Matrix near-rings had been defined by Meldrum and Van der Walt in 1986 and although a fair amount of results on the structure of these near-rings have been obtained since then, a satisfactory structure theory has yet to be developed for matrix d.g. near-rings. In this paper we give an alternate definition (in fact the dual definition) for matrix d.g. near-rings and develop a satisfactory structure theory for such d.g. near-rings.

Let f be a continuous real valued function defined on [−1, 1] and let En(f) denote the degree of best uniform approximation to f by algebraic polynomial of degree at most n. The supremum norm on [a, b] is denoted by ∥.∥[a, b] and the polynomial of degree n of best uniform approximation is denoted by Pn. We find a class of functions f such that there exists a fixed a ∈(−1, 1) with the following property

for some positive constants C and N independent of n. Moreover the sequence is optimal in the sense that if is replaced by then the above inequality need not hold no matter how small C > 0 is chosen.

We also find another, more general class a functions f for which

infinitely often.

We develop an isomorphism invariant for limit algebras: an extension of Power's strong algebraic order on the scale of the K0-group (Power, J. Operator Theory 27 (1992), 87–106). This invariant is complete for a certain family of limit algebras: inductive limits of digraph algebras (a.k.a. finite dimensional CSL algebras) satisfying two conditions: (1) the inclusions of the digraph algebras respect the order-preserving normalisers, and (2) the digraph algebras have chordal digraphs. The first condition is also used to show that the invariant depends only on the limit algebra and not the direct system. We give an intrinsic characterisation of the limit algebra and not the direct system. We give an intrinsic characterisation of the limit algebras satisfying both (1) and (2).

We show how the third integral homology of a group plays a role in determining whether a given group is isomorphic to an inner automorphism group. Various necessary conditions, and sufficient conditions, for the existence of such an isomorphism are obtained.

The objective of this article are sums S(M)=∑n;ψ(Mf(n/M)) where ψdenotes essentially the fractional part minus ½, f is a C4-function with fn nonvanishing, and summation is extended over an interval of order M. For S(M) an Ω-estimate and a mean-square bound is obtained. Applications to problems concerning the number of lattice points in large planar domains are discussed.

We give arithmetic characterizations which allow us to determine algorithmically when the semigroup ring associated to a simplicial affine semigroup is Cohen-Macaulay and/or Gorenstein. These characterizations are then used to provide information about presentations of this kind of semigroup and, in particular, to obtain bounds for the cardinality of their minimal presentations. Finally, we show that these bounds are reached for semigroups with maximal codimension.

In this work we tackle the Cartan determinant conjecture for finite-dimensional algebras through monoid gradings. Given an adequate ∑-grading on the left Artinian ring A, where ∑ is a monoid, we construct a generalized Cartan matrix with entries in ℤ∑, which is right invertitale whenever gl.dim A < ∞. That gives a positive answer to the conjecture when A admits a strongly adequate grading by an aperiodic commutative monoid. We then show that, even though this does not give a definite answer to the conjecture, it strictly widens the class of known graded algebras for which it is true.

We prove that the functions of the Bergman spaces Ap on tube domains may be written as Laplace transforms of functions when 1 ≤ p ≤ 2. We give in this context a generalization of the Hausdorff–Young inequality with the exact constant, and deduce from the case p = 2 the expression of the Bergman kernel as a Laplace transform.

Finite groups which are products of pairwise totally permutable subgroups are studied in this paper. The -residual, -projectors and -normalizers in such groups are obtained from the corresponding subgroups of the factor subgroups under suitable hypotheses.

We consider the half linear Sturm-Liouville problem

on the interval [0,1] subject to separated boundary conditions (which may be eigenparameter dependent at x = 1) and use Prüfer techniques to produce an oscillation theory for this problem. Both right definite (r > 0) and left definite (r of both signs) cases are discussed.

We investigate the behaviour of the classical (non-smooth) Hardy-Littlewood maximal operator in the context of Banach lattices. We are mainly concerned with end-point results for p = ∞. Naturally, the main role is played by the space BMO. We analyze the range of the maximal operator in BMOx. This turns out to depend strongly on the convexity of the Banach lattice . We apply these results to study the behaviour of the commutators associated to the maximal operator. We also consider the parallel results for the maximal fractional integral operator.

Let U be the quantized enveloping algebra associated to a simple Lie algebra g by Drinfel'd and Jimbo. Let λ be a classical fundamental weight for g, and ⋯(λ) the irreducible, finite-dimensional type 1 highest weight U-module with highest weight λ. We show that the canonical basis for ⋯(λ) (see Kashiwara [6, §0] and Lusztig [18, 14.4.12]) and the standard monomial basis (see [11, §§2.4 and 2.5]) for ⋯(λ) coincide.

The separating subspace of any Lie derivation on a semisimple Banach algebra A is contained in the centre of A.