It is known that the Unified Field Theories of Weyl [14] and of Einstein [4] give no indication of how Relativity and Quantum Theories should be connected into a comprehensive field theory of physics. Indeed, the only determined attempt to establish such a theory, due to Eddington [3] and [6], failed through lack of contact with the contemporary developments, especially in quantum electrodynamics and elementary particles. Its author tried to explain curvature of the space-time in terms of statistical fluctuations partly of a physical origin defined within a mechanical system, and partly of a geometrical origin of coordinates superimposed on the latter. It is clear however that both the fluctuations of Eddington refer to purely mathematical frames. The probabilistic nature of his theory takes no account of physical objects, such as particles or energy distributions. It is the author's belief that this is the cause of difficulties associated with the otherwise admirable work of Eddington.

This is the first of several papers which grew out of an attempt to provide C (X, Y), the family of all continuous functions mapping a topological space X into a topological space Y, with an algebraic structure. In the event Y has an algebraic structure with which the topological structure is compatible, pointwise operations can be defined on C (X, Y). Indeed, this has been done and has proved extremely fruitful, especially in the case of the ring C (X, R) of all continuous, real-valued functions defined on X [3]. Now, one can provide C(X, Y) with an algebraic structure even in the absence of an algebraic structure on Y. In fact, each continuous function from Y into X determines, in a natural way, a semigroup structure for C(X, Y). To see this, let ƒ be any continuous function from Y into X and for ƒ and g in C(X, Y), define ƒg by each x in X.

In the present paper we obtain first approximation formulae for the regions of dynamical instability of linear canonical systems. These formulae are analogous to the formulae for Hamiltonian systems stated by Krein and Jakubovič [5] and proved by Pittel' and Juzefovič [8]. Special cases were considered by Malkin [7] and Jakubovič [3]. Related papers are Hale [2] and Jakubovič [4]. However, our method differs from the methods used by these authors and seems to us to be both simpler and more general.

Recently A. L. Šmel'kin [14] proved that a product variety1 is generated by a finite group if and only if is nilpotent, is abelian, and the exponents of and are coprime. Alternatively, by the theorem of Oates and Powell [13], we may say that a Cross variety is decomposable if and only if it is of the above form.

Let Σn−0∞an, be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write

If

as n → ∞, then we say that the series is summable by the Nörlund method (N, pn) to σ And the series a,Σan, is said to be absolutely summable (N, pn) or summable |N, Pn| if {σn} is of bounded variation, i.e.,

This paper is taken from the author's doctoral dissertation at New York University; the work was supported by the National Science Foundation, Grant G19674. The author would like to thank Professor Gilbert Baumslag for suggesting the questions treated in this paper, and for his very helpful guidance.

This is a continuation of [5] and we begin by recalling two definitions and a result of that paper which are needed here. Let be a family of functions with domains contained in a set X and ranges contained in a set Y and let be a function with domain D()= Y and range with the property for each pair of elements ƒ and g of . Since the composition operation is associative, is a semigroup if for ƒ and g in , we define the product ƒg by .

Let Fx and Fy denote families of subsets of the nonempty sets X and Y respectively and let be a function mapping Y onto X with the property that [H] ∈ Fx for each H∈Fy. Then the family l of all functions mapping X into Y such that [H] ∈Fy for each H ∈Fx is a semigroup if the product g of two such functions and g is defined by g = o o g (i.e., (Fg)(x) = (f(g(x))) for each x in X). With some restrictions on the families Fx and Fy and the function , these are the semigroups mentioned in the title and are the objects of investigation in this paper. The restrictions on fxfy and are sufficiently mild so that the semigroups considered here include such semigroups of functions on topological spaces as semi- groups of closed functions, semigroups of connected functions, etc.

Let (x) be a continuous function with period 2π. It is well known that the Fourier series of (x) is summable Riesz of any positive order to (x). The aim of this paper is the proof of the following theorem.

Let Fp be the residue field modulo a prime number p. The mappings of Fp, into itself are viewed as functions in one variable over Fp. When the mapping is onto, the function is a permutation.

Let {f(n)} be a linear recurrence of order two, i.e.

for all positive integers n.

It is well known that the problem of determining the most economical covering of n-dimensional Eucidean space, by equal spheres whose centres form a lattice, may be formulated in terms of positive definite quadratic forms, as follows: Let

be positive definite, and

For real α, set

(the minimum being taken over integral x),

In his paper [8], N. Itô gives an elegant proof that the Sylow p-group of a finite solvable linear group of degree n over the field of complex numbers is necessarily normal if p > n+1. Moreover he shows that this bound on p is the best possible when p is a Fermat prime (i.e. a prime of the form 2sk + 1) but that the bound may be improved to p > n when p is not a Fermat prime.

A ring K is a radical extension of a subring B if for each x ∈ K there is aninteger n = n(x) > 0 such that xn ∈ B. In [2] and [3], C. Faith considered radical extensions in connection with commutativity questions, as well as the generation of rings. In this paper additional commutativity theorems are established, and rings with right minimum condition are examined. The main tool is Theorem 1.1 which relates the Jacobson radical of K to that of B, and is of independent interest in itself. The author is indebted to the referee for his helpful suggestions, in particular for the strengthening of Theorem 2.1.