We call a digraph “antisymmetrical” if there is an automorphism θ of its graph, of period 2, which reverses the direction of every edge and maps no edge or vertex onto itself. We construct a theory of flows invariant under θ for such a diagraph. This theory is analogous to the Max Flow Min Cut theory for ordinary flows in digraphs. It is found to include that part of the theory of undirected graphs which discusses the existence of spanning subgraphs with a specified valency at each vertex.

This paper deals with systems of pairwise equal and parallel cones in the affine n-space En, i.e. cones of a system are obtained one from another by means of translations. This subject is closely connected with the geometrical interpretation of the theory of relativity, so one may say that it belongs to “elementary chronogeometry.” The term “chronogeometry,” which is due, it seems, to A. D. Fokker, means the relativistic theory of space-time.

Bei der geometrischen Interpretation von gewissen Aufgaben z.B. aus dem Bereiche der Statistik einerseits oder der Matrizenalgebra andererseits treten Simplexe in Riemannschen Räumen konstanter Krümmung auf. Ein spezieller Typus eines solchen Simplexes ist das Orthoschem. Für dieses ergibt sich da nun die Aufgabe, anzugeben, in welcher Weise die Masse der Kanten und der verschiedensten Winkel eines Orthoschems miteinander zusammenhängen.

H. S. M. Coxeter (3) has recently studied the correspondence between two geometries the isomorphism of which was well known, but to which he was able to add some remarkable consequences. The two geometries are the inversive geometry of a plane E (the Euclidean plane completed with a single point at infinity or, what is the same thing, the plane of complex numbers to which ∞ is added) on the one hand, and the hyperbolic geometry of three-dimensional space S.

The decreasing rearrangement of a finite sequence a1, a2, … , an of real numbers is a second sequence aπ(1), aπ(2), … , aπ(n), where π(l), π(2), … , π(n) is a permutation of 1, 2, … , n and

(1, p. 260). The kth term of the rearranged sequence will be denoted by . Thus the terms of the rearranged sequence correspond to and are equal to those of the given sequence ak, but are arranged in descending (non-increasing) order.

The study of regular skew polyhedra was initiated in 1926 by Petrie's discovery of two infinite polyhedra in Euclidean three-space E3 which were free of false vertices; the only other regular skew polyhedron in E3 was found by Coxeter (1, pp. 33-34). The simplest of these is denoted {4, 6 | 4} and is derived from the space-filling of cubes by omitting half the faces.

In this paper we shall give necessary and sufficient conditions for three lines, passing respectively through the vertices of a proper triangle PQR in the real Euclidean plane, to be concurrent. Of course, the theorem of Ceva deals with this problem, but it is useful to have a criterion which involves only vectors localized at a point O of the plane, and the exterior products of these vectors. Applications are made to theorems which are not easily proved by other methods.

This paper deals with the algebraic classification of non-euclidean plane crystallographic groups (NEC groups, for short) with compact quotient space. The groups considered are the discrete groups of motions of the Lobatschewsky or hyperbolic plane, including those which contain orientation-reversing reflections and glide-reflections. The corresponding problem for Fuchsian groups, which contain only orientable transformations, is essentially solved in the work of Fricke and Klein (6).

If a convex polytope P is centrally symmetric, and has the property that all its faces (of every dimension) are centrally symmetric, then P is called a zonotope. Zonotopes have many interesting properties which have been investigated by Coxeter and other authors (see (4, §2.8 and §13.8) and (5) which contains a useful bibliography). In particular, it is known (5, §3) that a zonotope is completely characterized by the fact that all its two-dimensional faces are centrally symmetric.

Let P be a d-polytope (that is, a d-dimensional convex polytope in Euclidean space) and for 0 ≤ j ≤ d – 1 let (i = 1, . . . ,ƒj(P)) represent its j-faces. Associated with each face is a non-negative number ϕ(P, ), to be defined later, which is called the interior angle of P at the face .

G. Hessenberg (2) showed that euclidean plane geometry can be realized on the surface of the sphere without assuming the parallel axiom. This geometry will be called the pseudo-euclidean geometry due to G. Hessenberg. In the present paper we give a slightly different treatment, which is perhaps simpler than that of Hessenberg and which has a greater transparency.

Classical genetics studies the production of the descendants of a pair of parents with certain genetic characteristics. In the simplest case the genetic type of an individual is determined by a pair of genes each of which can be of one of two types, G and g. A given individual can have these genes in the combination GG (dominant person), or Gg (genetically not different from the combination gG; hybrid individual), or, finally, gg (recessive individual); GG- and Gg-individuals do not differ in appearance from one another, but a gg-individual is different from them.

The notion of a quasi-metric was introduced by Wilson (7) and has been studied by himself, Albert (1), and Ribeiro (6) among others. In this paper, we extend and unify some of their work, and connect it with results of Csaszar (2) and Pervin (4, 5) on quasi-uniformities.

If 0 → A → C → B → 0 is an exact sequence of abelian groups, if ƒ is a 2-cocyle for this extension, if α ∈ End A, and if β ∈ End B, then a necessary and sufficient condition that α extend to an endomorphism γ of C which induces β is that (M) αƒ and ƒβ be cohomologous ; see Montgomery (2). We shall extend this result to the case where 1 → A → G → B → 1 is an exact sequence of groups and A is abelian.

The difficulties encountered in the theory of mixed Abelian groups can become decidedly less complex, if it is possible to reduce the question to mixed groups whose torsion subgroup is 𝒫-primary. Call such a group a p-mixed group. In §1 we show that the splitting problem for a mixed group is reducible to the same problem for certain associated 𝒫-mixed groups. In §2 we look at groups which are a direct sum of 𝒫-mixed groups.

In achieving his celebrated results on the homology groups of fibre spaces, J. P. Serre used the exact couple of a fibring defined by J. Leray. One of his main tools was the so-called two-term condition on the E2 term of this exact couple, which, if satisfied, yielded exact sequences, such as those of Gysin and Wang (see (5), Chapter IX). H. Fédérer, in (3), defined an exact couple (X, F, v) on the mapping space M(X, Y) = {ƒ:X → Y|X, Y are spaces and ƒ is continuous} with the compact-open topology, where X is a locally finite CW complex and Y is arc-connected and n-simple for all n.

A ring R is called a J-ring if there exists an integer n > 1 such that xn = x for every x ∈ R. The following beautiful theorem due to Jacobson (4, 5) is a generalization of Wedderburn's theorem, which asserts that every finite division ring must be a field.

A valuation of a ring K is a function

where A is an archimedean ordered field and W has the properties of the absolute valuation; see (2, chap. X). The theory was extended in 1936 by K. Mahler (1), who introduced the concept of pseudovaluations. Whereas for a valuation we must always have

for a pseudovaluation it is sufficient that

The question of solvability of the differential equation

1

with x ranging over an interval (0, a], and with the boundary condition ƒ(0+) = 0, can be investigated as an initial-value problem at 0, which may be a singular point for the equation.

It is the purpose of this note to prove the following theorem: Let ƒ: G → R be a non-constant continuous function with G a locally compact connected topological group and with R the real numbers. Let C = ƒ(G) and suppose that F: C × C → C is a junction such that

Then ƒ is monotone and open and F is continuous.