Bulletin of the American Mathematical Society Volume 27 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1921 Excerpt: ...meeting, Professor Kasner showed that there are no four-dimensional manifolds obeying Einstein's equations Gik = 0 which can be immersed in a 5-flat; but that for a li-flat there exist an infinity of solutions, including in particular the solar field. He now examines this field, taken in the Schwarzschild form, in detail. The final result is where m is the mass of the sun, and r2 = x2 + y2 + z2. We thus have in finite form a model of the solar field. The model is situated in a flat space of six dimensions; or, using the more exact terminology of Weyl, in an affine-euclidean space of (2 + 4) dimensions. (18) Dr. Norbert Wiener: The average of an analytic functional. Dr. Wiener discusses in this paper the notion of the average value of a functional which can be expanded in a series of multiple integrals, and develops an average which, over a considerable range of cases, is identical with that treated in his other papers, but which will apply directly to functionals that are not bounded. (19) Dr. Norbert Wiener: The average of a functional. Dr. Wiener gives a definition of the average value of a functional which reduces to a special case of the type of Daniell integral already treated by him in his paper entitled The mean of a function of arbitrary elements, presented to this Society at its meeting in December, 1919. The average thus developed is in intimate relation with the theory of probabilities, for it is based on the assumption that the distribution of the values of F(T + r2), where F ranges over all continuous functions such that F(Ti) has a certain determinate value Xi, is what is known statistically as a normal distribution and is independent of Tx and Xi. (20) Dr. Norbert Wiener: Further properties of the average of a functional. Dr. Wiener defines t...