The asymptotics behaviour of solutions of the Becker-Döring cluster equations is determined for cases in which coagulation dominates fragmentation. We show that all non-zero solutions tend weak* to zero.

Earlier this year Professor W. L. Edge drew my attention to the paper (1). Since Black probably wrote his paper about 1890, the integral he considered must have been one of the earliest n-variate integrals to be evaluated. The present paper generalizes Black's result from a column vector as variate to a rectangular matrix—his integral is the case p = m = k = 1 of the integral J below.

This paper is concerned with the relationship between the properties of the subalgebra lattice ℒ(L) of a Lie algebra L and the structure of L. If the lattice ℒ(L) is lower semimodular, then the Lie algebra L is said to be lower semimodular. If a subalgebra S of L is a modular element in the lattice ℒ(L), then S is called a modular subalgebra of L. The easiest condition to ensure that L is lower semimodular is that dim A/B = 1 whenever B < A ≤ L and B is maximal in A (Lie algebras satisfying this condition are called sχ-algebras). Our aim is to characterize lower semimodular Lie algebras and sχ-algebras, over any field of characteristic greater than three. Also, we obtain results about the influence of two solvable modularmaximal subalgebras on the structure of the Lie algebra and some results on the structure of Lie algebras all of whose maximal subalgebras are modular.

In this study, the use of factorial moments and factorial moment generating functions as applied (2) to the Poisson frequency function and Charlier's Type B function is further extended towards developing a theory of these distributions for the case of two or more correlated variables.

On est amené à résoudre l'équation aux différences mêlées

lorsqu'on cherche, en coordonnées rectangulaires, une courbe telle que: M et M′ étant deux de ses points dont les abscisses différent d'une constante, la tangente de l'angle que fait avec OX la tangente en M soit en raison inverse de la sous-tangente en M′.

This paper has to deal with some topics suggested by a theorem which is notorious in the theory of quadratic forms, namely that the roots of the characteristic equation of any real quadratic form are all real. For a proof of this theorem, the reader may consult Fcrrar (1), p. 146.

§ 1. The object of this note is to establish the above inequality in as general a form as possible, and to prove by means of it two of the principal propositions in the theory of inequalities, one of which is usually proved by means of infinite series. The logical advantage in making the theory of inequalities independent of that of infinite series is obvious, when it is remarked that the discussion of the convergency of infinite series is strictly speaking a part of the theory of inequalities.

The idea of applying isoperimetric functions to group theory is due to M. Gromov [8]. We introduce the concept of a “bicombing of narrow shape” which generalizes the usual notion of bicombing as defined for example in [5], [2], and [10]. Our bicombing is related to but different from the combings defined by M. Bridson [4]. If they Cayley graph of a group with respect to a given set of generators admits a bicombing of narrow shape then the group is finitely presented and satisfies a sub-exponential isoperimetric inequality, as well as a polynomial isodiametric inequality. We give an infinite class of examples which are not bicombable in the usual sense but admit bicombings of narrow shape.

Certain properties of Ruled Surfaces relating to their Curves of Striction are suggested and immediately proved by the use of a particular kind of coordinates, known as Dual Coordinates, for the Generating Lines. These coordinates, were introduced by Prof. E. Study; the theory of them is fully explained and many beautiful applications are made in his treatise, Geometrie der Dynamen, Teubner, 1903, and in his article, Complexe Grössen in the Encyclopädie der math. Wissenschaften, Bd. I., pp. 147–183, and specially p. 166. In applying the method here I have ventured to regard the fundamental quantity c in a light which, if less rigorous, has the advantage of being familiar to most of us in other mathematical work.

The property is, that if such a decimal has 2k repeating figures, the first k are a1, a2, &c, in order, and the second k are b1, b2, &c, in order, and the a'a and b's are such that

for example,

“The following is, of course, substantially well-known, but it strikes me as rather pretty:—to find the Orthomorphic transformation of the circle

into itself. Assume this to be

Etherington introduced certain algebraic methods into the study of population genetics (6). It was noted that algebras arising in genetic systems tend to have certain abstract properties and that these can be used to give elegant proofs of some classical stability theorems in population genetics (4, 5, 9, 10).

Let E be a Banach sequence space with the property that if (αi) ∈ E and |βi|≦|αi| for all i then (βi) ∈ E and ‖(βi)‖E≦‖(αi)‖E. For example E could be co, lp or some Orlicz sequence space. If (Xn) is a sequence of real or complex Banach spaces, then E can be used to construct a vector sequence space which we will call the E sum of the Xn's and symbolize by ⊕EXn. Specifically, ⊕EXn = {(xn)|(xn)∈Xn and (‖xn‖)∈E}. The E sum is a Banach space with norm defined by: ‖(xn)‖ = ‖(‖xn‖)‖E. This type of space has long been the source of examples and counter-examples in the geometric theory of Banach spaces. For instance, Day [7] used E=lp and Xk=lqk, with appropriate choice of qk, to give an example of a reflexive Banach space not isomorphic to any uniformly conves Banach space. Recently VanDulst and Devalk [33] have considered Orlicz sums of Banach spaces in their studies of Kadec-Klee property.