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.

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.

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.

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.

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.

In this paper we consider a nonlinear periodic parabolic boundary value problem with a discontinuous nonmonotone nonlinearity. Using a lifting result for operators of type (S+), a general surjectivity theorem for operators of monotone type and an auxiliary problem defined by truncation and penalization we prove the existence of a solution in the order interval formed by an upper and lower solution. Moreover we show that the set of all such solutions is compact in Lp(T, (Z)).

The spectrum σ(Tφ) of a Toeplitz operator Tφ on the open unit disc D for a unimodular symbol φ is studied and many sufficient conditions for σ(Tφ)⊆∂D or σ(Tφ) = are given. In particular if φ is a unimodular function in H∞ + C, then σ(Tφ)⊆∂D or σ(Tφ) =

Let Γ be a torsion-free geometrically finite Kleinian group. In this paper, we investigate which systems of loxodromic conjugacy classes of Γ can be simultaneously made parabolic in a group on the boundary of the quasi-conformal deformation space of Γ. We shall prove that for this, it is sufficient that the classes of the system are represented by disjoint primitive simple closed curves on the ideal boundary of H3/Γ.

Provided that the weight function ω satisfies certain submultiplicative and decay conditions, the discrete convolution algebra ℓ1(ω) becomes a commutative radical Banach algebra with identity adjoined. There are obvious closed ideals in ℓ1(ω) and these are denoted standard ideals. Earlier results of Thomas, strengthened by Yakubovich and Domar, showed that if the weight ω is star-shaped then all closed ideals are standard. Consequently, the closed ideal generated by any element f in ℓ1(ω) must be standard.

The requirement that ω be star-shaped (essentially that ω(n)1/n must decrease to zero) is somewhat restrictive in that no local maxima of ω(n)1/n are allowed. We generalize this previous result to apply to the larger class of ε-star shaped weights (0 < ε ≤ 1) which allow such local maxima. If f is a non-zero element on ℓ1(ω) we let the integer α(f) = k0 denote the index of its first non-zero term. We introduce the concept of an ε-peak point for k0. If ε = 1 then ω is star-shaped in the usual sense and there are an infinite number of 1-peak points for any k0. Although this latter fact may fail if 0 < ε < 1, if ω(n)1/n tends to zero sufficiently quickly (dependent on k0 and ε) there will always be an infinite number of ε-peak points for k0.

Our main result is that if ω is an ε-star shaped weight, if f is an non-zero element of ℓ1(ω), if α(f) = k0, and if the number of ε-peak points for k0 is infinite, then the closed ideal generated by f is standard.

For N any member of a large class of finite abelian right centralizer near-rings, the subring of the ring End(N) of endomorphisms of (N, +) generated by the set of right multiplication maps on N is explicitly described as a generalized blocked triangular matrix ring, which in some cases turns out to be a structural matrix ring.

Let Gk, n, be the Grassmann manifold consisting in all non-oriented k-dimensional vector subspaces of the space Rk+n. In this paper we will show that any differentiable mapping f: Gk, n → Rm, has infinitely many critical points for suitable choices of the numbers m, n, k.