Rectangular-radial drawings of cubic plane graphs

Rectangular drawings and rectangular duals can be naturally extended to other surfaces. In this paper, we extend rectangular drawings and rectangular duals to drawings on a cylinder. The extended drawings are called rectangular-radial drawings and rectangular-radial duals. Rectangular-radial drawings correspond to periodic rectangular tilings of a 1-dimensional strip. We establish a necessary and sufficient condition for plane graphs with maximum degree 3 to have rectangular-radial drawings and a necessary and sufficient condition for triangulated plane graphs to have rectangular-radial duals. Furthermore, we present three linear time algorithms under three different conditions for finding a rectangular-radial drawing for a given cubic plane graph, if one exists.     

On the local structure of optimal measures in the multi-marginal optimal transportation problem

We consider an optimal transportation problem with more than two marginals. We use a family of semi-Riemannian metrics derived from the mixed, second order partial derivatives of the cost function to provide upper bounds for the dimension of the support of the solution.     

The Order of Automorphisms of Quasigroups

We prove quadratic upper bounds on the order of any autotopism of a quasigroup or Latin square, and hence also on the order of any automorphism of a Steiner triple system or 1‐factorization of a complete graph. A corollary is that a permutation σ chosen uniformly at random from the symmetric group will almost surely not be an automorphism of a Steiner triple system of order n, a quasigroup of order n or a 1‐factorization of the complete graph . Nor will σ be one component of an autotopism for any Latin square of order n. For groups of order n it is known that automorphisms must have order less than n, but we show that quasigroups of order n can have automorphisms of order greater than n. The smallest such quasigroup has order 7034. We also show that quasigroups of prime order can possess autotopisms that consist of three permutations with different cycle structures. Our results answer three questions originally posed by D.  Stones.     

The first 1-alkyl-3-perfluoroalkyl-4,5- dimethyl-1,2,4-triazolium salts

Syntheses of quaternary 1-alkyl-3-perfluoroalkyl-4,5-dimethyl-1,2,4-triazolium iodides have led to a variety of new quaternary salts via metathesis reactions. 1,4,5-Trimethyl-3-trifluoro-methyl-1,2,4-triazolium iodide (6) with LiN(SO(2)CF(3))(2), KSO(3)CF(3), AgClO(4), AgBF(4); 1-(3-fluoropropyl)-3-trifluoromethyl-4,5-dimethyl-1,2,4-triazolium iodide (7) with LiN(SO(2)CF(3))(2); and 1,4,5-trimethyl-3-perfluorooctyl-1,2,4-triazolium iodide (8) with LiN(SO(2)CF(3))(2), AgClO(4), AgBF(4) gave excellent yields of new thermally stable and relatively low melting quaternary salts. The structure of 1,4,5-trimethyl-3-perfluorooctyl-1,2,4-triazolium tetrafluoroborate (11c) was confirmed by single-crystal X-ray analysis. Although the molecular weight of 11c (cation) is 3-fold greater than that of the 3-trifluoromethyl derivative 9d, its melting point is 32 degrees C lower.     

Asymptotic enumeration of tournaments with a given score sequence containing a specified digraph

This paper studies the probability that a random tournament with specified score sequence contains a specified subgraph. The exact asymptotic value is found in the case that the scores are not too far from regular and the subgraph is not too large. An n‐dimensional saddle‐point method is used. As a sample application, we prove that almost all tournaments with a given score sequence (not too far from regular) have a trivial automorphism group. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 47–57, 2000     

On the geometry of spaces of oriented geodesics

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space, and consider the space L(M) of oriented geodesics of M. The space L(M) is a smooth homogeneous manifold and in this paper we describe all invariant symplectic structures, (para)complex structures, pseudo-Riemannian metrics and (para)Kähler structure on L(M).     

Evaluation of gas chromatography-tandem quadrupole mass spectrometry for the determination of organochlorine pesticides in fats and oils

A gas chromatography-tandem quadrupole mass spectrometry multi-residue method for the analysis of 19 organochlorine pesticides in fats and oils has been developed. Gel permeation chromatography was employed to remove lipid material prior to GC-MS/MS analysis. Average recoveries of the pesticides spiked at 10 and 50 microg kg(-1) into fish oil, pork fat, olive oil and hydrogenated vegetable oil were typically in the range 70-110% with relative standard deviations generally less than 10%. Calculated limits of detection are between 0.1 and 2.0 microg kg(-1) and results obtained for the analysis of proficiency test materials are in good agreement with assigned values. The higher selectivity of the GC-MS/MS compared to electron capture detection and GC-MS in selective ion monitoring mode allowed unambiguous identification and confirmation of all the target pesticides at low microg kg(-1) levels in fats and oils in a single analysis.     

Some generalizations of torsion-free Crawley groups

In this paper we investigate two new classes of torsion-free Abelian groups which arise in a natural way from the notion of a torsion-free Crawley group. A group G is said to be an Erd?s group if for any pair of isomorphic pure subgroups H,K with G/H ? G/K, there is an automorphism of G mapping H onto K; it is said to be a weak Crawley group if for any pair H,K of isomorphic dense maximal pure subgroups, there is an automorphism mapping H onto K. We show that these classes are extensive and pay attention to the relationship of the Baer-Specker group to these classes. In particular, we show that the class of Crawley groups is strictly contained in the class of weak Crawley groups and that the class of Erd?s groups is strictly contained in the class of weak Crawley groups.     

Assembly of a supramolecular cube, [(Cp*WS3Cu3)8 Cl8(CN)12Li4] from a preformed incomplete cubane-like compound [PPh4][Cp*WS3(CuCN)3

A supramolecular cube has been formed by linking WS3Cu3 clusters with bidentate ligands. Eight WS3Cu3 clusters, which have an incomplete cubane-like structure, serve as the three connecting nodes of the cube, while 12 bridging cyanide anions coordinated to copper centers lie along the edges of the cube. Eight chloride anions and four lithium cations are located inside the cube.