In this note we prove the Descendent Theorem (2) of Slepian, Moore, and Prange in an abstract form. Our proof shows that the theorem is valid in much more general settings than that of vector spaces over Z/2Z. Applications of the descendent theorem to coding theory may be found in (2), and a study of Prange's method of proof is carried out by Dade in (1).

Suppose that two men stand at the same elevation on opposite sides of a mountain range and begin to climb in such a way that their elevations remain equal at all times. Will they ever meet along the way? It is this question, restated in mathematical terms, that we shall consider. We replace the mountain range by the graph of a continuous, real-valued function f(x) defined for x ∈ [0, 1], where f(0) = f(1) = 0, and we ask whether there exist continuous mappings ϕ(t), ψ(t) from [0, 1] into [0, 1] such that

1

2

In this paper a simplified proof of a theorem of Sanov (4) is given. No mention is required of Lie elements or of the Baker-Hausdorff formula, both of which played central roles in Sanov's proof.

Let G be a group. Define, for all x, y ∈ G,

What conditions determine when a collection of points A lies on a collection of parallel lines each member of which intersects a set B? In order to describe these conditions the following notations and definitions are used. Also for earlier results see Robkin and Valentine (2).

In two recent papers by V. F. Cowling and C. L. Miracle (1; 2), the regularity of generalized Lototsky transformations, as well as their application to the geometric series, has been investigated. The main interest of the papers centres around (1, Theorems 3.1 and 4.1) and (2, Theorem A). In (1, Theorem 3.1) and in (2, Theorem A), the authors stated and proved a set of sufficient conditions for the regularity of Lototsky-type transformations. In (1, Theorem 4.1), they proved under certain additional conditions that these transformations sum the geometric series ∑zn to (1 — z)–l if Re z < 1.

With a holomorphic function f(z) defined in a domain H which includes the closed interval [—1, 1] we associate the Neumann series

1

where Pn(z), Qn(t) are, respectively, the nth Legendre polynomials of the first and second kind and γ is a closed and rectifiable Jordan curve which includes [— 1, 1] in its interior and is included, together with its interior, in H.

We shall consider a space S, a σ-algebra M of subsets of S, a measure μ defined on M, and the μ-integrals of certain μ-integrable functions f. To each point x of a certain set E of S we associate certain ones of the sets V ∈ M and form the quotients ∫ vf(x)dμ(x)/μ(V) for each such set V. In case these quotients tend to f(x) as the sets V converge to x in accordance with a definition we adopt in §2, then we say that the integral of f is differentiable or derivable at x. It is of interest to assert conditions that ensure the differentiability of a given integral or class of integrals at μ-almost all points of E.

An invertible operator T on a Hilbert space is a multiplicative commutator if there exist invertible operators A and B on such that T = ABA–1B–1. In this paper we discuss the question of which operators are, and which are not, multiplicative commutators. The analogous question for additive commutators (operators of the form AB — BA) has received considerable attention and has, in fact, been completely settled (2). The present results represent the information we have been able to obtain by carrying over to the multiplicative problem the techniques that proved efficacious in the additive situation. While these results remain incomplete, they suffice, for example, to enable us to determine precisely which normal operators are multiplicative commutators.

In this paper we discuss a generalization of the split exceptional Jordan algebra M38() of the 3 X 3 hermitian matrices with elements in the split Cayley-Dickson algebra (1). The generalization consists of replacing by the non-commutative Jordan algebra ≡ (A, ƒ, s, t) discussed in (2; 3) and forming the set of 3 X 3 hermitian matrices M3m() ≡ M with elements in the m-dimensional algebra . With the usual definition of multiplication X · Y = ½(XY + YX), M becomes a commutative algebra and we have the following theorem, which shows how the structure of M is reflected by that of .

By an undirected graph we mean a couple (X, R), where X is a set and R is a subset of X × X such that (x, y) ∈ R implies (y, x) ∈ R. The cardinal of X, denoted by |X|, will be called the cardinal of the graph.

A mapping f:X → X is called an endomorphism of (X, R) if (x, y) ∈ R implies that (f(x), f(y)) ∈ R for all x, y ∈ R.

An undirected graph (X, R) is called rigid if there is only one endomorphism of (X, R), namely the identity mapping of X.

P. Erdös communicated orally that, using probability methods, it is possible to prove that almost all finite undirected graphs are rigid.

The geometry of quadric varieties (hypersurfaces) in finite projective spaces of N dimensions has been studied by Primrose (12) and Ray-Chaudhuri (13). In this paper we study the geometry of another class of varieties, which we call Hermitian varieties and which have many properties analogous to quadrics. Hermitian varieties are defined only for finite projective spaces for which the ground (Galois field) GF(q2) has order q2, where q is the power of a prime. If h is any element of GF(q2), then = hq is defined to be conjugate to h.

The question posed by Sylvester (6) concerning the collinearity of a finite set of points in E2 having the property that each two together with some third be collinear has been the inspiration for numerous investigations. The original question was answered by the following theorem.

Let A be an n X n non-negative matrix, that is, a matrix whose entries are non-negative numbers. The permanent of A is the scalarvalued function of A defined by

where the summation extends over all permutations i1 … , in of the integers 1, … , n. The purpose of this paper is to prove several inequalities involving the permanent of A and the permanent of submatrices of A when suitable restrictions are placed on the row sums.

For two entire functions f(z) and g(z) the composition f(g(z)) may or may not be periodic even though g(z) is not periodic. For example, when f(u) = cos √u and g(z) = z2, or f(u) = eu and g(z) = p(z) + z, where p(z) is a periodic function of period 2πi, f(g(z)) will be periodic. On the other hand, for any polynomial Q(u) and any non-periodic entire function f(z) the composition Q(f(z)) is never periodic (2).

The general problem of finding necessary and sufficient conditions for f(g(z)) to be periodic is a difficult one and we have not succeeded in solving it. However, we have found some interesting related results, which we present in this paper.

In this paper, we give a connection between incidence matrices of affine resolvable balanced incomplete block designs and rectangular integer matrices subject to certain arithmetical conditions. The definition of these terms can be found in paper II of this series or in (2). For some necessary conditions on the parameters of affine resolvable balanced incomplete block designs and their properties see (2).

Let be the algebra of n X n matrices over the complex field C, X = ‖xrs‖ a matrix of , and f(X) = ‖(frs(x11, x12, … , xnn)‖ a function with domain and range in . If the frs are differentiable with respect to each of the xij on some open set R of , then the differential df(X) = ‖dfrs(xij)‖ exists, and, moreover, f(X) is Hausdorff-differentiable (HD) (1, 3, 7) i.e. df(X)is expressible in the form

where dX = ‖dxrs‖, and the matrices Ai Bi are independent of dX. The Hausdorff derivative fI(X) is defined to be

i.e. the value of df(X) for dX = I, the identity matrix (2).

The treatment of semigroups given in a previous paper (3) is based upon representations of a semigroup by means of transformations of a set (cf. also 12). In this paper we try to remove the assumption of the existence of a zero element proposed in (3). In accordance with our general programme explained at the beginning of (3) we utilize certain minimum conditions in order to gain more information on the structure of semigroups.

Our main results are structure theorems on primitive semigroups which have irreducible right ideals generated by idempotents (§§15-17). As we have shown in (5), these theorems permit the explicit construction of primitive semigroups.

Let G be a finite graph topologically embedded in the interior of a 3-manifold M. Doyle (4) and Debrunner and Fox (3) have noted that the following local homotopy condition at each point p ∈ G is necessary in order for the embedding of G to be tame:

For each sufficiently small open set U containing p, there is an open set V such that p ∈ V ⊂ U and if W is any connected open set such that p ∈ W ⊂ V, then the image under the inclusion homomorphism i*: π1(W — G) → π1(U — G) is a free group on n — 1 generators.

If a quasigroup (L, .) has finite order n, then there are n2 principal loop-isotopes. Some of these n2 loops may be isomorphic, and the main purpose of this paper is to obtain theorems that describe the isomorphism classes. Using these results and a computer, we have determined all the loops of order 6. These are listed (using the Fisher and Yates (2) designations) at the end of the paper.

By an Euler polyhedron of valence three or a trivalent convex polyhedron in Euclidean 3-space (4) we mean in the present paper an Euler polyhedron in the sense of Steinitz (8, p. 113), such that each vertex is incident with exactly three edges.

In the present paper we establish a theorem concerning the colouring of trivalent polyhedra. A specialization of this theorem solves the following problem implicit in Eberhard (1, p. 84): Does there exist a trivalent Euler polyhedron with an odd number of faces such that the number of edges incident with any face is divisible by three?