Symmetric inverse monoids of objects in arbitrary categories are studied. Necessary and sufficient conditions are given for such monoids to be E-unitary or else form (complete) inverse algebras. Particular attention is given to symmetric inverse monoids of objects in free categories.

Given a finite irredundant system of equations to be solved over the free group, one has four non-negative integers: the number of equations, the number of indeterminants, the rank of the system and the Abelian rank of the system. We show which four-tuples can actually occur.

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.

The Cartan determinant conjecture for left artinian rings was verified for quasihereditary rings showing detC(R) = detC(R/I), where I is a protective ideal generated by a primitive idempotent. This article identifies classes of rings generalizing the quasihereditary ones, first by relaxing the “projective” condition on heredity ideals. These rings, called left k-hereditary are all of finite global dimension. Next a class of rings is defined which includes left serial rings of finite global dimension, quasihereditary and left 1-hereditary rings, but also rings of infinite global dimension. For such rings, the Cartan determinant conjecture is true, as is its converse. This is shown by matrix reduction. Examples compare and contrast these rings with other known families and a recipe is given for constructing them.

We study two-parameter nonlinear Sturm-Liouville problems. We shall establish the continuity of the variational eigencurve λ(μ) and asymptotic formulas of λ(μ)as μ → ∞, μ → π2

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.

This paper contains some basic relations between Ganea strong category and Lusternik Schnirelmann category in proper homotopy theory. We focus our interest on the case of category 2 in order to show that ℚn is the unique open n-manifold with proper Lusternik-Schnirelmann category 2 (n ≠ 3).

In this paper, radial basis functions that are compactly supported and give rise to positive definite interpolation matrices for scattered data are discussed. They are related to the well-known thin plate spline radial functions which are highly useful in applications for gridfree approximation methods. Also, encouraging approximation results for the compactly supported radial functions are shown.

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).

Let M be an essentially finitely generated injective (or, more generally, quasi-continuous) module. It is shown that if M satisfies a mild uniqueness condition on essential closures of certain submodules, then the existence of an infinite independent set of submodules of M implies the existence of a larger independent set on some quotient of M modulo a directed union of direct summands. This provides new characterisations of injective (or quasi-continuous) modules of finite Goldie dimension. These results are then applied to the study of indecomposable decompositions of quasi-continuous modules and nonsingular CS modules.

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 space of Colombeau generalized functions is used as a frame for the study of hypoellipticity of a family of differential operators whose coefficients depend on a small parameter ε.

There are given necessary and sufficient conditions for the hypoellipticity of a family of differential operators with constant coefficients which depend on ε and behave like powers of ε as ε→0. The solutions of such family of equations should also satisfy the power order estimate with respect to ε.

This paper deals with Schauder decompositions of Banach spaces X2π of 2π-periodic functions by projection operators Pk onto the subspaces Vk, k = 0,1,…, which form a multiresolution of X2π,. The results unify the study of wavelet decompositions by orthogonal projections in the Hilbert space on one hand and by interpolatory projections in the Banach space C2π on the other. The approach, using “orthogonal splines”, is constructive and leads to the construction of a Schauder decomposition of X2π and a biorthogonal system for X2π, and its dual X2π. Decomposition and reconstruction algorithms are derived from the construction.

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 characterize tame automorphisms of finitely generated abelian groups via a simple determinant condition.

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.

Let R = ⊕n∈zRn be a ℤ-graded commutative Noetherian ring and let M be a ℤ-graded R-module. S. Goto and K. Watanabe introduced the graded Cousin complex *C(M)* for M, a complex of graded R-modules. Also one can ignore the grading on M and construct the Cousin complex C(M)* for M, discussed in earlier papers by the second author. The main results in this paper are that *C(M)* can be considered as a subcomplex of C(M)* and that the resulting quotient complex is always exact. This sheds new light on the known facts that, when M is non-zero and finitely generated, C(M)* is exact if and only if *C(M)* is (and this is the case precisely when M is Cohen-Macaulay).

Let C*(F∞) be the full C*-algebra associated to the free group of countably many generators and SnC*(F∞) be the class of all n-dimensional operator subspaces of C*(F∞). In this paper, we study some stability properties of SnC*(F∞). More precisely, we will prove that for any E0, E1 in SnC*(F∞), the Haagerup tensor product E0⊗hE1 and the operator space obtained by complex interpolation Eθ are (1 + ∈)-contained in C*(F∞) for arbitrary ∈>0. On the other hand, we will show an extension property for WEPC*-algebras.

We prove that any infinite Coxeter group has a finite index subgroup which surjects ℤ.

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.