Some of the most important G-Structures of the first kind [1] are those defined by linear operators satisfying algebraic relations. If the linear operator J acting on the complexified space of a differentiable manifold V satisfies a relation of the form

where I is the identity operator, the manifold has an almost complex structure ([2] [3]). The structures defined by

are the almost product structures ([3] [4]).

The following problem was brought to our attention by L. Moser: Locate eight points in the closed unit square so that the minimum of the distances between any two of the points should be as large as possible.

In [1] Dirac has determined the structure of all 3-connected graphs which do not contain two independent (i.e., disjoint) circuits. We shall here provide a short proof of this theorem by applying Tutte' s theory of 3-connected graphs [2].

By a group ring we mean in this paper a ring defined by a finite group G and an integral domain K:

such that A contains G and is freely generated by G over K, so that

The ring A = KG has a co-multiplication

defined by

so that A is a Hopf algebra.

A ring R with unity is called right (left) self injective if the right (left) R-module R is injective [7]. The purpose of this note is to prove the following: Let R be a prime ring with a maximal annihilator right (left) ideal. If R is right (left) self injective then R is a primitive ring with a minimal one-sided ideal. If R satisfies the maximum condition on annihilator right (left) ideals and R is right (left) self injective then R is a simple ring with the minimum condition on one-sided ideals.

Let K, k be fields and K|k a finite galois extension with galois group G. The multiplicative group K* of K is a G-module, that is, a module over the integral group ring ZG, the module action of an element σ ϵ G being its effect as an automorphism. It is shown in [2, p. 158] that the first cohomology group vanishes:

1

It is well known that the Laplacian operator is the infinitesimal generator of Brownian motion in Rn. Moreover, the classical harmonic measures are the hitting measures of Brownian motion. In other words, there is a natural correspondence between the Brownian motion and the classical harmonic functions. In this paper we will show that any family of abstract harmonic functions satisfying the axioms of M. Brelot [4] are annihilated by the infinitesimal generator of a diffusion, and that the corresponding harmonic measures are the hitting measures of the diffusion. This answers a question raised by P. A. Meyer [11] who has proved the existence of a Markov semi-group associated with a harmonic sheaf.

Let R, r, s represent respectively the circumradius, the inradius and the semiperimeter of a triangle with sides a, b, c. Let f(R, r) and F(R, r) be homogeneous real functions. Let q(R, r) and Q(R, r) be real quadratic forms. The latter functions are thus a special case of the former. Our main result is to derive the strongest possible inequalities of the form

1

with equality only for the equilateral triangle.

The principal object of this note is to determine the maximal order of Abelian subgroups of the symmetric group sn of degree n. We also discuss some related results and problems.

A largest Abelian subgroup of sn has order f(n) where

Given an integer k ≤ 2 and a finite set M of rational integers. Let vi (i = 1, 2, …, n) be m-dimensional (column-)vectors with all components from M and such that the kn sums

1.1

are all different. Then we shall say that {v1, v2, …, vn} is a detecting set of vectors.

This expository note was prompted by some questions asked by Professor P. Hilton during his lectures "Catégories non-abétiennes" at the University of Montréal, July 1964.

The descriptions of set functions as one to one and as onto can be characterized in terms of set function composition. A set function is one to one iff it has the left cancellation property, that is, f · g = f · h implies g = h.

It is well known [1] that the variational problem of minimizing

1

where F is positive homogeneous of degree one in (henceforth abbreviated to "plus-one" in ) leads to a Hamiltonian H{xi, pi} and corresponding Hamilton-Jacobi equation

2

Since the solution of Post' s Problem by Friedberg and Muchnik, a good deal of abstract knowledge about the semilattice of recursively enumerable degrees has been developed, especially in recent papers of G. E. Sacks. In this note, we show that one specific class of simple sets contains only sets of degree 0'; no contribution to the general theory is claimed. One of the sets belonging to the special class considered is the original simpie-but-not-hypersimple S of Post [1]. According to information received from Sacks, J. R. Myhill gave a proof, in 1953, of the completeness of S. However, as far as we know, Myhill' s proof (presented in a seminar) does not exist in published form.

In the ranking of a collection of p objects by the method of paired comparisons, a measure of consistency is provided by the relative number of transitive (or consistent) triples and cyclic (or inconsistent) triples. This point of view was introduced by Kendall and Babington Smith [4]. They found a formula for the maximum number of cyclic triples, thereby determining the greatest inconsistency possible. The purpose of this note is to extend the result to obtain the maximum number of "strongly connected" collections of n objects among the given p objects.

Let Ω be an index set and for each αϵΩ let fα:X → Xα be a function where X and Xα are sets. Assume that, for each α, a topology is given for Xα. Then, as is well-known, the functions fα and the topologies determine a topology for X. This is the so-called weak or initial topology, which is generated by .

In his paper [1] V. L. Klee gave an example of a smooth bounded convex body C, in En, with the property that extC s closed and extC - expC is dense in extC. As in [1] extC and expC denote the sets of extreme and exposed points of C respectively. It is the purpose of this note to exhibit a similar example in a general separable Banach space using a direct construction which involves Minkowski summation of convex sets.

Il s'agit d' étudier les automorphismes infinitésemaux (a. i.) de certaines structures presque cosymplectiques liées aux variétés finslériennes

If U and V are independent random variables, both drawn from the Normal distribution, then it is known that the distribution of U/V follows the Cauchy law, i. e. has frequency function 1/π(1+x2). Conversely if U and V are independently drawn from the same distribution and U/V is known to follow the Cauchy law of distribution must U and V be necessarily drawn from a Normal distribution? This question has been considered by several authors ([3], [4], [5], [7]) who have obtained several examples to show that the ratio property above is not confined to the Normal distribution.

The field of boundary limit theorems in analytic function theory is usually considered to have begun about 1906, with the publication of Fatou's thesis [8]. In this remarkable memoir a theorem is proved, that now bears the author's name, which implies that any bounded holomorphic function defined on the unit disk possesses an angular limit almost everywhere (Lebesgue measure) on the frontier. Outstanding classical contributions to this field can be attributed to F. and M. Riesz, R. Nevanlinna, Lusin, Privaloff, Frostman, Plessner, and others.