Lattice gas generalization of the hard hexagon model. II. The local densities as elliptic functions

In a previous paper we considered an extension of the hard hexagon model to a solvable two-dimensional lattice gas with at most two particles per pair of adjacent sites. Here we use various mathematical identities (in particular Gordon's generalization of the Rogers-Ramanujan relations) to express the local densities in terms of elliptic functions. The critical behavior is then readily obtained.Supported in part by the National Science Foundation Grant MCS 8201733.     

The isolation and structure determination of cryptomisrine,a novel indolo[3,2-b]quinoline dimeric alkaloid from cryptolepis sanguinolenta

The isolation and structure determination of cryptomisrine, a novel indolo[3,2-b]quinoline dimeric alkaloid obtained from extracts of the roots of the Ghanaian medicinal plant Cryptolepis sanguinolenta is reported. The structure determination was made via a consideration of the spectral data, including uv, ir, nmr, and mass spectra. In particular, one-dimensional proton/carbon nmr, one-dimensional nOe difference nmr, and a series of homonuclear (COSY) and inverse-detected heteronuclear two-dimensional (HMQC, HMBC) experiments were utilized, as well as high resolution FABMS. Cryptomisrine is most unusual in that its two monomeric parts apparently exist in such a C₂ symmetric environment that only one set of proton and carbon nmr resonances are observed. Cryptomisrine is the first example of a dimeric indolo-[3,2-b]quinoline alkaloid to have been isolated from nature.     

A combined theoretical and experimental study of the reaction products of laser-ablated thorium atoms with CO: first identification of the CThO, CThO(-), OthCCO, OTh(eta(3)-CCO), and Th(CO)(n) (n = 1-6) molecules

Laser-ablated thorium atoms have been reacted with CO molecules during condensation in excess neon. Absorptions at 617.7 and 812.2 cm(-1) are assigned to Th-C and Th-O stretching vibrations of the CThO molecule. Absorptions at 2048.6, 1353.6, and 822.5 cm(-1) are assigned to the OThCCO molecule, which is formed by CO addition to CThO and photochemical rearrangement of Th(CO)(2). The OThCCO molecule undergoes further photoinduced rearrangement to OTh(eta(3)-CCO), which is characterized by C-C, C-O, and Th-O stretching vibrations at 1810.8, 1139.2, and 831.6 cm(-1). The Th(CO)(n) (n = 1-6) complexes are formed on deposition or on annealing. Evidence is also presented for the CThO(-) and Th(CO)(2)(-) anions, which are formed by electron capture of neutral molecules. Relativistic density functional theory (DFT) calculations of the geometry structures, vibrational frequencies, and infrared intensities strongly support the experimental assignments. It is found that CThO is an unprecedented actinide-containing carbene molecule with a triplet ground state and an unusual bent structure ( angleCThO = 109 degrees ). The OThCCO molecule has a bent structure while its rearranged product OTh(eta(3)-CCO) is found to have a unique exocyclic structure with side-bonded CCO group. We also find that both Th(CO)(2) and Th(CO)(2)(-) are, surprisingly, highly bent, with the angleC-Th-C bond angle being close to 50 degrees; the unusual geometries are the result of extremely strong Th-to-CO back-bonding, which causes significant three-centered bonding among the Th atom and the two C atoms.     

Plane partitions IV: A conjecture of Mills—Robbins—Rumsey

In this paper we consider a problem posed by W Mills D Robbins and H Rumsey for a certain plane partition generating functionZ _n (x, m) The special caseZ _n (1,m) is the generating function that arose in the weak Macdonald conjecture Mills—Robbins—Rumsey conjectured thatZ _n (2,m) also possesses a nice finite product representation Their conjecture is proved as Theorem 1 The method of proof resembles that of the evaluation ofZ _n (1,m) given previously Many results for the₃ F ₂ hypergeometric function are required including Whipple's theorem, the Pfaff-Saalschutz summation and contiguous relations In passing we note that our Lemma 2 provides a new and simpler representation ofZ _n (2,m) as a determinant $$Z_n (2,m) = \det \left( {\delta _{ij} + \sum\limits_{i = 0}^l {\left( {\begin{array}{*{20}c} {m + j + t} \\ t \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {m + i} \\ {m + t} \\ \end{array} } \right)} } \right)_{0 \leqq ij \leqq n - 1} $$ Conceivably this new representation may provide new interpretations of the combinatorial significance ofZ _n (2,m) In the final analysis, one would like a combinatorial explanation ofZ _n (2,m) that would provide an algorithmic proof of the Mills Robbins—Rumsey conjecture     

Gallium metal mediated allylation of carbonyl compounds and imines under solvent-free conditions

Gallium metal is effective in mediating the allylation of various carbonyl compounds and imines under solvent-free conditions, with the application of sonic energy, affording the corresponding homoallylic alcohols and amines. The imines themselves were also prepared under solventless conditions in high yield, thereby establishing a two-step solvent-free synthesis of homoallylic amines. In comparison, indium metal produced a mixture of the desired homoallylic secondary amine and the bis-allylated species via an iminium ion intermediate.