Let V = V2n be the Segre product variety of two n–dimensional complex projective spaces. Then any Cremona transformation of Sn into (regarded as an irreducible algebraic system of ∞n ordered pairs of points) is represented on V by an irreducible n-dimensional subvariety H which satisfies (on V) the algebraic equivalence

where Si, j is a subvariety of V and m1, …, mn–1 are positive integers. We call m1, …, mn–1 the characters of T noting that, numerically,

Let E be a compact subset of R2, of Hausdorff dimension s > 0 and for each real number t let Ft be the linear set {x1 + tx2: (x1x2) ∈ E}. In this note we shall prove

THEOREM. If s ≤ 1 then Ft has dimension ≥ s, excepting numbers t in a set of dimension ≤ s. If s > 1 then Ft, has positive Lebesgue measure, excepting numbers t in a set of Lebesgue measure 0.

The Integral Equation. Consider a periodic wave moving with constant velocity c from right to left on the surface of an inviscid, incompressible fluid which is at rest at infinity. The motion is assumed to be irrotational and two-dimensional. The bottom is horizontal, and the depth of the undisturbed fluid is h.

As a consequence of the methods developed in the earlier papers of this series [1, 2, 3], an effective algorithm has recently been established for solving many Diophantine equations in two unknowns (see [4,5,6,7]). The algorithm leads to an explicit bound for the size of all the solutions, and, in principle therefore, it enables any specific equation of the type considered to be fully resolved by a finite amount of computation. On examining the various estimates occurring in the course of the exposition, however, it at once became apparent that the computation would involve a very large number of operations and would scarcely be practicable even with a modern machine. It was clear, on the other hand, that a modified version of the fundamental inequality involving the logarithms of algebraic numbers would much facilitate the computational work, and it was in the light of this observation that the researches discussed herein were begun. The object has been to obtain a theorem of an essentially practical nature which may be found useful in application to a wide variety of different problems. The result which we shall establish is neither the most precise nor the most general that can be obtained in this direction, but it would seem to be the most serviceable of its kind, and it would apparently make feasible many calculations which would otherwise have seemed quite out of the question.

Some diophantine equations in three variables with only finitely many solutions, 113–120. In formula (1) of Theorem 1 and in Corollary 1 read “min” fir “max”

Let T(n, k) denote the number of trees n with n labelled nodes of which exactly k have degree two. We shall derive a formula for T(n, k) and then determine the asymptotic behaviour of T(n,0); this will enable us to calculate the limiting distribution of Xn the number of nodes of degree two in a random tree n. Rényi [5] has treated the corresponding problem for nodes of degree one in random trees.

Let D be an integral domain with identity having quotient field K. A non-zero fractional ideal F of D is said to be divisorial if F is an intersection of principal fractional ideals of D[4; 2]. Equivalently, F is divisorial if there is a non-zero fractional ideal E of D such that

Divisorial ideals arose in the investigations of Van der Waerden, Artin, and Krull in the 1930's and were called v-ideals by Krull [9; 118]. The concept has played an important role in the development of multiplicative ideal theory.

The development of a population over time can often be simulated by the behavior of a birth and death process, whose transition probability matrix P(t) = (Pij(t), where X(t) denotes the number of individuals at time t, satisfies the differential equations and the initial condition

In the past a number of papers have appeared which give representations of abstract lattices as rings of sets of various kinds. We refer particularly to authors who have given necessary and sufficient conditions for an abstract lattice to be lattice isomorphic to a complete ring of sets, to the lattice of all closed sets of a topological space, or to the lattice of all open sets of a topological space. Most papers on these subjects give the conditions in terms of special elements of the lattice. We thus have completely join-irreducible elements — G. N. Raney [7]; join prime, completely join prime, and supercompact elements — V. K. Balachandran [1], [2]; N-sub-irreducible elements — J. R. Büchi [5]; and lattice bisectors — P. D. Finch [6]. Also meet-irreducible and completely meet-irreducible dual ideals play a part in some representations of G. Birkhoff & 0. Frink [4].

If (X, ) is a set X with topology we shall say that is connected if (X, ) is a connected topological space. We shall investigate the existence of and the properties of maximal connected topologies.

In a stationary GI/G/1 queueing system in which the waiting time variance is finite, it can be shown that the serial correlation coefficients {ρn} of a (stationary) sequence of waiting times are non-negative and decrease monotonically to zero. By means of renewal theory we find a representation for Σ∞0 ρn from which necessary and sufficient condition for its finiteness can be found. In M/G/1 rather more can be said: {ρn} is convex sequence, the asymptotic form of ρ n can be given in a nearly saturated queue, and a simple explicit expression for Σ∞0 ρn exists. For the stationary M/M/1 queue we find the ρn's explicitly, illustrate them numerically, and derive a representation which shows that {ρn} is completely monotonic.

We consider a queueing system with k identical servers in parallel, the services being negative exponential with parameter μ. The input is a natural generalisation of the usual general recurrent input. If we denote the sequence of arrival points by {An, n ≧ 0} then the inter-arrival intervals are given by where the ƒi: are (integrable) non-negative functions and {Ui} is a sequence of identically and independently distributed random variables. In the simplest case, p = 0, this is just a general recurrent input. We write U(·) for the probability distribution function of the Un.

Let Zn be the numer of individuals in the nth generation of a discrete branching process, descended from a single a singel ancestor, for which we put It is well known that the probability generating function of Zn is Fn(s), the n-th functional iterate of F(s), and that if m = EZ1 does not exceed unity, then lim (Harris [1], Chapter 1). In particular, extinction is certain.

A subset of a topological space which is both closed and open is referred to as a clopen subset. Here, a 0-dimensional space is a Hausdorff space which has a basis of clopen sets. Here, a 0-dimensional space is a Hausdorff space which has a basis of clopen sets. By a compactification αX of a completely regular Hausdorff space X, we mean any compact space which contains X as a dense subspace. Two compactifications αX and γX are regarded as being equivalent if there exists a homeomorphism from αX onto γX which keeps X pointwise fixed. We will not distinguish between equivalent compactifications. With this convention, we can partially order any family of compactifications of X by defining αX ≧ γX if there exists a continuous mapping from γX onto αX which leaves X pointwise fixed. This paper is concerned with the study of the partially ordered family [X] of all 0-dimensional compactifications of a 0-dimensional space X.

Although there is no need for a ‘distinguished’ submodule to be given a formal definition in the present paper, we like to indicate the meaning attached to this concept here. Perhaps the shortest way of doing so is to say that a distinguished submodule is a (covariant idempotent) functor from the category of (left) R-modules into itself mapping each R-module into its R-submodule specified by a family of left ideals of R. If is a family left ideals of R, then all elements of an R-module M of orders belonging to , do not, of course, in general form a submodule of M; but, there are certain families such that all the elements of orders from form a submodule in any R-module (distinguished submodules defined by ). Consequently, no particular structural properties of the R-module are involved in the definition of such submodules. In this way we can define radicals (in the sense of Kuroš [4]) of a module. In particular, we feel that an application of this method is an appropriate way in defining the (maximal) torsion submodule of a module.

In 1962, O. Frink [2] showed that in a pseudo-complemented semilattice 〈P; ∧, *, 0〉, the closed elements form a Boolean algebra. We shall consider an extension of this result to arbitrary commutative semigroups with zero.