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Morgan Ward pursued the study of elliptic divisibility sequences initiated by Lucas, and Chudnovsky and Chudnovsky suggested looking at elliptic divisibility sequences for prime appearance. The problem of prime appearance in these sequences is examined here from a theoretical and a practical viewpoint. We exhibit calculations, together with a heuristic… (More)

Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be… (More)

- Thomas Ward, Qing Zhang, QING ZHANG
- 1992

In this note we show that the entropy of a skew product action of a countable amenable group satisfies the classical formula of Abramov and Rokhlin.

The flow of viscous, particle-laden wetting thin films on an inclined plane is studied experimentally as the concentration is increased to the maximum packing limit. The slurry is a non-neutrally buoyant mixture of silicone oil and either solid glass beads or glass bubbles. At low concentrations ͑ Ͻ 0.45͒, the elapsed time versus average front position… (More)

- T. B. Ward, T. B. WARD
- 1992

We show that an expansive Z 2 action on a compact abelian group is measurably isomorphic to a two–dimensional Bernoulli shift if and only if it has completely positive entropy. The proof uses the algebraic structure of such actions described by Kitchens and Schmidt and an algebraic characterisation of the K property due to Lind, Schmidt and the author. As a… (More)

A general framework for investigating topological actions of Z d on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lower-dimensional subspaces of R d. Here we completely describe this expansive behavior for the class of algebraic Z d-actions given by commuting automorphisms of compact abelian groups. The… (More)

Analogues of the prime number theorem and Merten's theorem are well-known for dynamical systems with hyperbolic behaviour. In this paper a 3-adic extension of the circle doubling map is studied. The map has a 3-adic eigendirection in which it behaves like an isometry, and the loss of hyperbolicity leads to weaker asymptotic results on orbit counting than… (More)

- Thomas Ward, THOMAS WARD
- 2001

Let ξ 1 ,. .. , ξ r be complex numbers with K = Q(ξ 1 ,. .. , ξ r) having tran-scendence degree r − 1 over Q. Consider the equation a 1 x 1 + · · · + a k x k = 1, (1) in which the a i 's are fixed elements of K × , no proper subsum a i 1 x i 1 + · · · + a i j x i j vanishes, and we seek solutions x i ∈ Γ = ξ 1 ,. .. , ξ r. It is well–known that (1) has only… (More)

We introduce a class of group endomorphisms – those of finite combinatorial rank – exhibiting slow orbit growth. An associated Dirichlet series is used to obtain an exact orbit counting formula, and in the connected case this series is shown to to be a rational function of exponential variables. Analytic properties of the Dirichlet series are related to… (More)

- T. Ward
- 1997

Let α be a Z d –action (d ≥ 2) by automorphisms of a compact metric abelian group. For any non–linear shape I ⊂ Z d , there is an α with the property that I is a minimal mixing shape for α. The only implications of the form " I is a mixing shape for α =⇒ J is a mixing shape for α " are trivial ones for which I contains a translate of J. If all shapes are… (More)