ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR PROGRESS REPORT ON BASIC RESEARCH IN MATHEMATICS CONTRACTS NO. N8-ONR 71400 and NO. Nonr-330(00) By WILFRED KAPLAN Associate Professor of Mathematics Projects No. M801 and M960 OFFICE OF NAVAL RESEARCH, CHICAGO BRANCH 844 NORTH RUSH STREET CHICAGO9 ILLINOIS July 1, 1952

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN PROGRESS REPORT ON BASIC RESEARCH IN MATHEMATICS INTRODUCTION The preceding progress report, dated July 2, 1951, covered research from September 1, 1950, to June 1, 1951. The present report covers the research carried on from June 1, 1951, to June 15, 1952. On September 16, 1951, the contract N8 ONR 71400 (University of Michigan project M801) was terminated and the new contract Nonr-330(00) (ONR Project NR 045 087, University of Michigan Project M960) went into force as a continuation of the previous one. Thus the present report describes research under both contracts. A list of the personnel follows, June 1, 1951, to September 1, 1951 Professor S. B1. Myers, working on Banach spaces. Dr. H. C. Davis, working on quantum mechanicso Dr. A. J. Lohwater, working on complex variables, Dr. Sze-Tsen Hu, working on topological groups. Dr. D. J. Lewis, working on algebra. Mr. R. Barrar, working on partial differential equations under the direction of Professor E. HI. Rothe.

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Mr. J. G. Hocking, working on topology under the direction of Professor G. S. Young. Mr. W. E. Jenner, working on algebra under the direction of Professor R. Brauer. Mr. R. MacDowell, working on Banach spaces under the direction of Professor S. B. Myers. Mr. W. Smith, working on Banach spaces under the direction of Professor S. B. Myers. September 1, 1951, to June 15, 1952 Mr. E. Crisler, working on harmonic functions under the direction of Professor W. Kaplan. Mr. T. W. Hildebrandt, working on theory of elasticity, under the direction of Professor G. E. Hay. (Left project Feb. 1, 1952.) Mr. J. Kaiser, working on partial differential equations under the direction of Professor E. Rothe. (Left project FebO 1, 1952). Mr. R. G. Kuller, working on Banach spaces under the direction of Professor S, B. Myers. Mr. J. D. Miller, working on topology under the direction of Professor R. L. Wilder. (Joined project Febo 1, 1952). Mr. G. Prins, working on foundations of mathematics under the direction of Professor R. L. Wilder. Mr. R. Raimi, working on Banach spaces under the direction of Professor S. B. Myerso Mr. J. Shoenfield, working on foundations of mathematics under the direction of Professor R. L. Wilder. Mr, D. Wend, working on Complex variables, under the direction of Professor W. Kaplan. (Joined project Dec. 1, 1951). 2

- ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN DESCRIPTION OF INDIVIDUAL PROJECTS Professor S. B. Myers: Rotations of a Banach Space. A study was initiated of the group G of rotations (nor.preserving linear transformations) of a Banach space B. Various natural topologies are used on G, and an analysis of the structure of G is begun. B can be shown to be a Hilbert space if and only if for every pair o: points on the unit sphere of B there is a member of G which interchanges the pair. If B is finite-dimensional, G is a Lie group; transitivity of G over E is probably sufficient to make B a Hilbert space. An important problem is to determine for which infinite-dimensional B one can say that every compact subgroup of G is a Lie group. This relates to an unsolved problem concerning compact groups of homeomorphisms of a manifold. Dr. H. C. Davis: Second Quantization Dr. Davis began by examining, according to plan, certain apparent difficulties in the theory of second quantization. It quickly appeared that these were no more than imprecisions in statement, -.-iich could be made rigorous in a comparatively straightforward way. A more substantial problem remained in the fact that Fermi-Dirac quantum statistics has never been brought under the general quantum theory of fields, but is based on a purely ad hoc construction. Dr. Davis spent most of the summer looking for a formulation of the one-electron problem from which the correct second-quantized theory could be obtained by the general procedure used for other particles. This seemed to be feasible by the method used, but no conclusive results have yet been found. Dr. A. J. Lohwatero Boundary Values of Meromorphic and Harmonic Functions.The following theorems were established' Let w = f(z) be meromorphic in Izjl- and let if(rei@.) have limit 1, as r-l, for almost all @ of an arc A of |z| = 1. If p is a singular point of f(z) on A, then for every real. either eiX is in the range R(P ) of f(z) at P or else there exists a Jordan arc L in I zll terminating at a point. of [zl = 1 arbitrarily close to P, such that as z,43along L, f(z)-. ei (i.e., eiX is an asymptotic value of f(z) "near P"). If,. furthermore, f(z) has a finite number of zeros and poles in a neighborhood of the singular point p, then the union of R(P) and 35

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN the set of asymptotic values of f(z) "near P" is precisely one of the sets lw|1 1, Iw|I 1, or the extended plane. These theorems extend results of Carathgod.ory comm. Math. Helv. 19 (194I) p. 266J, Nevanlirma Ann. Acad. Sci. Fennicae (A) 32 (1929) No. 7, p. 2 and Seidel [Trans. Amer. Math. Soc. 36(1), (1934), p. 208. Let u(r,Q) be a harmonic function with bounded mean modulus for r <l. If u(r,~)->0 as r- 1 for almost all G and if on the remaining set (of measure 0) the radial limit, wherever it may exi.st, is finite, then u(r,Q) 0 in r<l; is on the remaining set (of measure 0) the radial limit is infinite only on a denumerable set F kol (.k = 1,2,...), then there exist real constants C, with_ ICkl<o such that, in r<l, c0( u(re) = Z Ck * - r22 k^^l K1 + r2 2r cos (Q - Q) k =1 These results include an unpublished uniqueness theorem of P. C. Rosenbloom and clear up a question raised by Zygmund EMath. Zs, 25 (1926) p. 28] Some of the results are included in a paper by Dr. Lohwater in the Proceedings -of the American Mathematical Society vol. 3 (1952) pp. 478-279' Dr. Sze-Tsen Hu: Topological Grop The following are'the accomplishments of this project(1) The preparation of a paper entitled "Cohomology Theory in Topological Groups" for publication in the Michigan Mathematical Journalo (2) The completion of the: major part of a research on finite dimensional compact connected groups. Let G be a finite dimensional compact connected group and Go a closed. connected subgroup. The purpose of this work is to reduce the Cech cohomology ring H(G/Go) of the homogeneous space G/Go with real coefficients to'that of the homogeneous space of some compact connected Lie group. In fact, it is well known that there is a totally disconnected closed central normal subgroup Z of G such that the Quotient group M = G/Z is a Lie group. Let h: G-M denote the natural projection and let M = h(Go) The main result of this work states that the cohomology rings H(G/G0) and H(M/Mo) are isomorphic. In particular, if Go reduces to the neutral element e, we obtain H(G)' H(M). This result enables us to generalize a number of theorems proved for compact connected Lie groups by H, Hopf and his school. A paper entitled "On Finite Dimensional Compact Connected 4

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Groups" has been prepared and submitted to the Annals of Mathematics. Dr. Do J. Lewis: Topics in Algebra Dr. Lewis made a concentrated study of class field theory by attending lectures, reading, and confering with Professors Artin and Brauer. He reworked his thesis of 1950 on "Cubic Homogeneous Polynomials over p-adic Number Fields" and prepared it for publication. Efforts were again made to classify quartic homogeneous polynomials over finite fields which have only singular solutions, some sharper results were obtained. This problem was reduced to the classification of homogeneous polynomials which assume only quadratic residues modulo a prime. All efforts to resolve the latter question proved fruitless. A number of results concerning the norm groups of algebras were obtained and are being prepared for publication. The principal result is that the norm group of a central simple algebra of dimension nine over an algebraic number field is exactly that field. Mro R. Barraro Parabolic Partial Differential Equations (Research Direction by Professor Eo H. Rothe) During the project period Mr. Barrar completed the work necessary to apply the method developed by Schauder for elliptic equations (Math. Zs. vol. 38) to linear parabolic partial differential equations. The existence theorems obtained are an improvement over those previously published. A first draft of a doctoral dissertation was prepared on the basis of these results. Mr. J. Hocking: Topology (Research Direction by Professor G. S. Young) Mro Hocking continued to work in the direction indicated in the progress report of July 2, 1951. He encountered difficulties in the proof of one of the key results and found that a much deeper analysis was requiredo Mr. W. E, Jenner: Arithmetic of Rings and Algebras (Research Direction by Professor R. Brauer) Mr. Jenner worked on his doctoral dissertation and nearly completed it. He studied the arithmetic of rings and algebraso In particular, he considered representations of ideals as direct intersections. An ideal is called a block ideal if it cannot be written as a direct intersection (except in a trivial manner). Under suitable assumptions, 5

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN every ideal has.a unique representation as a direct intersection of block ideals. In the case of an algebra A over an algebraic number field K, the block ideal components of a prime ideal p of K in A are studied. These block ideals generate an abelian group. The block ideal compenents of a prime ideal p of K are completely determined by the behavior of p in the center of K. The results are used to obtain information about multiplicative decomposition of ideals of K in A. Mr. R. MacDowell: Banach Spaces (Research Direction by Professor S. B. Myers) Mr. MacDowell continued work on his dissertation. His characterization of the Banach space of continuous functions over the product of two compact Hausdorff spaces has led him to a study of the space of continuous functions on a fiber bundle. Mr. W. Smith: Banach Algebras (Research Direction by Professor S. B. Myers) Mr. Smith continued his study of Banach algebras of continuous mappings of a compact Hausdorff space into a Banach algebra. Mr. E. Crisler: Harmonic Functions and Integrable Functions (Research Direction by Professor W. Kaplan) If f(G) is Lebesque integrable for 0 _ 0 L 2i, then the Poisson integral formula defines a corresponding function u(r,~), harmonic for r c 1, with radial limit equal to f(G) for almost all 0. Results of Kaplan, Boothby, and others provide certain decompositions of the family of level curves of u(r,~). From these decompositions one obtains decompositions of the function f(0). Mr. Crisler has been studying these decompositions with the purpose of obtaining new characterizations of the class of integrable functions. Mr. T. W. Hildebrandt: Stresses and Deflections in Preloaded Spherical Shells. (Research Direction by Professor G. E. Hay) The problem under consideration is the static behavior of a shallow spherical shell supporting the loads (1) a uniform normal pressure P1 (the preload), and (2) an axial force P uniformly distributed over a small circular area of the shell. - -- - ------------------------— 6

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Starting with the general differential equations for a spherical shell as given by Reissnerl, a system of linearized equations is developed, together with specific conditions for linearization. These conditions are as follows: EPI < 2E Ih2 t, 2 B 1 rr- Eh 2 ~-E 2-> f 2,a2 B laJ 4 Eo 2 e a = r s of te s, h = t where a = radius of the shell, h = thickness, = colatitude of edge of area supporting P, E = Young's Modulus, B l=.12(1 2 and = Poisson's ratio. In words, the maximum numerical value of the pressure due to the axial force P is independent of the preload.p for l21/2, but may increase linearly with p for^,* 1/2. The same linearized system is valid for both cases of the linearization conditions. The system has a closed solution for all values of 4 when the preload is directed outward (sgn p.= +1), and for/, < 1 when the preload is directed inward (sgn p = -1). For the case* <1, the solution appears in terms of Bessel functions of complex argument, while for.Al, sgn p = +1, the solution appears in terms of Bessel functions of pure imaginary argument (modified Bessel functions). Thus all functions which appear in the solutions are currently available in tabulated form. Mr. J. Kaiser: Bessel Functions in Connection with Elliptic Partial Differential Equations in a Space of Constant Curvature (Research Direction by Professor E. H. Rothe) The Laplacian operator can be defined in an invariant manner for an arbitrary Riemannian manifold with positive definite metric. This research concerns the equation A2u + Xu = o for such a generalized Laplacian A2. It is assumed that the space is three dimensional and that the curvature is constant. In appropriate coordinates a separation of variables is possible. One of the Sturm- Liouville problems arising through the separation is studied in detail and the solutions are expressed in terms of generalized Bessel functions. The results overlap recent ones of M. U. Olevsky. 1 E. Reissner, "On Axisymmetrical Defdrmations of Thin Shells of Revolution". Proc. Symp. on Appl. Math., 3, pp. 27-52, 1950. --------------------------- ^ -------------------------— ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Mr. R. G. Kuller: Banach Algebras (Research Direction by Professor S. B. Myers) Mr. Kuller has been studying Banach algebras, particularly the algebra of differentiable real-valued functions. He will probably need another year to complete his dissertation. Mr. J. M. Miller- Topology (Research Direction by Professor Ro L. Wilder) Mr. Miller has taken an active part in several seminars on an advanced levelo This has necessitated his doing a wide range of reading in topology, group theory, topological algebra, and function spaces. He should be ready to start on his doctoral dissertation in the near future. Mr. G. Prins: Foundations of Mathematics (Research Direction by Professor Ro L. Wilder) Mr. Prins has concentrated on laying a foundation for his research program by studying the works of Carnap, Church, G0odel, and other writers on symbolic and mathematical logic. In addition he has studied chapters of a forthcoming book by Beth, as well as other notes that have been made available to him. He should be about ready to corn mence his doctoral dissertationo Mr. Ro Raimi: Banach Spaces (Research Direction by Professor So B. Myers) Mr, Raimi has been working on various questions concerning linear transformations in Banach spaces, with possible applications to ergodic theory. He will probably need another year to complete his dissertation. Mro J. Shoenfield; Foundations of Mathematics (Research Direction by Professor Ro L. Wilder) Mro Shoenfield has written the first draft of his doctoral dissertation, entitled "Models of Formal Systems". Therein he investigates axiomatic systems and independence and consistency prooofs by use of models within formal systems. I —------ 8

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Mr. D. Wend: Rie.mann Surfaces (Research Direction by Professor W. Kaplan) By application of the Schwarz reflection principle, one can use a portion of a parabolic Riemann surface to generate, by reflections, a class of parabolic surfaces. Mr. Wend has been studying this geometric process, with special emphasis on Riemann surfaces of inverses of polynomials. --- 9