Let R be a topological (Hausdorff) ring such that for each a ∊ R, aR and Ra are closed subsets of R. We will prove that if the set of non - trivial right (left) zero divisors of R is a non-empty set and the set of all right (left) zero divisors of R is a compact subset of R, then R is a compact ring. This theorem has an interesting corollary. Namely, if R is a discrete ring with a finite number of non - trivial left or right zero divisors then R is a finite ring (Refer [1]).

Several authors have investigated "rings of quotients" of a given ring R . (See, for example, Johnson [7], Johnson and Wong [8], Utumi [11], Findlay and Lambek [5], Lambek [9], and Bourbaki [2].)

In a number of papers [1] - [7], successively simpler proofs were given for the following inequality of Opial [1], in case p=1.

Theorem 1. If x(t) is absolutely continuous with x(0)=0, then for any p ≧ 0,

(1)

Equality holds only if x(t) = Kt for some constant K.

A topological space is an R1 - space if and only if x, y ε X, and such that U∩V = ϕ and x ε U, y ε V, where Ā denotes closure of the set A. Let be an R1 - space.

The Weierstrass - respectively Weierstrass - Stieltjes transform of a function F(t) or μ(t) is defined by

1.1

and

1.2

for all x for which these integrals converge. In what follows we shall always assume that F(t) is Lebesgue integrable in every finite interval and that μ(t) is a function of bounded variation.

It is known that a strong tournament of order n contains a cycle of each length k, k=3,…, n, ([l], Thm. 7). Moon [2] observed that each vertex in a strong tournament of order n is contained in a cycle of each length k, k = 3,…, n. In this paper we obtain a similar result for each arc of a regular tournament, that is, a tournament in which all vertices have the same score.

Let f(z) be a function regular for | z | < R, and denote by L(r) the length of the curve Γ onto which the circle | z | = r is mapped by f(z), i. e.

If Dr is the image of | z | ≤ r on the Riemann surface of f(z) then its area S(r) is given by

Let ℓ, m, n be three mutually distinct lines in the projective plane. The (ℓ, m, n)-Pappus condition can be described as follows.

Let A, B, C, A', B', C' be any six mutually distinct points such that A, B, C lie on ℓ; A', B', C' lie on m; and none of these points lies on ℓ∩m, m∩n, or n∩ℓ. If the points AB'∩BA' and BC'∩CB' both lie on n, then the point AC'∩CA' also lies on n.

Hall and Puckett [2] have shown that an arc - preserving function defined on a locally connected continuum having no local separating points is a homeomorphism if its total image is not an arc or point. This note shows that their results can be extended to non-compact manifolds.

In [l] Melzak has posed the following problem: “A plane finite set Xn consists of n ≥ 3 points and contains together with any two points a third one, equidistant from them. Does Xn exist for every n ? Must it consist of points lying on some two concentric circles (one of which may reduce to a point)? How many distinct (that is, not similar) Xn are there for a given n ? …” We shall here provide a construction for uncountably many Xn for every n > 4, and a counterexample to the second question above.

In this note we prove what we believe to be a new result concerning matrices, namely, that if an n × n matrix with rational entries has a finite order then this order is bounded. We also give an estimate for this bound and an application.

From the plane R2 we remove the union of the sets Sk (k = 1, 2, …) defined as follows (using the notation z = x + iy):

Sk = {z: arg z = nπ2-k for some integer n; |z|≥k}.

The remaining connected open set Ω we call the spiny urchin.

Consider three externally tangent circles A, B, C, with radii a, b, c, respectively. Let D be the circle which is externally tangent to all three and enclosed by them. Pack circles so that Z1 = D and Zi is externally tangent to A, B, Zi-1.

In this note we study dependence in matroids as an exercise in combinational algebra. Because the work seems to have little connection with graph theory we will not use Tutte′s approach (1) which uses dual concepts. To define a matroid we use Edmunds (2).

Let X be a random variable whose frequency function is

1.1

Form (1.1) is Stacy′s [3] generalization of the gamma distribution. The familiar gamma, chi, chi-squared, exponential and Weibull variâtes are special cases, as are certain functions of normal variate - viz., its positive even powers, its modulus, and all positive powers of its modulus.

The “function” δ(x - xo) is known as the Dirac Delta function and may be defined as zero everywhere except at xo, where it is infinite in such a way that

1

having property that for every continuous function φ(x) on (a, b)

2

It is well known [2] δ(x-xo) can be approximated as a limit of a sequence of piecewise continuous functions, and there is an abundance of such sequences.