 27 Eddington, Sir Arthur, 15 global smoothing, 51 eigenvalue problem, 83 Glockner, P. G., 146 Einstein, Albert (1879–1955), 21, Goldstein, Andrew, 62 40, 55, 140–142 Golub, Gene H., 157, 159 electromagnetic waves, 13, 34, 38 gravitation effect, 12 Ellison, D. C., 160 gravitation theory, 12 E¨otv¨os Lorand University, 121 gravitational equations, 118, 133, E¨otv¨os, Baron Lorand, 2, 17, 18 137, 152 equivalence hypothesis, 14 gravitational mass, 14 ether, 13, 14, 134, 147 Green’s function, 37, 38 Greene, Brian, 150 Faddeev, D. K., 157 Gsponer, Andr´e A., 161 Faddeeva, V. N., 157 Guendelman, E. I., 152 Faraday, Michael (1791–1867), 12 Guth, Alan H., 151 Fast Fourier Transform (FFT), 61 Federal Bureau of Investigation Haar, Alfred (1885–1933), 2 (FBI), 89 Hajime, Hayashi, 145 field, 12, 14, 16, 19, 27, 36, 39, Hamilton, William Rowan 96–98, 118, 120 (1805–1865), 19, 39, 76, 95, field equations, 25, 27, 76, 95, 96, 103, 105, 138, 143, 158 98, 110, 117, 118, 120 hamiltonization, 95 field-theoretical representations, 37, Hanak, Peter, 151 135, 136, 153 Hardy, Godfrey Harold Flexner, Abraham, 158 (1877–1947), 47 Florides, Petros S., 159 Haruichi, Yabuki, 146 force of gravity, 14, 15 Havas, Peter, 152 Index 165

Hawking, Stephen, 161 Koenig, H., 155 heat radiation, 34 Krylov subspace, 87, 88, 157 Heisenberg, Werner Karl (1901–1976), 23, 35, 38, 39, Laderman, Jack, 63 152, 153 Lagrange multiplier, 95, 158 Hellinger, Ernst, 32 Lagrange, Joseph Louis Hersh, Reuben, 149 (1736–1813), 95 Hertz, Heinrich (1857–1894), 13 Lagrangian, 75, 97, 117, 139 Hestenes, Magnus R., 139, 157 Laguna-Castillo, P., 152 Highfield, Roger, 150 Lakatos, D´enes, 149 Hildebrand, Ilse, 90 Lanczos Method, 49, 79, 81, 87, 88, Hoffman, Paul, 154 100, 127 Hopkins, H. G., 125, 157 Lanczos tensor, 110 Householder, Alston S., 87 Lanczos, (born Rupp) M´aria, 41, Howard, D., 152 46, 53 Humming, R. W., 157 Lanczos, Alice, 155 Lanczos, Anna (1896–1944), 6, 43, ill-posed problems, 51 65 indefinite signature, 112 Lanczos, Elmar (1933–1982), 52, inertial mass, 14 90, 122 inflationary bubble, 26 Lark-Horovitz, Karl (1892–1958), inflationary cosmology, 26 41, 45, 46, 55, 59, 81, 154 inflationary model of the universe, lattice structure, 118, 120 25 Laue, Max von (1879–1960), 23, inner necessity, 57 118 Institute for Advanced Study of light deflection, 120 Princeton, 93 line element, 106, 107, 109, 110, Institute for Numerical Analysis, 112, 120, 121 85, 89, 157 Lorentz condition, 118 integral equations, 32, 36, 37, 101, Lorentzian line element, 107 134 L¨owy, (born Hahn) Ad´el, 6 Ipser, J., 152 L¨owy, K´aroly, 5 Isaacson, Eugene, 63 Lukacs, John, 149 Israel, Werner, 25, 107, 151, 159, 161 Madelung, Erwin, 31 Janick, Jules, 154 Magnus, Wilhelm, 31, 32, 152 John-Steiner, Vera, 149 Marchand, Ren´e, 145 Jordan, Camille (1838–1921), 100 Marx, George, 150 Jordan, Pascual (1902–1980), 39 Marx, Gy¨orgy, 125 Jungk, Robert, 151 material particle, 76–78, 98 Mathematical Association of Kahan, W., 159 America, 100, 157 Kardos, Istv´an, 157 Mathematical Tables Project, 50, Karman, Theodore von 63–65, 130 (1881–1963), 3 matrix inversion, 83 Karush, W., 139 matrix representation, 37 kernel function, 37, 39, 40 matter waves, 38, 76, 77, 89, 95, Koenig, G, 60 97, 110, 139, 156, 158 166 Index

Matzner, R. A., 152 Oppenheimer, Robert (1904–1967), Maxwell equations, 34, 96 24 Maxwell, James Clerk (1831–1879), O’Raifeartaigh, L., 145 13 Oregon State College, 130 McCarthy, Joseph (1908–1958), 89 Ortiz, Eduardo L., 50, 159, 161 McCarthyism, 89 Ortvay, Rudolf (1885–1945), 20, McCartney, Mark, 158, 159 154 McConnell, James R. (1915–1999), Ostrowski, Alexander M., 85, 86 103, 159, 161 Mehra, Jagdish, 154 Pais, Abraham, 40, 149 Merza, J´ozsef, 146 Pauli, Wolfgang (1900–1958), 23, metric, 28, 77, 107 39, 40 metrical plateau, 76, 77, 110 Peace Treaty of Trianon, 21 metrical tensor, 77, 117 Penrose, Roger, 100, 158 Michelson-Morley experiment, 13, photoelectric effect, 34 14 Planck’s constant, 34 Miller, J. J. H., 145 Planck, Max (1858–1947), 21, 34 minimized iterations, 84, 87, 140 Plemmons, R. J., 160 P´olya, Gy¨orgy (George) Minkowski, Hermann (1864–1909), (1887–1985), 2, 142 15, 106 positive definite signature, 107, 111 Minkowskian world, 121 power series, 48, 49, 126 Misner,C.W.,151 principal axes, 83, 84 Moore, E. H., 99, 158 principle of motion, 135, 152 Moore, Walter, 158 Prioli, Carmine A., 161 NASA (National Aeronautics and problem of motion, 26, 28, 152 Space Administration), 26 pseudo inverse, 159 Purdue University, 41, 45, 46, 64, National Bureau of Standards, 63, 68, 130, 154 65, 85, 86, 89, 130, 140, 157 National Defense Research Council, quadratic action principle, 75–77, 63 96, 116, 117, 144, 160 National Institute of Standards and quantum condition, 34, 37, 135 Technology, 157 quantum mechanics, 20, 31, 35, 38, National University of Ireland, 94, 40, 97, 122, 135, 153 131 quantum phenomena, 35, 74, 96, natural radiation, 118 110, 127 Nature’s Pythagorean theorem, quaternion calculus, 19 105–107 Neumann, John von (1903–1957), 3 random noise, 120 Newton’s equations, 12 random sequences, 118, 119, 122, Newton’s second law, 27 125, 126, 146, 161 Newton, Isaac (1642–1727), 150 Ravid, Benjamin, 149 Norris, Larry K., 161 Rechenberg, Helmuth, 153 North American Aviation, 89, 130 rectangular matrix, 98 North Carolina State University, reference system, 14, 110 114, 118, 130, 150, 160, 161 resonance effect, 77 Rhodes, Ida (1900–1986), 63–65, O’Connor, John J., 158 155 Index 167

Ridgeway, Don, 161 Szil´ard, Le´o (1898–1964), 3, 27 Riemann geometry, 109 Sztr´okay, K´alm´an, 145 Riemann, Bernhard (1826–1866), 109 Tangl, K´aroly (1869–1940), 18 Riemann-Christoffel tensor, 138 Tau method, 49, 50 Riemannian space-time, 75 Taussky-Todd, Olga (1906–1995), Riesz, Frigyes (1880–1956), 2 85, 86 Robertson, Edmund F., 158 Taylor expansion, 48–50 Rodin, Ervin Y., 126 Taylor series, 48, 49, 126 Rosser, J. B., 139 Teller, Edward (1908–2003), 3 Rotkin, Israel, 156 theory of relativity, 13–19, 23–25, rounding errors, 83, 84 27, 28, 31, 34, 47, 70, 73, 74, Royal Irish Academy, 94, 125, 131, 96, 105, 134–137, 140, 147, 150 145, 152 theory of special relativity, 19 Rudnick, Philip, 62, 155 Todd, John, 85, 86, 157 Runge, Karl David (1856–1927), trigonometric interpolation, 138, 60, 155 141, 160 Rutherford, Ernst (1871–1937), 34 trigonometric truncation, 50, 51, 140 Salam, Abdus, 145 Trinity College, 94, 95, 131, 140 Saltzer, Herbert E., 63 Tukey, John (1915–2000), 61, 155 Scaife, B. K. P., 152, 155, 161 Ugarov, V. A., 144 Scheiber, S´andor, 122 undulatory Riemannian spaces, Schr¨odinger, Erwin (1887–1961), 110, 111, 142, 160 38, 89, 94, 103, 153 unified field theory, 16, 24, 74, 75, Schwarzshild, Karl, 120 89, 95, 96, 120, 127 Siegel, Carl Ludwig (1896–1981), uniform gravitation field, 14 32, 152 University of Alberta, 151 Sikive P., 152 University of Budapest, 2, 16, 17, Singh, M. C., 146 129 Singular Value Decomposition, 100 University of Calgary, 122, 146 singular values, 159 University of California, 85, 130, smoothing of noisy data, 51, 140 150 Sommerfeld, Arnold (1868–1951), University of Frankfurt, 31, 32, 23 131, 152 Stachel, J., 152 University of Freiburg, 24, 129 Stanosz, Barbara, 144 University of Lancaster, 131 Stegun, Irene A., 50 University of Manchester Institute Stewart, G. W., 100, 159 of Science and Technology, 116 Stewart, William J., 161 University of Michigan, 130 successive orthogonalization, 84 University of Munich, 23 Sugar, Peter F., 151 University of Szeged, 129 Sullivan, Francis, 157 University of Washington, 84, 130 Sur´anyi, Istv´an, 149 University of Zurich, 38 Synge, John Lighton (1897–1995), 105 van der Vorst, Henk A., 157 Sz´asz, Ott´o, 2, 32, 85 van der Waerden, Bartel Leendert, Szeg¨o, G´abor (1895–1985), 2 39 168 Index van Loan, C. F., 157 variational principles, 108, 139, 143, 144 vector potential, 76, 96, 116, 118, 137, 145, 146, 160

Wartofsky, M. R., 143 wave equation, 38, 118 wave function, 38–40 wave mechanics, 38–41, 47, 138, 154 well-posed problems, 101 Wentzel, Gregor (1898–1978), 23 Weyl, Herman (1885–1955), 75 Wheeler, John, 151 Whitaker, Andrew, 158, 159 white noise, 119 Whitehead, Alfred North (1861–1947), 105, 106 Wigner, Eugene (1902–1995), 3 World War, 20, 53, 63 wrinkled plateau, 76, 78, 98

Yale University, 130, 154 York, James W., 161 Yourgrau, Wolfgang, 126, 161 This book recounts the extraordinary personal journey and scien- tific story of Hungarian-born mathematician and physicist Cornelius Lanczos. His life and his mathematical accomplishments are inextri- cably linked, reflecting the social upheavals and historical events that shaped his odyssey in 20th-century Hungary, Germany, the United States, and Ireland. In his life Lanczos demonstrated a remarkable ability to be at the right place, or work with the right person, at the right time. At the start of his scientific career in Germany he worked as Einstein’s assistant for one year and stayed in touch with him for years thereafter. Reacting to anti-Semitism in Germany in the 1930s, he moved to the United States, where he would work on some of the earliest digital computers at the National Bureau of Standards. After facing suspicion of Communist sympathies during the McCarthy era in the 1950s, Lanczos would relocate once again, joining Schrödinger at the Dublin Institute for Advanced Studies. Gellai’s biography analyzes a rich life and a body of work that reaches across many scientific disciplines. Lanczos made important contributions to several areas of mathematics and mathematical physics. His first major contribution was an exact solution of the Einstein field equations for gravity (in general relativity). He worked out the Fast Fourier Transform, but since there were no machines on which to run it, this accomplishment would be forgotten for 25 years. Once he had access to computers, Lanczos independently rediscovered what is now known as the singular value decomposition, a fundamental tool in numerical methods. Other significant contribu- tions included an important discovery about the Weyl tensor, which is now known as the Lanczos potential, and an important contribution on algorithms for finding eigenvalues of large matrices.

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