The problem of finding a suitable representation for a fractional power of an operator defined in a Banach space X has, in recent years, attracted much attention. In particular, Balakrishnan [1], Hovel and Westphal [3] and Komatsu [4] have examined the problem of defining the fractionalpower (–A)α for closed densely-defined operators A such that

A graph G is said to possess a perfect matching if there is a subgraph of G consisting of disjoint edges which together cover all the vertices of G. Clearly G must then have an even number of vertices. A necessary and sufficient condition for G to possess a perfect matching was obtained by Tutte (3). If S is any set of vertices of G, let p(S) denote the number of components of the graph G – S with an odd number of vertices. Then the condition

is both necessary and sufficient for the existence of a perfect matching. A simple proof of this result is given in (1).

Maximal left ideals in matrix rings were studied by Stone [10]. Similar results are not necessarily valid in the general near-ring case and one of the objectives of this paper is to study these differences. Furthermore, although much is known about 2-primitivity in general matrix near-rings (Van der Walt [11]), quite the opposite is true for 0-primitivity and the other objective of this paper is to present some results on 0-primitivity in matrix near-rings in certain restricted cases.

Non-degenerate monoids of skew type are considered. This is a class of monoids S defined by n generators and quadratic relations of certain type, which includes the class of monoids yielding set-theoretic solutions of the quantum Yang–Baxter equation, also called binomial monoids (or monoids of I-type with square-free defining relations). It is shown that under any degree-lexicographic order on the associated free monoid FMn. of rank n the set of normal forms of elements of S is a regular language in FMn. As one of the key ingredients of the proof, it is shown that an identity of the form xN yN = yN xN holds in S. The latter is derived via an investigation of the structure of S viewed as a semigroup of matrices over a field. It also follows that the semigroup algebra K[S] is a finite module over a finitely generated commutative subalgebra of the form K[A] for a submonoid A of S.

The earliest author to whom the discovery of the nine-pointcircle has been attributed is Euler, but no one has ever given a reference to any passage in Euler's writings where the characteristic property of this circle is either stated or implied. The attribution to Euler is simply a mistake, and the origin of the mistake may, I think, be explained. It is worth while doing this, in order that subsequent investigators may be spared the labour and chagrin of a fruitless search through Euler's numerous writings.

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.

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.

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

The present paper is concerned with the “logarithmetic”, or arithmetic of shapes of non-associative combinations as defined by Etherington in ref. (1). The shape of a non-associative product is defined as “the manner of association of its factors without regard to their identity”. Thus, for a binary non-communicative operation, the products ((AB)C)D and ((BA)C)D and ((AA)A)A all have the same shape, while D((AB)C) has a different shape. The sum of the two shapes a and b is defined as the shape of the product of two expressions, of shapes a and b respectively, in the original system of non-associative combination. The product of two shapes a and b is defined as the shape of any expression obtained by replacing every factor in an expression of shape b by an expression of shape a. It is readily shown that these definitions are unambiguous.

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.

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.

Let G be a group and K a field. We denote by (KG) the group of units of the group ring of G over K and for a group X we denote by T(X) the setof torsion elements of G i.e., the set of all elements of finite order.

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].

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 permanent of an m x n matrix A = (aij), m ≤ n, is defined by

where the summation is over all one-to-one functions σ from {1, … , m} to { 1, …, n}. In other words, the permanent of A is the sum of all the diagonal products of A, that is, all the products of m entries of A no two of which lie in the same row or in the same column. Thus the permanent of A may be evaluated by first multiplying all the row sums of A and then subtracting from the product all terms that contain as factors two or more entries from the same column of A. This is the idea behind the formulas of Binet and of Ryser.

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:

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 .