The sixty-second Summer Meeting of the American Mathematical Society was held at Pennsylvania State University, University Park, Pennsylvania, on Tuesday through Friday, August 27-30, 1957, in conjunction with meetings of the Mathematical Association of America, the Society for Industrial and Applied Mathematics, and Pi Mu Epsilon. The registration was 649 including 493 members of the Society. Professor N. E. Steenrod of Princeton University delivered the Colloquium lectures, entitled Cohomology operations, and obstructions to extending continuous functions on Tuesday at 2:00 P.M. and Wednesday, Thursday, and Friday at 9:00 A.M. Presiding officers were Professors Salomon Bochner, Samuel Eilenberg, Leo Zippin, and G. W. Whitehead. By invitation of the Committee to Select Hour Speakers for Annual and Summer Meetings, Professor A. P. Calderon of the Massachusetts Institute of Technology addressed the Society on Singular integrals and differential equations on Wednesday at 11:00 A.M., and Professor M. A. Rosenlicht of Northwestern University delivered an hour address, entitled Rationality questions on algebraic groups, on Thursday a t 10:30 A.M. Presiding officers were Professors Tibor Rado and A. M. Gleason, respectively. There were fifteen sessions for contributed papers. The presiding officers were Professors R. D. Anderson, J. H. Barrett, H. P. Evans, Orrin Frink, Jr., M. R. Hestenes, G. W. Mackey, N. H. McCoy, H. T. Muhly, E. D. Nering, Dr. W. H. Pell, Professor J. H. Roberts, Dr. R. L. San Soucie, Professors E. V. Schenkman, I. M. Sheffer, and Dr. W. C. Taylor. A reception in the Hetzel Union Building was held on Wednesday afternoon. There was a chicken barbecue on Thursday evening in Horticulture Woods. A banquet was held in the Hetzel Union Building on Wednesday evening. Professor H. B. Curry acted as toastmaster. The speakers were Dean Ben Euwema, Professor Orrin Frink, and President Eric A. Walker of Pennsylvania State University, Professors E. J. McShane, G. B. Price, J. S. Frame, and Dr. T. H. Southard. A resolution of thanks to Pennsylvania State University and to the Arrangements Committee, of which Professor Evan Johnson, Jr., was the chairman, was presented by Professor W. L. Hart . The Council met on Tuesday afternoon and evening, August 27, 1957.

s of the papers presented are listed below, those with a "t" after their numbers having been read by title. The joint papers numbered 340, 372, 373, 376, and 381 were read by Dr. Singer, Professor Miller, Mr. Crispin, Dr. Juncosa, and Dr. Lukacs, respectively. ALGEBRA AND THEORY OF NUMBERS 319. A. A. Albert: New power-associative algebras and their deriva-

Milton Abramowitz, M. I. Aissen, E. J. Akutowicz, C. B. Allendoerfer, R. D. Anderson, T. W. Anderson, R. G. Archibald, L. A. Aroian, Frederick Bagemihl, Joshua Barlaz, P. T. Bateman, F. P. Beer, E. G. Begle, Stefan Bergman, Lipman Bers, Garrett Birkhoff, D. W. Blackett, A. L. Blakers, A. A. Blank, S. G. Bourne, C. B. Boyer, Paul Brock, A. B. Brown, F. H. Brownell, L. J. Burton, Jewell H. Bushey, J. H. Bushey, W. R. Callahan, H. E. Campbell, P. G. Carlson, Jeremiah Certaine, K. T. Chen, Y. W. Chen, W. L. Chow, L. W. Cohen, A. J. Coleman, T. F. Cope, Natalie Coplan, L. M. Court, M. D. Darkow, Martin Davis, B. V. Dean, H. F. DeBaggis, J. L. Doob, Arnold Dresden, Aryeh Dvoretzky, Samuel Eilenberg, M. P. Epstein, R. M. Exner, J. M. Feld, F. A. Ficken, N. J. Fine, R. S. Finn, A. D. Fleshier, R. M. Foster, R. H. Fox, Gerald Freilich, Orrin Frink, G. N. Garrison, Hilda Geiringer, H. A. Giddings, J. B. Giever, B. P. Gill, G. H. Gleissner, H. E. Goheen, W. H. Gottschalk, Bernard Greenspan, Harriet Griffin, Emil Grosswald, E. J. Gumbel, V. H, Haefeli, F. C. Hall, F. S. Hawthorne, G. A. Hedlund, M. H. Heins, Alex Heller, Erik Hemmingsen, A. H. Henry, L. H. Herbach, A. A. Herschfeld, Einar Hille, Abraham Hillman, Joseph Hilsenrath, Banesh Hoffmann, T. R. Hollcroft, L. A. Hostinsky, E. M. Hull, Witold Hurewicz, L. C. Hutchinson, M. A. Hyman, Eugene Isaacson, Nathan Jacobson, Wenceslas Jardetzky, B^rge Jessen, S. A. Joffe, Fritz John, R. A. Johnson, F. E. Johnston, Shizuo Kakutani, Edward Kasner, W. H. Keen, M. E. Kellar, L. S. Kennison, J. F. Kiefer, H. S. Kieval, S. C. Kleene, J. R. Kline, E. G. Kogbetliantz, E. R. Kolchin, Horace Komm, B. O. Koopman, Jack Laderman, A. W. Landers, J. P. LaSalle, V. V. Latshaw, Solomon Lefschetz, Marguerite Lehr, Benjamin Lepson, Howard Levi, D. C. Lewis, M. A. Lipschutz, S. R. Lipsey, L. H. Loomis, E. R. Lorch, Lee Lorch, Janet McDonald, Brockway McMillan, L. A. MacColl, H, M. MacNeille, Irwin Mann, A. J. Maria, M. H. Maria, W. T. Martin, W. S. Massey, A. E. Meder, Paul Meier, A. N. Milgram, K. S. Miller, W. H. Mills, Don Mittleman, E. E. Moise, Deane Montgomery, Marston Morse, H. H. Mostafa, G. D. Mostow, T. S. Motzkin, F. J. Murray, David Nelson, C. V. Newsom, A. V. Newton, P. B. Norman, I. L. Novak, C. O. Oakley, J. C. Oxtoby, T. E. Peacock, A. J. Penico, I. D. Peters, B. J. Pettis, Everett Pitcher, E. L. Post, Walter Prenowitz, M. H. Protter, Hans Rademacher, Hans Râdstrom, G. N. Raney, H. E. Rauch, G. E. Raynor, M. S. Rees, Helene Reschovsky, Moses Richardson, J. F. Ritt, I. F. Ritter, J. E. Robinson, P. C. Rosenbloom, H. D. Ruderman, Raphael Salem, H. E. Salzer, Arthur Sard, Samuel Schecter, Henry Scheffé, Eugene Schenkman, M. M. Schiffer, Pincus Schub, Abraham Schwartz, G. E. Schweigert, Wladimir Seidel, Atle Selberg, D. B. Shaffer, Daniel Shanks, C. E. Shannon, H. N, Shapiro, LM. Sheffer, Seymour Sherman, James Singer, M. H. Slud, L. L. Smail, Ernst Snapper, Andrew Sobczyk, D. C. Spencer, George Springer, Fritz Steinhardt, Wolfgang Sternberg, J. J. Stoker, R R. Stoll, M. M. Sullivan, Olga Taussky, D. L. Thomsen, John Todd, P. M. Treuenfels, H. C. Tsang, Oswald Veblen, S. L Vrooman, H. V. Waldinger, J. L. Walsh, Alan Wayne, Alexander Weinstein, Louis Weisner, M. E. White, P. M. Whitman, D. V. Widder, Albert Wilansky, W. G. Wolfgang, Jacob Wolfowitz, Y. K. Wong,

This chapter presents a proof for the theorem: for any n x n stochastic matrix M of real numbers, there exists an integer p > 1 such that the powers Mpr converge to a stochastic matrix L and such that the p matrices - L, ML, M2L,…,Mp-l L, are distinct. Because the theorem was needed in the theory of automata, it was essential to produce a proof that is effective and provides finitistic methods for finding p and L. Experimenting with examples indicate that the integer p and the position of zeros in the matrix L depend only on the position of zeros in the matrix M and not on actual numerical values of its entries. This observation is correct and paves way to a finitistic treatment of a considerable part of the proof.

proofs of the gradient method for programming problems with concave and more general functions.

This Consulting Job is now open IPromoted only a short time ago, this man formerly worked as an IBM Applied Science Special Representative for the petroleum industry in Houston.Such service to the oil companies has become virtually essential.This man, for instance, counseled petroleum scientists and executives in the application of digital computers for exploration, production, and refining.He organized conferences and directed seminars . . .advising on coding systems and techniques.In addition to his other duties, he coordinated IBM contacts

The need for an axiomatic treatment of homology and cohomology theory has long been felt by topologists. Professors Eilenberg and Steenrod present here for the first time an axiomatization of the complete transition from topology to algebra. Originally published in 1952. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

We are concerned with polynomials of degree n of the typewhere x, ao, #i, • • •, a n are rea l quaternions (a^O for i = 0,1, • • •, n) and (x) is a sum of a finite number of similar monomials btfcbix • • • xbk, where k