The commutator has the following order theoretic properties: [α, β] ≦ α ∧ β, [α, β] = [β α],[α1 ∨ α2,β] = [α1, β] ∨ [α2, β] for congruences α, β ∈ Con A of an algebra A in a congruence modular variety generalising the original concept in group theory. A tolerance of a lattice L is a reflexive and symmetric sublattice of L2. We show that to every commutator [ , ] of Con A corresponds a ∧-subsemilattice of the lattice of tolerances of Con A. It can be shown that A in a congruence modular variety is nilpotent if |con A| > 2 and Con A is simple.

In this paper we study fixed points of sums of α-concave and (−α)-convex operators in γ-complete partially ordered linear spaces. As an application we obtain existence and uniqueness theorems for solutions of a certain type of nonlinear integral equation.

We prove that if X and Y are two (real) Banach spaces such that dim X ≥ 2 and dim Y ≥ 2, then the space K(X, Y) contains a convex compact subset C with dim C ≥ 2 (in the affine sense) which fails to be an intersection of balls. This improves two results of Ruess and Stegall.

It is shown that every minimal power-structure other than the singleton power-structure has at least two contenders; that is, any contender has at least one rival. The safe core of a power-structure is empty if and only if the power-structure is minimal.

This note presents a structure theorem for locally convex barrelled spaces. It is shown that, corresponding to any Hamel basis, there is a natural splitting of a barrelled space into a topological sum of two vector subspaces, one with its strongest locally convex topology. This yields a simple proof that a barrelled space has a dense infinite-codimensional vector subspace, provided that it does not have its strongest locally convex topology. Some further results and examples discuss the size of the codimension of a dense vector subspace.

It is shown that continuous self-mappings of a compact interval, chaotic in the sense of Li and Yorke, have generically, in the uniform topology, only scrambled sets which are nowhere dense and of zero Lebesgue measure.

Let { Xn, n ≥ 1 } be a sequence of independent Banach valued random variables and { an, n, ≥ 1 } a sequence of real numbers such that 0 < an ↑ ∞. It is shown that, under the assumption with some restrictions on φ, Sn/an → 0 a.s. if and only if Sn/an → 0 in probability if and only if Sn/an → 0 in L1. From this result several known strong laws of large numbers in Banach spaces are easily derived.

We study the following Dirichlet problem for the degenerate elliptic Monge-Ampère equation: Given , f ≥ 0 and , find a solution , t ≥ 2, satisfying in Ω and u = g on ∂Ω. Since f is nonnegative, we cannot apply any standard elliptic methods. In this paper, we use an iteration scheme of Nash-Moser type and a priori estimates for degenerate elliptic operators, and solve the Dirichlet problem for a certain class of f and g.

The aim of this paper is to give an inversion theorem for set-valued maps involving both some known results for functions and set-valued maps. To do this we introduce a notion of strict differentiability for set-valued maps and we use a Newton like method assuming the derivative to be surjective. Moreover we prove the pseudo-Lipschitz regularity of the inverse.

It will be proved that any mononomic variety of groups can be considered as a variety of (ρ ε) or (ρ, τ) or (ν, ε)-algebras, or as a variety of n-groupoids—which satisfy a single law, where: xyρ = x.y−1 = xτ = x−1, xyν = x−1.y−1, ε is the identity, and for certain interpretations of the n-ary operation. The problem is discussed for Ω-groups, too.

Boyle and Goodearl proved that if R is a left V-ring then R has left Krull dimension if and only if R is left Neotherian. In this paper we extend this result to arbitrary V-modules.

We show the existence, in c0 ⊗γc0, of a shrinking, subsymmetric basic sequence (zi) having no c0 subsequences. In particular, the coefficient functionals for (zi) are weakly null and the subspace fails to have the Dunford-Pettis property.

We prove a characterisation of locally s-regular manifolds using the theory of homogeneous structures.

A sequence of certain rational functions is determined where each member solves an ℓ2-minimisation problem on a “large” set of roots of unity. The sequence is compared with another sequence of L2-rational approximants. Our main result extends a result of Rivlin on Walsh type equiconvergence.

For a very large class of topological semigroups, we establish lower and upper bounds for the cardinality of the set of left invariant means on the space of left uniformly continuous functions. In certain cases we show that such a cardinality is exactly , where b is the smallest cardinality of the covering of the underlying topological semigroup by compact sets.

We characterise, by a property of roughness, the norms of a Banach space X such that the dual unit ball has no small combination of ω*-slices. Among separable Banach spaces, the existence of an equivalent norm for this new property of roughness characterises spaces which contain an isomorphic copy of ℓ1(N).

This paper gives some sufficient conditions under which upper, lower or both upper and lower quasi-continuity of multifunction in the process of semi-regularisation of a topological space are preserved. Analogous results for continuous maps are true.

An essential limitation for the method of weighting for equality constrained linear least-squares problems is the sub-optimality of the attainable convergence rate. In this paper, we propose a method, related to the method of iterated Tikhonov regularisation, that (under suitable conditions) gives rise to convergence rates which are arbitrarily near the optimal rate. As a by-product, we develop the theory of iterated Tikhonov regularisation for equations with unbounded linear operators.