Large antichains in the partition lattice

Consider the partially ordered set of all partitions of an n-element set, ordered by refinement. The sizes of the various ranks within this poset are the Stirling numbers of the second kind. We show that the ratio of the size of the largest antichain to the size of the largest rank exceeds n^1/35 for all n sufficiently large.     

Asymptotic behaviour of certain sets of prime ideals

Let be ideals of the commutative ring , let be a Noetherian -module and let be a submodule of ; also let be an Artinian -module and let be a submodule of . It is shown that, whenever is a sequence of -tuples of non-negative integers which is non-decreasing in the sense that for all and all , then Ass is independent of for all large , and also Att is independent of for all large . These results are proved without any regularity conditions on the ideals , and so (a special case of) the first answers in the affirmative a question raised by S. McAdam.

     

On Maximizing Discrimination in Multiple Criteria Decision Making

We examine an LP/DEA-based technique for establishing an overall ranking of alternatives that are ranked on multiple criteria, which themselves are ranked. This two-stage process involves one LP in the first stage, and N LPs in the second stage to rank N alternatives. We find that the information from N + 1 LPs can be obtained by solving two LPs. In many cases, the solution of one LP, which can be done by inspection, is almost as informative as the two-stage procedure. We also indicate when the second stage would be redundant. If maximum technical discrimination between the alternatives is sought, we consider how this might be achieved by aggressive cross-evaluation via N LPs. We also show how to identify a subset of the alternatives that would be ranked in the first place under any ordering of the criteria, and thus play an important role in the evaluation procedure.     

Comparison of high-order many-body perturbation theory and configuration interaction for H2O

Diagrammatic many-body perturbation theory, coupled with a recursive computational procedure, is employed to obtain the correlation energy of H₂O within a 39-STO basis set by evaluating all double-excitation diagrams through twelfth order without any approximations. This provides, for the first time, the complete double-excitation diagrams contributions to the correlation energy, which is ?0.28826 hartree, compared with a correlation energy of ?0.27402 hartree obtained from a configuration interaction calculation which includes all double excitations. The difference of 0.0142 hartree includes the “size consistency” correction to the all-double-excitations CI energy, due to the “pathological” unliked-diagram terms remaining in that result, but also involves certain fourth- and higher-order rearrangement diagrams. Contrary to conventional belief, the unshifted, or Møller-Plesset partitioning of the hamiltonian provides a much more rapid convergence of the perturbation series that does the shifted, or Epstein-Nesbet partitioning. In both cases. Padé approximants enhance the convergence of the series considerably. A simple variation-perturbation scheme based on the first-order MBPT wavefunction is sufficient to provide 97.5% of the all-doubles CI correlation energy.     

PCC-mediated novel oxidation reactions of homobenzylic and homoallylic alcohols

A new PCC-mediated carboncarbon bond cleavage reaction during oxidation of homobenzylic alcohols leading to the formation of benzylic carbonyl compounds has been observed. Homobenzylic alcohols with no benzylic substitution (R¹=H) gave benzylic aldehydes without further oxidation, while those with benzylic substitution (R¹=Me, Et, Ar) gave benzylic ketones. In contrast, homoallylic alcohols gave products arising from double bond migration, cis- to trans-olefin isomerization and/or allylic oxidation.     

Solvent-shared radical ion pairs [pyrene·⊖Na⊕(C2H5)2]∞: ESR evidence for two different aggregates in solution,room temperature crystallization,and structural proof of another polymorphic modification

The reduction of pyrene with sodium in aprotic diethyl ether allows to crystallize the extremely air-sensitive radical ion pair pyrene-sodium-diethylether. The single-crystal structure determination at 130 K shows that each sodium counter cation, solvated by one diethyl-ether molecule, is η³- and η⁶-coordinated to one of the short-axis six-membered rings of two pyrene radical anions. The resulting dibenzene-sodium sandwiches form a string, in which the hydrocarbon planes are canted to each other by 62°. In the pyrene radical-anion skeleton, no distortion due to its negative charge can be detected relative to that of the neutral molecule. From the temperature-dependent signal multiplets of preceding ESR investigations, the solvent-separated pyrene radical anion as well as two different contact radical-ion pairs had been identified and their structures in solution approximated by potential-energy estimates. Referring to the recently discovered long-axis Na⊕ contact ion pair polymorph, crystallized at lower temperatures, the structure reported here represents the second and probably thermodynamically more stable one. Both the ESR and the structural results provide some insight into the multidimensional networks of equilibria in aprotic solution, which are activated by alkali-metal reduction of unsaturated organic compounds.     

The superposition of the backward and forward processes of a renewal process

A fixed sampling point O is chosen independently of a renewal process

on the whole real line. The distances Y₁, Y₂, … from O to the renewal points of

, when they are measured either forwards or backwards in time, define a point process

. The process

is a folding over of the past of

onto its future. It is the superposition of two equilibrium renewal processes which are known to be independent only when

is a Poisson process. The joint distribution of Y₁, Y₂, …, Y_k is found. The marginal distribution of 2Y_k is shown to be the same as that of the distance from O to the k^th following point of

. The intervals of

are shown to have a stationarity property, and it is proved that if any pair of adjacent intervals of

are independent, then

is a Poisson process.     

Many-body perturbation theory applied to hydrogen fluoride

Pair contributions to the electronic correlation energy of the hydrogen fluoride molecule are computed using many-body perturbation theory (MBPT) with a basis set of Slater orbitals on both atomic centers. Summation of all two-body diagrams through third order gives ≈ 93% of the total correlation energy at the equilibrium internuclear separation. However, including certain higher-order effects, by means of denominator shifts, gives a correlation energy of ?0.3691 hartree, or ≈ 97% of the correlation energy.