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.

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.

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.

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.

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.

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.

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.

Recently Guerrero and the first author (Diaz Carrillo) proved an anologue to Daniell's extension process which works for arbitrary nonnegative linear functionals, without any continuity conditions. With the aid of Schäfke's local integral metrics we generalise this extension process and prove convergence theorems using a suitable local mean convergence, which can be traced back to Loomis.

In the present paper a theorem on the spectral radius of the sum of linear operators is established. The application of this theorem to a functional differential equation of neutral type is also given.

We prove some existence theorems for solutions of a semilinear two point boundary value problem at resonance in which the nonlinear terms can have arbitrary polynomial growth in one of the directions ∞ or –∞, and are bounded in the other. The results are based on degree theoretic arguments and the Borsuk odd mapping theorem.

Let G be a locally compact Vilenkin group. We study multipliers which satisfy a generalised Hörmander condition from power-weighted Hardy space (G) to (G) with 0 < p ≤ q < ∞, 0 < p ≤ 1, −1 < β, β′.

In this note, we establish some new generalisations of an H-KKM type theorem which unify and generalise the corresponding results of Horvath, Bardaro-Ceppitelli, Tarafdar, Shioji, Park and others. As applications of our H-KKM type principle, we obtain some new generalisations of the Ky Fan type geometric properties of. H-spaces, minimax inequalities and coincidence theorems in Horvath's abstract setting.

This paper presents a construction of a digital convex 2k–gon (for a given natural number k), which can be inscribed into a square grid of the minimal possible size F(2k). An explicit expression for the function F(2k) is also given.

Consider the following neutral delay differential equation

where p ∈ R, τ ∈ (0, ∞), δ ∈ R+ = (0, ∞) and Q ∈ (C([t0, ∞), R). We show that if

then Equation (*)has a nonoscillatory solution when p ≠ –1. We also deal in detail with a conjecture of Chuanxi, Kulenovic and Ladas, and Györi and Ladas.

A sharp Landau-Kolmogorov inequality on a finite interval is proved. The proof yields the known Landau-Kolmogorov inequality on R as a limiting case, and thus provides a new proof for that result.

A group G ≤ GLK(V) is called K-primitive if there exists no non-trivial decomposition of V into a sum of K-spaces which is stabilised by G. We show that if V is a finite vector space and G a K-primitive subgroup of GLK(V) whose order is coprime to |V|, we can bound the order of G by |V|log2(|V|) apart from one exception. Later we use this result to obtain some lower bounds on the number of p–singular elements in terms of the group order and the minimal representation degree.