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.     

Modules of piecewise polynomials and their freeness

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

A novel photoaddition of 2-phenylbenzimidazole

Irradiation of 2-phenylbenzimidazole ( 3 ) in the presence of methyl acrylate (20 equivalents) using a medium pressure mercury lamp gives methyl 2-(2-phenylbenzimidazol-1-yl)propionate ( 4 ) (75%) whose structure has been confirmed by an X-ray analysis. A similar reaction using acrylonitrile gave only a low yield (1%) of the corresponding propionitrile derivative 5 . A mechanism involving photoexcitation of 2-phenylbenzimidazole ( 3 ) followed by sequential single electron transfer, proton transfer and radical coupling is proposed to account for the formation of the novel photoadducts 4 and 5 .     

Scratch and Abrasion Resistant Coatings on Plastic Lenses—State of the Art,Current Developments and Perspectives

The manufacturing of ophthalmic lenses is one of the most important markets worldwide and, therefore, strong research efforts are undertaken to continuously improve the quality of the products, either silicate glasses or organic polymer lenses. Hybrid sol-gel based materials play a major role in this highly competitive field and have contributed significantly to the commercial success of the organic base materials. Recent developments concern fast curing and patternable coatings that might soon become part of this business. The compatibility of hybrid sol-gel materials either with organic dyes or with inorganic vacuum borne coatings offers further possibilities to develop highly sophisticated lenses meeting not only customer needs like perfect corrective function, high optical quality and protection, but also high durability as well as cosmetic and decorative aspects. An overview and a few recent developments are outlined below.     

Functionalizable polycyclic aromatics through oxidative cyclization of pendant thiophenes

We present a general strategy for obtaining large sulfur-containing polycyclic aromatics from thienyl precursors through iron(III) chloride mediated oxidative cyclizations. By placing thienyl moieties in close proximity to adjacent arenes, we have directed the oxidized intermediates into controlled cyclization pathways, effectively suppressing polymer formation. Utilizing these cyclized compounds and their thienyl precursors, we have studied cyclization/polymerization pathways of polymers such as poly(2). The unsubstituted positions alpha to the sulfur atoms within these aromatic cores allowed for efficient halogenation and further functionalization. As a demonstration, we prepared a series of arylene-ethynylene polymers with varying degrees of chromophore aromatization and used them to probe the effects of synthetically imposed rigidity on polymer photophysical behavior. The symmetries and effective conjugation pathways within the monomers play a key role in determining photophysical properties. We observed that rigid, aromatized chromophores generally led to increased excited-state lifetimes by decreasing radiative rates of fluorescence decay.     

Thermal effects of contamination,adsorbed water and annealing on the DTA of powdered quartz

During the grinding of quartz, water is adsorbed from the atmosphere, but this water gave no detectable thermal effect on DTA because it was evolved uniformly up to 1000°. However, in conjunction with an iron contaminant, the adsorbed water was involved in an oxidation reaction causing noticeable thermal effects for quartz powdered in a vibration mill constructed of steel parts. When powdered so finely that the - inversion peak at 573° had disappeared, annealing caused a partial redevelopment of the peak but recrystallization of disrupted quartz was too small an effect to give any detectable exothermic peak on DTA.

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Zusammenfassung Während des Vermahlens von Quartz wird Wasser aus der Atmosphäre adsorbiert, jedoch ergab dieses Wasser keinen nachweisbaren thermischen Effekt in der DTA, da es bis zu 1000° gleichmässig freigesetzt wird. Im Zusammenhang mit einer Eisen-Verunreinigung wurde jedoch das adsorbierte Wasser in eine Oxidationsreaktion einbezogen, welche beim in einer aus Stahlteilen gefertigten Vibrationsmühle zerpulvertem Quartz nachweisbare thermische Effekte verursachten. Wenn so fein pulverisiert wurde, daß der --Inversions-Peak bei 573° verschwunden war, verursachte eine Wärmebehandlung eine teilweise Wiederentwicklung des Peaks, jedoch war die Rekristallisation des zerstörten Quartzes zu geringfügig um einen nachweisbaren exothermen Peak in der DTA-Kurve zu ergeben.

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Résumé Lors du broyage du quartz, il se produit une adsorption de l'eau atmosphérique mais cette eau ne donne pas d'effet thermique décelable par ATD car elle se dégage régulièrement jusqu'à 1000°. Cependant, en présence de fer comme contaminant, l'eau adsorbée intervient dans une réaction d'oxydation qui donne des effets thermiques perceptibles dans le cas de quartz pulvérisé dans un vibrobroyeur en acier. Si le quartz est en poudre assez fine pour que le pic d'inversion - à 573° disparaisse, le recuit provoque la réapparition partielle du pic. La recristallisation du quartz est cependant trop faible pour donner un effet exothermique décelable par ATD.

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, , 1000°. , , . - 573° , , , - .

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We are grateful to the Science Research Council for supporting this work.     

Counterion Exchange Selectivity in Detergent–Polymer Aggregates

In aqueous solution, the interaction between sodium dodecyl sulfate (SDS) and poly(ethylene glycol) (PEG) results in the formation of small aggregates or clusters of SDS attached to the PEG polymer chain. Selectivity coefficients for exchange of two monovalent (N-methyl-4-cyanopyridinium cation and Tl⁺) and two divalent (methylviologen cation and Cu²⁺) counterions at the surface of SDS–PEG clusters, determined employing photophysical techniques, are similar, but not identical, to those for exchange at the surface of SDS micelles in the absence of PEG. The principal factor affecting ion exchange selectivity in SDS–PEG clusters does not appear to be aggregate size or surface charge density but rather the presence of poly(oxyethylene) subunits at the aggregate surface.