Synthesis of trialkylaluminum derivatives by the reaction of non-solvated aluminum hydride with α-olefins

Hydroalumination of -olefins by non-solvated polymeric aluminum hydride (AlH₃) _n occurs at 120—140 °C. Mechanochemical activation accelerates this reaction. The addition of catalytic amounts of the prepared R₃Al forms to the reaction system decreases the temperature of the process to 90—100 °C. The greatest initiation effect is observed when ate-complexes of the MAlR₄ type (M = Li, Na) are used: the reaction occurs with a higher rate already at 60—90 °C affording R₃Al free of admixtures of carbalumination products and dimers of -olefins.     

Completion of Linearly Ordered Metabelian Groups

We prove a theorem saying that in finitely generated linearly ordered metabelian groups there exists a finite system of normal convex subgroups satisfying orderability conditions for groups, and an embedding theorem for linearly ordered metabelian groups whose initial linear orders extend to -divisible linearly ordered metabelian ones. As a consequence, it is stated that orderable metabelian groups are embedded, with extension of all their linear orders, in -divisible orderable metabelian groups.     

Reaction of 3H-perfluoroalk-2-enes with ammonia

3H-Perfluoroalk-2-enes react with aqueous ammonia to form polyfluorinated iminoenamines, from which hexafluoroacetylacetone and its analogs were synthesized.     

Zinc(II)-porphyrin Receptors in Multi-point Molecular Recognition: Recent Progress

The tremendous efforts made in order to control the coordination chemistry of hemoprotein models have considerably enhanced the synthesis of functionalized porphyrins, whose carefully designed architectures allowed for selective bindings of exogenic substrates. The common use of zinc(II) in place of pentacoordinated iron(II) has induced the use of zinc(II) porphyrins as building blocks for selective receptors. These receptors offer a convenient combination of multi-point recognition of substrates, and monitoring of the complexation due to the chromophoric nature of the tetrapyrrolic unit. This review is dedicated to recent progress made in the field of molecular recognition involving multi-point selective binding processes, in whichthe establishment of a strong coordination bond is finely tuned by one or more weak interactions adequately introduced in the architecture of a functionalized zinc(II)-porphyrin.     

Shapley value for constant-sum games

It is proved that Youngs [4] axiomatization for the Shapley value by marginalism, efficiency, and symmetry is still valid for the Shapley value defined on the class of nonnegative constant-sum games with nonzero worth of grand coalition and on the entire class of constant-sum games as well.The research was supported by NWO (The Netherlands Organization for Scientific Research) grant NL-RF 047-008-010.I am thankful to Theo Driessen, Natalia Naumova and Elena Yanovskaya for interesting discussions and comments. The useful remarks of two anonymous referees are also appreciated.     

On finding curb sets in extensive games

We characterize strategy sets that are closed under rational behavior (curb) in extensive games of perfect information and finite horizon. It is shown that any such game possesses only one minimal curb set, which necessarily includes all its subgame perfect Nash equilibria. Applications of this result are twofold. First, it lessens computational burden while computing minimal curb sets. Second, it implies that the profile of subgame perfect equilibrium strategies is always stochastically stable in a certain class of games.I am grateful to J. Kamphorst, G. van der Laan and X. Tieman, who commented on the earlier versions of the paper. I also thank an anonymous referee and an associate editor for their helpful remarks. The usual disclaimer applies.     

Expected Sums of General Parking Functions

A (u₁, u₂, . . . )-parking function of length n is a sequence (x₁, x₂, . . . , x_n) whose order statistics (the sequence (x₍₁₎, x₍₂₎, . . . , x_(n)) obtained by rearranging the original sequence in non-decreasing order) satisfy x_(i) u_(i). The Gonarov polynomials g _n (x; a₀, a ₁, . . . , a _n-1) are polynomials biorthogonal to the linear functionals (a _i) Dⁱ, where (a) is evaluation at a and D is differentiation. In this paper, we give explicit formulas for the first and second moments of sums of u-parking functions using Gonarov polynomials by solving a linear recursion based on a decomposition of the set of sequences of positive integers. We also give a combinatorial proof of one of the formulas for the expected sum. We specialize these formulas to the classical case when u _i=a+ (i-1)b and obtain, by transformations with Abel identities, different but equivalent formulas for expected sums. These formulas are used to verify the classical case of the conjecture that the expected sums are increasing functions of the gaps u_i+1 - u_i. Finally, we give analogues of our results for real-valued parking functions.AMS Subject Classification: 05A15, 05A19, 05A20, 05E35.     

Approximation and Prediction of the Numerical Solution of Some Burgers Problems

The Aitken's ²-prediction of Brezinski has already been used by Morandi Cecchi et al. in order to compute a numerical approximation of the solution of a parabolic initial-boundary value problem. This method consists in two consecutive steps: the first one is the approximation with a finite elements method, where the solution of the involved nonlinear system is computed by Gauss–Seidel method; the second one is a prediction of further terms with Aitken's ²-process. By comparison with this method, we use other methods of prediction in another way. First, we consider a generalization of ²-prediction, the so-called -prediction. In this paper, we only use vector prediction which is more stable than the scalar one. Then, the methods of prediction presented can be used in order to predict the starting vector of the Gauss–Seidel method.     

Influence of the Structure of an Inhomogeneously Broadened NMR Line on the Shape of the Free Precession Signal

We have obtained theoretically an expression for the nuclear free precession signal in an inhomogeneously broadened spin system with a complex structure of the NMR spectrum. It is shown that in this case in the nuclear precession signal there appear maxima the moments of whose formation depend on the detuning of the pulse carrier frequency from the central frequencies of the NMR lines, their widths, and the Rabi frequency. The theoretical results are in good agreement with the experimental data on the observation of proton free precession in toluene. We have constructed a physical model explaining the appearance of maxima in the free precession signal of the spin system.