# Henry Berthold Mann

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1905-2000 Henry B. Mann, our friend and former colleague, passed away on February 1, 2000 in Tucson, Arizona. A mathematician of international fame, Mann, in a career of more than fifty years, made significant contributions to algebra, number theory, statistics, and combinatorics.

Henry Mann was born October 27, 1905, in Vienna to Oscar and Friedrike (Schönnhof) Mann. He received his Ph.D. degree in mathematics in 1935 from the University of Vienna where. as a student of Philipp Furtwängler, he wrote his dissertation in algebraic number theory. After a year of teaching school in Vienna and a couple of years spent in research and tutoring, he emigrated in 1938 to the United States.

In New York he earned his living for several years primarily by tutoring. He had by then developed an interest in mathematical statistics, particularly in the analysis of variance, and in the problem of designing experiments with a view to their statistical analysis. He later contributed to this subject in a number of research papers and in his book (1949) "Analysis and Design of Experiments."

One of Mann's most remarkable achievements was his discovery in 1941 of a proof of a celebrated conjecture of Schnirelmann and Landau in additive number theory. This conjecture had its origin in the work of L. Schnirelmann in the early 1930s. Let $A (B,C)$ be sets of positive integers. Form $A^0$, $B^0$ by adjoining $0$ to $A$ and $B$ respectively.  Let $A(n)$ be the number of positive integers in $A$ that are $\le n$. The greatest lower bound of the quotients $A(n)/n$ is called the density of $A$. Let $C^0$ consist of all integers of the form $a+b$ ($a\in A^0$, $b\in B^0$).

Let $\alpha$ be the density of $A$, $\beta$ the density of $B$, and $\gamma$ the density of $C$. It had been conjectured by E. Landau, I. Schur, and A. Khintchine that
$$\gamma \ge \alpha + \beta \,\, or \,\, = 1\qquad (*)$$

Approximations to this inequality had been obtained by Landau in 1930, who showed that $\gamma \ge \alpha + \beta -\alpha \beta$, and by A. Brauer in 1941, who showed that $\gamma \ge (9/10)(\alpha + \beta)$. Schnirelmann had shown that

1. $\alpha + \beta \ge \alpha+\beta - \alpha \beta$.
2. $C$ contains all positive integers if $\alpha + \beta \ge 1$.

From these two rules Schnirelmann obtained (readily) the result that any set having positive density is a basis for the integers (that is, if $\alpha > 0$, then the sum of $A$ with itself sufficiently many times contains all positive integers). As an application of these ideas, Schnirelmann proved (for the first time) the existence of a value $k$ such that every integer greater than 1 is the sum of at most $k$ primes. This he did by showing that $P + P - P$ is the set of primes together with 1 - has positive density, hence is a basis of the integers.

Out of further study of these ideas by Schnirelmann and by E. Landau, there arose the conjecture that (1) and (2) may be replaced by the much stronger statement $(*)$: Either $\gamma \ge \alpha + \beta$ or $C$ contains all positive integers.

This conjecture, appealing in its apparent simplicity, soon attracted wide attention. Many distinguished mathematicians attempted to find a proof; indeed, partial results were obtained over the next decade by E. Landau, A. Khintchine, A. Besicovitch, I. Schur, and A. Brauer.

It was this conjecture that Mann succeeded in proving in 1941. His interest in the problem had been aroused through the lectures of A. Brauer at New York University. Actually, he proved the still sharper statement:

$$\frac{C(n)}{n}\geq\min_{ \substack{0 < m \le n \\ m\not\in C} }\left(1,\frac{A(m)+B(m)}{m}\right).$$

For his proof he was awarded the Cole Prize in Number Theory by the American Mathematical Society in 1946. The technique that Mann introduced in his proof, and its various modifications, have led to further important results in additive number theory and have also proved useful in the more general setting of additive problems in groups.

In 1942 Mann was the recipient of a Carnegie Fellowship for the study of statistics at Columbia University. At Columbia he had the opportunity of working with Abraham Wald in the department of economics, which at that time was headed by Harold Hotelling. He taught for a year (1943-1944) in the Army Specialized Training Program at Bard College; he spent a year (1944-1945) as research associate at Ohio State University, and six months as research associate at Brown University. In 1946 he returned to Ohio State to join the mathematics faculty where, as associate professor (1946-1948) and full professor (1948-1964), he was actively engaged in teaching and research for many years. After retiring from Ohio State, he held professorships at the University of Wisconsin Mathematics Research Center from 1964 to 1971, and at the University of Arizona from 1971 until his second retirement in 1975.

Mann's research interests in algebra and combinatorics covered a wide range. He had a special fondness, though, for algebraic number theory and Galois theory. and imparted his enthusiasm for these subjects to many students over the years. Besides his dozen or so papers that contribute directly to these subjects, several of his papers on difference sets and coding theory contain beautiful applications of theorems on algebraic numbers and Galois theory.

Mann married Anna Löffler on July 19, 1935, and had one son Michael.

## Bibliography of Henry B. Mann

### Research Papers

1. Ein Satz Ãber Normalteiler, Anz. Ã–sterreich. Akad. Wiss. Math.-Naturwiss. Kl. (1935), Nr. 6, 49-50.
2. Ãœber eine notwendige Bedingung fïr die Ordnung einfacher Gruppen, Anz. Ã–sterreich. Akad. Wiss. Math.-Naturwiss. Kl.(1935). Nr. 19. 209-210.
3. Untersuchungen Ãber Wabenzellen bei allgemeiner Minkowskischer Metrik, Mh. Math. Phys. 42 (1935), 417-424.
4. Ãœber die Erzeugung von Darstellungen von Gruppen durch Darstellungen von Untergruppen, Mh. Math. Phys. 46 (1937), 74-83.
5. A proof of the fundamental theorem on the density of sums of sets of positive integers. Ann. of Math. 43 (1942), 523-527.
6. On the choice of the number of class intervals in the application of the chi square test. Ann. Math. Stat. 13 (1942), 306-317 (with A. Wald).
7. The construction of orthogonal Latin squares, Ann. Math. Stat. 13 (1942), 418-423.
8. Quadratic forms with linear constraints, Amer. Math. Monthly 50 (1943), 430-433.
9. On stochastic limit and order relationships, Ann. Math. Stat. 14 (1943), 217-226 (with A. Wald).
10. On the statistical treatment of linear stochastic difference equations. Econometrica 11 (1943), 173-220 (with A. Wald).
11. On the construction of sets of orthogonal Latin squares, Ann. Math. Stat. 14 (1943), 401-414.
12. On orthogonal Latin squares, Bull. Amer. Math. Soc. 50 (1944), 249-257.
13. On certain systems which are almost groups, Bull. Amer. Math. Soc. 50 (1944), 879-881.
14. On a problem of estimation occurring in public opinion polls, Ann. Math. Stat. 16 (1945), 85-90. [A correction appears in Ann. Math. Stat. 17 (1946), 87-88.]
15. On a test for randomness based on signs of differences, Ann. Math. Stat. 16 (1945), 193-199.
16. Note on a paper by C. W. Cotterman and L. U. Snyder, Ann. Math. Stat. 16 (1945), 311-312.
17. Nonparametric tests against trend, Econometrica 13 (1945), 245-259.
18. Correction of G-M counter data, Phys. Rev. 68(1945), 40-43 (with J. D. Kurbatov).
19. A note on the correction of Geiger Mãller counter data. Quart. J. Mech. Appl. Math. 4 (1946), 307-309.
20. On a test of whether one of two random variables is stochastically larger than the other, Ann. Math. Stat. 18 (1947), 50-60 (with D. R. Whitney).
21. Integral extensions of a ring, BulL Amer. Math. Soc. 55(1949). 592-594 (with H. Chatland).
22. On the field of origin of an ideal, Canad. J. Math. 2 (1950), 16-21.
23. On the number of integers in the sum of two sets of positive integers, Pacific J. Math. 1(1951), 249-253.
24. On the realization of stochastic processes by probability distributions in function spaces, Sankhya 11(1951), 3-8.
25. The estimation of parameters in certain stochastic processes, Sankhya 11(1951), 97-106.
26. On simple difference sets, Sankhya 11(1951), 357-364 (with T. A. Evans).
27. On products of sets of group elements, Canad. J. Math. 4 (1952), 64-66.
28. Some theorems on difference sets, Canad. J. Math. 4 (1952), 222-226.
29. On the estimation of parameters determining the mean value function of a stochastic process, Sankhya 12 (1952), 117-120.
30. An addition theorem for sets of elements of Abelian groups, Proc Amer. Math Soc. 4(1953), 423.
31. Systems of distinct representatives, Amer. Math. Monthly 60 (1953), 397-401 (with H. J. Ryser).
32. On the moments of stochastic integrals, Sankya 12(1953), 347-350 (with A. P. Calderãn).
33. On integral closure, Canad. J. Math. 6 (1954), 471-473 (with H. S. Butts and M. Hall Jr.).
34. On an exceptional phenomenon in certain quadratic extensions. Canad. J. Math. 6 (1954), 474-476.
35. A generalization of a theorem of Ankeny and Rogers, Rend. Circ. Mat. Palermo 3(1954), 106-108.
36. A theory of estimation for the fundamental random process and the Ornstein Uhlenbeck process, Sankhya 13 (1954), 325-350.
37. On the efficiency of the least square estimates of parameters in the Ornstein Uhlenbeck process, Sankhya 13 (1954). 351-358 (with P. B. Moranda).
38. Corresponding residue systems in algebraic number fields, Pacific J. Math. 6 (1956), 211-224 (with H. S. Butts).
39. On integral bases, Proc. Amer. Math. Sac. 9 (1958), 167-172.
40. A note to the paper "On integral bases" by H. B. Mann, Proc Amer. Math. Soc. 9(1958), 173-174 (with V. Hanly).
41. Some applications of the Cauchy-Davenport theorem. Norske Vid. Selsk. Forh. (Trondheim) 32 (1959), 74-80 (with S. Chowla and F. G. Straus).
42. The algebra of a linear hypothesis. Ann. Math. Stat. 31(1960), 1-15.
43. A refinement of the fundamental theorem on the density of the sum of two sets of integers, Pacific J. Math. 10 (1960), 909-915.
44. Intrablock and interblock estimates, in "Contributions to Probability and Statistics," pp. 293-298. Stanford Univ. Press, Stanford, California, 1960 (with M. V. Menon).
45. On modular computation, Math. Comput. 15 (1961), 190-192.
46. An inequality suggested by the theory of statistical inference, Illinois J. Math. 6 (1962), 131-136.
47. On the number of information symbols in Bose-Chaudhuri codes, information and Control 5(1962), 153-162.
48. Main effects and interactions, Sankhya Ser. A 24 (1962), 185-202.
49. Balanced incomplete block designs and Abelian difference sets, Illinois J. Math. 8 (1964), 252-261.
50. On the casus irreducibilis, Amer. Math. Monthly 71(1964), 289-290.
51. Decomposition of sets of group elements, Pacific J. Math. 14 (1964), 547-558 (with W. B. Lalfer).
52. On multipliers of difference sets, Canad. J. Math. 17 (1965), 541-542 (with R. L. McFarland).
53. Difference sets in elementary Abelian groups, Illinois J. Math. 9(1965), 212-219.
54. On linear relations between roots of unity, Mathematika 12 (1965), 107-117.
55. Recent advances in difference sets, Amer. Math. Monthly 74 (1967), 229-235.
56. On canonical bases of ideals, J. Combinatorial Theory Ser. 0 2 (1967), 71-76 (with K. Yamamoto).
57. Sums of sets in the elementary Abelian group of type (p,p), J. Combinatorial Theory 2 (1967), 275-284 (with J. E. Olson).
58. Two addition theorems, J. Combinatorial Theory 3 (1967), 233-235.
59. Properties of differential forms in n real variables. Pacific J. Math. 21(1967), 525-529 (with J. Mitchell and L. Schoenfeld). [A correction appears in Pacific J. Math. 23 (1967), 631.]
60. On the p-rank of the design matrix of a difference set, Information and Control 12 (1968), 474-488 (with F. J. MacWilliams).
61. On orthogonal m-pods on a cone, J. Combinatorial Theory Ser. 0 5 (1968), 302-307.
62. A new proof of the maximum principle for doubly-harmonic functions, Pacific J. Math. 27 (1968). 567-571 (with J. Mitchell and L. Schoenfeld).
63. On canonical bases for subgroups of an Abelian group. in "Combinatorial Mathematics and its Applications" (Proc. Conf., Univ. North Carolina, Chapel Hill, 1967), 38-54, Univ. of North Carolina Press, Chapel Hill, 1969.
64. A note on balanced incomplete block designs, Ann. Math. Stat. 40 (1969), 679-680.
65. On multipliers of difference sets, Illinois J. Math. 13(1969), 378-382 (with S. K. Zaremba).
66. On the difference between the geometric and the arithmetic mean of n quantities, Advances in Math. 5 (1970), 472-473 (with C. Loewner).
67. Linear equations over a commutative ring, J. Algebra 18(1971), 432-446 (with P. Camion and L. S. Levy).
68. Antisymmetric difference sets, J. Number Theory 4 (1972), 266-268 (with P. Camion).
69. Representations by kth powers in GF(q), J.Number Theory 4 (1972), 269-273 (with G. T. Diderrich).
70. A necessary and sufficient condition for primality and its source, J. Combinatorial Theory Ser. A. 13(1972), 131-134 (with D. Shanks).
71. Combinatorial problems in finite Abelian groups, in "A Survey of Combinatorial Theory" (Proc. Internat. Symp. Combinatorial Math, and Its Appl., Colorado State Univ.. Fort Collins, Colo. 1971), pp.95-100. North-Holland. Amsterdam, 1973 (with G.T. Diderrich).
72. On Hadamard difference sets, in "A Survey of Combinatorial Theory" (Proc. Internat. Symp. Combinatorial Math. and Its Appl. Colorado State Univ., Fort Collins, 1971). pp. 333-334. North-Holland, Amsterdam, 1973 (with R. L. McFarland).
73. Prãfer rings, J. Number Theory 5(1973), 132-138 (with P. Camion and L. S. Levy).
74. Additive group theory- a progress report, Bull. Amer. Math. Soc. 79 (1973), 1069-1075.
75. The solution of equations by radicals, J. Algebra 29(1974), 551-554.
76. Lectures on error correcting codes, The University of Arizona Department of Mathematics Lecture Note Series, University of Arizona, Tucson, Ariz., 1974, iii+88 pp. (with D. K. Ray-Chaudhuri).
77. On normal radical extensions of the rationals, Linear and Multilinear Algebra 3 (1975), 73-80 (with W. Y. Vélez).
78. Prime ideal decomposition in F, Monatsh. Math. 81 (1976), 131-139 (with W. Y. Vélez).
79. A short proof of Fermat's theorem for n = 3, Math. Student 46 (1978), 103-104 (with W. A. Webb).
80. An addition theorem for the elementary abelian group of type (p,p), Monatsh. Math. 102(1986), 273-308 (with Y. F. Wou).

### Books

1. "Analysis and Design of Experiments," Dover, New York, 1949.
2. "Introduction to Algebraic Number Theory," Ohio State Univ. Press, Columbus, 1955.
3. "Addition Theorems: The Addition Theorems of Group Theory and Number Theory," Wiley (Interscience), New York, 1965.

### Ph. D. Students of Henry B. Mann

• Donald Ransom Whitney, Ohio State University, 1949
• George Marsaglia, Ohio State University, 1951
• Hubert Spence Butts, Jr., Ohio State University, 1953
• Walter Wilson Hoy, Ohio  Ohio State University, 1953
• Chio-Shih Lin, Ohio State University, 1955
• Leon Royce McCulloh, Ohio State University, 1959
• Manavazhi Vijaya Krishna Menon, Ohio State University, 1959
• Walter Ball Laffer, I, Ohio State University, 1963
• George T. Diderrich, University of Wisconsin-Madison, 1972
• William Yslas Vélez, University of Arizona, 1975
• Ying Fou Wou, University of Arizona, 1980