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.     

Thermal analysis of glasses for proecological applications

Thermal behaviour and biochemical activity of potassium-silicate-phosphate glasses modified by addition of Ca, Mg, acting as ecological fertilisers of controlled release rate of the nutrients for plants were studied. It has been found that the biochemical activity of silicate-phosphate glasses depends on the nature and the number of components forming the glassy framework and is related to the thermal stability of glass demonstrated by the formation of new compounds during the process of crystallisation. It is proved that these seemingly different properties are determined by the same parameters which are the strengths bonds of glass network-formers and modifiers as well as their chemical affinity. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Sulfur adsorption on Fe(1 1 0): a DFT study

The adsorption of atomic S on the Fe(1 1 0) surface is examined using density functional theory (DFT). Three different adsorption sites are considered, including the atop, hollow and bridge sites and the S is adsorbed at a quarter monolayer coverage in a p(2 × 2) arrangement. The hollow site is found to be the most stable, followed by the bridge and atop sites. At all three sites, S adsorption results in relatively minor surface reconstruction, with the most significant being that for the hollow site, with lateral displacements of 0.09 Å. Comparisons between S-adsorbed and pure Fe surfaces revealed reductions in the magnetic moments of surface-layer Fe atoms in the vicinity of the S. At the hollow site, the presence of S causes an increase in the surface Fe d-orbital density of states between 4 and 5 eV. However, S adsorption has no significant effect on the structure and magnetic properties of the lower substrate layers.     

Novel antibiotics: second generation macrocyclic peptides designed to trap Holliday junctions

Described are the syntheses of 15 macrocyclic peptides designed to trap Holliday junctions (HJs) in bacteria during site-specific and homologous recombination. This leads to inhibiting bacterial growth. These second generation macrocycles were based on the C-2 symmetrical HJ. They were synthesized using a strategy that permits elucidation of the amino acid role in binding HJs. The syntheses of these macrocycles are an important step in the development of a new class of antibiotics.