To George Szekeres on his ninetieth birthday

We give new proofs of some sum–to–product identities due to Blecksmith, Brillhart and Gerst, as well as some other such identities found recently by us.

Dedicated to George Szekeres on his 90th birthday

For a transcendental entire function f let M(r) denote the maximum modulus of f(z) for |z| = r. Then A(r) = log M(r)/logr tends to infinity with r. Many properties of transcendental entire functions with sufficiently small A(r) resemble those of polynomials. However the dynamical properties of iterates of such functions may be very different. For instance in the stable set F(f) where the iterates of f form a normal family the components are preperiodic under f in the case of a polynomial; but there are transcendental functions with arbitrarily small A(r) such that F(f) has nonpreperiodic components, so called wandering components, which are bounded rings in which the iterates tend to infinity. One might ask if all small functions are like this.

A striking recent result of Bergweiler and Eremenko shows that there are arbitrarily small transcendental entire functions with empty stable set—a thing impossible for polynomials. By extending the technique of Bergweiler and Eremenko, an arbitrarily small transcendental entire function is constructed such that F is nonempty, every component G of F is bounded, simply-connected and the iterates tend to zero in G. Zero belongs to an invariant component of F, so there are no wandering components. The Julia set which is the complement of F is connected and contains a dense subset of “buried’ points which belong to the boundary of no component of F. This bevaviour is impossible for a polynomial.

Dedicated to George Szekeres on the occasion of his 90th Birthday

In 1990, new statistics on partitions (called cransk) were found which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11 and 25. The methods are extended to find cranks for Ramanujan's partition congruence modulo 49. A more explicit form of the crank is given for the modulo 25 congruence.

In honour of George Szekeres on his 90th birthday

If X is a variety over a number field K, the set of K-rational points on X is contained in the subset of the adelic points cut out by the Brauer group; we call this set the set of Brauer points on the variety. If S is a set of valuations of K, we also define S-Brauer points in a natural way. It is natural to ask how good a bound on the rational points is provided by the Brauer (or S-Brauer) points.

Let p > 3 be a prime number, and let X be the Fermat curve of degree p, xp + yp = 1. Let K be the field of p-th roots of unity, and let r be the p-rank of the class group of K. In this paper we show that if r < (p + 3)/8, then the set of p-Brauer points on X has cardinality at most p. We construct elements of the Brauer group of X by relating it to the Weil-Chatelet group of the jacobian of X, then use the method of Coleman and Chabauty to bound the points cut out by these elements.

Dedicated to George Szekeres on the occasion of his 90th birthday

Necessary and sufficient conditions for equality over the pseudovariety CR of all finite completely regular semigroups are obtained. They are inspired by the solution of the word problem for free completely regular semigroups and clarify the role played by groups in the structure of such semigroups. A strengthened version of Ash's inevitability theorem (κ-reducibility of the pseudovariety G of all finite groups) is proposed as an open problem and it is shown that, if this stronger version holds, then CR is also κ-reducible and, therefore, hyperdecidable.

Dedicated to George Szekeres on his ninetieth birthday

In this paper we prove that a bipartite graph with parts of sizes M and N, having no cycles of even length less that or equal to 2(2k + 1), where k is a positive integer, has at most (NM)(k+1)/(2k+1) + Dk(N + M) edges, where Dk only depends on k.

In particular, we show that when k = 1, D1 = 1 is possible.

Criteria for smooth points, very smooth points and strongly smooth points in Musielak-Orlicz sequence spaces equipped with the Luxemburg norm are given.

In this paper the subgroup-closed saturated Fitting formations of radical locally finite groups with min-p for all p are fully characterised. Moreover the study of a class of generalised nilpotent groups introduced by Ballester-Bolinches and Pedraza is continued.

We obtain lower bounds on the degrees of polynomials representing the Diffie-Hellman mapping (gx, gy) → gxy, where g is a primitive root of a finite field q of q elements. These bounds are exponential in terms of log q. In particular, these results can be used to obtain lower bounds on the parallel arithmetic complexity of breaking the Diffie-Hellman cryptosystem. The method is based on bounds of numbers of solutions of some polynomial equations.

By the classical Banach-Stone Theorem any surjective isometry between Banach spaces of bounded continuous functions defined on compact sets is given by a homeomorphism of the domains. We prove that the same description applies to isometries of metric spaces of unbounded continuous functions defined on non compact topological spaces.

The existence of travelling wave solutions to equations of a viscous, heat-conducting combustible fluid is proved. The reactions are assumed to be one step exothermic reactions with a natural discontinuous reaction rate function. The problem is studied for a general gas. Instead of assuming the ideal gas conditions we consider a general thermodynamics which is described by a fairly mild set of hypotheses. The existence proof of travelling waves for Chapman-Jouguet detonation reduces to finding specific heteroclinic orbits of a discontinuous system of ordinary differential equations: these heteroclinic orbits connect a rest point corresponding to unburnt state to that of the burnt state. The existence proof for heteroclinic orbits corresponding to Chapman-Jouguet detonation waves is carried out by some general topological arguments in ordinary differential equations theory.

This paper treats a rarefied Knudsen gas flow between two infinite plates, with boundary reflexion ruled by a reflexive chaotic law called “Arnold's cat map”. It is shown that the limiting behaviour, when the distance between the plates goes to 0, is described by an (anisotropic) diffusion equation in the norm topology.

In this paper the generalised Hamiltonian approach to the modelling of dynamical systems is developed no via the standard formalism of symplectic geometry but rather via Poisson manifolds and evolution equations. This alternative approach has the merit of being available in a wider context than the former. Application is made to three biomechanical models, one in which the symplectic–geometry approach also applies (the motion of a body segment) and two in which it does not (Schwan's model of blood and lymph circulation and Davydov's molecular model of muscle contraction).