The greatest line joining two points in the perimeter of a polygon is a side or a diagonal of the polygon.

For (fig. 2) PS is obviously less than one or other of the lines PQ, PR. Hence if AB, CD (fig. 3) are any two finite lines, L and M any points, one in each of these lines, LM is less than LC, or LD and LC is less than CA or CB. From this the theorem follows at once. The theorem and the above proof apply to crossed and gauche polygons.

Let T: A → B be a linear operator between two Banach algebras A and B. The basic problem in the theory of automatic continuity is to find algebraic conditions on T, A, and B which ensure that T is continuous. As a means to study continuity properties of T the separating space of T has played a crucial role. It is defined as

Two possible concepts of rank in inverse semigroup theory, the intermediate I-rank and the upper I-rank, are investigated for the finite aperiodic Brandt semigroup. The so-called large I-rank is found for an arbitrary finite Brandt semigroup, and the result is used to obtain the large rank of the inverse semigroup of all proper subpermutations of a finite set.

Many problems involving the solution of partial differential equations require the solution over a finite region with fixed boundaries on which conditions are prescribed. It is a well known fact that the numerical solution of many such problems requires additional conditions on these boundaries and these conditions must be chosen to ensure stability. This problem has been considered by, amongst others, Kreiss [11, 12, 13], Osher [16, 17], Gustafsson et al. [9] Gottlieb and Tarkel [7] and Burns [1]

It is well known that sufficient conditions for the existence of a positive vector u which satisfies the matrix equation Au = λu are that A should be non-negative and irreducible. This result, the qualitative part of the Perron-Frobenius theorem, has been proved in a variety of ways, one of the most attractive of which is that given by Alexandroff and Hopf in their treatise “ Topologie ”. The aim of this note is to show how their method can be adapted to deal with the generalised eigenvalue problem defined by Au = λBu where A and B are square matrices.

B. Maddox [15] defined absolutely pure modules and derived some interesting properties of these modules. C. Megibben [17] continued the study of these modules and found more interesting properties. We introduce in this paper co-absolutely co-pure modules as dual to absolutely pure modules. We first prove that over a commutative classical ring these modules are precisely the flat modules. As a biproduct we get a projective characterization of flat modules over a commutative co-noetherian ring. Secondly, over a quasi-Frobenius ring R, co-absolutely co-pure right R-modules turn out to be projective modules. Finally we get a characterization of almost Dedekind domains in terms of co-absolutely co-pure modules.

§1. The coordinates considered are linear, i.e. in a plane the equation of a straight line, and in space the equation of a plane, is linear in the coordinates. We shall first consider point-coordinates in plane geometry, taking elliptic geometry as typical, with space-constant unity.

Given a ring R we consider the category Ŕ of R-rings (rings A with given ring-homomorphisms R → A), and R-homomorphisms (ring-homomorphisms that form commutative triangles with the given maps from R). All rings are associative and have 1, all homomorphisms send 1 to 1.

We define a c-R-ring as an object A in Ŕ with a family of maps {ρx ∈ homŔ(A,A)|x∈A}. Equivalently, a c-R-ring is an R-ring A with a binary operation a · b(= aρb) on A satisfying

The object of this communication is simply to show how a formula may be obtained which will indicate the most economical speed for a steamer in relation to cargo carried and coals consumed on the voyage.

I have begun to study the laws of the struggle for life by a group of species living in the same environment in such a way that some devour others. I have used for this purpose the so called principle of encounter, considering the encounters of the individuals of the various species, and what follows by reason of the actions that the individuals exercise on one another.

In (1) numbers related to the Stirling numbers are defined. Later in (2) these numbers were called Lah-numbers (cf. 2, p. 43, Ex. 16). According to (1) these numbers are of importance in Mathematical Statistics. In this paper we shall generalise the method and apply it to generalised Stirling numbers as defined in (3).

The theorem in question is the following:—if φ(x) is a function that either always increases or else always decreases as x increases from a to b, then

where a < ζ < b

R. A. Rankin [2] and J. Lehner [1] considered the non-vanishing of Poincaré series for the classical modular matrix group and for an arbitrary fuchsian group, respectively.

In this paper we consider the non-vanishing of Poincaré series for the congruence group

In this paper we continue our study of the Frattini p-subalgebra of a Lie p-algebra L. We show first that if L is solvable then its Frattini p-subalgebra is an ideal of L. We then consider Lie p-algebras L in which L2 is nilpotent and find necessary and sufficient conditions for the Frattini p-subalgebra to be trivial. From this we deduce, in particular, that in such an algebra every ideal also has trivial Frattini p-subalgebra, and if the underlying field is algebraically closed then so does every subalgebra. Finally we consider Lie p-algebras L in which the Frattini p-subalgebra of every subalgebra of L is contained in the Frattini p-subalgebra of L itself.

Let G be one of the following compact simply connected Lie groups: SU(3), Sp(2), G2. In the first two cases there is a well known stable decomposition of G as Q ∨ Sd where d = dim G and Q is a certain subspace of G. For SU(3), Q is the stunted complex quasiprojective space Σ(ℂP2/ℂP1) which fits into a cofibration sequence S3→Q→S5 with stable attaching map η:S5 → S4 For Sp(2), Q is the quaternionic quasi-projective space ℍℚ1 and fits into a cofibration sequence S3→Q→S7 with stable attaching map 2ν:S7→S4 (Here η and ν are generators of respectively.)

Cayley has shown that if the family of surfaces φ(x, y, z) = λ is one of an orthogonal triad, φ satisfies a differential equation of the third order. If, however, the parameter λ is involved implicitly in the equation of the family, the condition requires modification: for example, although

is one of an orthogonal triad, φ does not satisfy Cayley's equation.