The finite element method with non-uniform mesh sizes applied to the exterior Helmholtz problem

Summary The finite element method with non-uniform mesh sizes is employed to approximately solve Helmholtz type equations in unbounded domains. The given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary. This approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases. This results in a great reduction in the number of equations to be solved. It is proved that optimal error estimates hold inL ²,H ¹ andL , provided that certain relationships hold between the frequency, mesh size and outer radius.     

Ergodic properties of a system in contact with a heat bath: A one dimensional model

We consider a one-dimensional model of a system in contact with a heat bath: A particle (the system ormolecule) of massM, confined to the unit interval [0, 1], is surrounded by an infinite ideal gas (thebath of atoms) of point particles of massm with which it interacts via elastic collisions. The atoms are not affected by the walls at 0 and at 1. We obtain convergence to equilibrium for the molecule, from essentially any initial distribution on its position and velocity. The infinite composite system of molecule and bath has very good ergodic properties: it is a Bernoulli system.Supported in part by NSF Grants PHY 78-03816 and PHY 78-15920     

Mixed-metal hydride complexes containing ZrHA1 bridges. synthesis and relation to transition-metal-catalyzed reactions of aluminum hydrides

Dialkylaluminum hydrides and Cp₂ZrX₂ (Cp = π⁵-C₅H₅; X = Cl, H, alkyl) were found to react by aggregation to give mixed ZrAl trihydride complexes containing ZrHAl bridges. Reactions of these mixed-metal hydrides are discussed and structural parallels between them and known aluminum hydride complexes are drawn.     

Formation of a cyclobutylidene ring: intramolecular [2 + 2] cycloaddition of allyl and vinylidene C=C bonds under mild conditions

Intramolecular [2 + 2] cycloaddition of two C=C bonds in vinylidene complexes [Ru(eta5-C9H7){=C=C(R)H}(PPh3){kappa1-(P)-PPh2(C3H5)][BF4] affords cyclobutylidene complexes [Ru(eta5-C9H7){kappa2-(P,C)-(=CC(R)HCH2CHCH2PPh2)}(PPh3)][BF4], which can be also obtained by reaction of terminal alkynes with [Ru(eta5-C9H7)(PPh3){kappa3-(P,C,C)-PPh2(C3H5)}][PF6]. The reaction proceeds under mild conditions via vinylidene complexes, and the activation parameters were determined by kinetic studies.     

A nonequilibrium entropy for dynamical systems

It is proposed to define entropy for nonequilibrium ensembles using a method of coarse graining which partitions phase space into sets which typically have zero measure. These are chosen by considering the totality of future possibilities for observation on the system. It is shown that this entropy is necessarily a nondecreasing function of the timet. There is no contradiction with the reversibility of the laws of motion because this method of coarse graining is asymmetric under time reversal. Under suitable conditions (which are stated explicitly) this entropy approaches the equilibrium entropy ast+ and the fine-grained entropy ast–. In particular, the conditions can always be satisfied if the system is aK-system, as in the Sinai billiard models. Some theorems are given which give information about whether it is possible to generate the partition used here for coarse graining from time translates of a finite partition, and at the same time elucidate the connection between our concept of entropy and the entropy invariant of Kolmogorov and Sinai.Research supported in part by NSF grants PHY78-03816 and PHY78-15920.