Let Sn be the set of n2 points with integer coordinates n (x, y), 1 ≤ x, y <n. Let fn be the maximum cardinal of a subset T of Sn such that no three points of T are collinear. Clearly fn < 2n.

For 2 ≤ n ≤ 10 it is known ([2], [3] for n = 8, [ 1] for n = 10, also [4], [6]) that fn = 2n, and that this bound is attained in 1, 1, 4, 5, 11, 22, 57, 51 and 156 distinct configurations for these nine values of n. On the other hand, P. Erdös [7] has pointed out that if n is prime, fn ≥ n, since the n points (x, x2) reduced modulo n have no three collinear. We give a probabilistic argument to support the conjecture that there is only a finite number of solutions to the no-three-in-line problem. More specifically, we conjecture that

Let G be an undirected graph, without loops or multiple edges. An automorphism of G is a permutation of the vertices of G that preserves adjacency. G is vertex transitive if, given any two vertices of G, there is an automorphism of the graph that maps one to the other. Similarly, G is edge transitive if for any two edges (a, b) and (c, d) of G there exists an automorphism f of G such that {c, d} = {f(a), f(b)}. A graph is regular of degree d if each vertex belongs to exactly d edges.

In this note we consider the enumeration of unrestricted and restricted minimal lattice paths from (0, 0) to (m, n), with the following (μ + 2) moves, μ being a positive integer. Let the line segment between two lattice points on which no other lattice point lies be called a step. A lattice path at any stage can have either (1) a vertical step denoted by S0, or (2) a diagonal step parallel to the line x = ty (t = 1,…, μ), denoted by St, or (3) a horizontal step, denoted by Sμ+1.

In [3] H. Schwerdtfeger embedded the one-dimensional affine group over the real numbers in the projective plane. The relationship between group-theoretical properties and geometrical concepts was studied.

In this paper the methods of [3] are used to prove Pappus' theorem. In the final section we give a similar theorem for (4n+2)-gons.

This paper is a generalization of part of my master's thesis, written under the direction of Professor H. Schwerdtfeger.

The main result of this paper is the determination of all nonisomorphic local near-rings <N, +, •> with <N, +> =<C(p) × C(p), +> which are not near-fields. Together with the fundamental paper [6] by Zassenhaus on near-fields and the corollary to Theorem 1 of [2], this 2 paper gives a complete description of all local near-rings of order p2.

We recall that a unitary near-ring N is called local if the subset L of elements in N without left inverses is an (N, N)-subgroup and N ≠ J(N). (J(N) denotes the radical of N given in [1].) In [3] it was proved that N ≠ J(N) whenever L is an ideal of N. (For previous result s concerning local near-rings we refer the reader to [3].)

In [10], it is defined that a right (or left) ideal I of a ring R is very large if the cardinality of R/I is finite. It is also proven in [10, Theorem 3.4] that if R is a prime ring with 1 such that its characteristic is zero, then R is a right order in a simple ring with the minimum condition on one sided ideals if every large right ideal of R is very large. In the present note, we shall prove that if R is a prime ring with 1 such that its characteristic is zero and R is also a compact topological ring, then R is a right and left order in a simple ring with the minimum condition on one sided ideals, which is also a non-discrete locally compact topological ring if and only if every large right ideal of R is open. In particular, if R is an integral domain with 1 (not necessarily commutative) such that its characteristic is zero, then R is openly embeddable [13, p. 58] in a locally compact (topological) division ring if and only if every large right ideal of R is open. Following S. Warner [13], we shall say R is openly embeddable in a quotient ring Q(R) if there is a topology on Q(R) which is compatible with its structure, which induces on R its given topology and for which R is an open subset.

This note proves another special case of a conjecture of U. Oberst. Oberst considered [1], for any small category , the X abelian category of abelian group-valued functors on , and the X functor Colim: which takes each diagram to its colimit. The question is, when is Colim exact? For its relationships, see [1], It is a sufficient condition that each component of is upward filtered. Oberst conjectured that it is also necessary, and proved this under some conditions. He mentioned particularly the case that is a monoid, i.e. a category with one object. We shall verify the conjecture in that case.

The General Selection Principle referred to in the title is really Tychonoff's theorem on products of compact spaces, but in a somewhat disguised form (c.f. Theorem 2.1, below). It is believed that this form is one which lends itself very well to many applications. More specifically, one of the immediate corollaries of the main theorem is a theorem due to Rado (c.f. Cor 2.1.1.), which has been used by Erdos and de Bruijn [2], Luxemburg [7], and the author [9] to give simple proofs of a variety of results.

In this paper we prove that if {sm, n(x, y)} is the sequence of partial sums of the Fourier series of a function f(x, y), which is periodic in each variable and of bounded variation in the sense of Hardy-Krause in the period rectangle, then {sm, n(x, y)} converges uniformly to f(x, y) in any closed region D in which this function is continuous at every point. This result is then used to prove that the regular Hausdorff means of the Fourier series of such a function also converge uniformly in such a region.

It might be thought that, as far as the oscillation of solutions is concerned, the behaviour of

and

would be the same as long as t - α(t) → ∞ as t→∞. To motivate the theorem presented in this note, we show first that this is not the case. Consider the above equation with α(t) = 3t/4, a(t) = l/2t2 i.e.

This equation has the non-oscillatory solution y(t) = t1/2 although all solutions of

are oscillatory [1, p. 121].

Given a sequence of r distinct real numbers such that the number of terms of every decreasing subsequence is at most m, then there exists an increasing subsequence of more than n terms, where n is the largest integer less than r/m.

An extremely simple and elegant proof of the theorem was given by A. Seidenberg [2]. This note is intended to point out that a result analogous to the above holds under a more general setting.

The spaces involved in the theory of equations of evolution (that is, the theory of semi-groups) are such that the infinitesimal generators are only densely defined. For the infinitesimal generator to be everywhere defined and smooth (that is, differentiable), one must work with a Frechet space. This is especially important in the non linear theory of Moser (see Moser [3] and Marsden [2]). If the spaces were Banach spaces, the theory would reduce to the classical Picard theory for ordinary differential equations. See Lang [1], for a discussion of the classical theory. In fact in this case the existence theory is both simpler and more comprehensive, because we obtain the important fact that the flow is a diffeomorphism (the solutions depend smoothly, and not merely continuously on the initial data). This has other advantages too, since the theory for smooth flows on Banach manifolds is well developed. (For example, see Marsden [2], theorem. 6.10 for an application we have in mind.)

Certain theorems of recent interest [1, 2] concerning sequential spaces may be deduced from the fact that the category of sequential spaces, is a coreflective subcategory of the category of topological spaces, J. A space is said to be sequential if it has the finest topology that permits the convergence of its convergent sequences.

In a recent note of Kolodner [2], the Cantor Intersection Theorem is used to give an alternative proof of the well known Contraction Mapping Principle. Kolodner applied Cantor's theorem first to a bounded metric space and then reduced the general case to this special case. Sometime ago, we found a somewhat different proof of the Contraction Mapping Principle using Cantor's theorem. Since our proof seems somewhat more direct we propose to present it here.

Let (E, E') be a dual pair and G be a collection of σ(E, E') bounded sets in E with the following properties:

1. i) G contains all singletons.

2. ii) If A, B are in G there is a C in G with A ∪B ⊆ C.

3. iii) If A is in G then λ A ε G for any scaler λ.

Then sets of the form the polar of Ai, form base of neighbourhoods in E' for the polar topology TG of G convergence.