In 1948 P. Lévy formulated the following theorem: If U is an open subset of the complex plane and f:U → ℂ is a nonconstant analytic function, then f maps a 2-dimensional Brownian motion Bt (up to the exit time from U) into a time changed 2-dimensional Brownian motion. A rigorous proof of this result first appeared in McKean [22]. This theorem has been used by many authors to solve problems about analytic functions by reducing them to problems about Brownian motion where the arguments are often more transparent. The survey paper [8] is a good reference for some of these applications. Lévy's theorem has been generalized, first by Bernard, Campbell, and Davie [5], and subsequently by Csink and Øksendal [7]. In Section 1 of this note we use these generalizations of Lévy's theorem to extend some results about BMO functions in the unit disc to harmonic morphisms in ℝn to holomorphic functions in ℂn and to analytic functions on Riemann surfaces. In Section 2, we characterize the domains in ℝn which have the property that the expected exit time of elliptic diffusions is uniformly bounded as a function of the starting point. This extends a result of Hayman and Pommerenke [15], and Stegenga [24] about BMO domains in the complex plane.

The purpose of this paper is to expose a method which will match a function f(z) existing in a domain D to a formal series whose radius of convergence may be zero. This matching process has to be done in a “natural way”, and has to be “regular”, which means that if a power series converges absolutely in the circle E = {z | |z|<r} then the summability function f(z) produced by our method in the domain D and matched to will coincide with in the domain E∩D. Euler, in his time, matched the function to the power series .

A barrel in a locally convex Hausdorff space E[τ] is a closed absolutely convex absorbent set. A σ-barrel is a barrel which is expressible as a countable intersection of closed absolutely convex neighbourhoods. A space is said to be barrelled (countably barrelled) if every barrel (σ-barrel) is a neighbourhood, and quasi-barrelled (countably quasi-barrelled) if every bornivorous barrel (σ-barrel) is a neighbourhood. The study of countably barrelled and countably quasi-barrelled spaces was initiated by Husain (2).

Let S be a regular semigroup and be its congruence lattice. For ρ ∈ , we consider the sublattice Lρ of generated by the congruences pw where w ∈ {K, k, T, t}* and w has no subword of the form KT, TK, kt, tk. Here K, k, T, t are the operators on induced by the kernel and the trace relations on . We find explicitly the least lattice L whose homomorphic image is Lρ for all ρ ∈ and represent it as a distributive lattice in terms of generators and relations. We also consider special cases: bands of groups, E-unitary regular semigroups, completely simple semigroups, rectangular groups as well as varieties of completely regular semigroups.

Soit ABCD un rectangle, AC l'une de ses diagonales; si l'on prend un point M sur la diagonale et qu'on mène les parallèles aux côtés, on a deux rectangles AEFD, AGHB équivalents.

Soit en effet AB une force, AD une autre force, AC sera la résultante; et si l'on prend les moments des deux forces par rapport à un point M de la résultante on aura

ce qui démontre le théorème.

The present paper is mainly concerned with decomposition theorems of the Jordan, Yosida-Hewitt, and Lebesgue type for vector measures of bounded variation in a Banach lattice having property (P). The central result is the Jordan decomposition theorem due to which these vector measures may alternately be regarded as order bounded vector measures in an order complete Riesz space or as vector measures of bounded variation in a Banach space. For both classes of vector measures, properties like countable additivity, purely finite additivity, absolute continuity, and singularity can be defined in a natural way and lead to decomposition theorems of the Yosida-Hewitt and Lebesgue type. In the Banach lattice case, these lattice theoretical and topological decomposition theorems can be compared and combined.

The object of this paper is to introduce certain functions analogous to the circular functions. The functions will be denoted by

Formulæ analogous to

will be obtained, and the use of the functions in symbolical solutions of certain differential equations exemplified. The connection of the functions with generalised Bessel-functions of order half an odd integer will be shown.

The first part of this paper depends on the theorem that if a, b, c are three positive quantities such that

then abc is a maximum when a=b=c; with the corollary that if a, b, c are three negative quantities such that

then abc is a minimum when a = b = c.

Schouten and van Kampen (1) have studied the deformation of a . Applying the methods of that paper to the tangent vectors , which exist by hypothesis at all points of a certain region Vm′ (m′ > m) of Vn, we shall have

Whence we define the differentials

In the application of the to the lower index is treated as an ordinal index only. We shall not be concerned with any extension of the to indices other than those of the general Vb (see 1, equ. 3.24).

We suppose throughout that f(t) is periodic with period 2π, and Lebesgue-integrable in (− π, π).

We write

and suppose that the Fourier series of φ(t) and ψ(t) are respectively cos nt and sin nt. Then the Fourier series and allied series of f(t) at the point t = x are respectively and , where A0 = ½a0, An = ancos nx + bnsin nx, Bn = bncos nx − ansin nx and an, bn are the Fourier coefficients of f(t).

In this paper we classify all infinite metacyclic groups up to isomorphism and determine their non-abelian tensor squares. As an application we compute various other functors, among them are the exterior square, the symmetric product, and the second homology group for these groups. We show that an infinite non-abelian metacyclic group is capable if and only if it is isomorphic to the infinite dihedral group

In this paper we prove five structure theorems for groups with dihedral 3-normalisers. The interest in these theorems lies not so much in the results themselves as in what can be proved from them. The original versions of the results are contained in our doctoral thesis (1) where they are used to prove the following theorem, of which this paper, together with (2), (3) and other papers in preparation, will constitute a published proof:

The important inequality of which Prof. Chrystal has given so many examples may be called the “Power Inequality.”

There is a simple theorem of the Differential Calculus which is to a general function f(x), what the Power Inequality is to the function xm.