“I have received from Einstein himself a copy of his recent papers, and I shall always consider it an honour to have done so. But still I am at liberty to say that the theory does not impress me as holding the secret of the laws of nature. I feel much more disposed to trust the classical theory, modified perhaps so as to recognise the principle of equivalence. Einstein's theory is wonderful—wonderful in its complexity and in the mathematical difficulties overcome. Newton's is even more wonderful—wonderful in its simplicity and in its agreement with nearly all the experimental evidence of two and a half centuries. I cannot agree with those who would make nature more akin to the complex than to the simple.”

George Szekeres was born in Budapest on 29th May, 1911 the second of three sons to wealthy Jewish parents. As a youth he was shy and retiring, but early it become clear that his gifts lay in the direction of science and mathematics. At high school George was greatly influenced by his teacher in mathematics and physics, K. (Charles) Novobátzky, who worked actively in the theory of relativity and was in 1945 to become a professor of theoretical physics at the University of Budapest. Small wonder that George's first great mathematical interest was relativity. The other major formative influence at high school was the journal ‘Koözeépiskolai Matematikai és Fizikai Lapok’. The names of problem solver were published with each solution and with the completion of the year's volume, photographs of the main contributors were reproduced.

It is well-known that, in any semigroup S, all the left and right Schützenberger groups of the ℋ-class contained in a fixed D class D of S are isomorphic to one group. We prove a sharper result: that, up to actionpreserving isomorphisms, all these Schützenberger groups and their classical isomorphisms are but one group and its identity isomorphism; thus, one group is essentially sufficient to describe not only all the Schützenberger groups of ℋ-classes in D, but their action on these ℋ-classes and classical isomorphisms as well.

A theory T is said to κ-stable if, given a pair of models U ⊂ B of T with U of power κ, there are only κ types of elements of B over U (types are defined below). This notion was introduced by Morley (1965) who gave a powerful analysis of ω-stable theories. Shelah (1971) showed that there are only four possibilities for the set of κ in which a countable theory is stable. This partition of all theories into four classes (ω-stable, superstable, stable, and unstable theories) has proved to be of great value. However, most familiar examples of theories are unstable.

In determining geometrical conditions on a Banach space under which a Chebychev set is convex, Vlasov (1967) introduced a smoothness condition of some interest in itself. Equivalent forms of this condition are derived and it is related to uniformly weak differentiability of the norm and rotundity of the dual norm.

This paper investigates the logical stability of various groups. Theorem 1: If a group G is stable and locally nilpotent then it is solvable. Theorem 2: Every non-Abelian variety of groups is unstable.

All groups in this paper are Abelian. The notation is similar to Fuchs (1970) and Maclane (1963).

Various investigations on (E, M)-categories are given by many authors. Recently, Herrlich (to appear) and Strecker (1972) gave interesting results on relations between factorizations (E, M) on nice categories and epireflective subcategories of . Factorization (E, M) treated by them satisfy that E ⊂ Epi. In this paper we shall consider factorization structures (E, M) such that E⊃Epi in the category of all topological spaces and study relations between these structures and some subcategories of . Our results are quite different from those in Herrlich (to appear) and Strecker (1972) and are rather related with the results in Herrlich (1969). Herrlich (1969) defined limit operators on and gave a one-to-one correspondence between suitable limit operators and bicoreflective subcategories of . We shall consider two kinds of factorization structures on and show that the first ones correspond to bicoreflective subcategories which are closed under closed embeddings, the second ones correspond to bireflective subcategories which satisfy that the classes of all reflections are closed under closed embeddings (see Proposition 11) and these correspondences are one-to-one.

A group will be called full if it is the Galois group of an algebraic closure of a field. In this paper we first investigate full Abelian groups and classify them. Then we examine full groups from the point of view of how we can operate on them and still maintain the property of being full. Of course, by the fundamental theorem of the infinite Galois theory, closed subgroups (with standard profinite topology) of full groups are full. In general, products of full groups are notfull (for example, Z2 × Z2 is not full, by a theorem of Artin and Schreier (1927)); however we produce a set of groups which can always be attached as direct factors to full groups and still retain full groups. For the definition and basic properties of profinite groups we refer the reader to Cassels and Frohlich (1967). It has been shown by Leptin (1955) and independently by the author that profinite groups are Galois groups. The author (1971) has shown that if G is profinite, then G = Gdl(k/L) where L may have any desired characteristic and contains all primitive nth roots of unity, for all n. We will denote by p the pro-p-group which is the inverse limit of cyclic p-groups. (This is the group of p-adic integers).

The author investigates M(n)= min¦Σηkk−1¦ where the minimum is over all sets of signs η1 = ±1 and shows M(n) < n½−(1−e)log2n.

The purpose of this note is to apply the work of Kasteleyn (1967) on the enumeration of I-factors of a graph to derive a quick proof of a theorem of Szekeres (1973). In the following, G is understood to be a finite, directed graph. If u, v are adjacent vertices of G, we denote by (u, v) an edge of G directed from u to v. Let{f1, f2,…, fm} be a set of 1-factors of G, and for all i write where n is half the number of vertices of G. (Here we regard a 1-factor as a set of edges). Then Kasteleyn associates with fi a plus sign if ui1vi1ui2vi2…uinvin is an even permutation of u11v11u12v12…u1nv1n, and a minus sign otherwise. The symmetric difference of two 1-factors is a collection of circuits, called alternating circuits. An alternating circuit of G is said to be clockwise even if the number of its edges that are directed in agreement with the clockwise sense is even; otherwise it is clockwise odd. Since the length of any alternating circuit is even, these definitions are not dependent on the sense designated as clockwise. It follows from the work of Kasteleyn that two given 1-factors in a directed graph agree in sign if and only if the number of clockwise even alternating circuits in their symmetric difference is even.

The biharmonic Green's function of a simply supported plate is generalized to Riemannian manifolds and shown to exist if and only if the harmonic measure of the ideal boundary is square integrable.

Profinite groups (that is, compact totally disconnected topological groups) have been characterized as Galois groups in Krakowski (1971) and Leptin (1955). They can, in fact, be realized as groups of permutations on sets of transcendentals where every transcendental has a finite orbit under the group action and the fixed field is generated by the invariant rational functions on these orbits. If G = {G←x ¦ α∈A} we define the cardinality of G as ¦ G¦ = ¦A¦. In this paper we construct, for every cardinal number c and profinite group G with ¦G¦ ≦ c two universal fields kc and Kc so that G ≃ Gal (KcKG) and G ≃ (KGKc) for fields KG and kG which depend on G. See Cassels and Frohlich (1967) for a description of the basic properties of profinite groups.

The author investigates M(n, α) = min ¦α – Σηkk−1¦ where the minimum is over all sets of signs η1 = ± 1 and shows .

Free groups and free inverse semigroups are characterized by certain properties which are usually called universal. Category-theoretically they arise from left adjoints of forgetful functors which assign to each structure its underlying set. The purpose of this note is to give a construction of a wealth of inverse semigroups with certain universal properties which subsumes free groups and free inverse semigroups and incidentally elucidates some well-known constructions of free inverse semigroups (Scheiblich (1972)). Categorytheoretical terminology will be freely taken from (Mac Lane (1971)).

Since it is no longer fashionable to publish per se results which depend only on algebraic manipulation, many useful and complicated results of quite general interest languish unseen hidden as lemmata in specialist papers. In particular, the theory of transcendental numbers is rich in ingenious techniques for evaluating determinants. These techniques are apparently not well known even to workers in the field of transcendental numbers, let alone to researchers on other areas where the results might find application. This note accordingly discusses a variety of interesting results on determinants and is to be viewed as an appendix to the encyclopaedic volumes of Muir (1911, 1933) which the researcher might approach in order to obtain information in this area. For reasons of motivation a brief mention is made of the context in which the determinants arise.

Let 1 ≦ a1 < … < ak ≦ x; b1 < … b1 ≦ x. Assume that the number of solutions of a1b1 = m is less than c. The authors prove that then . They also give a simple proof of Szemerédi's theorem: If the products aibj are all distinct then . They conjecture that (2) holds for c2 = 1 + ε if x > x0(ε). Several other unsolved problems are stated.

In this paper we examine the linear differential equations x′ = Ax + P(φ)x, φ′ = ω where x ∈ Rn, φ ∈ Rm, A and ω are constant and P(φ) is real analytic and periodic in φ. We use the method of accelerated convergence to overcome the small divisors problem and reduce this system to the system y′ = By, φ′ = ω with constant coefficients.

This problem has already been examined by Mitropolśki and Samolenko but the calculations and details in their work are formidable and difficult to follow. Besides being simpler our method provides more precise estimates at all stages and can be extended to the differentiable case to provide a significant improvement over previous results.

Let k be a fixed integer ≧ 2. A positive integer n is called unitarily k-free, if the multiplicity of each prime factor of n is not a multiple of k; or equivalently, if n is not divisible unitarily by the k-th power of any integer > 1. By a unitary divisor, we mean as usual, a divisor d> 0 of n such that (d, n/d) = 1. The interger 1 is also considered to be unitarily k-free. The concept of a unitarily k-free integer was first introduced by Cohen (1961; §1). Let denote the set of unitarily k-free integers. When k = 2, the set coincides with the set Q* of exponentially odd integers (that is, integers in whose canonical representation each exponent is odd) discussed by Cohen himself in an earlier paper (1960; §1 and §6). A divisor d > 0 of the positive integer n is called a unitarily k-free divisor of n if d ∈ . Let (n) denote the number of unitarily k-free divisors of n.

Every lattice generated by three unordered elements contains a finite sublattice generated by three unordered elements. A list ℒ of twelve finite lattices, each generated by a three-element unordered set, is given. It is proved that every lattice generated by a three-element unordered set contains a sublattice isomorphic to onė of the lattices in ℒ moreover, ℒ is the smallest such list.