Let E be a real Banach space. The set of all continuous linear mappings of E into E is a Banach algebra under the usual algebraic operations and the operator bound as norm. We denote this Banach algebra by ℒ, if E is a separate Hilbert space.

Let G be a group. The Fitting subgroup F(G) of G is defined to be the set union of all normal nilpotent subgroups of G. Since the product of two normal nilpotent subgroups is again a normal nilpotent subgroup (see [10] p. 238), F(G) is the unique maximal normal, locally nilpotent sungroup of G. In particular, is G is finite, then F(G) is the unique maximal normal nilpotent subgroup of G. If G is a notrivial solvable group, then clearly F(G) ≠1.

Let G be a locally compact Abelian Hausdorff group (abbreviated LCA group); let X be its character group and dx, dx be the elements of the normalised Haar measures on G and X respectively. If 1 < p, q < ∞, and Lp(G) and Lq(G) are the usual Lebesgue spaces, of index p and q respectively, with respect to dx, a multiplier of type (p, q) is defined as a bounded linear operator T from Lp(G) to Lq(G) which commutes with translations, i.e. τxT = Tτx for all x ∈ G, where τxf(y) = f(x+y). The space of multipliers of type (p, q) will be denoted by Lqp. Already, much attention has been devoted to this important class of operators (see, for example, [3], [4], [7]).

Let G be a group and let Ω(G) denote the semigroup of all mappings of G into G with the usual composition of mappings as multiplication, namely g(θ1θ22) = (gθ1)θ2.

It is well-known that the real number system can be characterised as a topological space [1], [3], as an ordered set [2], and as an ordered field [4]. It is the aim of this note to give two characterisations of the system purely as a field (see Theorems 4 and 9) without any extra notion of order, topology, et cetera.

Suppose that the real-valued function ƒ(t) is positive, continuous and monotonic increasing for t ≧ t0. If x = x (t) is a solution of the equation for for t ≧ t0, it is known that the solution x(t) oscillates infinitely often as t → ∞ and that the successive maxima of |x(t)| decrease, with increasing t. In particular x(t) is bounded as t → ∞.

It is customary to define a vector space in some such manner as: A vector space over a field F is a set V of elements a, b, … called vectors, having the following properties: For arbitrary a, b, c ∈ V; λ, μ ∈ F.

A sequence of integers 0 <a1 < a2 <… no term of which divides any other will be called a primitive sequence. Throughout this paper c1, c2,… will denote suitable positive absolute constants. Behrend [1] proved that for every primitive sequence

A method of deriving a class of solutions of Einstein's field equations with symmetry of de Sitter type is investigated, and the most general solution of this kind is derived.

The iterated equation of generalized axially symmetric potential theory (GASPT) [1] is defined by the relations (1) where (2) and Particular cases of this equation occur in many physical problems. In classical hydrodynamics, for example, the case n = 1 appears in the study of the irrotational motion of an incompressible fluid where, in two-dimensional flow, both the velocity potential φ and the stream function Ψ satisfy Laplace's equation, L0(f) = 0; and, in axially symmetric flow, φ and satisfy the equations L1 (φ) = 0, L-1 (ψ) = 0. The case n = 2 occurs in the study of the Stokes flow of a viscous fluid where the stream function satisfies the equation L2k(ψ) = 0 with k = 0 in two-dimensional flow and k = −1 in axially symmetric flow.

In 1923 Nörlund [1] considered the difference equation

and showed that the formal solution of (1)

obtained by iteration, although in general divergent, is in fact Abel summable to a solution of (1). He writes the arbitary constant c as

and thus the “principal solution” of the difference equation is

In 1959, Professor N. A. Court [2] generated synthetically a twisted cubic C circumscribing a tetrahedron T as the poles for T of the planes of a coaxal family whose axis is called the Lemoine axis of C for T. Here is an analytic attempt to relate a normal rational curve rn of order n, whose natural home is an n-space [n], with its Lemoine [n—2] L such that the first polars of points in L for a simplex S inscribed to rn pass through rn anf the last polars of points on rn for S pass through L. Incidently we come across a pair of mutually inscribed or Moebius simplexes but as a privilege of odd spaces only. In contrast, what happens in even spaces also presents a case, not less interesting, as considered here.

In this paper we show that whether or not a group admits a lattice-order often depends upon whether or not it possesses a set of subgroups that satisfy certain algebraic conditions. Using these techniques we are able to determine large classes of groups that can be lattice-ordered.

In this paper a number of questions about locally compact groups are studied. The structure of finite dimensional connected locally compact groups is investigated, and a fairly simple representation of such groups is obtained. Using this it is proved that finite dimensional arcwise connected locally compact groups are Lie groups, and that in general arcwise connected locally compact groups are locally connected. Semi-simple locally compact groups are then investigated, and it is shown that under suitable restrictions these satisfy many of the properties of semi-simple Lie groups. For example, a factor group of a semi-simple locally compact group is semi-simple. A result of Zassenhaus, Auslander and Wang is reformulated, and in this new formulation it is shown to be true under more general conditions. This fact is used in the study of (C)-groups in the sense of K. Iwasawa.

In the first paper of this series [1] which will be designated I, particular solutions of various kinds have been found for the iterated equation of generalized axially symmetric potential theory (GASPT) which, in the notation defined in I, is (1) where the operator is defined by

Let Xi, ι = 1, 2, 3,… be a sequence of independent and identically distributed random variables and write S0 = 0, Sn = ∑ni=1Xi, n ≧ 1. Nn is the number of positive terms in the sequence S0, S1, S2,…, Snn ≧ 0.

Consider an m × n rectangular array whose m rows are permutations of 1, 2, …, n. Such an array will be called a constant-sum array if the sum of the elements in each column is the same (and equal to ½m (n+1)). An example of a 3×9 constant-sum array is In contrast to a Latin rectangle, elements in the same column of a constantsum array may be equal. It will be convenient to assume arrays normalised in the sense that the columns are arranged so that, as in (1), the first row is in the standard order 1, 2, …, n.

The iterated equation of generalized axially symmetric potential theory (GASPT), in the notation of the first paper of this series [1] which will be designated I, is the equation where the operator Lk is defined by

Let E be a real Banach space. A mapping f of E into E is said to be bounded if f maps every bounded set into a bounded set.