The Arrow of Time in the Equations of Motion

It is argued that time's arrow is present in all equations of motion. But it is absent in the point particle approximations commonly made. In particular, the Lorentz-Abraham-Dirac equation is time-reversal invariant only because it approximates the charged particle by a point. But since classical electrodynamics is valid only for finite size particles, the equations of motion for particles of finite size must be considered. Those equations are indeed found to lack time-reversal invariance, thus ensuring an arrow of time. Similarly, more careful considerations of the equations of motion for gravitational interactions also show an arrow of time. The existence of arrows of time in quantum dynamics is also emphasized.     

On the hydrodynamic limit of a Ginzburg-Landau lattice model

Summary A lattice system of interacting diffusion processes is investigated. The evolution is attractive and time reversible, the spin satisfies a conservation law. It is shown that the rescaled spin field converges in probability to the corresponding solution to a nonlinear diffusion equation.Supported in part by the Hungarian National Foundation for Scientific Research, grant No. 819/1, and by the Mathematical Department of Rutgers University, N.S.F. grant DMR 8612369     

An explicit characterization of Calogero-Moser systems

Combining theorems of Halphen, Floquet, and Picard and a Frobenius type analysis, we characterize rational, meromorphic simply periodic, and elliptic KdV potentials. In particular, we explicitly describe the proper extension of the Airault-McKean-Moser locus associated with these three classes of algebro-geometric solutions of the KdV hierarchy with special emphasis on the case of multiple collisions between the poles of solutions. This solves a problem left open since the mid-1970s.

     

Infinite time optimal control and periodicity

For smooth nonlinear systems

Synthesis and molecular structure of [Al(CH3)2]2[C10H22N4] [Al(CH3)3]2: Condensation product of trimethylaluminum and a multidentate open-chain amine

The crystalline condensation product [Al(CH₃)₂]₂[C₁₀H₂₂N₄][Al(CH₃)₃]₂ is prepared from reaction of the multidentate open-chain amine 1,4-bis(3-aminopropyl)piperazine with trimethylaluminum in methylene chloride. The compound crystallized in the monoclinic space groupC2/c with unit cell parametersa=25.166(12),b=7.268(4),c=17.316(4) Å,=115.3(4)°, andD _c =1.06 g cm^–3 forZ=4. Least-squares refinement based on 1054 observed reflections with intensitiesI 3(I) in the range 2<2<45° led to a finalR factor of 0.053 (R _w =0.064). The title compound resides on a crystallographic center of symmetry with independent Al-N distances of 1.889(5), 2.058(5), and 2.030(5) Å.     

Miniaturized solid-phase extraction with resin disks

The properties and advantages of membrane disks for solid-phase extraction (SPE) are described. Miniaturization is a trend in SPE, as well as chemical analysis in general. A semimicro method is reviewed in which an extraction disk 4 mm in diameter is used for SPE. Even smaller scale separations are possible with a device in which a membrane 0.7 mm in diameter was incorporated into the needle of a 50-micro syringe. Aqueous samples containing 10 ppb of eight substituted benzenes were passed through the miniaturized-SPE syringe automatically using a single-syringe infusion pump. Elution volumes of 5 microl provided 500-fold concentrations and delivered average recoveries greater than 90% and an average relative standard deviation (RSD) of 4.6% for the analytes. Direct injection of the 5 microl eluate from the miniaturized-SPE syringe into a gas chromatograph also produced average recoveries greater than 90% and an average RSD of 6.2%.