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.

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.

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

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.

Let G be a transitive permutation group on a set Ω of n points, and let P be a Sylow p-subgroup of G for some prime p dividing ∣G∣. If P has t long orbits and f fixed points in Ω, then it is shown that f ≦ tp − ip(n), where ip(n) = p – rp(n), rp(n) denoting the residue of n modulo p. In addition, groups for which f attains the upper bound are classified.

Conditions are found for the existence of supplementary difference sets consisting of cosets of e-th power residues modulo a prime p. For e = 4, 6 and 8 all known pairs of sets are listed in the summary.

Let PA(n) denote the number of partitions of n into summands chosen from the set A = {a1, a2, …}. De Bruijn has shown that in Mahler's partition problem (aν = rν) there is a periodic component in the asymptotic behaviour of PA(n). We show by example that this may happen for sequences that satisfy aν ν and consider an analogous phenomena for partitions into primes. We then consider corresponding results for partitions into distinct summands. Finally we obtain some weaker results using elementary methods.

We present in this paper a discussion on some stability properties of line graphs. After relating the semi-stability properties of the line graph of a graph to a concept of Sheehan, we proceed to deduce that, with fully characterised lists of exceptions, the line graphs of trees and unicyclic graphs are semi-stable. We then discuss the problem of deciding which line graphs are stable. Via a discovery of the finite number of graphs G such that both G and its complement have stable line graphs, we show that P4 is the only self-complementary graph whose line graph is stable.

Let G be a group, and associate with G a graph 1 γ = Γ (G) as follows: the vertices of Γ are the elements of G, and two vertices g, h of Γ are joined by an undirected edge if, and only if, g and h do not commute as elements of G, that is [g, h] ≠ 1 [where [g, h] is the commutator g−1h−1gh, and 1 is the unit element of the groups that occur as well as the integer, according to context].We are interested in complete subgraphs of Γ, or equivalently in sets of elements of G no two of which commute.

Erdös, Harary and Tutte have defined the chromatic number of the plane to be the least number of sets partitioning the plane such that no set contains two points at unit distance apart. By analogy, the chromatic number χ(S1), of the sphere, Sr of radius r is defined to be the least number of sets partitioning the surface of Sr such that no set contains two points at unit chordal distance apart. In this paper it is proven that and that this bound is best possible since .

Let p be a prime ≡ 1 (mod 4) and put v = p(p + 1)/2. It is proved in this paper that there exist four symmetric circulant matrices A, B, C, D of order υ such that where Iv is the identity matrix of order υ. This result is used to construct Hadamard matrices of order 4υ that are of the type originally prescribed by Williamson.

Schur's matrix Mn is ordinarily defined to be the n by n matrix (εjk), 0 ≦ j, k < n, where ε = exp (2 πi/n). This matrix occurs in a variety of areas including number theory, statistics, coding theory and combinatorics. In this paper, we investigate Pn, the permanent of Mn, which is define by where π ranges over all n! permutations on {0,1, …, n – 1}. Pn occurs, for example, in the study of circulants. Specifically, let Xn denote the n by n circulant matrix (xi, j) with xi, j = xi, j, where the subscript is reduced modulo n. The determinant of Xn is a homogeneous polynomial of degree n in the xi and can be written as Then Pn = A (1,1, … 1). Typical of the results established in this note are: (i) P2n = 0 for all n, (ii) Pp ≡ p ! (mod p3) for p a prime >3. (iii) If pa divides n then divides Pn. Also, a table of values of Pn is given for 1 ≦ n ≦ 23.

Let A be a complex sequentially complete, locally convex (not necessarily commutative) topological algebra with the defining family {pα}α∈D of seminorms in which (*): for each sequence xn → 0 there exists xm ∈ {xn} such that xmk → 0 as k → ∞. Then each multiplicative linear functional on a Fréchet algebra satisfying the above condition (*) is Continuous.

These results answer open questions (1) and (2) (Mem. Amer. Math. Soc. 11, 1953) in the affirmative for Fréchet algebras in which (*) holds. It is also shown that a positive linear functional on such algebras with identity and continuous involution is continuous, thus partially generalizing Shah's result (1959).

Let S be a closed bounded convex set in d-dimensional Euclidean space Ed. The width w(S) of S is the minimum distance between supporting hyperplanes of S, and L(S) is the number of integral lattice points in the interior of S.

A composite number N is called a pseudoprime for the base a in case . An odd pseudoprime N is called strong for the base a.