We explore the problem of finding the Hausdorff dimension of the set of points that recur to shrinking targets on a self-affine fractal. To be exact, we study the dimension of a certain related symbolic recurrence set. In many cases, this set is equivalent to the recurring set on the fractal.

The determination of the harmonic functions of elliptic and hyperbolic cylinders depends on the solution of Mathieu's differential equation. This equation, it has been remarked by Professor Whittaker, is the one which naturally comes up for study after the hypergeometric equation has been disposed of. Its solution presents difficulties which do not arise in connection with the hypergeometric equation or its degenerate cases, and it cannot, I think, be said that any discussion of the equation has yet been given with which the student of analysis can rest content. The treatment given below, though certainly incomplete at some points, seetns to follow the lines along which a thoroughly successful theory may be hoped for.

The Fibonacci groups are a special case of the following class of groups first studied by G. A. Miller (7). Given a natural number n, let θ be the automorphism of the free group F = 〈x1, …, xn |〉 of rank n which permutes the subscripts of the generators in accordance with the cycle (1, 2, …, n). Given a word w in F, let R be the smallest normal subgroup of F which contains w and is closed under θ. Then define Gn(w) = F/R and write An(w) for the derived factor group of Gn(w). Putting, for r ≦ 2, k ≦ 1,

with subscripts reduced modulo n, we obtain the groups F(r, n, k) studied in (1) and (2), while the F(r, n, 1) are the ordinary Fibonacci groups F(r, n) of (3), (5) and (6). To conform with earlier notation, we write A(r, n, k) and A(r, n) for the derived factor groups of F(r, n, k), and F(r, n) respectively.

We classify generically transitive actions of semi-direct products on ℙ2. Motivated by the program to study the distribution of rational points on del Pezzo surfaces (Manin's conjecture), we determine all (possibly singular) del Pezzo surfaces that are equivariant compactifications of homogeneous spaces for semi-direct products .

Let Fn = 〈 a1,a2,…,an〉 denote the free group of rank n, and let θ denote the automorphism of Fn which permutes the generators cyclically, in other words:

In this paper we discuss various aspects of invariant measures for nonlinear Hamiltonian partial differential equations (PDEs). In particular, we show almost-sure global existence for some Hamiltonian PDEs with initial data of the form ‘a smooth deterministic function + a rough random perturbation’ as a corollary to the Cameron–Martin theorem and known almost-sure global existence results with respect to Gaussian measures on spaces of functions.

In 1962 Gilbert Baumslag introduced the class of groups Gi, j for natural numbers i, j, defined by the presentations Gi, j = < a, b, t; a−1 = [bi, a] [bj, t] >. This class is of special interest since the groups are para-free, that is they share many properties with the free group F of rank 2.

Magnus and Chandler in their History of Combinatorial Group Theory mention the class Gi, j to demonstrate the difficulty of the isomorphism problem for torsion-free one-relator groups. They remark that as of 1980 there was no proof showing that any of the groups Gi, j are non-isomorphic. S. Liriano in 1993 using representations of Gi, j into PSL(2, pk), k ∈ ℕ, showed that G1,1 and G30,30 are non-isomorphic. In this paper we extend these results to prove that the isomorphism problem for Gi, 1, i ∈ ℕ is solvable, that is it can be decided algorithmically in finitely many steps whether or not an arbitrary one-relator group is isomorphic to Gi, 1. Further we show that Gi, 1 ≇ G1, 1 for all i > 1 and if i, k are primes then Gi, 1 ≅ Gk, 1 if and only if i = k.

A subgroup Q of a group G is called quasinormal in G if Q permutes with every subgroup of G. Of course a quasinormal subgroup Q of a group G may be very far from normal. In fact, examples of Iwasawa show (for a convenient reference see [8]) that we may have Q core-free and the normal closure QG of Q in G equal to G so that Q is not even subnormal in G. We note also that the core of Q in G, QG, is of infinite index in QG in this example. If G is finitely generated then any quasinormal subgroup Q is subnormal in G [8] and although Q is not necessarily normal in G we have that |QG:Q| is finite and |QG:Q| is a nilpotent group of finite exponent [5].

If A is a complex Banach algebra (not necessarily unital) and x∈A, σ(x) will denote the spectrum and spectral radius of x in A. If I is a closed two-sided ideal in A let x + I denote the coset in the quotient algebra A/I containing x. Then

We shall denote by ω the natural partial order on the idempotents E = E(S) of a regular semigroup S, so that in E,

A partially ordered semigroup S(≦) is called naturally partially ordered [9] if the imposed partial order ≦ extends ω in the sense that

No assumption is made about the reverse implication.

In this paper we prove that if the strong dual of an echelon space fulfils the Mackey convergence condition the echelon space is quasi-normable. Also we give a characterisation of the quasi-normable echelon spaces and we deduce that every non-quasi-normable echelon space is the strong dual of a non-complete (LB)-space.

New spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajłasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

AMS 2000 Mathematics subject classification: Primary 42B35. Secondary 46E35; 58J35; 43A99

A cotorsion theory is defined as a pair of classes Ext-orthogonal to each other. We give a hereditary condition (HC) which is satisfied by the (flat, cotorsion) cotorsion theory and give properties satisfied by arbitrary cotorsion theories with an HC. Given a cotorsion theory with an HC, we consider the class of all modules having a special precover with respect to the first class in the cotorsion theory and show that this class is closed under extensions. We then raise the question of whether this class is resolving or coresolving.

AMS 2000 Mathematics subject classification: Primary 18G15. Secondary 18E40

In this paper, using the idea of upper and lower faithful d.g. near-rings, introduced in (6), we show that the category of all faithful d.g. near-rings is a reflective as well as coreflective subcategory of the category of all d.g. near-rings. We also prove that both and are complete and cocomplete categories.

A module of a ring of differential operators $\mathcal{D}$ over a smooth surface has order $1$ if it is isomorphic to a factor module of $\mathcal{D}$ by a cyclic ideal generated by an operator of order $1$. Let $k$ be a positive integer. We give conditions under which an indecomposable $\mathcal{D}$-module of order $1$ is GK-critical of length $k$. We also give examples of indecomposable, non-critical, $\mathcal{D}$-modules whose subfactors have order $1$.

AMS 2000 Mathematics subject classification: Primary 16S32. Secondary 37F75

Let ξ be the set of all finite groups that have efficient presentations. In this paper we give sufficient conditions for the standard wreath product of two ξ-groups to be a ξ-group.

This paper is concerned with d.g. near-rings and their relationship to faithful d.g. near-rings. For general definitions and results, we refer to Pilz [5]. We use left near-rings where he uses right near-rings, but otherwise there is little difference. This work follows earlier work [3], [4] and Mahmood [2]. Before outlining the contents of the paper we present a précis of the definitions.

Let W be a class of not necessarily associative rings which is universal in the sense that it is closed under homomorphic images and is hereditary to subrings. All rings considered will be assumed to belong to W. The notation I ⊲ R will mean I is an ideal of R. A relation σ on W will be called an H-relation if σ satisfies the properties:

(1) I σ R implies I is a subring of R.

(2) If I σ R and ø is a homomorphism of R, then IØ σ Rø.

(3) If I σ R and J is an ideal of R, then I ∩ J σ J.

Many fundamental properties of graph C*-algebras may be determined directly from the structure of the underlying graph, and because of this, they have been celebrated as C*-algebras that can be seen. This paper shows how permutative endomorphisms of graph C*-algebras can be represented by labelled directed multigraphs that give visual representations of the endomorphisms and facilitate computations. This formalism provides a useful calculus for permutative automorphisms and allows efficient exhaustive construction of such automorphisms.

We give a necessary and sufficient condition for a sequence {ak}k in the unit ball of ℂn to be interpolating for the class A–∞ of holomorphic functions with polynomial growth. The condition, which goes along the lines of the ones given by Berenstein and Li for some weighted spaces of entire functions and by Amar for H∞ functions in the ball, is given in terms of the derivatives of m ≥ n functions F1, …,Fm ∈ A–∞ vanishing on {ak}k.