Synthesis and Reactivity of Phenylimidorhenium(V) Complexes with N‐[(N′,N′‐Dialkylamino)(thiocarbonyl)]benzamidines. Observation of a pH‐Dependent Isomerisation of Related Oxorhenium(V) Compounds

Reactions of [Re(NPh)Cl₃(PPh₃)₂] with N‐[(N′,N′‐dialkylamino)(thiocarbonyl)]benzamidines (H₂R₂tcb) (R₂ = Et₂, (CH₂)₂O(CH₂)₂) in methanol give mono‐chelates of the composition [Re(NPh)Cl₂(PPh₃)(HR₂tcb)] as the sole products independent of the amount of the added H₂R₂tcb. Addition of a supporting base such as NEt₃ results in hydrolysis of the Re=NPh bonds and partial hydrolysis of the thiocarbamoylbenzamidines. Orange‐brown, cationic oxorhenium(V) compounds of the formula [ReO(HR₂tcb)₂]Cl were isolated from such reaction mixtures in good yields, and the formation of small amount of the unusual sulfido/persulfido‐bridged Re^V dimer [{ReO(HEt₂tcb)}₂(μ‐S)(μ‐S₂)] give evidence for a considerable degree of ligand decomposition under such conditions. The products have been characterized by spectroscopic methods and X‐ray crystallography. Acidification of orange‐brown solutions of the five‐coordinate Re^V oxo complex [ReO(HEt₂tcb)₂]Cl causes an immediate change of the color and deep blue crystals of the neutral, six‐coordinate [ReOCl(HEt₂tcb)₂] can be isolated from the resulting mixture. Alternatively, the product can be prepared by a ligand‐exchange protocol starting from (NBu₄)[ReOCl₄] and H₂Et₂tcb. The pH‐dependent isomerization between [ReO(HEt₂tcb)₂]Cl and [ReOCl(HEt₂tcb)₂] is reversible.     

Mathematical Analysis of a Soccer Game. Part I: Individual and Collective Behaviors

Detailed time series analysis of a soccer match is given based on the detailed data of the 2D motions of all 22 players and of the ball for the match. The whole analysis includes two parts. In Part I, the individual and collective behaviors of the players of the two teams as well as the motion of the ball are presented as various time series. Geometrical centers, radii, expansion speeds, possession functions of the two teams are defined and calculated as functions of time. Major ranges of all players as well as of different groups of players (defenders, midfielders, forwards) of the two teams during the entire first half, the attacking phase of team A and the attacking phase of team B are calculated, respectively, showing the structures of the two teams during different phases. Distance coverage of each player and the mean distances covered by different groups of players (defenders, midfielders, forwards) during different phases are calculated. The time portions of possession of the ball by each team and the time portions of different phases are also calculated. In Part II, energy and spectral analysis and various correlations will be derived. The relation between various parameters and potential indicators will be discussed. The major purpose of the present study is to offer some general mathematical tools for the detailed analysis and to reveal some general features of soccer match when the detailed 2D data are available. The results would offer the raw materials for various potential indicators which may eventually be used in the coaching process to enhance the performance and in the prediction of the results of soccer matches.     

Sparse geometric graphs with small dilation

Given a set S of n points in , and an integer k such that 0k<n, we show that a geometric graph with vertex set S, at most n−1+k edges, maximum degree five, and dilation O(n/(k+1)) can be computed in time O(nlogn). For any k, we also construct planar n-point sets for which any geometric graph with n−1+k edges has dilation Ω(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position.     

The flow complex: A data structure for geometric modeling

We study a special case of the critical point (Morse) theory of distance functions namely, the gradient flow associated with the distance function to a finite point set in . The fixed points of this flow are exactly the critical points of the distance function. Our main result is a mathematical characterization and algorithms to compute the stable manifolds, i.e., the inflow regions, of the fixed points. It turns out that the stable manifolds form a polyhedral complex that shares many properties with the Delaunay triangulation of the same point set. We call the latter complex the flow complex of the point set. The flow complex is suited for geometric modeling tasks like surface reconstruction.     

Zur Chemie der Chlorthionitrenkomplexe des Rheniums Die Kristallstruktur von [N(SCl)2]⊕ [Re2Cl9]⊖

Chemistry of Chlorothionitrene Complexes of Rhenium. Crystal Structure of [N(SCl)₂]⊕ [Re₂Cl₉]? By reaction of S₃N₂Cl₂ with ReCl₅ the chlorothionitrene complex [ReCl₃(NSCl)₂]₂ is obtained in good yield; it has a dimer structure with chloro bridges. By the same reaction in POCl₃ solution the solvate [ReCl₃(NSCl)₂OPCl₃] is obtained. Instead, when a molar ratio of ReCl₅ and S₃N₂Cl₂ of 2:1 is taken, the product is [N(SCl)₂][Re₂Cl₉]. [ReCl₄(NSCl)OPCl₃] and excess PPh₃ react to give the nitrido complex [ReNCl₂(PPh₃)₂]. The crystal structure of [N(SCl)₂][Re₂Cl₉] was determined and refined with X-ray diffraction data (1021 independent reflexions, R = 0.031). It crystalizes in the space group C2/c with four formula units per unit cell (a = 1197, b = 1288, c = 1144 pm, ß = 107.83°). The [N(SCl)₂]⊕ cations have exactly C₂ and approximately C_2v symmetry; the NS bond lengths of 162 pm and the bond angles SNS (133.6°) and NSCl (117.6°) deviate considerably from the values of known [N(SCl)₂]⊕ structures. The [Re₂Cl₉]? anion consists of two face sharing octahedra and has a Re—Re distance of 270 pm. I. r. spectra of all compounds are reported and discussed.