Parabolic subgroups in FC-type Artin groups

Parabolic subgroups are the building blocks of Artin groups. This paper extends previous results of Cumplido, Gebhardt, Gonzales-Meneses and Wiest, known only for parabolic subgroups of finite type Artin groups, to parabolic subgroups of FC-type Artin groups. We show that the class of finite type parabolic subgroups is closed under intersection. We also study an analog of the curve complex for mapping class group constructed by Cumplido et al. using parabolic subgroups. We extend the construction of this complex, called the complex of parabolic subgroups, to FC-type Artin groups. We show that this simplicial complex is, in most cases, infinite diameter and conjecture that it is δ-hyperbolic.     

Patterns—a fundamental idea of mathematical thinking and learning

Taking advantage of patterns is typical of our everyday experience as well as our mathematical thinking and learning. For example a working day or a morning at school displays a certain structure, which can be described in terms of patterns. On the one hand regular structures give us the feeling of permanence and enable us to make predictions. On the other hand they also provide a chance to be creative and to vary common procedures. School students usually encounter patterns in math classes either as number patterns or geometric patterns. There are also patterns that teachers can find in analyzing the errors students make during their calculations (error patterns) as well as patterns that are inherent to mathematical problems. One could even go so far as to say that identifying and describing patterns is elementary for mathematics (cf. Devlin 2003). Practising good interacting with patterns supports not only the active learning of mathematics but also a deeper understanding of the world in general. Patterns can be explored, identified, extended, reproduced, compared, varied, represented, described and created. This paper provides some examples of pattern utilization and detailed analyses thereof. These ideas serve as “hooks” to encourage the good use of patterns to facilitate active learning processes in mathematics.     

Measurements of edge-uncolorability

Cubic bridgeless graphs with chromatic index four are called uncolorable. We introduce parameters measuring the uncolorability of those graphs and relate them to each other. For k=2,3, let c_k be the maximum size of a k-colorable subgraph of a cubic graph G=(V,E). We consider r₃=|E|−c₃ and . We show that on one side r₃ and r₂ bound each other, but on the other side that the difference between them can be arbitrarily large. We also compare them to the oddness ω of G, the smallest possible number of odd circuits in a 2-factor of G. We construct cyclically 5-edge connected cubic graphs where r₃ and ω are arbitrarily far apart, and show that for each 1c<2 there is a cubic graph such that ωcr₃. For k=2,3, let ζ_k denote the largest fraction of edges that can be k-colored. We give best possible bounds for these parameters, and relate them to each other.     

Modules of piecewise polynomials and their freeness

Partially supported by NSF Grant DMS-9004123 and ARO through MSI Cornell (DAAG 29-85-C-0018)     

The coexistence of γ(γ′) and θ alumina observed by STM and LEED on top of oxide layer grown on CoAl(1 0 0)

We report the coexistence of γ(γ′) and θ alumina grown on the CoAl(1 0 0) surface. Both phases were observed after annealing of alumina at 1150 K by means of scanning tunnelling microscopy (STM). The crystal structures of γ(γ′) and θ alumina were indexed based on low energy electron diffraction (LEED) and STM experimental data.     

Strategies for the development of a voltammetric sensor for the determination of hydrogen peroxide at high concentrations

Different strategies are presented for the development of a voltammetric sensor for in-line measurement and monitoring of H₂O₂ in high concentrations (up to 30 g/l) as used in textile bleaching systems. When using macroelectrodes, signal saturation is observed even at medium concentrations. Some improvement can be achieved by hydrodynamic electrodes like ring-disk electrodes. Satisfactory results up to the desired level in practical solutions (possessing only limited conductivity) are obtained with Pt-microelectrodes.