Asymmetric equilibria in dynamic two-sided matching markets with independent preferences

A fundamental fact in two-sided matching is that if a market allows several stable outcomes, then one is optimal for all men in the sense that no man would prefer another stable outcome. We study a related phenomenon of asymmetric equilibria in a dynamic market where agents enter and search for a mate for at most n rounds before exiting again. Assuming independent preferences, we find that this game has multiple equilibria, some of which are highly asymmetric between sexes. We also investigate how the set of equilibria depends on a sex difference in the outside option of not being mated at all.     

High molar mass ethene/1‐olefin copolymers synthesized with acenaphthyl substituted metallocene catalysts

The influence of ligand structure on copolymerization properties of metallocene catalysts was elucidated with three C₁‐symmetric methylalumoxane (MAO) activated zirconocene dichlorides, ethylene(1‐(7, 9)‐diphenylcyclopenta‐[a]‐acenaphthadienyl‐2‐phenyl‐2‐cyclopentadienyl)ZrCl₂ ( 1 ), ethylene(1‐(7, 9)‐diphenylcyclopenta‐[a]‐acenaphthadienyl‐2‐phenyl‐2‐fluorenyl)ZrCl₂ ( 2 ), and ethylene(1‐(9)‐fluorenyl‐(R)1‐phenyl‐2‐(1‐indenyl)ZrCl2 ( 3 ). Polyethenes produced with 1 /MAO had considerable, ca. 10% amount of trans‐vinylene end groups, resulting from the chain end isomerization prior to the chain termination. When ethene was copolymerized with 1‐hexene or 1‐hexadecene using 1 /MAO, molar mass of the copolymers varied from high to moderate (531–116 kg/mol) depending on the comonomer feed. At 50% comonomer feed, ethene/1‐olefin copolymers with high hexene or hexadecene content (around 10%) were achievable. In the series of catalysts, polyethenes with highest molar mass, up to 985 kg/mol, were obtained with sterically most crowded 2 /MAO, but the catalyst was only moderately active to copolymerize higher olefins. Catalyst 3 /MAO produced polyethenes with extremely small amounts of trans‐vinylene end groups and relatively low molar mass 1‐hexene copolymers (from 157 to 38 kg/mol) with similar comonomer content as 1 . These results indicate that the catalyst structure, which favors chain end isomerization, is also capable to produce high molar mass 1‐olefin copolymers with high comonomer content. In addition, an exceptionally strong synergetic effect of the comonomer on the polymerization activity was observed with catalyst 3 /MAO. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 373–382, 2008     

The Steiner quadruple systems of order 16

The Steiner quadruple systems of order 16 are classified up to isomorphism by means of an exhaustive computer search. The number of isomorphism classes of such designs is 1,054,163. Properties of the designs—including the orders of the automorphism groups and the structures of the derived Steiner triple systems of order 15—are tabulated. A consistency check based on double counting is carried out to gain confidence in the correctness of the classification.     

Scheduling a triple round robin tournament with minitournaments for the Finnish national youth ice hockey league

This paper presents a method to schedule a triple round robin tournament, which involves minitournaments, each hosted by one team. A key issue is that at the end of the season the number of home games should be balanced over the teams, despite the fact that in minitournament matches only the host team plays at home. This format is played in the Finnish national ice hockey league for players under the age of 20 years, where the problem is further complicated by many other constraints, for example, preassigned matches resulting from away tours that should limit the distances travelled by the teams. To obtain a schedule for this league, we sequentially solve four distinct combinatorial problems. This method allows us to construct a schedule for the 2009–2010 season, which is superior to the official schedule: it has no hard constraint violations, and outperforms the official schedule on three of five soft constraints.     

Limited resolution in complex network community detection with Potts model approach

According to Fortunato and Barthélemy, modularity-based community detection algorithms have a resolution threshold such that small communities in a large network are invisible. Here we generalize their work and show that the q-state Potts community detection method introduced by Reichardt and Bornholdt also has a resolution threshold. The model contains a parameter by which this threshold can be tuned, but no a priori principle is known to select the proper value. Single global optimization criteria do not seem capable for detecting all communities if their size distribution is broad.     

Bentley’s conjecture on popularity toplist turnover under random copying

Bentley et al. studied the turnover rate in popularity toplists in a ‘random copying’ model of cultural evolution. Based on simulations of a model with population size N, list length ? and invention rate μ, they conjectured a remarkably simple formula for the turnover rate: $\ell \sqrt{\mu}$ . Here we study an overlapping generations version of the random copying model, which can be interpreted as a random walk on the integer partitions of the population size. In this model we show that the conjectured formula, after a slight correction, holds asymptotically.     

Determination of talc in geological samples by infrared spectrometry

Talc is determined by infrared spectrometry in talc ore and mineral mixtures containing clinochlore chlorite, dolomite and magnesite as major minerals. When the Si-O-Si vibration at 668 cm^?1 is used, the calibration graph is linear up to about 0.5 mg of talc. Band overlap from chlorite is corrected by using the ratio of absorbances at 668 and 3620 cm^?1. Talc (?15%) in talc/carbonate/chlorite mixtures and ore was determined with relatively good precision and accuracy (recovery 91.2–97.4%). The sample sizes were 0.5–1 mg for 40–60% talc and 0.5–1.5 mg for 7.5–15% talc. The recoveries were worse (84.0–94.6%) for < 15% talc. The precision for unknown samples of talc ore varied from 1.5 to 4.0% (n=4).     

Local equivalence relations

In this paper, we investigate the concept of local equivalence relation, a notion suggested by Grothendieck. A local equivalence relation on a topological space X is a global section of the sheaf of germs of equivalence relations on X. We investigate the extent to which a local equivalence relation can be described by a global one and analogously when can a global equivalence relation be recovered from its associated local one. We also look at the notion of a fiber map, which sheds further light on these concepts. A motivating example is that of a foliation on a manifold.     

Lecture Hall Partitions II

For a non-decreasing integer sequence a=(a₁,...,a_n) we define L_a to be the set of n-tuples of integers = (₁,...,_n) satisfying . This generalizes the so-called lecture hall partitions corresponding to a_i=i and previously studied by the authors and by Andrews. We find sequences a such that the weight generating function for these a-lecture hall partitions has the remarkable form In the limit when n tends to infinity, we obtain a family of identities of the kind the number of partitions of an integer m such that the quotient between consecutive parts is greater than is equal to the number of partitions of m into parts belonging to the set P, for certain real numbers and integer sets P. We then underline the connection between lecture hall partitions and Ehrhart theory and discuss some reciprocity results.