Typical curvature behaviour of bodies of constant width

It is known that an n-dimensional convex body, which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical n-dimensional convex body of constant width 1 (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to 1. (In contrast, note that a ball of width 1 has radius 1/2, hence all its curvatures are equal to 2.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.     

A nonstandard mixed finite element family

We show that standard mixed finite element methods, of second or higher degree, for second order elliptic equations can be modified by imposing additional continuity conditions for the flux, which reduces the dimension of the space. This reduced space still gives a stable method with an optimal order of convergence. We recall our postprocessing method and the a posteriori error estimator based on this.     

Non-commutative Markov chains and multi-analytic operators

We study a model of repeated interaction between quantum systems which can be thought of as a non-commutative Markov chain. It is shown that there exists an outgoing Cuntz scattering system associated to this model which induces an input-output formalism with a transfer function corresponding to a multi-analytic operator, in the sense of multivariate operator theory. Finally we show that observability for this system is closely related to the scattering theory of non-commutative Markov chains.     

Swedish employment offices: A new model for evaluating effectiveness

We measure how well Swedish employment offices perform in delivering the services required of them by the Swedish government. In contrast to earlier studies we use a dynamic efficiency framework, which allows us to better model the intertemporal nature of these services, explicitly allowing for placements of intermediate nature across periods. Rather than using second stage analysis to assess the effects of varying local labor market conditions and differences in client characteristics on performance, we include a measure of the office’s expected work load directly in the model. This measure, derived from duration analysis, is designed to capture the variation across offices in resources needed before an average individual can obtain employment. It is estimated from the characteristics of all unemployed individual and local labor market conditions.     

Approximation of the dual problem for error estimation in inelastic problems

In numerical simulations with the finite element method the dependency on the mesh – and for time-dependent problems on the time discretization – arises. Adaptive refinements in space (and time) based on goal-oriented error estimation [1] become more and more popular for finite element analyses to balance computational effort and accuracy of the solution. The introduction of a goal quantity of interest defines a dual problem which has to be solved to estimate the error with respect to it. Often such procedures are based on a space-time Galerkin framework for instationary problems [2]. Discretization results in systems of equations in which the unknowns are nodal values. Contrary, in current finite element implementations for path-dependent problems some quantities storing information about the path-dependence are located at the integration points of the finite elements [3], e.g. plastic strains etc. In this contribution we propose an approach – similar to [4] for sensitivity analysis – for the approximation of the dual problem which mainly maintains the structure of current finite element implementations for path-dependent problems. Here, the dual problem is introduced after discretization. A numerical example illustrates the approach. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Small faces in stationary Poisson hyperplane tessellations

We consider the tessellation induced by a stationary Poisson hyperplane process in d‐dimensional Euclidean space. Under a suitable assumption on the directional distribution, and measuring the k‐faces of the tessellation by a suitable size functional, we determine a limit distribution for the shape of the typical k‐face, under the condition of given size and this tending to zero. The limit distribution is concentrated on simplices. This extends a result of Gilles Bonnet.     

Research on the reasoning, teaching and learning of probability and uncertainty

In this editorial, we set out the aims in the call to publish papers on informal statistical inference, randomness, modelling and risk. We discuss how the papers published in this issue have responded to those aims. In particular, we note how the nine papers contribute to some of the major debates in mathematics and statistics education, often taking contrasting positions. Such debates range across: (1) whether knowledge is fractured or takes the form of mental models; (2) heuristic or intuitive thinking versus operational thinking as for example in dual process theory; (3) the role of different epistemic resources, such as perceptions, modelling, imagery, in the development of probabilistic reasoning; (4) how design and situation impact upon probabilistic learning.     

On the role of arbitrary order Bessel functions in higher dimensional Dirac type equations

This paper exhibits an interesting relationship between arbitrary order Bessel functions and Dirac type equations. Let be the Euclidean Dirac operator in the n-dimensional flat space the radial symmetric Euler operator and α and λ be arbitrary non-zero complex parameters. The goal of this paper is to describe explicitly the structure of the solutions to the PDE system in terms of arbitrary complex order Bessel functions and homogeneous monogenic polynomials. Received: 27 October 2005     

An Efficient Exact Algorithm for Constraint Bipartite Vertex Cover

The constraint bipartite vertex cover problem (CBVC for short) is as follows: given a bipartite graph G with n vertices and two positive integers k₁, k₂, is there a vertex cover taking at most k₁ vertices from one and at most k₂ vertices from the other vertex set of G? CBVC is NP-complete. It formalizes the spare allocation problem for reconfigurable arrays, an important problem from VLSI manufacturing. We provide a nontrivial so-called fixed parameter algorithm for CBVC, running in O(1.3999^{k₁ + k₂} + (k₁ + k₂)n) time. Our algorithm is efficient and practical for small values of k₁ and k₂, as occurring in applications. The analysis of the search tree is based on a novel bonus point system: after the processing of the search tree (which takes time exponential in k), a polynomial-time final analysis follows. Parts of the computation that would be normally done within the search-tree phase can be postponed; nevertheless, knowledge about the size of those parts can be used to reduce the length of the search paths (and hence the depth of the search tree as a whole) by a sort of bonus points.