Let S(N(t)) be defined by

where {N(t), t ≥ 0} is a Poisson process with intensity parameter 1/μ > 0 and {Xi i ≥ 1} is a family of independent random variables which are also independent of {N(t), t ≥ 0}.

1. Introduction. In (2) we associated with a commutative ring R with identity the lattice ℒ(R) of rings S which satisfy (i) R ⊆ S ⊆ T, where T is the total quotient ring of R, and (ii) dS ⊆ R for some d ∈ R which is not a zero-divisor. The rings of ℒ(R) can be constructed by blowing up ideals of the set of ideals A of R which satisfy (i) 0: A = 0, and (ii) there exist integers m1n ≥ 1 and α ∈ An such that αAm = Am+n (see Propositions 2·3 and 2·5 of (2)).

A class of representations on Fock space is associated to a representation of the *-algebra structure of a cocommutative graded bialgebra with an involution. We prove that the Gelfand–Naimark–Segal (GNS) representation given by the convolution exponential of a conditionally positive linear functional can be embedded into a representation of this class. Our theory generalizes a well-known construction for infinitely divisible positive definite functions on a group. Applying our general result, we obtain a complete characterization of the GNS representations given by infinitely divisible states on involutive Lie superalgebras.

We characterize those domains Ω in ℝN for which the positive harmonic functions on Ω separate the points of Ω.

An S-system over a monoid S is a set A together with a function ø: A × S → A such that (a, 1) ø = a and (a, st) ø = ((a, s) ø, t) ø for all a ∈ A and for all s, t ∈ S. So an S-system is a set on which S acts unitarily on the right. For (a, s) ø we write simply as: the notions of S-homomorphism, S-subsystem etc. are defined in the obvious manner. Many papers have been written characterizing monoids by properties of their S-systems; here we are concerned with injectivity and a related concept, absolute purity. Further details of the terms we use are given in Section 2.

A simple formula is derived for transforming the slowly convergent series involving Bessel functions which arise in axially symmetric potential problems into more rapidly convergent series. Using it, sums of series previously found by elaborate special methods may be obtained by direct substitution. The necessary numerical coefficients are given.

In 1974, Erdős posed the following problem. Given an oriented graph H, determine or estimate the maximum possible number of H-free orientations of an n-vertex graph. When H is a tournament, the answer was determined precisely for sufficiently large n by Alon and Yuster. In general, when the underlying undirected graph of H contains a cycle, one can obtain accurate bounds by combining an observation of Kozma and Moran with celebrated results on the number of F-free graphs. As the main contribution of the paper, we resolve all remaining cases in an asymptotic sense, thereby giving a rather complete answer to Erdős’s question. Moreover, we determine the answer exactly when H is an odd cycle and n is sufficiently large, answering a question of Araújo, Botler and Mota.

1. A paper with the same title was published with Professor H. F. Baker in 1920; that paper was severely compact and would seem to deserve elucidation, more detail, and some revision. We use a symbolism which, while leaving the proofs geometrical, readily converts itself into ordinary analysis, as we shall show in § 12.

A recently developed theory of the electromagnetic field in nearly periodic structures is used to calculate the field in an Alvarez type linear accelerator. The problem is reduced to the solution of the transmission line equations for the ‘voltage’ and ‘current’ of the dominant mode. Formulae are given for the coefficients in these equations, and also for the ratio of mode current to accelerating field, as functions of the cavity dimensions of the accelerator; numerical values are given for the Berkeley accelerator.

Introduction. The object of this note is to give a proof of the following theorem on the solution of an important functional equation.

We give a precise description of combed trees in terms of Kelly–Mac Lane graphs. We show that any combed tree is uniquely expressed as an allowable Kelly–Mac Lane graph of a certain shape. Conversely, we show that any such Kelly–Mac Lane graph uniquely defines a combed tree. We show how to apply this to the construction of opetopes.

Let f be a primitive, cuspidal Hilbert modular form of parallel weight. We investigate the Rankin convolution L-values L(f,g,s), where g is a theta-lift modular form corresponding to a finite-order character. We prove weak forms of Kato's ‘false Tate curve’ congruences for these values, of the form predicted by conjectures in non-commmutative Iwasawa theory.

Owing to a chance observation some interesting phenomena were observed during the splashing of water in an electric field. If a water drop falls into water, there is a characteristic splash of which the details were investigated photographically by Professor A. M. Worthington in the latter years of the last century. The final account of this work was given in two papers (1) and also in a book entitled A Study of Splashes (2). The splash cannot be examined with the unaided eye because the various stages of the splash follow one another too quickly. For the moment it may be said that an important stage of the splash consists of the rising up of a cylindrical column of water to a considerable height. The radius of the column is comparable with the radius of the drop.

Numerical integration of the partial differential equations shows that rigid boundaries promote the development of a shock from a finite-amplitude disturbance. A simple step-by-step method agrees with one that also utilizes the characteristics.

In two previous notes I have given the various types of non-singular surfaces with sectional genera 4 and 5. In this paper the methods previously adopted are applied to the non-singular surfaces of sectional genus 6. For purposes of classification these may be divided into the four classes:

(1) Regular, non-rational.

(2) Rational.

(3) Irregular, non-referable.

(4) Referable.

Necessary and sufficient conditions for the symbolic dynamics of a given Lorenz map to be fully embedded in the symbolic dynamics of a piecewise continuous interval map are given. As an application of this embedding result, we describe a new algorithm for calculating the topological entropy of a Lorenz map.

We study the existence of semi-classical bound states of the nonlinear Schrödinger equation

\begin{linenomath}$$ -\varepsilon^2\Delta u+V(x)u=f(u),\quad x\in {\bf R}^N,$$\end{linenomath}

1·1. In this paper we shall be concerned with the equations

where K is a compact (completely continuous) linear operator in a Hilbert space , K is the adjoint of K, I is the identity operator, x and y are elements of ∥ x ∥ denotes the norm of x, and κ and σ are complex numbers.

Suppose that λ > − 1 and that

We give an elegant simple proof of a result concerning the first normal bundle of ideal Lagrangian immersions. This enables us to simplify and clarify several of the results in an earlier paper.