What power of two divides a weighted Catalan number?

Given a sequence of integers b = (b_0,b_1,b_2,...) one gives a Dyck path P of length 2n the weight wt(P) = b_{h_1} b_{h_2} ... b_{h_n}, where h_i is the height of the ith ascent of P. The corresponding weighted Catalan number is C_n^b = sum_P wt(P), where the sum is over all Dyck paths of length 2n. So, in particular, the ordinary Catalan numbers C_n correspond to b_i = 1 for all i >= 0. Let xi(n) stand for the base two exponent of n, i.e., the largest power of 2 dividing n. We give a condition on b which implies that xi(C_n^b) = xi(C_n). In the special case b_i=(2i+1)^2, this settles a conjecture of Postnikov about the number of plane Morse links. Our proof generalizes the recent combinatorial proof of Deutsch and Sagan of the classical formula for xi(C_n).Comment: Fixed reference

Flux surface shaping effects on tokamak edge turbulence and flows

Shaping of magnetic flux surfaces is found to have a strong impact on turbulence and transport in tokamak edge plasmas. A series of axisymmetric equilibria with varying elongation and triangularity, and a divertor configuration are implemented into a computational gyrofluid turbulence model. The mechanisms of shaping effects on turbulence and flows are identified. Transport is mainly reduced by local magnetic shearing and an enhancement of zonal shear flows induced by elongation and X-point shaping.Comment: 10 pages, 11 figures. Submitted to Physics of Plasma

Dialing in single-site reactivity of a supported calixarene-protected tetrairidium cluster catalyst.

A closed Ir4 carbonyl cluster, 1, comprising a tetrahedral metal frame and three sterically bulky tert-butyl-calix[4]arene(OPr)3(OCH2PPh2) (Ph = phenyl; Pr = propyl) ligands at the basal plane, was characterized with variable-temperature 13C NMR spectroscopy, which show the absence of scrambling of the CO ligands at temperatures up to 313 K. This demonstration of distinct sites for the CO ligands was found to extend to the reactivity and catalytic properties, as shown by selective decarbonylation in a reaction with trimethylamine N-oxide (TMAO) as an oxidant, which, reacting in the presence of ethylene, leads to the selective bonding of an ethyl ligand at the apical Ir site. These clusters were supported intact on porous silica and found to catalyze ethylene hydrogenation, and a comparison of the kinetics of the single-hydrogenation reaction and steady-state hydrogenation catalysis demonstrates a unique single-site catalyst-with each site having the same catalytic activity. Reaction orders in the catalytic ethylene hydrogenation reaction of approximately 1/2 and 0 for H2 and C2H4, respectively, nearly match those for conventional noble-metal catalysts. In contrast to oxidative decarbonylation, thermal desorption of CO from silica-supported cluster 1 occurred exclusively at the basal plane, giving rise to sites that do not react with ethylene and are catalytically inactive for ethylene hydrogenation. The evidence of distinctive sites on the cluster catalyst leads to a model that links to hydrogen-transfer catalysis on metals-involving some surface sites that bond to both hydrocarbon and hydrogen and are catalytically engaged (so-called "*" sites) and others, at the basal plane, which bond hydrogen and CO but not hydrocarbon and are reservoir sites (so-called "S" sites)

Computational Methods and Results for Structured Multiscale Models of Tumor Invasion

We present multiscale models of cancer tumor invasion with components at the molecular, cellular, and tissue levels. We provide biological justifications for the model components, present computational results from the model, and discuss the scientific-computing methodology used to solve the model equations. The models and methodology presented in this paper form the basis for developing and treating increasingly complex, mechanistic models of tumor invasion that will be more predictive and less phenomenological. Because many of the features of the cancer models, such as taxis, aging and growth, are seen in other biological systems, the models and methods discussed here also provide a template for handling a broader range of biological problems

Drug Use, Dependence, and Addiction at a British Columbia University: Good News and Bad News

Two studies were undertaken to determine the perceived and actual prevalence of drug use, dependence, and addiction among students at Simon Fraser University. In thefirst, 144 students estimated the percentage of their fellow students that used each of seven drugs daily. Subsequently, each student reported how many days he or she had personally used each drug in the previous month. Estimated daily use (attributed tofellow students) was far higher than reported daily use. In a second study, detailed individual interviews of another group of students confirmed the unexpectedly low frequencies of daily use found in the first study: However, this does not mean that dependence and addiction were not a problem; of 107 students interviewed in the second study, 31% reported current dependence and about 5% current addiction. Surprisingly, the most common drugs in compulsive use were caffeine and nicotine followed, at a distance, by cannabis and alcohol. A new orientation towards drug problems among Canadian undergraduates in the 1980's is proposed, in which exaggerated concern over exotic, illicit drugs is replaced by greater awareness of serious problems of dependence and addiction to familiar drugs.Deux études ont été menées afin de déterminer la perception des étudiante)s de l'Université Simon Fraser en matière d'usage de drogue, de dépendance à celle-ci, de toxicomanie, et leur importance réelle. Dans la première étude, 144 étudiant(e)s ont évalué le pourcentage de leurs collègues qui consommaient quotidiennement chacune des sept drogues proposées. Ensuite, chaque étudian-te) a précisé sa propre consommation quotidienne de ces drogues dans le mois précédent. Les suppositions d'usage quotidien (par les collègues) étaient bien plus importantes que ce qui a été effectivement rapporté. Dans la deuxième étude, des entrevues individuelles dans un autre groupe d'étudiant(e)s ont confirmé le faible taux de consommation quotidienne relevé dans la première étude. Cependant, ceci ne signifie pas que la dépendance et la toxicomanie ne posent pas de problème. Des 107 étudiant(e)s interviewé(e)s dans la seconde étude, 31% ont confirmé une dépendance et 5% une toxicomanie. Fait surprenant, les drogues les plus utilisées de façon régulière étaient la caféine et la nicotine, suivies d'assez loin par le cannabis et l'alcool. Une nouvelle approche des problèmes de drogue parmi les étudiante)s canadien(ne)s dans les années 80 est proposée, dans laquelle l'inquiétude quelque peu exagérée au sujet des drogues exotiques illicites est remplacée par une plus grande mise en garde contre les problèmes sérieux de dépendance, voire même de toxicomanie, à l'égard de drogues plus familières