A particle is initially at the origin in the (X, Y) plane and each successive step it takes is of unit length and parallel either to the X-axis or to the Y-axis. Its path of n steps is called a spiral if (i) the particle never occupies the same position twice, (ii) any turns the path makes are all counter-clockwise or all clockwise and (iii) for every m > n, the path can be continued to m steps without violating (i) or (ii).

A graph G is defined as a set X = {x1, …, xn} of elements xi called vertices, and a collection Γ of (not necessarily distinct) unordered pairs of distinct vertices, called edges. An edge (xi, xj) is said to be incident to xi and xj which are its end-vertices.

The celebrated criterion of Kuratowski [2] for the planarity of a graph G involves the determination of whether G contains a subgraph homeomorphic to K5 or K3, 3 shown in Figure 1.

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.

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.

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.

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

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.

The Cantor function C [2; p. 213], which appears in analysis as a simple example of a continuous increasing function which is not absolutely continuous, has the following properties:

1. (i) C is defined on [0,1], with C(0) = 0, C (l) = l;

2. (ii) C is continuous and non-decreasing on [0,1];

3. (iii) C is constant on each interval contiguous to the perfect Cantor set P;

4. (iv) C fails to be constant on any open interval containing points of P;

5. (v) The set of points at which C is non-differentiable is non-denumerable.

G. de Rham [1] has shown that certain remarkable curves, defined by limits of sequences of inscribed polygons, are solutions of functional equations of the form

If the functions FP (z) (p = 1,2) are contraction mappings of the complex plane into itself, i.e. for fixed r < 1

Egoroff' s theorem [1] was extended by Kvačko [3] to functions with values in a separable metric space; and, as is easily seen, this result applies also to separable pseudometric spaces. In the present note we shall use this theorem to obtain some propositions on iterated limits, which, despite their simplicity, seem not yet to be known in the proposed generality.

Let F(x) be the continuous distribution function of a random variable X and Fn(x) be the empirical distribution function determined by a random sample X1, …, Xn taken on X. Using the method of Birnbaum and Tingey [1] we are going to derive the exact distributions of the random variables

and and where the indicated sup' s are taken over all x' s such that -∞ < x < xb and xa ≤ x < + ∞ with F(xb) = b, F(xa) = a in the first two cases and over all x' s so that Fn(x) ≤ b and a ≤ Fn(x) in the last two cases.

A well-known theorem of Copping [2] states that a conservative matrix with a bounded left inverse cannot evaluate a bounded divergent sequence. (Definitions are given in the next paragraph.) A proof was given by Parameswaran [3, Theorem 6. 1], using only the simplest Banach-space ideas. This proof, however, is valid only for co-regular methods; it was stated in [3, Theorem 6. 2] that a co-null matrix cannot have a bounded left inverse, but the proof there given is incorrect, as it uses for co-null methods a theorem established only for co-regular. It would be desirable to have a short independent proof of this known result, which excludes co-null matrices from consideration in Copping' s theorem. This is furnished by the slightly more general result given below.

A ring R is called a prime ring [1] if and only if a·R·b = 0 implies that a = 0 or b = 0 for all a, b ϵ R. Hence if R is a prime ring and a is a non-zero element of R, a·R ≠ 0 and R·a ≠ 0. In the present note we prove that a prime ring with a maximal annihilator right ideal has no non-zero nil right or left ideal.