We investigate the relationship between the peripheral spectrum of a positive operator T on a Banach lattice E and the peripheral spectrum of the operators S dominated by T, that is, ]Sx] ≤ T]x] for all x ε E. This can be applied to obtain inheritance results for asymptotic properties of dominated operators.

If Sp(V) is the symplectic group of a vector space V over a finite field of characteristic p, and r is a positive integer, the abelian p-subgroups of largest order in Sp(V) whose fixed subspaces in V have dimension at least r were determined in the preceding paper, in the case p ≠ 2. Here we deal with the case p = 2. Our results also complete earlier work on the orthogonal groups.

The coproduct of two frames A, B (equivalently the product of two locales, see Isbell (1972a)) will be shown to be given by the lattice of Galois connections between A and B (the connection product in the sense of Isbell (1972b)). We show the inverse limit of a direct system of locales to be given by the set inverse limit of the underlying lattices under the bonding antimaps (see Dowker and Strauss (1975)). This implies the existence of infinite locale products.

We investigate parabolic Higgs bundles and $\Gamma $-Higgs bundles on a smooth complex projective variety.

For Γ a revursively enumerable set of formulae, a structure U on a recursive universe is said to be “Γ-recursively enumerable” if the satisfaction predicate restricted to Γ is recursively enumerable (equivalently, if the formulae of Γ uniformulae of Γ uniformly denote recursively enumerable relations on U).

For recursively enumerable sets Γ1 ⊆ Γ2 of formulae we shall, under certain conditions, characterize structures U with the following properties.

1) Every isomorphism form U to a Γ1-recursively enumerable structure is a recursive isomorphism.

2) Every Γ1-recursively enumerable structure isomorphic to U is recursively isomorphic to U.

3) Every Γ1-recursively enumerable structure isomorphic to U is Γ2-recursively enumerable.

Let Ω be the set of the analytic functions F(z), regular in some neighbourhood of the origin with the expansion There may exist a function F(s, z) arndytic in s and satisfying the following conditions (s and s′ are any complex numbers): and the ƒ k(s) are polynomials in s.

We extend the normal surface Q-theory to non-compact 3-manifolds with respect to ideal triangulations. An ideal triangulation of a 3-manifold often has a small number of tetrahedra resulting in a system of Q-matching equations with a small number of variables. A unique feature of our approach is that a compact surface F with boundary properly embedded in a non-compact 3-manifold M with an ideal triangulation with torus cusps can be represented by a normal surface in M as follows. A half-open annulus made up of an infinite number of triangular disks is attached to each boundary component of F. The resulting surface , when normalized, will contain only a finite number of Q-disks and thus correspond to an admissible solution to the system of Q-matching equations. The correspondence is bijective.

In [5] James defined an nth order Perron integral, the Pn- ntegral, and developed its properties. His proofs are often indirect, using properties of the CkP-integrals of Burkill, [3]. In this paper a simpler definition of the Pn-integral is given — the original and not completely equivalent definition, was probably chosen as James considered this integral as a special case of one defined in terms of certain symmetric derivatives, [5], when end points of the interval of definition had naturally to be avoided. We then give direct proofs of the basic results, give a characterization of Pn-primitives, and connect the integral with certain work of Denjoy, [4].

In this paper we give a sufficient condition of quasi-pseudometrization for bitopological spaces. From this condition we obtain, as immediate corollaries, some known results.

This paper is a continuation of [2] and will use the terminology and notation introduced therein. Basically, we seek to investigate the structure of indexed lattices in which the set of lengths of the minimal indexing chains is bounded.

An early extension of Lindeberg's central limit theorem was Bernstein's (1939) discovery of necessary and sufficient conditions for the convergence of moments in the central limit theorem. Von Bahr (1965) made a study of some asymptotic expansions in the central limit theorem, and obtained rates of convergence for moments. However, his results do not in general imply that the moments converge. Some better rates have been obtained by Bhattacharya and Rao for moments between the second and third. In this paper we give improved rates of convergence for absolute moments between the third and fourth.

Let Q(x, y, z, t, u) be a real indefinite 5-ary quadratic form of type (3,2) and determinant D(> 0). Then given any real numbers x0, y0, z0, t0, u0 there exist integers x, y, z, t, u such that 0 < Q(x+x0,y+y0,z+z0,t+t0,u+u0)≦(16D)1/5. All the critical forms are also determined.

In 1964 Frink [8] generalized Wallman's method [17] of compactification and asked the question: “Is every Hausdorff compactification of a Tychonoff space a Wallman compactification?”. This problem, which is as yet unsolved, has led to the discovery of a number of necessary and/or sufficient conditions for a Hausdorff compactification to be Wallman; (see Alò and Shapiro [2], [3], Banasĉhewski [6], Njåstad [13] and Steiner [15]). Recently Alò and Shapiro [4], [5] have generalized the Wallman procedure to discuss what they call Z*-realcompact spaces and Z*-realcompactification n(Z) corresponding to a countably productive (c.p.) normal base Z on X. Some further work has been done by Steiner and Steiner [14].

The aim of this article is to review the progress made in the last few years in the representation theory of solutions of parabolic systems in the sense of Petrowskii.

The theme of the paper is a Möbius inversion principle for infinite sums. We deal with the origins and unprincipled use of this idea in the nineteenth century, its rigorous justification under minimal hypotheses and some applications to a problem in numerical integration.

Every lattice generated by three unordered elements contains a finite sublattice generated by three unordered elements. A list ℒ of twelve finite lattices, each generated by a three-element unordered set, is given. It is proved that every lattice generated by a three-element unordered set contains a sublattice isomorphic to onė of the lattices in ℒ moreover, ℒ is the smallest such list.

A class of totally disconnected groups consisting of partial direct products on an index set is examined. For such a group, the scale function is found, and for automorphisms arising from permutations of the index set, the tidy subgroups are characterised. When applied to the case where the index set is a finitely-generated free group and the permutation is translation by an element x of the group, the scale depends on the cyclically reduced form of x and the tidy subgroup on the element which conjugates x to its cyclically reduced form.

A non-empty subset I of a semigroup S is called an ideal if ab, ba ∈ I whenever a ∈I, b∈S. A subset R of S will be called a retract if there exsists a retraction of S onto R, that is a homomorphism of S onto R which leaves each element of R fixed. The purpose of this paper is to study semigroups in which every ideal is a retract. For convenience we shall call such semigroups retractable. Such semigroups seem to arise naturally; for example, it is easy to show that if the lattice of congruence relations on S is a complemented lattice then S is retractable.

The theory of groups acting on trees due to Bass and Serre (1969) is applied to simplify some results of Burns (1972, 1973) giving conditions under which an amalgamated free product or HNN extension has the properties that any finitely generated subgroup containing an infinite subnormal subgroup must have finite index and that the intersection of two finitely generated subgroups is finitely generated.

There is no single generalization of distributivity to semilattices. This paper investigates the class of mildly distributive semilattices, which lies between the two most commonly discussed classes in this area—weakly distributive semilattices and distributive semilattices. Particular attention is paid to describing and characterizing congruence distributive mildly distributive semilattices, in contrast to distributive semilattices, whose lattice of join partial congruences is badly behaved and which are difficult to describe.