A categorification of non-crossing partitions

We present a categorification of the non-crossing partitions given by crystallographic Coxeter groups. This involves a category of certain bilinear lattices, which are essentially determined by a symmetrisable generalised Cartan matrix together with a particular choice of a Coxeter element. Examples arise from Grothendieck groups of hereditary artin algebras.Comment: 34 pages. Substantially revised and final version, accepted for publication in Journal of the European Mathematical Societ

Crosshair challenge : recapturing sustainable competitive advantages through restructuring

Includes bibliographical references.The purpose of this report is to examine the potential to reorganize a company was started in 2006, Crosshair Challenge (CC), in a way that will provide sustainable competitive advantage. The company previously hosted paintball events at local fields in the area and attracted enthusiasts from the surrounding states. This feasibility analysis will look at opportunities to restructure the current operations into an industry-unique league formatting. In the beginning, the competitive advantage CC had was the scheduling of the league as well as the indoor operations. Since the first season, several indoor paintball facilities have closed down leaving the winter season inoperable and destroying CC's competitive advantages. In order to stay in business, CC moved its operations to outdoor fields. This move resulted in CC being trapped into a highly competitive season, staggering capital requirements to reestablish desired competitive advantages, several competitors with economies of scale, and because of a loss of all competitive advantages; several failed instances of operations. I will be reviewing several critical topics that will influence my decision and determine the scope and breadth of my pursuant action. These topics include operational feasibility, market favorability, and a financial outlook of the proposed operations.B.S. (Bachelor of Science

Turing conditions for pattern forming systems on evolving manifolds

The study of pattern-forming instabilities in reaction-diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction-diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace-Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach

Amplitude death criteria for coupled complex Ginzburg-Landau systems

Amplitude death, which occurs in a system when one or more macroscopic wavefunctions collapse to zero, has been observed in mutually coupled solid-state lasers, analog circuits, and thermoacoustic oscillators, to name a few applications. While studies have considered amplitude death on oscillator systems and in externally forced complex Ginzburg-Landau systems, a route to amplitude death has not been studied in autonomous continuum systems. We derive simple analytic conditions for the onset of amplitude death of one macroscopic wavefunction in a system of two coupled complex Ginzburg-Landau equations with general nonlinear self- and cross-interaction terms. Our results give a more general theoretical underpinning for recent amplitude death results reported in the literature, and suggest an approach for tuning parameters in such systems so that they either permit or prohibit amplitude death of a wavefunction (depending on the application). Numerical simulation of the coupled complex Ginzburg-Landau equations, for examples including cubic, cubic-quintic, and saturable nonlinearities, is used to illustrate the analytical results.Comment: 7 pages, 4 figure

Plasticity of morphology, chemoreception, and predatory behavior in garter snakes (Thamnophis Sirtalis)

Successful predation is a key component to the survival of snakes. Snakes that encounter periodic fluctuations in prey availability, or that move into novel feeding niches, must be behaviorally and morphologically equipped to adjust to new environmental conditions. Generalists, like the common garter snake (Thamnophis sirtalis), thrive in a wide variety of environments. The wide geographic distribution and considerable inter-population variability of T. sirtalis highlight their adaptability as a species. While microevolutionary change is known to contribute to the morphological and behavioral diversity of T. sirtalis, phenotypic plasticity is prevalent as well. In contrast to prey specialists, T. sirtalis feeds on a great diversity of prey species, each of which places different demands on the snakes\u27 sensory and behavioral capacities. In such a generalist, relatively high levels of morphological and behavioral plasticity, can be expected. The purpose of my study was to examine the relationship between diet and morphological variation in garter snakes, and how diet influences the ontogeny of chemosensory responses and predatory behavior. I examined the association between diet and morphological and behavioral differences in T. sirtalis inhabiting two sites on Beaver Island in Lake Michigan. Body and head size variation in 457 garter snakes from two ecologically dissimilar habitats were measured over the course of two field seasons (1998-1999). At one of the sites. Miller\u27s marsh, the snakes eat a wide diversity of amphibian species, as well as earthworms. Snakes at the second site, McCafferty farm, consume earthworms almost exclusively. The dietary differences between the two sites are due to differences in prey availability among sites. Probably owing to dietary differences among sites, adult snakes at Miller\u27s marsh were larger than snakes at McCafferty farm. However, only adult females significantly differed in body size between sites. There was a significant sex by site interaction for body length, suggesting differences in the degrees of sexual dimorphism between the two sites. Controlling for body size, relative head sizes differed among the sites, but this was significant for only one of the four measurements, interocular distance. In addition to morphological measurements, snakes from both sites were tested on their abilities to consume live frog, fish, and worm prey. Overall, adult snakes from both sites did not substantially differ in their abilities to capture, handle and consume these prey items. I conducted developmental studies of neonates born to mothers from both sites during both years. Females from Miller\u27s marsh had larger litters than females from McCafferty farm; however, the regressions of maternal snout-vent length (SVL) on litter size and neonatal length and mass were low and non-significant (n = 33 litters). I tested for the effects of litter, sex, and site on morphological and behavioral traits. Postpartum morphological analyses revealed significant sex and litter differences in SVL, body weight, and head size. Males had greater SVLs and tail lengths than females, but females were heavier and had larger heads. Site did not influence neonatal SVL or weight. However, neonates from Miller\u27s marsh had significantly longer jaw lengths and interocular distances than neonates from McCafferty farm. Neonates were reared on diets of fish, worms, or both, and growth rates were measured at 80-day intervals until the snakes reached 240 days. Diet had a significant effect on SVL and mass, but did not influence relative head sizes. Snakes reared on a mixture of fish and worms grew longer and were heavier at 240 days than snakes feeding on single diets. Sex-based differences in head size persisted through 240 days. I also examined the influence of diet and diet switching on the development of chemosensory responses to prey. Neonatal snakes were divided into diet groups comprising live fish, worms, or both. Tests for chemosensory responses to surface extracts of fish and worms were done prior to feeding experience and at two 80-day intervals following feeding experience. Snakes that had fish in their diet significantly increased their responses to this prey after feeding experience, whereas snakes reared on worms did not reveal a bias toward worm or fish extract. When the diets of the snakes in, the fish and worm groups were switched at 160 days, chemosensory responses to fish and worm stimuli were not significantly different for either group when re-tested at 240 days. Three experiments examined the role of learning and memory in the development of predatory skills. In Experiment I, the snakes used in the growth study were tested for their abilities to approach, capture, handle, and swallow prey at their first feeding, and were twice re-tested at 80 day intervals (11-12 feedings per interval). Diets were then reversed for the groups feeding exclusively on fish or worms, and the same behavioral measures as above were recorded for the first feeding on the new prey, and again after 11-12 feedings. A final trial tested the snakes\u27 retention for consuming the prey comprising their initial diets. Snakes in all three diet-groups decreased their overall latencies to consume prey (e.g., capture, handle, and swallow) after feeding experience. However, snakes feeding initially on worms were slow when consuming fish after diet switching, whereas snakes that initially fed on fish rapidly consumed worms upon their first feeding. Snakes who had switched to fish decreased their total consumption times after 11-12 feedings. Feeding skills for initial prey were retained following the diet-switching phase. In Experiment II, the amount of feeding experience prior to diet switching and after diet switching was reduced (6 feedings per group). This was done to assess the effects of shorter durations on each diet. No effect of diet switching or decrement in feeding skills was detected. However, the number of snakes completing the study was small and variation was very high. Individual differences were important contributors to this variation and are described in some detail. Experiment III was conducted to determine the long-term effects of feeding experience on prey consumption times. Snakes from McCafferty farm and Miller\u27s marsh were tested on their abilities to approach, capture, handle, and swallow frogs, fish, and worms. Although total consumption times for all three prey did not differ among the adult snakes from both sites, there were differences among the four predatory phases measured. Actual feeding experience may be less important for adults than for neonates. Morphological and behavioral plasticity accounted for much of the variation that I observed. Results from the morphological studies and four behavioral experiments revealed phenotypically plastic responses to varying environments, although important exceptions were found. Behavioral differences due to microevolutionary change were not detected, probably due to either the close proximity of the two sites that I studied, or to larger intra site variability. Plasticity is known to buffer the effects of natural selection and allows organisms to adjust to environmental variability. The high level of morphological and behavioral plasticity found in garter snakes is a viable explanation of their wide distribution and success as a species