The Fourier transform F(y) of a function f(t) in L1(Ek) where Ek is the k-dimensional cartesian space will be defined by .

Consider a Markov process defined in discrete time t = 1, 2, 3, hellip on a state space S. The state of the Process at time time t will be specifies by a random varable Vt, taking values in S. This paper presents some results concerning the behaviour of the saquence V1, V2, V3hellip, considered as a time series. In general, S will be assumed to be a Borel subset of an h-dimensional Euclideam space, where h is finite. The results apply, in particular, to a continuous state space, taking S to be an interval of the realine, or to discrete process having finitely or enumerably many states. Certain results, which are indicated in what follows, apply also to more general (infinite-dimensional) state spaces.

We prove that factor groups of cartesian powers of finite non-abelian simple groups cannot be countably infinite. Thisis not our main result, but it had been our original aim. The proof is based on a similar fact concerning σ-complete Boolean algebras, and on a representation of certain subcartesian powers of a group in its group ring over a Boolean ring. This representation, to which we give the name “Boolean power”, will be our central theme, and we begin with it.

Although the harmonic series diverges, there is a sense in which it “nearly converges”. Let N denote the set of all positive integers, and S a subset of N. Then there are various sequences S for which converges, but for which the “omitted sequence” N–S is, in intuitive sense, sparse, compared with N. For example, Apostol [1] (page 384) quotes, without proof the case where S is the set of all Positive integers whose decimal representation does not invlove the digit zero (e.g. 7∈S but 101 ∉ S); then (1) converges, with T < 90.

In this paper we shall use the integral operator method of Bergman, B[1–6], to investigate solutions of the partial differential equation where s > −1. In particular, information concerning the growth, and location of singularities, of solutions of (1.1) will be obtained. Equations of the form (1.1) with s = 1, 2, arise from the (n+k+1)-dimensional Laplace equation Δn+k+1u = 0 in the “axially symmetric” coordinates x1, …xn, p where the relationship between cartesian and “axially symmetric” coordinates is given by

Equations describing a possible mode of propagation of plane waves in isotropic plastic solids are solved numerically.

Let K(y) be a known distribution function on (−∞, ∞) and let {Fn(y), n = 0, 1,…} be a sequence of unknown distribution functions related by subject to the initial condition If the sequence {Fn(y)} converges to a distribution function F(y) then F(y) satisfies the Wiener-Hopf equation

Let E be a given field and G some (not necessarily the total) group of automorphisms of E. We introduce a topology in G by saying that a net of elements Tα in G converges to T in G if for each x∈E, Tα(x) coincides with Tα(x) ultimately. We shall call this the Krull topology on G.

Let ω(x) be a non-decreasing function defined in the interval [a, b]. We extend the definition to all x by taking ω(x) = ω(a) for x < a and ω(x) = ω(b) for x > b. R. L. Jeffery [2] has denoted by the class of functions F(x) defined as follows:

Let Ω be the set of the analytic functions F(z), regular in some neighbourhood of the origin with the expansion There may exist a function F(s, z) arndytic in s and satisfying the following conditions (s and s′ are any complex numbers): and the ƒ k(s) are polynomials in s.

It is shown algebraically that the full defining relation for the σκ(img)ν is effectively contained in its symmetric part.

The object of this note is to show that under suitable restrictions some results on the wreath product of groups can be carried over to topological groups. We prove in particular the following analogue of the well-known theorem of Krasner and Kaloujnine (see for example [2] Theorem 3.5): Theorem. Let A and B be two locally compact topological groups, and let (C, ε) be an extension of A by B. If there exists a continuous left inverseof ε, that is to say a continuous mapping τ: B → C such that re is the identity on B, then there exists a continuous monomorphism of C into the topological standard wreath product of A by B.