Abstracts of Papers Submitted for Presentation to the Society

S OF PAPERS SUBMITTED FOR PRESENTATION TO THE SOCIETY The following papers have been submitted to the Secretary and the Associate Secretaries of the Society for presentation at meetings of the Society. They are numbered serially throughout this volume. Cross references to them in the reports of the meetings will give the number of this volume, the number of this issue, and the serial number of the abstract. 353. Stefan Bergman : On a generalized Green's function and certain of its applications, A theorem of Blaschke states: 2Z*=.i lg \av\ > — oo, \av\ < 1 , is the necessary and sufficient condition for the existence of a nonnegative harmonic function H(z), i7(0) = 1, 2, • • • , is regular at z = a„. Let 9tt 4 = E [ | s 2 | < 1 , zx e 33 fe)], Sö2(02)==E[01 = ^(z 2 , X), 0 ^ s < l , 0 ^ \ ^ 2 T T ] , h(z2, X), h(z2, 0)=h(z2, 2w), being for every X an analytic function of z2, z2 e QÊ . In the paper (Compositio Mathematica, vol. 6 (1939), pp. 307-335) there was introduced for every domain 9tt an "extended class of functions" possessing properties analogous to those of harmonic functions. If g&u; z2) is in 90? 4 an analytic function, it is then possible to construct the "Green's function of the extended class" T(zi, s2; g; SO? ) vanishing on the boundary surface E [ | z 2 | = 1 , zi=»h(z2, X), 0 ^ X ^ 2 T T ] such that T(zh z2; g; $? )+lg |g(si , z2)\ is a function of extended class. Let gv(z\, z>>) designate a set of functions analytic in #ft 4 satisfying a certain additional restriction concerning the distribution of zero surfaces. The c o n d i t i o n X X i r ( 0 , 0; gv;W(l*) < oo is necessary and sufficient for the existence of a nonnegative function H(zh z2), H(zh z2)< oo for (zi, z2) e SOî —Sv®l, ©Ï^Efóyfei, z2) = 0, (si, z2) e ffll ] such that for every n, H(zh Z 2 ) + ] L ? I lg I &( h 22) | , belongs to the extended class of functions of Tlno, no^n, where limn+«> ffli—till*. (Received March 28, 1940.) 354. L. M. Blumenthal: A new concept in distance geometry, with applications to spherical subsets. If o-C 5, a semimetric space, then S has o--relative congruence indices {n, k} with respect to a given class (2) of semimetric spaces provided any space 2 of (2) with more than n-\-k pairwise distinct points is congruent ly contained in S whenever each n of its points is congruently contained in <r. This concept gives rise to new problems in distance geometry. For <r — S the indices {n, k] are called merely congruence indices of S with respect to (2) . The notions of congruence and quasi congruence orders introduced by Menger correspond to congruence indices {n, 0 } , {n, l } , respectively, while {n, 2} signifies a property weaker than quasi congruence order n but stronger than quasi congruence order w-j-1. Let K2,P denote a cap with spherical radius p of the sphere S2,r. It is shown that (1) if a = K2tirr/2, with base circle removed, then S2,r has (r-relative congruence indices {4, 2 } with respect to all semimetric spaces (2) ; (2) the closed cap K2)P, p < x r / 2 , has congruence indices {4, 2} with respect to (2) ; (3) /c2(P, p<r cos 1 1/3, has indices {4, 1} . Analogous theorems for caps of Sn,r are obtained. (Received April 12, 1940.)