Deterministic random walks on regular trees

Jim Propp's rotor–router model is a deterministic analog of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order. Cooper and Spencer (Comb Probab Comput 15 (2006) 815–822) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid ?^d and does a simultaneous walk in the Propp model, then at all times and on each vertex, the number of chips on this vertex deviates from the expected number the random walk would have gotten there by at most a constant. This constant is independent of the starting configuration and the order in which each vertex serves its neighbors. This result raises the question if all graphs do have this property. With quite some effort, we are now able to answer this question negatively. For the graph being an infinite k‐ary tree (k ≥ 3), we show that for any deviation D there is an initial configuration of chips such that after running the Propp model for a certain time there is a vertex with at least D more chips than expected in the random walk model. However, to achieve a deviation of D it is necessary that at least exp(Ω(D²)) vertices contribute by being occupied by a number of chips not divisible by k at a certain time. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Synthesis of 2,4-disubstituted piperidines via radical cyclization: unexpected enhancement in diastereoselectivity with tris(trimethylsilyl)silane

A novel approach to 2,4-disubstituted piperidines is reported, involving the radical cyclization of 7-substituted-6-aza-8-bromooct-2-enoates. Cyclization with tributyltin hydride affords the trans piperidines with trans/cis diastereomeric ratios ranging typically from 3:1 to 6:1. Cyclization with tris(trimethylsilyl)silane affords the same products with diastereomeric ratios of up to 99:1 in certain cases. The enhancement in diastereoselectivity results from the selective rearrangement of the minor stereoisomer through a cascade process involving radical cyclization to the piperidine radical, 1,5-radical translocation, and attack of the translocated radical onto the sulfonamide with extrusion of SO2 in a Smiles-type rearrangement. Slower trapping of the piperidine radical by tris(trimethylsilyl)silane compared to tributyltin hydride accounts for the occurrence of the rearrangement cascade in the former case.     

Sulfur adsorption on Fe(1 1 0): a DFT study

The adsorption of atomic S on the Fe(1 1 0) surface is examined using density functional theory (DFT). Three different adsorption sites are considered, including the atop, hollow and bridge sites and the S is adsorbed at a quarter monolayer coverage in a p(2 × 2) arrangement. The hollow site is found to be the most stable, followed by the bridge and atop sites. At all three sites, S adsorption results in relatively minor surface reconstruction, with the most significant being that for the hollow site, with lateral displacements of 0.09 Å. Comparisons between S-adsorbed and pure Fe surfaces revealed reductions in the magnetic moments of surface-layer Fe atoms in the vicinity of the S. At the hollow site, the presence of S causes an increase in the surface Fe d-orbital density of states between 4 and 5 eV. However, S adsorption has no significant effect on the structure and magnetic properties of the lower substrate layers.     

Hydroformylation of cyclic dienes catalysed by acetatocarbonylbis(triphenylphosphine)rhodium(I)

The hydroformylation of cyclic dienes in benzene using Rh(CO₂Me)(CO)(PPh₃)₂ as catalyst was studied. The formation of mono- or di-aldehydes was critically dependent on ring size, the smaller cyclic dienes favouring dihydroformylation. Intermediate alkene-aldehydes could be isolated under appropriate conditions, indicating that initial attack on the diene was by hydroformylation. Under hydroformylation conditions, these intermediates underwent either hydroformylation or hydrogenation, depending on the ring size. Linear dienes gave complex mixtures of products.     

Synthesis of chiral titanium-containing phosphinoamide ligands for enantioselective heterobimetallic catalysis

The synthesis of six chiral titanium-containing phosphinoamide ligands is discussed. These ligands assemble chiral heterobimetallic Pd–Ti complexes, enable enantioselective intramolecular allylic aminations with hindered amine nucleophiles and achieve selectivity up to 53% ee. Mechanistic studies demonstrate the reversibility of the enantio-determining C–N bond forming step, which leads to a gradual increase in the % ee of the reaction over time. These results represent a rare example of enantioselective heterobimetallic catalysis and suggest that these new ligands could find broad application in enantioselective transition metal catalysis.     

Lattice systems with a continuous symmetry

We prove that the expansion in powers of the temperatureT of the correlation functions and the free energy of the plane rotator model on ad-dimensional lattice is asymptotic to all orders inT. The leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series. We also prove the inverse power decay of the pair correlation at low temperatures ford=3.Supported by NSF Grant No. MCS 78-01885Supported by NSF Grant No. PHY 78-15920Supported by NSF Grant No. DMR 73-04355Supported by NSF Grant No. PHY-7825390 A01On leave from: Institut de Physique Théorique, Université de Louvain, BelgiumAlso: Department of Physics     

Euclidean ramsey theorems. I

The general Ramsey problem can be described as follows: Let A and B be two sets, and R a subset of A × B. For a?A denote by R(a) the set {b?B | (a, b) ?R}. R is called r-Ramsey if for any r-part partition of B there is some a?A with R(a) in one part. We investigate questions of whether or not certain R are r-Ramsey where B is a Euclidean space and R is defined geometrically.     

Practical scheme for polarization-insensitive and frequency-conversion interval independent four-wave mixing in semiconductor optical amplifiers

A practical scheme for polarization-insensitive and frequency-conversion interval-independent four-wave mixing in an optical loop configuration is proposed that uses one conventional semiconductor optical amplifier device and two perpendicular polarized pump sources. The significant advantages of this technique are simplicity of configuration, efficient conjugate power generation, effective suppression of the signal and one pump wave at the output port, and reduction in noise.     

A [3+2] annulation procedure for methylenecyclopentanes

A two step sequence converts certain cyclic enones to fused methylenecyclopentanes using the annulation reagent 2-chloromethyl-3-trimethylsilylpropene.