It is a well-known fact in the theory of Appell's hypergeometric function of two variables F4, defined by

1

where |x|1/2 + |y|1/2 < 1, that it can be expressed interms of products of ordinary hypergeometric functions when γ + γ' = α + β + 1.

It is well-known that in a Hausdorff space, a sequence has at most one limit, but that the converse is not true. The condition that every sequence have at most one limit will be called the semi-Hausdorff condition. We will prove that the semi-Hausdorff condition is strictly stronger than the T1 -axiom and is thus between the T1 and T2 axioms. In this note, we investigate into some properties of the spaces satisfying the semi-Hausdorff condition.

We consider one-parameter families of distributions which are of the general exponential type, the general series distribution type, or which can be transformed into the exponential type by a one-to-one transformation. In this paper we establish theorems to the effect that such distributions may be characterized by a simple differential equation involving the mean function. It is illustrated that almost all the classical one-parameter families of distributions are characterized by these theorems. Multivariate generalizations are given, and it is also noticed that the functional form of the normalizing factor determines the specific distribution in each general family.

The solution of a linear non-homogeneous differential equation whose non-homogeneous term is of the form tkeαt can be obtained by what is usually called the method of undetermined coefficients. The application of this method may be justified in several different ways (see for example [1, pp. 114–117], [2, pp. 94–99], [3]).

We shall consider the analogous problem for a system of differential equations. It turns out that we can solve this problem using only elementary techniques of linear algebra. The solution has essentially the same form as in the case of a single equation, but may contain terms which would not be expected and may lack terms which would be expected in a straightforward extension of the theory to systems. Our method of obtaining the solution is constructive, in the sense that while our results give only the form of the solution, the solution itself may be found by substitution of this form into the system of differential equations.

Let T be a tetrahedron and let V(T) and L(T) denote its volume and the sum of its edge-lengths. In this note we prove

Theorem 1.

with equality if and only if the tetrahedron T is regular.

By a semi-lattice we mean a system <L,.> where L is a set and. is a binary operation in L that is idempotent, commutative and associative. In a recent article [2] D.H. Potts considers the problem of reducing the number of axioms for a semi-lattice. His result was that the following two axioms viz. (1) xx=x, (2) (uv)((wx)(yz)) = ((uv)(xw))(zy) are sufficient to give a semi-lattice. But the second identity contains six elements instead of the original three. In the following we give a set of two simple identities with just three elements for a semi-lattice. This improves the above mentioned result.

The theorem that the homotopy sequence is exact splits into six statements. Scherk ([4]) obviates the use of homotopy extension in the proof of one of these statements. The purpose of this note is to show that the method can be adapted to give a direct proof of the corresponding statement in the theorem that the Eckmann-Hilton homotopy sequence ([l]) is exact. The note is based on Eckmann' s exposition ([2]). We are concerned with the proof of b2, pp. 34–35.

Let be an arbitrary family of continuous complex-valued functions defined on a compact Hausdorff space X. A closed subset B ⊆ X is called a boundary for if every attains its maximum modulus at some point of B. A boundary, B, is said to be minimal if there exists no boundary for properly contained in B. It can be shown that minimal boundaries exist regardless of the algebraic structure which may possess. Under certain conditions on the family , it can be shown that a unique minimal boundary for exists. In particular, this is the case if is a subalgebra or subspace of C(X) where X is compact and Hausdorff (see for example [2]). This unique minimal boundary for an algebra of functions is called the Silov boundary of .

It is well known that the determination of a (non-isotropic) curve in the euclidean 3-space with given curvature κ(S) and torsion τ(s), where s represents the arc-length, depends upon the integration of a Riccati equation; and that this can be performed only if a particular integral of the equation is known.

In an n-dimensional manifold Vn, coordinates xi for i = 1, …, n, let each curve x(t) for t0≤t≤t1 of class C1 define a corresponding λ by means of the integral equation

1

It is a commonplace that F is continuous on the cartesian square of the range of f if f is continuous and satisfies

1

say, for all real x, y (cf. e.g. [2]). A.D. Wallace has kindly called my attention to the fact, that this is trivial only if f is (constant or) strictly monotonic and asked for a simple proof of the strict monotonicity of f. The following could serve as such: if on an interval f is continuous, nonconstant and satisfies (1), then f is strictly monotonic there.

This note is concerned with the extension to locally convex spaces of a theorem of J. Y. Barry [ 1 ]. The basic assumptions are as follows. E is a separated locally convex topological vector space, henceforth assumed to be barreled. E' is its strong dual. For any subset A of E, we denote by w(A) the closure of A in the σ-(E, E')-topology. See [ 2 ] for further information about locally convex spaces. By a projection we shall mean a continuous linear mapping of E into itself which is idempotent.

If Fn is the distribution function of a distribution n with moments up to order n equal to those of the standard normal distribution, then from Kendall and Stuart [1, p.87],

We call k integers x1 < x2 … < xk chosen from 1, 2, …, n} a k-choice (combination) from n. With 1, 2, …, n arranged in a circle, so that 1 and n are consecutive, we have a circular k-choice from n. A part of a k-choice from n is a sequence of consecutive integers not contained in a longer one. Let denote the number of circular k-choices from n with exactly r parts all ≤ w.

Let n be an arbitrary but fixed positive integer. Let Tn be the set of all monotone - increasing n-tuples of positive integers:

1

Define

2

In this note we prove that ϕ is a 1–1 mapping from Tn onto {1, 2, 3,…}.

The primordial problems of linear algebra are the solution of a system of linear equations

and the solution of the eigenvalue problem

for the eigenvalues λk, and corresponding eigenvectors of a given matrix A.

Planarity in graphs implies relatively small valencies and numbers of edges. In this note we find the maximum sum of valencies and the maximum number of incident edges for a set of n vertices in a planar graph with v vertices. Graphs considered are without multiple edges or loops.

Let w = f(z) be holomorphic on the unit disk D = { z: | z | < 1}, with the additional restrictions that | f ( z ) | < l and , where denotes the (outer) angular limit of f (z) at z = 1. Let us now define and then focus our attention on the behaviour of g(z) in an arbitrary angular neighbourhood of z = 1. Whenever exists, this limit is commonly referred to as the angular derivative of f(z) at z = 1.

In [2, p. 511] Sanderson has shown that if every Large left ideal of a ring R with identity contains a regular element, and if the regular elements in R satisfy Ore's condition, then the complete (Utumi's) ring of quotients coincides with the classical ring of quotients. He conjectured that the above conditions are also necessary. The following is a counter example.