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.

In this paper we obtain some permanence properties of a class of locally convex spaces located between quasi-suprabarrelled spaces and quasi-totally barrelled spaces, for which a closed graph theorem is given.

Some extensions to the non reflexive case of continuity results for the Legendre-Fenchel transform are presented following an approach due to J.-L. Joly. We compare the topology introduced by J.-L. Joly and the Mosco-Beer topology introduced by G. Beer. In particular, in the case of the space of closed proper convex functions defined on the dual of a normed vector space they coincide.

Recently the authors extended to positive semi-definite matrices converses of matrix inequalities corresponding to ratios and differences of means and of Hölder's inequality. Here, further generalisations are given to the case of several vectors with appropriate weights.

Let sln denote the set of all n × n complex matrices with trace zero. Suppose that ø: sln → sln is a bijective linear mapping preserving square-zero matrices. Then ø is either of the form ø(A) = cUAU-1 or ø(A) = cUAtU-1 where U is an invertible n × n matrix and c is a nonzero complex number. The same result holds if we assume that ø is a linear mapping preserving square-zero matrices in both directions. Applying this result we prove that a linear mapping ø defined on the algebra of all n × n matrices is an automorphism if and only if it preserves zero products in both directions and satisfies ø(I) = I. An extension of this last result to the infinite-dimensional case is considered.

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.

In this paper we present a characterization of baric algebras. In particular we study those in which the identity x3 = w(x)x2 holds. Moreover, for every field K, we prove that this identity guarantees that the annihilator of Ker (w) is an ideal in A and we give example of a subspace of Ker (w) whose annihilator is not an ideal.

We characterise in terms of free topological groups those Banach-Lie algebras with finite-dimensional centre coming from Lie groups.

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.

Let k be an imaginary quadratic number field and let k1 be the 2-Hilbert class field of k. If Ck,2, the 2-Sylow subgroup of the ideal class group of k, is elementary and |Ck,2|≥ 8, we show that Ck1,2 is not cyclic. If Ck,2 is isomorphic to Z/2Z × Z/4Z and Ck1,2 is elementary we show that k has finite 2-class field tower of length at most 2.

It is proved in this paper that if a vector field is both C structurally stable and topologically stable, then it satisfies Axiom A.

We show that a bounded linear operator on a dual Banach space X may be perturbed by a compact operator of arbitrarily small norm to yield an operator which attains its numerical radius provided the weak star and norm topologies coincide on the unit sphere of X.

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.

In this note we present a new result which is equivalent to the celebrated Ekeland's variational principle, and a set of implications which includes a new non-convex minimisation principle due to Takahashi.

We give sufficient and necessary conditions for simple modules of the quantum group or the quantum enveloping algebra Uq(g) to have weight space decompositions, where g is a semisimple Lie algebra and q is a nonzero complex number. We show that

(i) if q is a root of unity, any simple module of Uq(g) is finite dimensional, and hence is a weight module;

(ii) if q is generic, that is, not a root of unity, then there are simple modules of Uq(g) which do not have weight space decompositions.

Also the group of units of Uq(g) is found.

In this paper we introduce the concept of quasimonotone maps and prove that a lower semicontinuous function on an infinite dimensional space is quasiconvex if and only if its generalised subdifferential or its directional derivative is quasimonotone.

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 description is obtained of the free semifields with both fundamental operations commutative.

We show that the normal version (R, R.) of the Banach simplicial cohomology of operator algebras can be expressed in terms of the functor Ext on the category of Banach R-bimodules. As an application, we prove that (R, R*) = Hn(R, R*) and thus the latter space vanishes for some von Heumann algebras for positive τ.