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Biographical Memoirs: Volume 61 This page in the original is blank.
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Biographical Memoirs: Volume 61 NORBERT WIENER November 26, 1894-March 18, 1964 BY IRVING EZRA SEGAl NORBERT WIENER was one of the most original mathematicians and influential scientists of the twentieth century. He developed a new, purely mathematical theory, an integral calculus for functions of infinitely many variables known as functional integration. It has been of great importance for probability and theoretical physics. Wiener made huge strides in the harmonic analysis of functions of real and complex variables. In a unified way, this resolved old problems, produced new challenges, and provided a prototype for key aspects of harmonic analysis on topological groups. In part concurrently, he developed applications of his mathematical ideas in engineering, biology, and other fields. In later life he developed a synthesis of such applications with diverse ideas represented by central parts of the work done in the twenties and thirties by Vannevar Bush, Walter B. Cannon, Alan Turing, and others. This synthesis, which he called ''cybernetics," has since been a productive unifying philosophy in science and engineering. In the United States, it primarily epitomized his earlier contributions to communication engineering; in Britain, it had a notable impact on neurophysiology, and its delayed, but eventually enthusiastic, acceptance in the Soviet Union stimulated important mathematical developments
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Biographical Memoirs: Volume 61 in control and ergodic theory. Before and while cybernetics was being developed, Wiener was a prime mover in multidisciplinary groups in these subjects. As much as anyone, he showed the importance of higher mathematics for fundamental applications, and the general scientific effectiveness of the mathematical way of thinking. At the same time, in association with his work, he elaborated philosophical and social ideas that influenced world culture. ORIGIN Norbert's paternal grandfather, Solomon Wiener, was a journalist and teacher of German background who worked in Poland. Norbert wrote of him that he sought to encourage the replacement of Yiddish by German among the Jews there. Norbert's father, Leo Wiener, was born in Bialystok, Poland, in 1862. Leo was related through his mother, Freda Wiener, to Leon Lichtenstein, a well-known German mathematician, as first cousin. Norbert later met Lichtenstein in Europe, and it is interesting that Lichtenstein's central interests of applied mathematics and potential theory came to be important ones for Norbert. Leo Wiener studied engineering in Berlin and medicine in Warsaw. At the age of eighteen he emigrated to the United States. He had had a plan to join in an undertaking to found a utopian community along Tolstoyan lines in Central America. which fell through when his partner backed out. However, in this connection, apparently, he disembarked at New Orleans. After a succession of employments and travels, he became professor of modern languages at the University of Missouri. Norbert's maternal grandfather, Henry Kahn, had a department store in Missouri, to which he had emigrated from Germany. Kahn's wife came from a family named Ellinger, which had been settled in the United States for
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Biographical Memoirs: Volume 61 some time. Their daughter, Bertha Kahn, married Leo Wiener in 1893. Their first child, Norbert, was born a year later on November 26, 1894, in Columbia, Missouri. The name Norbert was taken from a work of Robert Browning, the couple being thought to have met at a Browning club. Within the following year, Leo Wiener lost his position, apparently as a result of university politics. He decided to move to the Boston area in search of employment, and found an apartment in Cambridge. After a variety of positions, he obtained a part-time instructorship in Slavic languages at Harvard University. In conjunction with ancillary positions at Boston University and the New England Conservatory, among others, this provided a livelihood during the earlier years in Cambridge. Eventually he became a tenured professor of Slavic languages at Harvard University, a position he held until his retirement. In the first volume of his unusually intimate autobiography, Norbert gave a portrait of his parents, and especially of his father. He was highly adulatory, but at the same time displayed some intellectual but principally emotional reservations He indicated that his father had placed somewhat excessive pressure on him and had not given him sufficient credit for his intrinsic merits. Instead, he felt, his father attributed his son's precocity and brilliance to his upbringing in accordance with the educational and social ideas he had espoused. By other accounts, Leo Wiener was exceptionally original, imaginative, and productive intellectually. At the same time, he was a fine teacher and socially very broadly involved. His primary profession was that of a linguist and philologist, and he attained very high distinction in these fields. However, he developed original ideas in quite different areas, for example geology, but his theories had few followers in his day.
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Biographical Memoirs: Volume 61 There is no question that Leo Wiener was quite concerned about the intellectual development of Norbert and his other children. Early on, he taught Norbert mathematics, languages, and other subjects. He put Norbert in touch with many outstanding intellects that later influenced him. A typical example of this was his taking Norbert to visit the laboratory of his friend, Walter B. Cannon. Cannon's concept of homeostasis was later to form one of the crucial pillars of Wiener's Cybernetics. In 1898 the family had a second child, Constance. She later married Professor Philip Franklin, a mathematician at the Massachusetts Institute of Technology. A boy was born in 1900, but died in infancy. In 1901, the family visited Europe, following which Norbert entered third grade in a public school in Cambridge. However, after quickly advancing to the fourth grade, he was removed from the school by his father until he entered high school, two years later. Meanwhile, his second sister, Bertha, was born in 1902. She later married Professor Carroll W. Dodge, a botanist at Washington University in St. Louis, Missouri. Besides these two girls, the family eventually included four boys, of whom two died in infancy. EDUCATION In 1906, Norbert graduated from high school in Ayer, Massachusetts, and entered Tufts College to study biology and mathematics. His brother Frederick was born in the same year. In 1909, Norbert was graduated from Tufts College with a cum laude A.B. degree. He then entered Harvard Graduate School with the intention of studying zoology. However, the emphasis on laboratory work in this subject turned out to be inappropriate for him, and a year later he transferred to Cornell University, where he had been given a scholarship in the Sage School of Philosophy. He stud-
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Biographical Memoirs: Volume 61 ied there with Frank Thilly, a friend of his father's from the Missouri days who had facilitated Norbert's transfer to Cornell, and Walter A. Hammond and Ernest Albee. But the work there again did not proceed really well, and a year later Norbert transferred back to the Harvard Graduate School. However, he stayed with the subject of philosophy. He studied with Edward V. Huntington, G. H. Palmer, Josiah Royce, and George Santayana, some of the well-known philosophers of the time. He received an M.A. degree in 1912, but because of Royce's diminishing health, worked for his doctorate with Karl Schmidt of Tufts, who as a young professor was interested in mathematical logic. In 1913, Norbert was graduated from Harvard with a Ph.D. in philosophy. In the meantime, the last child to be born to the family (in 1911) died in infancy. Norbert was living at home while at graduate school, and had responsibility for the care of his brother Frederick. Family pressures were strong and burdensome for Norbert. He was quite pleased when the traveling fellowship for which he had applied was awarded to him by Harvard. He contemplated going to Cambridge, England, to work with Russell, and to Turin to work with Peano. Hearing of Peano's decline in scientific activity and considering Russell to be quite active, he decided to go to Cambridge to pursue studies in mathematical logic. POSTDOCTORAL YEARS Norbert was not quite as satisfied with Russell's lectures and their meetings as he had expected. At the same time he found that despite his limited background in mathematics, he was able to pick up quickly on the mathematics lectures with which he supplemented his philosophical studies. This was particularly true of the lectures of G. H. Hardy, which Norbert found absorbing. Hardy was probably the leading
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Biographical Memoirs: Volume 61 English mathematician of his day. Hardy showed less of the reserve that Wiener was sensitive to in some of his other teachers and fellow students, The support and encouragement given by Hardy to the ambitious but uncertain young man seeking a challenging direction in which he could display his intellectual prowess was probably an important factor in Norbert's becoming a mathematician. Hardy's lectures and writings were virtual works of art as much as of science, displaying personal enthusiasm, richness of content, and unsurpassed lucidity. He showed a sincere and effective interest in talented young mathematicians, and he was not put off by eccentricity —most notably in the case of the Indian mathematician, Ramanujan, but also that of Wiener. In effect, he converted Norbert from a relatively diffuse interest in issues of broader relevance to one of concern for mathematical penetration and perfection. Hardy became somewhat in loco parentis to Norbert, and played this role intermittently for two decades afterward. However, Hardy seems never to have understood the side of Wiener that was deeply attracted to the broad issues of science and to possible applications; he even raised the question of whether Wiener's apparent concern about the latter was not a pose. From Hardy's position, somewhat that of a gentlemanly, latter-day scientific aesthete, such a pose would have been acceptable, while a true interest in applications would have been quite irrelevant. But however much Wiener's professional career depended on his prowess in pure mathematics, his later work was to display quite convincingly and effectively a profound concern for issues of external relevance. With Hardy's support, Wiener was to become better known for his work on Tauberian theorems than for his earlier and probably more innovative work on Brownian motion, which was outside the mainstream of mathematics during the twenties.
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Biographical Memoirs: Volume 61 During the second semester of Norbert's fellowship year, Russell was away. As a consequence, he went to Göttingen for an extended visit. There Wiener attended the lectures of David Hilbert and Edmund Landau in mathematics, and those of Edmund Hüsserl in philosophy, before returning to Cambridge. The outbreak of World War I led to his return to the United States, where he completed a second traveling fellowship he had been awarded at Columbia University. He studied there with John Dewey, among others. Following this, Norbert received a junior position at Harvard University. During 1915-16 he lectured there on the logic of geometry. During 1916 he served with Harvard's reserve regiment at the Officer's Training Camp in Plattsburg, New York. In 1917 he served with the Cambridge R.O.T.C. During these years he also worked variously as an instructor in mathematics at the University of Maine in Orono (1916-17); as an apprentice engineer in the Turbine Department of the General Electric Corporation in Lynn, Massachusetts (1917); and as a staff writer for the Encyclopedia Americana in Albany, New York (1917-18). Both Norbert and his father were strongly and publicly for the Allied cause, despite their ties to German culture. Norbert expressed opposition to "Prussian militarism." He wrote frequently to individual members of his family, and in January 1918 wrote to his father as follows (in part): I think the time has come for me to make a last try to get into military service, and I am writing to ask you for permission.... It is not that I am dissatisfied with my work nor that I have any particular love for a military career, but I hate to think of myself as less of a man than those of my friends who are in the army, and I do not care after this war to look back on myself as a slacker. I cannot be anything but ashamed of myself when I advocate a war that I do not share in. In 1918, Norbert accepted an invitation from Oswald Veblen, who was then an officer in the Army in charge of
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Biographical Memoirs: Volume 61 the computation of ballistic tables, to join him in this work at the Aberdeen Proving Ground, Maryland, as a civilian employee. Veblen's letter noted, in apparent response to Norbert's expressed inclinations, that should he prefer to do the work in a military capacity, such an arrangement might later prove possible. Indeed, Norbert enlisted in the Army as a private some months before the war ended, and continued in the same work. Two months after the war ended, Norbert felt that he was no longer needed. He sought his discharge, and this came through in February 1919. Veblen, who was already a leading mathematician, returned to Princeton University, bringing with him various younger mathematicians who had displayed talent. Norbert would have liked to go with him but did not receive the call. Instead, he was recommended for a position in the Department of Mathematics at the Massachusetts Institute of Technology by Professor W. F. Osgood of Harvard University. Although the Institute had world renown as a school of engineering in 1919, it was then far from being a leading center of mathematical research. Norbert did not regard the recommendation as evidence of attainment of much standing as a mathematician, but he accepted the position that was offered. He remained at the Institute up to the time of his death, and his scientific interaction with it was to prove a great mutual benefit. CENTRAL DECADE I: 1919-29 Wiener's early mathematical papers concerned mathematical logic and its relations to space, time, and measurement. In their physical and empirical concerns they foreshadow some of his mature interests. They display notable seminality and independence, and are quite interesting from a historical perspective. Their publication, in considerable
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Biographical Memoirs: Volume 61 number in the Proceedings of the Cambridge Philosophical Society, was facilitated by Hardy, whose lectures Wiener was attending at the time of his first publication (in mathematics) in 1913. Whether despite or in part because of the turbulence of the war years, the seed of Wiener's scientific innovation began to sprout vigorously soon after. His first major mathematical salient, in what is now called functional integration, began around 1919. In August 1920, in one of his many intimate letters to his sister Constance, he wrote from Paris as follows: I have not been able so far to get in touch with Frechet. I have wired him that I am here and awaiting an answer. I find that I am making a little headway with my problem—integration in function space —and in a way that may have practical application. I define the measure of an interval in it in a way that hitches up with probability theory as it is applied in statistical mechanics, and I have been living in hopes that the Lebesgue integral which I can get from it will be good for something. At any rate, when I meet Frechet, I shall have a peach of a problem to work on. This was the beginning of his work on a mathematical theory of Brownian motion, essentially the theory of ''Wiener space," as others have called it, and the prime example for the modern theory of functions of functions, or for functions of infinitely many variables. The physical theory of Brownian motion had earlier been studied by Einstein and Smoluchowski and was proposed to Wiener as a topic for investigation by Russell; coincidentally and again serendipitously, the problem of integration in function space had been proposed to Wiener by I. A. Barnett, a former student of E. H. Moore, one of the founders of modern American mathematics, who had initiated research on this problem. These earlier approaches were different from Wiener's and had no special relation to Brownian motion, which Russell had earlier suggested as a topic for investiga-
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Biographical Memoirs: Volume 61 tion. Wiener was quick to see that the conjunction of the integration in function space idea with the normal probability law established in the physical theory of Brownian motion led to an extremely incisive and interesting mathematical development, which at the same time dovetailed beautifully with the qualitative aspects of the physical theory noted by Perrin. During the early 1920s, Wiener sank his roots deeper into functional integration, while at the same time making significant contributions in a variety of other parts of analysis. The novelty of his Brownian motion theory was such that it was not at all widely appreciated at the time, and the few who did, such as H. Cramer in Sweden and P. Levy in France, were outside the United States. He became somewhat better known for his work in potential theory. This was a traditional field, unlike functional integration, and although his work connected most closely with work of Lebesgue, Perron, and others in Europe, it seems to have been precipitated by his attendance at the lectures of O. D. Kellogg at Harvard, which both informed Wiener and aroused his interest. In a remarkably short period of time, of the order of two years, Wiener made a series of brilliant contributions that fundamentally altered the subject, which was never the same thereafter. He developed a fruitful concept of generalized solution to the Dirichlet problem (that of the solution of the Laplace equation attaining prescribed values on the boundary of the region in question). He was led thereby to a general notion of capacity that has been essential for modern potential theory. In a beautiful epilogue to this work, he gave a precise geometrical criterion for the regularity of a boundary point relative to the Dirichlet problem. The theory of almost periodic functions burst on the scene in the twenties with the work of H. Bohr in Copenhagen;
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Biographical Memoirs: Volume 61 Wiener's collected works, exclusive of his books, have been published with commentaries by MIT Press, Cambridge, Massachusetts, edited by P. Masani, in four volumes, 1976-85. A special issue of the Bulletin of the American Mathematical Society, vol. 72, no. 1, part II (1966), was dedicated to Wiener and includes reviews of his works, organized by field. CHRONOLOGICAL SUMMARY 1894 Born November 26 in Columbia, Missouri 1906 H.S. diploma, Ayer High School, Ayer, Massachusetts 1909 B.A. cum laude in mathematics, Tufts College 1909-10 Harvard University 1910-11 Cornell University 1911-13 Harvard University—M.A., 1912; Ph.D., 1913 1913-14 Travelling Fellow, Harvard University; study with Bertrand Russell, Cambridge, England, and with David Hilbert, Göttingen, Germany; awarded Bowdoin Prize (1914) 1914-15 Travelling Fellow, Harvard; study with Bertrand Russell and G. H. Hardy, Cambridge, England, and at Columbia University 1915-16 Harvard University, Docent Lecturer, Department of Philosophy 1916-17 University of Maine, Instructor in Mathematics 1917-18 General Electric Corporation, Lynn, Massachusetts 1918 Staff Writer, Encyclopedia Americana, Albany, New York 1918-19 U.S. Army Aberdeen Proving Ground, Maryland 1919 Editorial Writer, Boston Herald 1919-24 MIT, Instructor in Mathematics 1925-29 MIT, Assistant Professor of Mathematics 1929-32 MIT, Associate Professor of Mathematics; Bocher Prize, American Mathematical Society (1933) 1932-59 MIT, Professor of Mathematics; Lord and Taylor American Design Award (1949); Hon. Sc.D., Tufts College (1946), University of Mexico (1951), and Grinnell College (1957)
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Biographical Memoirs: Volume 61 1959-60 MIT, Institute Professor; ASTME Research Medal (1960) 1960-64 MIT, Institute Professor Emeritus; National Medal of Science (1963) 1964 Died in Stockholm, Sweden, March 18
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Biographical Memoirs: Volume 61 SELECTED BIBLIOGRAPHY 1913 On a method of rearranging the positive integers in a series of ordinal numbers greater than that of any given fundamental sequence of omegas Messenger Math. 43:97-105. 1914 A simplification of the logic of relations. Proc. Cambridge Philos. Soc. 17:387-90. A contribution to the theory of relative position. Proc. Cambridge Philos. Soc. 17:441-49. 1915 Studies in synthetic logic. Proc. Cambridge Philos. Soc. 18:14-28. Is mathematical certainty absolute? J. Philos. Psych. Sci. Method 12:568-74. 1916 The shortest line dividing an area in a given ratio. Proc. Cambridge Philos. Soc. 18:56-8. 1917 Certain formal invariances in Boolean algebras. Trans. Am. Math. Soc. 18:65-72. 1920 Bilinear operations generating all operations rational in a domain Ann. Math. 21:157-65. A set of postulates for fields. Trans. Am. Math. Soc. 21:237-46. Certain iterative characteristics of bilinear operations. Bull. Am. Math. Soc. 27:6-10. On the theory of sets of points in terms of continuous transformations G R. Strasboug Math. Congr., pp. 312-15. The mean of a functional of arbitrary elements. Ann. Math. 22(2): 66-72. 1921 A new theory of measurement: A study in the logic of mathematics. Proc. Lond. Math. Soc. 19:181-205.
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Biographical Memoirs: Volume 61 The isomorphisms of complex algebra. Bull. Am. Math. Soc. 27:443-45 . The average of an analytical functional. Proc. Natl. Acad. Sci. USA 7: 253-60 . The average of an analytic functional and the Brownian movement. Proc. Natl. Acad. Sci. USA 7:294-98 . With F. L. Hitchcock. A new vector method in integral equations. J. Math. Phys. 1:1-20 . 1922 The relation of space and geometry to experience. Monist 32:12-60. 200-47, 364-94 . The group of the linear continuum. Proc. Lond. Math. Soc. 20:329-46 . Limit in terms of continuous transformation. Bull. Soc. Math. France 50:119-34 . With J. L. Walsh. The equivalence of expansions in terms of orthogonal functions. J. Math. Phys. 1:103-22 . A new type of integral expansion. J. Math. Phys. 1:167-76 . 1923 With H. B. Phillips. Nets and the Dirichlet problem. J. Math. Phys. 2:105-24 . Discontinuous boundary conditions and the Dirichlet problem. Trans. Am. Math. Soc. 25:307-14 . Differential-space. J. Math. Phys. 2:131-174 . Note on the series ± 1/n . Bull. Acad. Polon. Ser. A 13:87-90 . Note on a new type of summability. Am. J. Math. 45:83-6 . Note on a paper of M. Banach. Fund. Math. 4:136-43 . 1924 Certain notions in potential theory. J. Math. Phys. 3:24-51 . The Dirichlet problem. J. Math. Phys. 3:127-146 . Une condition nécessaire et suffisante de possibilité pour le problème de Dirichlet. C.R. Acad. Sci. Paris 178:1050-1054 . The average value of a functional. Proc. Lond Math. Soc. 22:454-67 . Un problème de probabilités denom brables. Bull. Soc. Math. France 11:569-78 . The quadratic variation of a function and its Fourier coefficients J. Math. Phys. 3:72-9 .
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Biographical Memoirs: Volume 61 1925 Note on a paper of O. Perron. J. Math. Phys. 4:21-32 . The solution of a difference equation by trigonometrical integrals J. Math. Phys. 4:153-63 . On the representation of functions by trigonometrical integrals. Math. Z. 24:575-616 . Verallgemeinerte trigonometrische Entwicklungen, Göttingen Nachr., pp. 151-58 . A contribution to the theory of interpolation. Ann. Math. 26(2):212-16 . Note on quasi-analytic functions. J. Math. Phys. 4:193-99 . 1926 With M. Born. Eine neue formulierung de Quantengesetze für periodische und nicht periodische Vorgänge. Z. Physik 36:174-87 . With P. Franklin. Analytic approximations to topological transformations. Trans. Am. Math. Soc. 28:762-85 . The harmonic analysis of irregular motion (second paper). J. Math. Phys. 5:158-89 . The operational calculus. Math. Ann. 95:557-84 . 1927 The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical functions, Part I. J. Math. Phys. 6:145-57 . A new definition of almost periodic functions. Ann. Math. 28(2):365-67 . On a theorem of Bochner and Hardy. J. Lond. Math. Soc. 2:118-23 . Une methode nouvelle pour la démonstration des théorème s de M. Tauber. C. R. Acad. Sci. Paris 184:793-95 . On the closure of certain assemblages of trigonometrical functions Proc. Natl. Acad. Sci. USA 13:27-29 . Laplacians and continuous linear functionals. Acta Sci. Math. (Szeged ) 3:7-16 . Une généralisation des fonctions à variation bornee. C.R. Acad. Sci. Paris 185:65-67 . 1928 The spectrum of an arbitrary function. Proc. Lond. Math. Soc. 27(2): 483-96 .
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Biographical Memoirs: Volume 61 A new method in Tauberian theorems. J. Math. Phys. 7:161-84 . Coherency matrices and quantum theory. J. Math. Phys. 7:109-25 . 1929 Harmonic analysis and group theory. J. Math. Phys. 8:148-54 . A type of Tauberian theorem applying to Fourier series. Proc. Lond. Math. Soc. 30(2):1-8 . Hermitian polynomials and Fourier analysis. J. Math. Phys. 8:525-34 . Harmonic analysis and the quantum theory. J. Franklin Inst. 207: 525-34 . 1930 Generalized harmonic analysis. Acta Math. 55:117-258 . 1931 With E. Hopf. Über eine Klasse singularer integralgleichungen. Sitzber. Preuss. Akad. Wiss. Berlin, Kl. Math. Phys. Tech. 1931 , pp. 696-706 . A new deduction of the Gaussian distribution. J. Math. Phys. 10 : 284-88. 1932 Tauberian theorems. Ann. Math. 33:1-100, 787 . 1933 With R. E. A. C. Paley and A. Zygmund. Notes on random functions. Math. Z. 37:647-68 . A one-sided Tauberian theorem. Math. Z. 36:787-89 . With R. E. A. C. Paley. Characters of Abelian groups. Proc. Natl. Acad. Sci. USA 19:253-57 . With R. C. Young. The total variation of g(x + h)-g(x). Trans . Am. Math. Soc. 35:327-40 . With R. E. A. C. Paley. Notes on the theory and application of Fourier transforms I, II. Trans. Am. Soc. 35:348-55; III, IV, V, VI, VII . Trans. Am. Math. Soc. 35:761-91 . The Fourier Integral and Certain of Its Applications . New York: Cambridge University Press. 1934 Random functions. J. Math. Phys. 14:17-23 .
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Biographical Memoirs: Volume 61 A class of gap theorems. Ann. Scuola Norm. Sup. Pisa E( 1934-36): 1-6 . With R. E. A. C. Paley. Fourier Transforms in the Complex Domain . Am. Math. Soc. Colloq. Publ. 19. Providence, R.I.: American Mathematical Society. 1935 Fabry's gap theorem. Sci. Rep. Natl. Tsing Hua Univ. Ser. A 3:239-45 . 1936 A theorem of Carleman. Sci. Rep. Natl. Tsing Hua Univ. Ser. A 3: 291-98 . With S. Mandelbrojt. Sur les séries de Fourier lacunaires. Théorèmes directs. C.R. Acad. Sci. Paris 203:34-36 ; Théorèmes inverses, 233-34 . Gap theorems. C.R. Congr. Intl. Math. , pp. 284-96 . A Tauberian gap theorem of Hardy and Littlewood. J. Chin. Math. Soc. 1:15-22 . 1937 With W. T. Martin. Taylor's series of entire functions of smooth growth. Duke Math. J. 3:213-23 . 1938 The homogeneous chaos. Am. J. Math. 60:897-936 . With H. R. Pitt. On absolutely convergent Fourier-Stieltjes transforms. Duke Math. J. 4:420-40 . With A. Wintner. Fourier-Stieltjes transforms and singular infinite convolutions. Am. J. Math. 60:513-22 . With W. T. Martin. Taylor's series of functions of smooth growth in the unit circle. Duke Math. J. 4:384-92 . The historical background of harmonic analysis. In Semicentennial Addresses . Am. Math. Soc. Semicentennial Publ. Vol. II . Providence, R.I.: American Mathematical Society, pp. 513-22 . With D. V. Widder. Remarks on the classical inversion formula for the Laplace integral Bull. Am. Math. Soc. 44:573-75 . 1939 The ergodic theorem. Duke Math. J. 5:1-18 . With A. Wintner. On singular distributions. J. Math. Phys. 17:233-46 .
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Biographical Memoirs: Volume 61 With R. H. Cameron. Convergence properties of analytic functions of Fourier-Stieltjes transforms. Trans. Am. Math. Soc. 46:97-109 ; Math. Rev. 1(1940):13; rev. 400. With H. R. Pitt. A generalization of Ikehara's theorem. J. Math. Phys. 17:247-58 . 1941 With A. Wintner. Harmonic analysis and ergodic theory. Am. J. Math. 63:415-26 . With A. Wintner. On the ergodic dynamics of almost periodic systems. Am. J. Math. 63:794-824 . 1942 With G. Polya. On the oscillation of the derivatives of a periodic function. Trans. Am. Math. Soc. 52:249-56 . 1943 With A. Wintner. The discrete chaos. Am. J. Math. 65:279-98 . With A. Rosenblueth and J. Bigelow. Behavior, purpose, and teleology. Philos. Sci. 10:18-24 . 1946 With A. E. Heins. A generalization of the Wiener-Hopf integral equation. Proc. Natl. Acad. Sci. USA 32:98-101 . With A. Rosenblueth. The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch. Inst. Cardiol. Mexicana 16:205-65 ; Bol. Soc. Mat. Mexicana 2(1945); 37-42. 1947 With S. Mandelbrojt. Sur les fonctions indéfiniment dérivables sur une demidroite. C.R. Acad. Sci. Paris 225:978-80 . 1948 With L. K. Frank, G. E. Hutchinson, W. K. Livingston, and W. S. McCulloch. Teleological mechanisms. Ann. N.Y. Acad. Sci. 50: 187-278 . With A. Rosenblueth, W. Pitts, and J. Garcia Ramos, and F. Webster. An account of the spike potential of axons. J. Cell. Comp. Physiol. 32:275-318; 33:787 .
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Biographical Memoirs: Volume 61 Cybernetics, or Control and Communication in the Animal and the Machine . Actualités Sci. Ind. No. 1053. Paris: Hermann et Cie. Cambridge, Mass.: The MIT Press; New York: Wiley. 1949 With A. Rosenblueth, W. Pitts, and J. Garcia Ramos. A statistical analysis of synaptic excitation. J. Cell. Comp. Physiol. 34:173-205 . 1950 Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications . Cambridge, Mass.: The MIT Press; New York: Wiley; London: Chapman & Hall. With L. Geller. Some prime-number consequences of the Ikehara theorem. Acta Sci. Math. (Szeged) 12:25-28 . The Human Use of Human Beings . Boston: Houghton-Mifflin. 1951 Problems of sensory prosthesis. Bull. Am. Math. Soc. 57:27-35 . 1953 Ex-Prodigy: My Childhood and Youth . New York: Simon and Schuster. 1955 On the factorization of matrices. Comment. Math. Helv. 29:97-111 . 1956 I Am a Mathematician . The Later Life of a Prodigy . Garden City, N.Y.: Doubleday. 1957 With E. J. Akutowicz. The definition and ergodic properties of the stochastic adjoint of a unitary transformation. Rend. Circ. Mat. Palermo 6(2):205-17 ; Addendum, 349. With P. Masani. The prediction theory of multivariate stochastic processes, Part I. Acta Math. 98:111-50 . 1958 With P. Masani. The prediction theory of multivariate stochastic processes, Part II. Acta Math. 99:93-137 .
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Biographical Memoirs: Volume 61 Nonlinear Problems in Random Theory . Cambridge, Mass.: The MIT Press; New York: Wiley. 1961 Cybernetics, 2nd ed. (revisions and two additional chapters). Cambridge, Mass.: The MIT Press; New York: Wiley. 1962 The mathematics of self-organizing systems. In Recent Developments in Information and Decision Processes, pp. 1-21. New York: MacMillan. DOCTORAL STUDENTS OF NORBERT WIENER 1930 Shikao Ikehara Ph.D. Sebastian Littauer Sc.D. Dorothy W. Weeks Ph.D. 1933 James G. Estes Ph.D. 1935 Norman Levinson Sc.D. Henry Malin Ph.D. 1936 Bernard Friedman Ph.D. 1939 Brockway McMillan Ph.D. 1940 Abe M. Gelbart Ph.D. 1959 Donald G. Brennan Ph.D.
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