A concept of convergence in geodesic spaces

A CAT(0) space is a geodesic space for which each geodesic triangle is at least as ‘thin’ as its comparison triangle in the Euclidean plane. A notion of convergence introduced independently several years ago by Lim and Kuczumow is shown in CAT(0) spaces to be very similar to the usual weak convergence in Banach spaces. In particular many Banach space results involving weak convergence have precise analogues in this setting. At the same time, many questions remain open.     

Smoothed hat functions in subdivision

Following the general idea of smoothing through convolution, we discuss schemes for smoothing hat functions on the three-directional and the four-directional mesh. The smoothing masks–which in the four-directional case are matrix masks–have the same support as the refinement mask for the hat functions. Several smoothing schemes are presented, and in some cases the smoothness properties of the smoothed functions are analysed. In particular, the results for the four-directional case follow new and original lines of research.     

A second-order difference scheme for a parameterized singular perturbation problem

In this paper we consider a singularly perturbed quasilinear boundary value problem depending on a parameter. The problem is discretized using a hybrid difference scheme on Shishkin-type meshes. We show that the scheme is second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results.     

Reflexive ideals in Iwasawa algebras

Let G be a torsionfree compact p-adic analytic group. We give sufficient conditions on p and G which ensure that the Iwasawa algebra ΩG of G has no non-trivial two-sided reflexive ideals. Consequently, these conditions imply that every non-zero normal element in ΩG is a unit. We show that these conditions hold in the case when G is an open subgroup of SL₂(Zp) and p is arbitrary. Using a previous result of the first author, we show that there are only two prime ideals in ΩG when G is a congruence subgroup of SL₂(Zp): the zero ideal and the unique maximal ideal. These statements partially answer some questions asked by the first author and Brown.     

Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics

We propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the Ricci flow, as well as another flow, in Kähler geometry. We introduce and study dynamical systems related to the Ricci operator on the space of Kähler metrics that arise as discretizations of these flows. We pose some problems regarding their dynamics. We point out a number of applications to well-studied objects in Kähler and conformal geometry such as constant scalar curvature metrics, Kähler-Ricci solitons, Nadel-type multiplier ideal sheaves, balanced metrics, the Moser-Trudinger-Onofri inequality, energy functionals and the geometry and structure of the space of Kähler metrics. E.g., we obtain a new sharp inequality strengthening the classical Moser-Trudinger-Onofri inequality on the two-sphere.     

On the connected component of compact composition operators on the Hardy space

We show that there exist non-compact composition operators in the connected component of the compact ones on the classical Hardy space H². This answers a question posed by Shapiro and Sundberg in 1990. We also establish an improved version of a theorem of MacCluer, giving a lower bound for the essential norm of a difference of composition operators in terms of the angular derivatives of their symbols. As a main tool we use Aleksandrov-Clark measures.     

Higher Hopf formulae for homology via Galois Theory

We use Janelidze's Categorical Galois Theory to extend Brown and Ellis's higher Hopf formulae for homology of groups to arbitrary semi-abelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the Barr-Beck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A is the category Gp of all groups and B is the category Ab of all abelian groups, this yields a new proof for Brown and Ellis's formulae. We also give explicit formulae in the cases of groups vs. k-nilpotent groups, groups vs. k-solvable groups and precrossed modules vs. crossed modules.     

A note on some chain conditions

We answer a question raised by Othman Echi: Is an E ₁ (resp., a C ₁) ring an E (resp., a C) ring? We construct a C ₁ (thus E ₁) ring which is not an E ₂ (thus not a C ₂) ring. Received: 11 June 2007     

A simplicial branch and duality bound algorithm for the sum of convex–convex ratios problem

This article presents a simplicial branch and duality bound algorithm for globally solving the sum of convex–convex ratios problem with nonconvex feasible region. To our knowledge, little progress has been made for globally solving this problem so far. The algorithm uses a branch and bound scheme where the Lagrange duality theory is used to obtain the lower bounds. As a result, the lower-bounding subproblems during the algorithm search are all ordinary linear programs that can be solved very efficiently. It has been proved that the algorithm possesses global convergence. Finally, the numerical experiments are given to show the feasibility of the proposed algorithm.