Our object of study is the arithmetic of the differential modules (l) (l ∈ ℕ – {0}), associated by Dwork's theory to a homogeneous polynomial f (λ,X) with coefficients in a number field. Our main result is that (1) is a differential module of type G, c'est-à-dire, a module those solutions are G-functions. For the proof we distinguish two cases: the regular one and the non regular one.

Our method gives us an effective upper bound for the global radius of (l), which doesn't depend on “l” but only on the polynomial f (λ,X). This upper bound is interesting because it gives an explicit estimate for the coefficients of the solutions of (l).

In the regular case we know there is an isomorphism of differential modules between (1) and a certain De Rham cohomology group, endowed with the Gauss-Manin connection, c'est-à-dire, our module “comes from geometry”. Therefore our main result is a particular case of André's theorem which assert that at least in the regular case, all modules coming from geometry are of type G.

A process is described for enumerating the Cayley graphs isomorphic to a binary d-cube for small values of d. There are 4 Cayley graphs isomorphic to the 3-cube, 14 isomorphic to the 4-cube, 45 isomorphic to the 5-cube and 238 isomorphic to the 6-cube. A similar method may be used for any graph with a prime power number of vertices.

In this paper, common right factors (in the sense of composition) of p1 + p2F and p3 + p4F are investigated. Here, F is a transcendental meromorphic function and pi's are non-zero polynomials. Moreover, we also prove that the quotient (p1 + p2F)/(p3 + p4F) is pseudo-prime under some restrictions on F and the pi's. As an application of our results, we have proved that R (z) H (z)is pseudo-prime for any nonconstant rational function R (z) and finite order periodic entire function H (z).

Given a group N of Heisenberg type, we consider a one-dimensional solvable extension NA of N, equipped with the natural left-invariant Riemannian metric, which makes NA a harmonic (not necessarily symmetric) manifold. We define a Fourier transform for compactly supported smooth functions on NA, which, when NA is a symmetric space of rank one, reduces to the Helgason Fourier transform. The corresponding inversion formula and Plancherel Theorem are obtained. For radial functions, the Fourier transform reduces to the spherical transform considered by E. Damek and F. Ricci.

The link between a Lie colour algebra and a corresponding Lie superalgebra is clarified in the case of the general linear algebras. The Lie superalgebra inherits from the colour cocommutative coproduct of the corresponding Lie colour algebra, a super coproduct which differs from the usual one, and is not supercocommutative. It is associated with a new R-matrix satisfying the super Yang-Baxter equation.

In 1987, Sullivan determined when a partial transformation α of an infinite set X can be written as a product of nilpotent transformations of the same set: he showed that when this is possible and the cardinal of X is regular then α is a product of 3 or fewer nilpotents with index at most 3. Here, we show that 3 is best possible on both counts, consider the corresponding question when the cardinal of X is singular, and investigate the role of nilpotents with index 2. We also prove that the nilpotent-generated semigroup is idempotent-generated but not conversely.

A group is covered by a collection of subgroups if it is the union of the collection. The intersection of an irredundant cover of n subgroups is known to have index bounded by a function of n, though in general the precise bound is not known. Here we confirm a claim of Tompkinson that the correct bound is 16 when n is 5. The proof depends on determining all the ‘minimal’ groups with an irredundant cover of five maximal subgroups.

An Abelian group G is called an SI-group if for every ring R with additive group R+ = G, every subring S of R is an ideal in R. A complete description is given of the torsion SI-groups, and the completely decomposable torsion free SI-groups. Results are obtained in other cases as well.

We find the convex planar set of maximal area which is partitioned into subsets of equal area by a given propellor. The argument depends on the characterisation of an n-sided polygon which is circumscribed about a regula n-gon Pn, and inscribed in a regula n-gon Qn, where Pn and Qn are homothetic about their common centre.

Given a fixed curve C0 in Rn of length L0 and a variable curve C of fixed length L ≤ L0, the thread problem seeks a least-area surface bounded by C0 + C. We show that an extreme case is a circular arc and its chord. We provide some counterexamples and generalisations to higher dimensions.

In this short note we give a new characterisation of weak compactness.

We study the mod 2 homology of the moduli space of instantons associated with the prinicpal Sp (n) bundle over the four-sphere and the classifying space of the gauge group using the Serre spectral sequence and the homology operations.

We prove a Schwarz Lemma for conformal mappings between two complete Riemannian manifolds when the domain manifold has Ricci curvature bounded below in terms of its distance function. This gives a partial result to a conjecture of Chua.

We describe a virtually Abelian group G generated by a finite set X such that there is no regular language of geodesic X-words that surjects to G by evaluation.

If the Poisson kernel of the unit disc is replaced by its square root, it is known that normalised Poisson integrals of Lp boundary functions converge almost everywhere at the boundary, along approach regions wider than the ordinary non-tangential cones. The sharp approach region, defined by means of a monotone function, increases with p. We make this picture complete by determining along which approach regions one has almost everywhere convergence for L∞ boundary functions.