A single valued function D(x, y) is a metric for a topological space provided that for points x, y, z of the space:

1. D(x, y) ≽ 0, the equality holding if and only if x = y,

2. D(x, y) = D(y, x) (symmetry),

3. D(x, y) + D(y, z) ≽ D(x, z) (triangle inequality),

4. x belongs to the closure of the set M if and only if D(x, m) (m element of M) is not bounded from 0 (preserves limit points).

If

is the prime power decomposition of an integer v, and we define the arithmetic function n(v) by

then it is known, MacNeish (10) and Mann (11), that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. We shall denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v. Then the Mann-MacNeish theorem can be stated as

MacNeish conjectured that the actual value of N(v) is n(v).

A graph G consists of a non-null set V of objects called vertices together with a set E of objects called edges, the two sets having no common element. With each edge there are associated just two vertices, called its ends. Two or more edges may have the same pair of ends.

Each irreducible representation [λ] of the symmetric group Sn may be identified by a partition [λ] of n into non-negative integral parts λ1 ≥ λ2 ≥ … λn ≥ 0, of which the first λ'j parts are ≥j, or by a right (Young) diagram also called [λ], that contains λi nodes in its ith row and λ'j nodes in its jth column.

Given a set E of n elements we denote by S(l, m, n), (l ≤ m ≤ n) a system of subsets of E, having m elements each, such that every subset of E having l elements is contained in exactly one set of the system S (l, m, n).

It is clear (3), that a necessary condition for the existence of S (l, m, n) is that

1

is the number of elements of S(l, m, n) and

is the number of those elements of S (l, m, n) which contain h fixed elements of E.

It is known that condition (1) is not sufficient for S(l, m, n) to exist. It has been proved that no finite projective geometry exists with 7 points on every line. This implies non-existence of S(2, 7, 43).

Since the classical work of Minkowski and Jensen it is well known that many of the inequalities used in analysis may be considered as consequences of the convexity of certain functions. In several of these inequalities pairs of “conjugate” functions occur, for instance pairs of powers with exponents a and a related by 1/a + 1/a = 1. A more general example is the pair of positively homogeneous convex functions denned by Minkowski and known as the distance (or gauge) function and the function of support of a convex body. The purpose of the present paper is to explain the general (by the way rather elementary) idea underlying this correspondence.

Let X be a countable fundamental set in a Hilbert space H, and let T be the operator Whenever T is well-defined and bounded, X is said to be a Bessel sequence. If, in addition, ran T is closed, then X is a frame. Finally, a frame whose corresponding T is injective is a stable basis (also known as a Riesz basis). This paper considers the above three properties for subspaces H of L2(ℝd), and for sets X of the form with Φ either a singleton, a finite set, or, more generally, a countable set. The analysis is performed on the Fourier domain, where the two operators TT* and T* T are decomposed into a collection of simpler "fiber" operators. The main theme of the entire analysis is the characterization of each of the above three properties in terms of the analogous property of these simpler operators.

Let Tn denote a tree with n nodes that is rooted at node r. (For definitions not given here see [4] or [10].) The altitude of a node u in Tn is the distance α = α (u, Tn) between r and u in Tn. The width of Tn at altitude is the number Wk = Wk(Tn) of nodes at altitude in Tn, where = 0, 1, …

In classical dynamics one has usually supposed that when one has solved the equations of motion one has done everything worth doing. However, with the further insight into general dynamical theory which has been provided by the discovery of quantum mechanics, one is lead to believe that this is not the case. It seems that there is some further work to be done, namely to group the solutions into families (each family corresponding to one principal function satisfying the Hamilton-Jacobi equation). The family does not have any importance from the point of view of Newtonian mechanics; but it is a family which corresponds to one state of motion in the quantum theory, so presumably the family has some deep significance in nature, not yet properly understood.

An edge of a 3-connected graph G is called essential if the 3-connection of G is destroyed both when the edge is deleted and when it is contracted to a single vertex. It is known (1) that the only 3-connected graphs in which every edge is essential are the “wheel-graphs.” A wheel-graph of order n, where n is an integer ⩾3, is constructed from an n-gon called its “rim” by adding one new vertex, called the “hub,” and n new edges, or “spokes” joining the new vertex to the n vertices of the rim; see Figure 4A.

A matroid can be regarded as a generalized graph. One way of developing the theory of matroids is therefore to generalize known theorems about graphs. In the present paper we do this with the theorem stated above.

A projective plane geometry π is a mathematical system composed of undefined elements called points and undefined sets of points (at least two in number) called lines, subject to the following three postulates:

(P1) Two distinct points are contained in a unique line.

(P2) Two distinct lines contain a unique common point.

(P3) Each line contains at least three points.

A single-valued function f(z) is said to be univalent in a domain if it never takes on the same value twice, that is, if f(z1) = f(z2) for implies that z1 = z2. A set is said to be starlike with respect to the line segment joining w0 to every other point lies entirely in . If a function f(z) maps onto a domain that is starlike with respect to w0, then f(z) is said to be starlike with respect to w0. In particular, if w0 is the origin, then we say that f(z) is a starlike function. Further, a set is said to be convex if the line segment joining any two points of lies entirely in . If a function f(z) maps onto a convex domain, then we say that f(z) is a convex function in .

An interesting set of geometric figures is composed of the convex polyhedra in Euclidean 3-space whose faces are regular polygons (not necessarily all of the same kind). A polyhedron with regular faces is uniform if it has symmetry operations taking a given vertex into each of the other vertices in turn (5, p. 402). If in addition all the faces are alike, the polyhedron is regular.

That there are just five convex regular polyhedra—the so-called Platonic solids—was proved by Euclid in the thirteenth book of the Elements (10, pp. 467-509). Archimedes is supposed to have described thirteen other uniform, “semi-regular” polyhedra, but his work on the subject has been lost.

Geometrically, a graph is a collection of points (or vertices) together with a set of edges (or curves) each of which joins two distinct vertices of the graph, and no two of which have points in common except possibly end points. Two given vertices of the graph may be joined by no edge or one edge, but may not be joined by more than one edge. From an abstract point of view, a graph G is a collection of elements ﹛x1, x2, …﹜ called points or vertices, together with a second collection of certain pairs (xα, Xβ) of distinct points of G. It is helpful to retain the geometric language, and refer to any pair in as an edge (or a curve) of G that joins the points xα and Xβ.

1. Let be a finite projective plane (8, §17), i.e. a projective space of dimension 2 over a Galois field γ. We suppose that γ has characteristic p ≠ 2, hence order q = pn, where p is an odd prime and h is a positive integer. It is well known that every straight line and every non-singular conic of then contains q + 1 points exactly.

We define a graph as a set V of objects called vertices together with a set E of objects called edges, the two sets having no common element. With each edge there are associated just two vertices, called its ends. We say that an edge joins its ends. Two vertices may be joined by more than one edge.

Let R be the discrete group ring of the group G over the ring A. In this paper we attempt to find necessary and sufficient conditions on G and A so that R will have some standard ring-theoretic property ; among the properties considered are those of being artinian, regular, self-injective, and semi-prime.

The contents of this paper form essentially the author's doctoral thesis. The author would like to thank his supervisor Dr. J. Lambek for his generous encouragement and continued interest.

1. Introduction. The following elementary logical problem was a question in the William Lowell Putnam Mathematical Competition held in March 1953 (1): Six points are in general position in space (no three in a line, no four in a plane). The fifteen line segments joining them in pairs are drawn, and then painted, some segments red, some blue. Prove that some triangle has all its sides the same color.

The graphs considered in this paper are assumed to be finite, with no edge joining a vertex to itself and with no two distinct edges joining the same pair of vertices. An undirected graph will be denoted by G or (V, E), where V is the set of vertices and E is the set of edges. An edge joining the vertices i,j ∊ V will be denoted by the unordered pair (i,j).

An orientation of G = (V, E) is an assignment of a unique direction i → j or j → i to every edge (i,j) ∊ E. The resulting directed image of G will be denoted by G→ or (V, E→), where E→ is now a set of ordered pairs E→ = {[i,j]| (i,j) ∊ E and i → j}. Notice the difference in notation (brackets versus parentheses) for ordered and unordered pairs.

Let X be a topological space, that is, a space with open sets such that the union of any collection of open sets is open and the intersection of any finite number of open sets is open. A covering of X is a collection of open sets whose union is X. The covering is called countable if it consists of a countable collection of open sets or finite if it consists of a finite collection of open sets ; it is called locally finite if every point of X is contained in some open set which meets only a finite number of sets of the covering. A covering is called a refinement of a covering U if every open set of X is contained in some open set of . The space X is called countably paracompact if every countable covering has a locally finite refinement.