A short proof that N 3 is not a circle containment order

A partially ordered set P is called a circle containment order provided one can assign to each xP a circle C _x so that . We show that the infinite three-dimensional poset N ³ is not a circle containment order and note that it is still unknown whether or not [n]³ is such an order for arbitrarily large n.     

Self-diffusion in a non-uniform one dimensional system of point particles with collisions

Summary We generalize the results of Spitzer, Jepsen and others [1–4] on the motion of a tagged particle in a uniform one dimensional system of point particles undergoing elastic collisions to the case where there is also an external potential U(x). When U(x) is periodic or random (bounded and statistically translation invariant) then the scaled trajectory of a tagged particle converges, as A , to a Brownian motion W _D (t) with diffusion constant , where is the average density, is the mean absolute velocity and ^–1 the temperature of the system. When U(x) is itself changing on a macroscopic scale, i.e. , then the limiting process is a spatially dependent diffusion. The stochastic differential equation describing this process is now non-linear, and is particularly simple in Stratonovich form. This lends weight to the belief that heuristics are best done in that form.Dedicated to Frank Spitzer on the occasion of his 60th birthdayWork supported in part by NSF Grants No. PHY 8201708 and No. DMR 81-14726Heisenberg-fellowAlso Department of Physics     

Forward reactions and the exploration of particle dynamics

Since the polarization structure of reactions in the collinear directions is greatly simplified, since reactions at very high energies tend to be predominantly close to collinear, and since the one-particle exchange mechanisms are more pronounced in the forward direction, tests of one-particle-exchange (OPE) mechanisms are formulated for collinear reactions. The number of amplitudes are derived for a reaction with arbitrary spins, and examples for the practically occurring cases are given. The results are applied to the reaction 1/2+1/21+1 and in particular ton+pd+ (or to its inverse). A detailed procedure is outlined for a step-by-step testing of the various OPE processes in this reaction through a series of polarization experiments. In particular, the tests can separate out theJ=0 andJ=1/2 OPE processes.     

Porous metal–organic alloys based on soluble coordination cages

Diverse strategies for the preparation of mixed-metal three-dimensional porous solids abound, although many of them lend themselves only moderate levels of tunability. Herein, we report the design and synthesis of surface functionalized permanently microporous coordination cages and their use in the isolation of mixed metal solids. Judicious alkoxide-based ligand functionalization was utilized to tune the solubility of starting copper(ii)-based cages and their resulting compatibility with the mixed-cage approach described here. We further prepared a family of isostructural molybdenum(ii) cages for a subset of the ligands. The preparation of mixed-metal cage solids proceeds under facile conditions where solutions of parent cages are mixed and product phases isolated. A suite of spectroscopic and characterization tools confirm the starting cages are intact in the amorphous product. Finally, we show that utilization of precise ligand functional groups can be used to prepare mixed cage solids that can be easily and cleanly separated into their constituent components through simple solvent washing or solvent extraction techniques.

Surface-functionalized porous coordination cages can be used to create homogeneous mixed-cage alloys with high levels of tunability and processability.     

On the Nature of the 7-Norbornadienyllithiums. Preliminary Communication

Lithium reductions of 7-chloronorbornadiene and of bis(7-norbornadienyl)mercury both provide (C₇H₇)₂Li₂ ( 5a ). This product is accompanied by C₇H₇Li₂Cl ( 5c ) in the first case, and by C₇H₇Li ( 5b ) in the second. The theoretically anticipated properties of all three organolithiums are apparent in the consistent C_s symmetry of their hydrocarbon ligands, their protolytic destruction by 12-crown-4, and their significant J(C(7), Li) ( 5a , 7.6; 5b , 16.0; 5c 8.9 Hz).     

How to Control the Chemoselectivity of the Catalytic Formation of Chiral γ-Lactams or 2,3-Disubstituted Pyrroles by the Choice of Solvent

Summary. The reaction of the unsaturated imine methyl(3-phenylallylidene)amine with ethylene and carbon monoxide in the presence of catalytical amounts of Ru₃(CO)₁₂ leads to the formation of two heterocyclic products. One of the products is a chiral γ-lactam, the other one a 2,3-disubstituted pyrrole derivative, in which only the carbon atom from carbon monoxide is incorporated. The selectivity in the formation of the products may be controlled by the choice of solvent. In general, in nonpolar solvents the formation of the lactam is preferred whereas the use of more polar solvents enhances the yield of the pyrrole. For most of the solvents used there is a linear dependence of the product ratio on the relative permittivity of the corresponding solvent. Typically, polar aprotic solvents do not follow this rule.     

Synthesis and structural characterisation of a novel tris-methylene bridged compound (NO) 4Fe2 Se(μ-CH2)3

The tris-methylene bridged compound (NO)₄Fe₂Se(μ-CH₂)₃ has been isolated. It has been characterised by IR and ¹H, ¹³C, and ⁷⁷Se NMR spectroscopy. Its structure has been determined by single-crystal X-ray diffraction methods. The structure consists of a heavy atom triangle consisting of one Se and two Fe atoms. The Fe-Fe and the two Fe-Se edges are bridged by methylene groups.