In this paper we show that if f (X) ∈; Z [X ] is a nonzero polynomial, then ω(n)/f(n) holds only on a set of n of asymptotic density zero, where for a positive integer n the number ω(n) counts the number of distinct prime factors ofn.

In this paper we consider groups in which every subgroup has finite index in the nth term of its normal closure series, for a fixed integer n. We prove that such a group is the extension of a finite normal subgroup by a nilpotent group, whose class is bounded in terms of n only, provided it is either periodic or torsion-free.

We show that the operator-valued Marcinkiewicz and Mikhlin Fourier multiplier theorem are valid if and only if the underlying Banach space is isomorphic to a Hilbert space.

Let p be a prime, a field of pn elements, and G a finite p-group. It is shown here that if G has a quotient whose commutator subgroup is of order p and whose centre has index pk, then the group of normalized units in the group algebra has a conjugacy class of pnk elements. This was first proved by A. Bovdi and C. Polcino Milies for the case k = 2; their argument is now generalized and simplified. It remains an intriguing question whether the cardinality of the smallest noncentral conjugacy class can always be recognized from this test.

It is known that 4 ≤ x(ℝ2) ≤ 7, where x(ℝ2) is the number of colour necessary to colour each point of Euclidean 2-space so that no two points lying distance 1 apart have the same colour. Any lattice-sublattice colouring sucheme for R2 must use at least 7 colour to have an excluded distance. This article shows that at least 6 colours are necessary for an excluded distance when convex polygonal tiles (all with area greater than some positive constant) are used as the colouring base.

We prove a new formula for the number of integral points on an elliptic curve over a function field without assuming that the coefficient field is algebraically closed. This is an improvement on the standard results of Hindry-Silverman.

We develop a theory of ergodicity for unbounded functions ø: J → X, where J is a subsemigroup of a locally compact abelian group G and X is a Banach space. It is assumed that ø is continuous and dominated by a weight w defined on G. In particular, we establish total ergodicity for the orbits of an (unbounded) strongly continuous representation T: G → L(X) whose dual representation has no unitary point spectrum. Under additional conditions stability of the orbits follows. To study spectra of functions, we use Beurling algebras L1w(G) and obtain new characterizations of their maximal primary ideals, when w is non-quasianalytic, and of their minimal primary ideals, when w has polynomial growth. It follows that, relative to certain translation invariant function classes , the reduced Beurling spectrum of ø is empty if and only if ø ∈ . For the zero class, this is Wiener's tauberian theorem.

Pointwise bounds for characters of representations of the compact, connected, simple, exceptional Life groups are obtained. It is a classical result that if μ is a central, continuous measure on such a group, then μdimG is absolutely continuous. Our estimates on the size of characters allow us to prove that the exponent, dimension of G, can be replaced by approximately the rank of G. Similar results were obtained earlier for the classical, compact Lie groups.

We study a convolution semigroup satisfying Gaussian estimates on a group G of polynomial volume growth. If Q is a subgroup satisfying a certain geometric condition, we obtain high order regularity estimates for the semigroup in the direction of Q. Applications to heat kernels and convolution powers are given.

The optimality of dualities on a quasivariety , generated by a finite algebra , has been introduced by Davey and Priestley in the 1990s. Since every optimal duality is determined by a transversal of a certain family of subsets of Ω, where Ω is a given set of relations yielding a duality on , an understanding of the structures of these subsets—known as globally minimal failsets—was required. A complete description of globally minimal failsets which do not contain partial endomorphisms has recently been given by the author and H. A. Priestley. Here we are concerned with globally minimal failsets containing endomorphisms. We aim to explain what seems to be a pattern in the way endomorphisms belong to these failsets. This paper also gives a complete description of globally minimal failsets whose minimal elements are automorphisms, when is a subdirectly irreducible lattice-structured algebra.