This article consists of a list of problems connected with the theory of convex sets in the Euclidean space En, although in some cases the connection is indirect and/or inessential. The assumptions may often be relaxed or otherwise changed, thus giving rise to a wide variety of questions.

The configuration formed by N points in general position in space, together with the lines joining them in pairs will be called an N-clique. The N-clique is coloured by assigning to each edge exactly one colour from a set of t possible colours. A theorem due to Ramsey [4] ensures the existence of a least integer M(q1, q2, …, qt) such that if N ≥ M, any such colouring of the N-clique must contain either a q1-clique entirely of colour 1, or a q2-clique of colour 2, …, or a qt-clique of colour t. Another proof of Ramsey's theorem is given by Ryser [5].

A subset consisting of k elements chosen, from n distinct ordered elements, with given restrictions is called are stricted choice. For example, one restriction on the k elements may be that no two consecutive elements appear, while another may be that no two alternate elements appear. Certain restricted choices may be used to obtain solutions to permutation problems ([1, p. 349]; [4]). Each restricted choice corresponds to a "restricted sequence of Bernoulli trials" as described in [1], In this paper an elementary method of obtaining more general types of restricted choices is given. Some special cases of the restricted choices and restricted Bernoulli trials are presented in the form of examples.

Up to a duality, the known finite projective planes which are not translation planes all are equivalent to affine planes which contain the type of structure defined below to be a "derivable net". (Insofar as the known finite planes are concerned, this means that the intimate connection between projective geometry and linear algebra still holds for non-Desarguesian planes.)

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

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.

Let f(t) be integrable L(-π, π) and periodic with period 2π, and let

1.1

be its Fourier series. The series

1.2

obtained by the term by term differentiation of the series (1. 1), is called the Derived Fourier series of (1. 1). The series conjugate to (1. 2),

1.3

is called the Derived conjugate series of the Fourier series of f.

We consider a system of differential equations of second order given by

1

(' = d/dt) where P and Q have continuous first partial derivatives with respect to x and y in some open and simply connected set R containing O = (0, 0) which we assume to be the only singular point in R. In fact, let R be the whole plane; for if not then the following discussion can be modified.

Recently A. Haimovici [1] has proved a general fixed point theorem of transformations in metric spaces from which he obtained existence theorems for certain types of ordinary and partial differential equations. However, both the result and the proof are given for a rather special case. One of the purposes of this present note is to put his result on a more concrete basis and give a stronger characterization of the kind of transformations used in [l]. (Theorem 3).

If R is a simple ring with 1 which contains a maximal annihilator right ideal then R is the endomorphism ring of a unital torsion-free module over an integral domain.

We first prove the following:

Let R be a ring with 1. If a ϵ R such that (a)r = {r ϵ R|ar = 0} is a maximal annihilator right ideal then HomR(aR, aR) is an integral domain.