A Banach space X has the asymptotic-norming property if and only if the Lebesgue-Bochner function space Lp (μ, X) has the asumptotic-norming property for p with 1 < p < ∞. It follows that a Banach space X is Hahn-Banach smooth if and only if Lp (μ, X) is Hahn-Banach smooth for p with 1 < p < ∞. We also show that for p with 1 < p < ∞, (1) if X has the compact Mazur intersection property then so does Lp(μ, X); (2) if the measure μ is not purely atomic, then the space Lp(μ, X) has the Mazur intersection property if and only if X is an Asplund space and has the Mazur intersection property.

Relations between lp-type estimates of Khamsi and a uniform version of the Kadec-Klee property are studied. Khamsi's result on normal structure is strengthened.

By using zeros of elliptic integrals we establish an upper bound for the number of limit cycles that emerge from the period annulus of the Hamiltonian XH in the system Xε = XH + ε(P, Q), where H = y2 + x4 and P, Q are polynomials in x, y, as a function of the degrees of P and Q. In particular, if (P, Q) = , with N = 2k + 1 or 2k + 2, this upper bound is k − 1.

The theory of monotone flows and operators is applied to study stable equilibria of autonomous cooperative systems and stable periodic solutions of periodic perturbations of these systems. The connection between analyticity and the property of asymptotic stability is established.

We survey the present trends in theory of universal arrows to forgetful functors from various categories of topological algebra and functional analysis to categories of topology and topological algebra. Among them are free topological groups, free locally convex spaces, free Banach-Lie algebras, and more. An accent is put on the relationship of those constructions with other areas of mathematics and their possible applications. A number of open problems is discussed; some of them belong to universal arrow theory, and other may become amenable to the methods of this theory.

Let R be a ring. A right R-module M is called an SC-module if every M-singular right R-module is continuous. The purpose of this note is to give some characterisations of SC-modules.

A ring R is called right co-FPF if every finitely generated cofaithful right R-module is a generator in mod-R. This definition can be carried over from rings to modules. We say that a finitely generated projective distinguished right R-module P is a co-FPF module (quasi-co-FPF module) if every P-finitely generated module, which finitely cogenerates P, generates σ[P] (P, respectively). We shall prove a result about the relationship between a co-FPF module and its endomorphism ring, and apply it to study some co-FPF rings.

We prove that (a) if R is a commutative coherent ring, the weak global dimension of R equals the supremum of the flat (or (FP–)injective) dimensions of the simple R-modules; (b) if R is right semi-artinian, the weak (respectively, the right) global dimension of R equals the supremum of the flat (respectively, projective) dimensions of the simple right R-modules; (c) if R is right semi-artinian and right coherent, the weak global dimension of R equals the supremum of the FP-injective dimensions of the simple right R-modules.

The quantum double construction is applied to the group algebra of a finite group. Such algebras are shown to be semi-simple and a complete theory of characters is developed. The irreducible matrix representations are classified and applied to the explicit construction of R-matrices: this affords solutions to the Yang-Baxter equation associated with certain induced representations of a finite group. These results are applied in the second paper of the series to construct unitary representations of the Braid group and corresponding link polynomials.

Let X be a completely regular space, E a Banach space, Cb(X, E) the space of all continuous, bounded and E-valued functions defined on X, M(X, L(E, F)) the space of all L(E, F)-valued measures defined on the algebra generated by zero subsets of X. Weakly compact and β0-continuous operators defined from Cb(X, E) into a Banach space F are represented by integrals with respect to L(E, F)-valued measures. The strict Dunford-Pettis and the Dunford-Pettis properties are established on (Cb(X, E), βi), where βi denotes one of the strict topologies β0, β or β1, when E is a Schur space; the same properties are established on (Cb(X, E), β0), when E is an AM-space or an AL-space.

Let Fq be the finite field of order q, an odd number, Q a non-degenerate quadratic form on , O(n, Q) the orthogonal group defined by Q. Regard O(n, Q) as a linear group of Fq -automorphisms acting linearly on the rational function field Fq(x1, …, xn). We shall prove that the invariant subfield Fq(x1,…, xn)O(n, Q) is a purely transcendental extension over Fq for n = 5 by giving a set of generators for it.

An elementary proof is given of the fact that an n–tuple A = (A1, …, An) of self-adjoint matrices in a 2-dimensional Hilbert space consists of mutually commuting matrices Aj, 1 ≤ j ≤ n, if and only if γ(A) is non-empty. Here γ(A) ⊆ ℝn is the joint spectrum of A (in the sense of McIntosh and Pryde) consisting of those points β ∈ ℝn for which the matrix is not invertible.

The general structure of Taylor expansions of functions of solutions of continuous Stieltjes differential equations is established. A compact formalism involving hierarchical sets of multi-indices and their associated remainder sets is used. The corresponding multiple Riemann-Stieltjes integrals of time and of the driving functions in the Stieltjes terms of the differential equations, necessarily of bounded variation and continuous here, appear in the expansions and their remainders.

In his paper on knot cobordism groups in codimension 2, Levine develops conditions for a knotted Sn in Sn+2 to bound a disc in Bn+3 In this paper some of his methods are extended to introduce a necessary condition for a classical link in S3 to bound a surface of specified genus in B4. In particular, this answers a question of Zeemann's about some links related to the ‘Mazur link’.

Let X be a tree and f be a continuous map from X into itself. Denote by P(f) and R(f) the set of periodic points and the set of recurrent points of f respectively. We show in this note that the centre is and the depth of the centre is at most 3. Furthermore we have .