ALFRED T. BRA UER: TEACHER, MATHEMATICIAN, AND DEVELOPER OF LIBRARIES

RICHARD D. CARMICHAEL

Made in Uniled SImes 0/America Reprinted from The JOURNAL OF THE ELISHA MITCHELL SOCIETY

Vol. 102, No.3, FalJ 1986

The Journal ofthe Elisha Mitchell SCientific Society. 102(3), 1986, pp. 88-106

ALFRED T. BRAUER: TEACHER, MATHEMATICIAN, AND DEVELOPER OF LIBRARIES

RICHARD D. CARMICHAEL

Department ofMathematics and Computer Science, Wake Forest University, Winston-Salem, NC 27109

Abstract: Alfred Theodor Brauer (1894-1985), a native of Germany who came to the United States in 1939 and became an American citizen in 1944, was a member of the Department of Mathematics at the University of North Carolina at Chapel Hill for twenty-four years and, after his retirement there, served the Department of Mathematics at Wake Forest University for ten years. Brauer's contributions to mathematics are many including teaching; research and publi­cation in two fields, number theory and matrix theory; and library development. This biographical survey includes a complete list of his Ph.D. and Master's degree students and a complete list of his publications.

Key Words: Alfred T. Brauer; biography; mathematician.

INTRODUCTION

Alfred Theodor Brauer (Fig. I) was born 9 April 1894 in Berlin. Twenty years later in 1914 he began to study engineering, the subject his father had encouraged. After only three months, however, World War I interrupted his studies. He served in the German army four years and was decorated twice, once for being wounded and again with Das Eiserne Kreuz (The Iron Cross) for outstanding service.

During the short time he studied engineering, Brauer decided that mathematics was the discipline he really enjoyed. Following the war and after taking some time to recover from the experience, he entered the University of Berlin in 1920 as a mathematics student. Eight years later he obtained the degree of Doctor Philo­sophiae (Dr. Phil.) under Issai Schur.

Brauer's mathematical research and publications span more than fifty-two years with his first paper appearing in 1925 and his last in 1977. His brother, Richard Brauer, was co-author of the first publication, and his son-in-law, Richard H. Hudson, was co-author of the final one. Brauer served five educational institutions in his roles of teacher, research mathematician, and developer of mathematical libraries.

Brauer's publications number 113. He directed 21 Ph.D. dissertations and advised 38 Master's degree students at the University ofNorth Carolina at Chapel Hill (UNC), and he advised 13 Master's degree students at Wake Forest University (WFU). He touched thousands of students with his ability as a teacher. He de­veloped or helped to develop four mathematical research libraries, those of the University of Berlin, the Institute for Advanced Study at Princeton, UNC, and WFU; and he served as a consultant to others. To fully appreciate the career of Alfred T. Brauer is to acknowledge his contributions as a teacher; as a research mathematician in two fields, number theory and matrix theory; and as a developer of mathematical libraries.

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89 CARMICHAEL: ALFRED T. BRAUER, MATHEMATICIAN

FIG. I. Alfred Theodor Brauer.

In 1984 the article "Alfred T. Brauer as a Mathematician and Teacher" by Richard H. Hudson, Brauer's son-in-law, and Thomas L. Markham, a former student of Brauer at UNC, appeared in the mathematics journal Linear Algebra and Its Applications. Hudson's and Markham's article (1984) and this biographical survey are complementary.

CAREER IN EDUCAnON

After attaining the degree Doctor Philosophiae in 1928 for his work "Uber diophantische Gleichungen mit endlich vielen L6sungen," which is the reference [B-ll] in the list of publications given later, Brauer served at the University of Berlin as an assistant to Issai Schur until 1932. During this time he and his colleague, Hans Rohrbach, initiated the MAPHA (Matematische Physikalische Arbeitsgemeinschaft), a student organization to provide academic support for its members and to promote sociability among them.

In 1932 Brauer was named "Privat Dozent" at the University of Berlin after giving a special lecture required for this appointment and for the "venia legendi," the "permission to give lectures" or "right to read" at a German university; he remained in this position until early 1936. For several years prior to 1936, the Jews were enduring increasing pressures imposed by the Nazi government. As Hitler's power grew, all Jewish professors who were not war veterans lost their university positions. Included among them was Richard Brauer, Alfred's younger brother by seven years, who came to the United States in 1933. At first, Jewish war veterans were permitted to keep their positions at the University of Berlin, but the policy soon changed. Brauer was forced to leave in early 1936.

Brauer's second cousin operated a private school in Berlin; from 1936 to 1939 Brauer taught mathematics there at the junior high school level. He continued to do mathematical research, and he continued to live in the city. He married the

91 CARMICHAEL: ALFRED T. BRAUER, MATHEMATICIAN

Wake Forest. During his final two years of teaching at UNC, Brauer held a half­time position; and, according to a technicality in the UNC system, he could not be paid more than half salary from the Air Force grant. Thus, Brauer would have to give up the grant if it were continued to be administered at UNe. Knowing of this situation, W. M . Whyburn, Brauer's colleague at UNC, suggested to Gentry that Brauer be hired at WFU in the summer of 1965 and that WFU assume the administration of the grant. At the same time at WFU, more money was becoming available for the purchase oflibrary materials, and Gentry knew of Brauer's ability in establishing mathematical libraries. Wake Forest offered Brauer the Visiting Professorship in the summer of 1965, and the Air Force grant was transferred to WFU. Gentry's motivation was for Brauer to build the mathematics library at Wake Forest. In addition to his library work, Brauer taught two mathematics courses and began a research collaboration with Gentry. Gentry maintains that bringing Brauer to Wake Forest to build the mathematics library was his single most important act during the twenty-five years that he was chairman. Gentry believes that a good mathematics library is the foundation for developing faculty and programs. In 1965 the Brauers thought that they would be traveling from Chapel Hill to Winston-Salem for one academic year; yet they did so for ten years, much to Wake Forest's benefit.

RESEARCH AND LIBRARY WORK

Alfred Brauer attained world-wide recognition in two areas of mathematical research, number theory and matrix theory. He began working in number theory when he was a graduate student and continued throughout his career, with all of his research efforts going into this area until 1946 when he began also to work in matrix theory. While E. T. Browne, Brauer's colleague at UNC, was abroad in 1945-1946 teaching at the American Army University at Shrivenham, England, Brauer had to substitute for him, teaching the matrix theory course as noted in [B-84]. Having never taught the course before nor having published a paper on the subject, Brauer wanted to read some of the more recent research publications in the field, and he chose to study some of Browne's papers. The results in one of Browne's papers (1930) particularly caught Brauer's attention, and using a method from a paper by Hans Rohrbach (1931), his friend and colleague in Berlin, Brauer improved some of Browne's results. This work was the beginning ofBrauer' s research in matrix theory, and he continued working in this area, as well as in number theory, for the remainder of his research career. Brauer always acknowl­edged the influence of Browne's work upon his own [B-84] .

From 1946 until 1965 Brauer published papers in matrix theory, both singly­authored and jointly with his students. When Brauer came to WFU in 1965, he began to collaborate with Ivey e. Gentry, who was teaching a course in algebra and matrix theory. Brauer asked Gentry whether he would be interested in meeting regularly to discuss matrices. They agreed to meet each Wednesday night from about 7:00 to 9:00 p.m. Together they published seven papers. (During the ten years that Brauer was on the faculty at Wake Forest, he and Mrs. Brauer would drive to Winston-Salem from Chapel Hill on Wednesday of each week. Brauer taught one course in the afternoon, met with Gentry in the evening, taught another course on Thursday morning, and then worked on orders for the library. The Brauers would return to Chapel Hill on Thursday afternoons.)

93 CARMI CHAEL: ALFRED T. BRAUER, MATHEMATICIAN

FIG. 2. Mrs. Brauer (center) introduces Librarian Betty A. Davis (left) to Dr. and Mrs. Ivey C. Gentry (right) during the reception for the dedication of The Alfred T. Brauer Library for mathematics, physics, and computer science at UNC on I December 1976. Alfred T . Brauer is on the extreme left.

Brauer received many honors and awards for teaching, mathematical research, and service to the profession. A chronological list appears below.

1949 Oak Ridge Institute of Nuclear Studies Science Research Award for "Sig­nificant Contributions to Science in the South."

1959 Kenan Professor at UNC. 1965 Tanner Excellence in Teaching Award at UNC. 1971 G. W. F. Hegel Medal from Humboldt University (formerly the University

of Berlin) for devotion and service to the University. 1972 Honorary degree, Doctor of Laws (LL.D.), from UNC. 1975 Wake Forest University established The Alfred T. Brauer Instructorship in

mathematics. (Dr. Ellen E. Kirkman, presently Associate Professor ofMath­ematics at Wake Forest, was the first Brauer Instructor.)

1976 UNC named its mathematics and physics library "The Alfred T. Brauer Library."

1984 The Alfred Brauer Fund was established at UNC with its purposes including support of the A. T. Brauer Algebra Prize which is awarded annually to an undergraduate who demonstrates outstanding ability and promise in algebra or number theory.

1985 The Alfred Brauer Lectures were established by the Department of Math­ematics at UNC.

95 CARMICHAEL: ALFRED T. BRAUER, MATHEMATICIAN

knew him had the highest respect and warm feelings for this man who did more for mathematical scholarship at UNC than anyone else."

A memorial service was held at the Judea Reform Synagogue in Durham, N.C., on Friday, 27 December 1985. At the service, Fred B. Wright, Professor at UNC, former chairman of the Department of Mathematics at UNC, and former student of Brauer, delivered an eloquent eulogy entitled "Alfred Theodor Brauer. 1894­1985" which is reproduced in full below.

Alfred Brauer came to Chapel Hill and to the University of North Carolina in 1942. His official service to the University ended with his retirement in 1966, but his real contributions to UNC and to the Department of Mathematics have not yet run their full course, and in truth probably will continue for as long as there is a mathematical community within the University.

The magnificant mathematical library which he built on the fragile base which he found when he came here has been named in his honor, and the University awarded him an honorary doctorate in 1972 for his efforts in building this collection, which is one of the treasures of this State, and indeed of this nation .

Alfred Brauer the mathematician began his career in Germany, as the student of the revered Issai Schur, and Alfred felt it a great honor to be the editor of the collected works of Schur. His research in mathematics was principally done in the fields of number theory and algebra. The quality of this research is indicated by his receipt of the Oak Ridge Science Award, and of the G . W. F. Hegel Medal from the University of Berlin , and by his appointment in 1959 as Kenan Professor at the University of North Carolina.

Alfred had many graduate students whose work was formally under his direction . His impact on these students, it need hardly be said, was enormous. But his influence on mathematicians goes far beyond these graduate students, and reaches from his own family circle throughout the mathematical community, to the mathematicians, statisticians, and computer scientists who were inspired by him, and who profited from his freely given help and genuine interest in their problems.

Alfred Brauer the teacher is remembered with affection and admiration by the many generations of non-mathematical students who were fortunate enough to find their way into one of Alfred's elementary classes. He was determined that these students would learn something from his classes no matter how inept they thought themselves, and no matter how elementary the course. We have heard it many times: "Oh, I was never very good at mathematics, but I had a class with Dr. Brauer, and-"; a beginning which usually has as its point a memory of a patient and respected teacher, whose character and personality remain with the student long after all the mathematics is forgotten .

When the last and inevitable time for retirement from UNC came, Alfred was not ready to retire; he had unfinished business with the world . For some years after, he worked at Wake Forest University, teaching and helping with the construction ofa sound graduate program there. This work has been memorialized by Wake Forest in the Alfred T. Brauer Instructorship at that school. During these years, he remarkably continued to produce publishable research in math­ematics.

His concern for the well being of the graduate students of course endeared him to these impoverished and downtrodden souls. Many years ago, with the help of the family of his friend and colleague, the late Professor Edward T . Browne, he established a fund which over the years had provided considerable simple financial assistance to graduate students, and even to younger faculty in times of hardship. He ran this fund on his own recognizance; it was never an official University matter. When he felt compelled to relinquish this task a very short time ago, the Department of Mathematics created the Alfred T. Brauer gift trust fund , which will do a great deal. The Alfred Brauer Lectures will periodically remind us all of this scholar who once worked among us, and the Alfred Brauer Prize in Algebra and Number Theory annually recognizes the undergraduate student who has shown the greatest promise during the year, in those fields in which Alfred was most interested, and will remind us of the teacher who taught beside us.

I was one of his earlier students here, and wrote a Master's Thesis under his direction. I took every advanced course which he taught during my years here as a student, and I still have the notes drafted from these lectures. These courses ranged far beyond those fields in which he was a comfortable expert. Even today, so many years later, these notes have the power to astonish

97 CARMICHAEL: ALFRED T. BRAUER, MATHEMATICIAN

7. Losung der Aufgabe 32. lahresbericht der Deutschen Mathematiker Ver­einigung, 35 (1926), Abt. 2, pp. 95-96.

8. Losung der Aufgabe 46. lahresbericht der Deutschen Mathematiker Ver­einigung, 36 (1927), pp. 90-92. (with R . Brauer)

9. Uber Sequenzen von Potenzresten. Sitzungsberichte der Preussischen Aka­demie der Wissenschaften, physikalisch-mathematischen Klasse, 1928, pp. 9-16.

10. Problem 263, solution and note. American Mathematical Monthly, 35 (1928), pp. 494-495.

11. Uber diophantische Gleichungen mit endlich vielen Losungen . 10urnal fur die reine und angewandte Mathematik, 160 (1929), pp. 70-99. (Dissertation, University of Berlin, submitted to the Faculty of Philosophy of the Uni­versity of Berlin on 12 luly 1927)

12. Uber die Approximation algebraischer Zahlen durch algebraische Zahlen. lahresbericht der Deutschen Mathematiker Vereinigung, 38 (1928), Abt. 2, p. 47. (Abstract)

13. Uber diophantische Gleichungen der Form g2(X,y) - ILh2(X,y) = TIn. 10urnal fUr die reine und angewandte Mathematik, 161 (1929), pp. 1-13.

14. Uber den kleinsten quadratischen Nichtrest. Mathematische Zeitschrift, 33 (1931), pp. 161-176.

15. Uber eine zahlentheoretische Behauptung von Legendre. Sitzungsberichte der Berliner Mathematischen Gesellschaft, 29 (1930), pp. 116-125. (with H. Zeitz)

16. Uber den kleinsten quadratischen Nichtrest. lahresbericht der Deutschen Mathematiker Vereinigung, 40 (1931), Abt. 2, p. 32. (Abstract)

17. Uber Sequenzen von Potenzresten, II. Sitzungsberichte der Preussischen Akademie der Wissenschaften, physikalisch-mathematische Klasse, 1931 , pp. 329-341.

18. Uber die Verteilung der Potenzreste. Mathematische Zeitschrift, 35 (1932), pp. 39-50.

19. Uber die Nullstellen der Hermiteschen Polynome. Mathematische Annalen, 107 (1932), pp. 87-89.

20. Questions concerning the maximum term in the diatomic series. American Mathematical Monthly, 40 (1933), pp. 409-410. (a reply to a problem posed by A. A. Bennett)

21. Bemerkungen zu einem Satze von Herrn G . P6lya. lahresbericht der Deutschen Mathematiker Vereinigung, 43 (1933), pp. 124-129.

22. NachrufaufHermann Zeitz. Sitzungsberichte der Berliner Mathematischen Gesellschaft, 33 (1934), pp. 2-6.

23 . Uber Irreduzibilitatskriterien von 1. Schur und G. P61ya. Mathematische Zeitschrift, 40 (1935), pp. 242-265. (with Richard Brauer)

24. Uber eine Erweiterung des kleinen Fermatschen Satzes. Mathematische Zeit­schrift, 42 (1937), pp. 255-262.

25. Uber die Dichte der Summe von Mengen positiver ganzer Zahlen. 1. Annals of Mathematics, 39 (1938), pp. 322-340.

26. Uber die Dichte der Summe zweier Mengen , deren eine von positiver Dichte ist. Mathematische Zeitschrift, 44 (1938), pp. 212-232.

CARMICHAEL: ALFRED T. BRAUER, MATHEMATICIAN 99

51. Limits for the characteristic roots of a matrix . III. Duke Mathematical Journal, 15 (1948), pp. 871-877.

52. On the approximation of irrational numbers by the convergents of their continued fractions. American Journal of Mathematics, 71 (1949), pp. 349­361. (with Nathaniel Macon)

53. On the approximation of irrational numbers by the convergents of their continued fractions , II. American Journal of Mathematics, 72 (1950), pp. 419-424. (with Nathaniel Macon)

54. A criterion for a common root of k algebraic equations. American Mathe­matical Monthly, 57 (1950), pp. 322-324.

55 . Some number theory and its applications to elementary mathematics. Math­ematics at Work-High Lights of the Ninth Annual Mathematics Institute, Duke University, 1949, pp. 79-82.

56. On algebraic equations with all but one root in the interior of the unit circle. Mathematische Nachrichten, 4 (1951), pp. 250-257.

57. On the irreducibility of polynomials with large third coefficient. II. American Journal of Mathematics, 73 (1951), pp. 717-720.

58. On a theorem of Aubry-Thue. Canadian Journal of Mathematics, 3 (1951), pp. 367-374. (with T . L. Reynolds)

59. Fun with numbers. Mathematics at Work-High Lights of the Tenth Annual Mathematics Institute, Duke University, 1950, pp. 16-17.

60. On algebraic equations with all but one root in the interior of the unit circle. Proceedings of the International Congress of Mathematicians, 1950, Cam­bridge, Massachusetts, Vol. 1(1952), p. 330.

61. Limits for the characteristic roots of a matrix. IV: Applications to stochastic matrices. Duke Mathematical Journal, 19 (1952), pp. 75-91.

62. The proof of the law of sines. American Mathematical Monthly, 59 (1952), p. 319.

63 . Limits for the characteristic roots of a matrix. V. Duke Mathematical Jour­nal, 19 (1952), pp. 553-562.

64. Matrices with all their characteristic roots in the interior of the unit circle. Journal of the Elisha Mitchell Scientific Society, 68 (1952), pp. 180-183.

65. Letter to the Editor. Econometrica, 21 (1953), pp. 218-219. 66. On a new class of Hadamard determinants. Mathematische Zeitschrift, 58

(1953), pp. 219-225 . 67. On the distribution of the Jacobian symbols. Mathematische Zeitschrift, 58

(1953), pp. 226-231. 68. Bounds for characteristic roots of matrices. Simultaneous Linear Equations

and the Determination of Eigenvalues, National Bureau of Standards Ap­plied Mathematics Series, Vol. 29 (1953), Chapter 14, pp . 101-106.

69 . Uber die Lage der charakteristischen Wurzeln einer Matrix. Journal fUr die reine und angewandte Mathematik, 192 (1953), pp. 113-116.

70. On a problem of partitions. II. American Journal ofMathematics, 76 (1954), pp. 343-346. (with Ben M . Seelbinder)

71. Elementary estimates for the least primitive root. Studies in Mathematics and Mechanics Presented to Richard von Mises, Academic Press, New York, 1954, pp. 20-29.

CARMICHAEL: ALFRED T. BRAUER, MATHEMATICIAN 101

in Mathematical Analysis and Related Topics, Stanford University Press, Stanford, California, 1962, pp. 48-55.

92. A system of diophantine equations: problem E 1555. American Mathe­matical Monthly, 69 (1962), p. 1009. (with Aubrey Kempner)

93 . On a problem of Frobenius. Journal flir die reine und angewandte Mathe­matik, 211 (1962), pp. 215-220. (with James E. Shockley)

94. On the characteristic roots of non-negative matrices. Recent Advances in Matrix Theory, edited by Hans Schneider, University of Wisconsin Press, Madison, Wisconsin, 1964, pp. 3-38.

95. Some elementary results on the distribution of the quadratic residues. Pro­ceedings of the 1963 Number Theory Conference, University of Colorado, Boulder, Colorado, 1964, pp. 11-12.

96. A method for the computation of the greatest root of a nonnegative matrix. SIAM Journal on Numerical Analysis, 3 (1966), pp. 564-569.

97. A problem of partitions: solution of problem E 1811. American Mathe­matical Monthly, 74 (1967), pp. 88-89.

98. On the characteristic roots of stochastic matrices. Journal of the Elisha Mitchell Scientific Society, 84 (1968), pp. 382-383. (with Ivey C. Gentry)

99. On the characteristic roots oftournament matrices. Bulletin of the American Mathematical Society, 74 (1968), pp. 1133-1135. (with Ivey C. Gentry)

100. A new proofofa theorem by H. G. Landau on tournament matrices. Journal of Combinatorial Theory, 5 (1968), pp. 289-292. (with Ivey C. Gentry and Kay Shaw)

10 1. Einige Anwendungen der Matrizentheorie auf algebraische Gleichungen. Journal fur die reine und angewandte Mathematik, 236 (1969), pp. 11-25. (with Hans Rohrbach)

102. Combinatorial methods in the distribution of kth power residues. Combi­natorial Mathematics and its Applications, University of North Carolina Press, Chapel Hill, North Carolina, 1969, pp. 14-36.

103. K-sequences with small divisors: solution of problem E 2182. American Mathematical Monthly, 77 (1970), pp. 407-408.

104. A remark on the paper of H. H. Schaefer: Abschatzung der nichttrivalen Eigenwerte stochastischer Matrizen. Numerische Mathematik, 17 (1971), pp. 163-165.

105. On stochastic matrices whose absolute smallest characteristic root is real. Duke Mathematical Journal, 39 (1972), pp. 265-266. (with Ivey C. Gentry)

106. Some remarks on tournament matrices. Linear Algebra and its Applications, 5 (1972), pp. 311-318. (with Ivey C. Gentry)

107. An unpublished number theoretical paper by I. Schur. Proceedings of the 1972 Number Theory Conference, University of Colorado, Boulder, Col­orado, 1972, p. 19.

108. Gedenkrede auflssai Schur. Issai Schur-Gesammelte Abhandlungen, Band I, 1973, pp. V-XIV.

109. Bounds for the greatest characteristic root of an irreducible nonnegative matrix. Linear Algebra and its Applications, 8 (1974), pp. 105-107. (with Ivey C. Gentry)

110. A lower bound for the greatest root of an irreducible non-negative matrix.

CARMICHAEL: ALFRED T. BRAUER, MATHEMATICIAN 103

1961 Richard F. McCoart, Jr. "Irreducibility of certain classes of Legendre polynomials."

1962 James E. Shockley, Jr. "On the best bound for the solvability of a linear diophantine equation by positive integers."

1963 Richard J. Painter "Extensions of theorems of Ostrowski on the zeros of certain classes of polynomials. "

1964 Frank A. Roescher "Some properties of Euler's ¢ function ."

1964 Clifton T. Whybum "On the second smallest quadratic non-residue."

1965 Douglas E. Crabtree "Some new results on the characteristic roots of matrices."

1965 Emanuel Vegb " Estimates for the minimum distance between the zeros of a polynomial and a conjecture by Leo Moser."

MASTER OF ARTS THESES DIRECTED BY ALFRED T. BRAUER AT THE UNIVERSITY OF NORTH CAROLINA

1944 Irwin Stoner "The lemma of Schwarz: Its generalizations and applications."

1945 Gertrude Ehrlich "The irreducibility of polynomials which take certain values for integral arguments ."

1948 Hubert C. Dixon "Weisner's results on irreducibility of polynomials."

1948 Walter T. Graybeal "Multiply perfect numbers."

1948 Nathaniel Macon "The theorems of Hurwitz, Vahlen, Borel, and Fujiwara on the approxi­mation of irrational numbers."

1948 Eleanor L. Rankin "Quadrature of lunes."

1948 Fred B. Wright, Jr. 'The inverse of Euler's function."

1949 William J. Graves "The exact number of prime numbers below a given limit."

1949 Annye E. WeIch "The number ofgenera of binary quadratic forms with even discriminant."

1950 Martha K. Grogan "Elementary estimates for least quadratic nonresidues."

1950 Owen H. Hoke "Quadratic fields with Euclidean algorithms."

1950 Betty Lide "Number theory in the quadratic field k(5 'h)."

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1963 Anne E. Britton "Mahler's matrices."

1964 Sally T. Dalton "Liouville transcendental numbers."

1964 Thomas L. Markham "On bounds for the greatest characteristic root of a positive matrix."

1965 Judith H. Hallman "On Brauer's method for the computation of the greatest characteristic root of a positive matrix."

1966 Miriam A. Leiva "Elementary estimates for the least positive primitive root modulo pr."

MASTER OF ARTS THESES DIRECTED BY ALFRED T. BRAUER AT WAKE FOREST UNIVERSITY

1966 Linda R. May "A new proof of the theorems of Perron and Frobenius on positive ma­trices. "

1966 Edith V. Sloan "An elementary proof of the theorem of Tchebycheff."

1966 Nancy B. Smith "Proofs for the reality ofthe characteristic roots ofa real symmetric matrix."

1967 Danny L. Boyter "On the theorems of Emilie Haynsworth on determinants with dominant main diagonal."

1967 George W. Dinolt "Bounds of the characteristic roots of non-negative matrices."

1967 Martha Kay Shaw "On matrices of zeros and ones."

1968 David C. Kurtz "Bounds for the least posi ti ve quadratic non-residue obtained by elementary methods."

1968 Daniel J. Richman "Some aspects of the theory of minimal Gerschgorin sets."

1968 Emory M. Underwood II "The separation theorem for the characteristic roots of an Hermitian ma­trix . "

1969 Lois M. Robertson Louden "On some results of Romanovsky on stochastic matrices."

1969 Safar Mohammad N azari "Elementary estimates for the least primitive roots."

1971 Roger L. Cordle "Some major results about odd perfect numbers."

1973 Nancy G. Bassett "Fermat's last theorem for N = 3, N = 4 and other diophantine equations."

Acknowledgments: The author gratefully acknowledges the following persons for their contributions to the development of this article: Hilde W. Brauer, Jane A.

ALFRED THEDOR BRAUER