Convergence in Conformal Field Theory

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory. The author reviews some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. He also reviews the related analytic extension results, conjectures and problems. He discusses the convergence and analytic extensions of products of intertwining operators (chiral conformal fields) and of q-traces and pseudo-q-traces of products of intertwining operators. He also discusses the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result. He then explains conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras.     

Left invariant special Kähler structures

We construct left invariant special Kähler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. We introduce the twisted cartesian product of two special Kähler Lie algebras according to two linear representations by infinitesimal Kähler transformations. We also exhibit a double extension process of a special Kähler Lie algebra which allows us to get all simply connected special Kähler Lie groups with bi-invariant symplectic connections. All Lie groups constructed by performing this double extension process can be identified with a subgroup of symplectic (or Kähler) affine transformations of its Lie algebra containing a nontrivial 1-parameter subgroup formed by central translations. We show a characterization of left invariant flat special Kähler structures using étale Kähler affine representations, exhibit some immediate consequences of the constructions mentioned above, and give several non-trivial examples.     

An existence result for linear partial differential equations with coefficients in an algebra of generalized functions

We prove the existence of solutions for essentially all linear partial differential equations with -coefficients in an algebra of generalized functions, defined in the paper. In particular, we show that H. Lewy's equation has solutions whenever its right-hand side is a classical -function.

     

Normsets and determination of unique factorization in rings of algebraic integers

The image of the norm map from to (two rings of algebraic integers) is a multiplicative monoid . We present conditions under which is a UFD if and only if has unique factorization into irreducible elements. From this we derive a bound for checking if is a UFD.

     

Design of a new molecule that exhibits observable CD spectra under σ-σ* excitation: 2,6-Disilyl-2,6-disilaspiro[3.3]heptanes

Abstract

We previously designed and prepared the first molecules to exhibit observable CD spectra by n-σ* excitation, 2,6-dithiaspiro[3.3]heptane 2,6-dioxide. Spiro[3.3]heptane frameworks possess axial asymmetry due to puckering of 2 four-membered rings; the ring bonds are rich in p-character due to acute bond angles, which lowers the σ* energy levels. In contrast, the lone pairs are rich in s-character, which results in a good donor with conformational fixation. We expected that, instead of lone pairs as donating orbitals, the use of σ-electron-donating Si-Si bonds should result in UV absorption in the observable range (>180?nm), so that the Cotton effect could appear, at least partially, in that range. We designed 2,6-disilyl-2,6-disilaspiro[3.3]heptanes as models, and performed theoretical calculations to confirm our idea.     

New dynamic axial compression columns with means for monitoring floating adapter position. Column design and utilisation in low pressure LC on soft packings

Summary The design of new dynamic, axial-compression columns with a system for continous packed bed adjustment and monitoring of the floating adapter position is described. The columns are meant for liquid chromatography at low pressures (up to 8 bar) in aqueous and organic media with stationary phases of all types. The columns have adapter position pickups for continuous automatic monitoring of the bed height (original “swellographic” monitoring). The column described with gas pressurisation was tested with soft Sephadex G-10 and G-25. In spite of the reduction in external porosity there was no dramatic increase in back-pressure. The column proved to provide long-term stability of the packed bed and improvement in resolution. Presented at the 21^st ISC held in Stuttgart, Germany, 15^th–20^th September, 1996     

Electronic and Spatial Structure of Piperidine and Its Substituted Derivatives from the Results of ab initio Calculations

The results of non empirical quantum-chemical calculations using the RHF/6-31G(d) and MP2/6-31G(d) methods do not agree with proposals for the axial position of the H atom on the N atom in the piperidine molecule. According to RHF/6-31G(d) calculations for the N-methylpiperidine molecule and its chloro-substituted derivatives an equatorially placed methyl group is energetically more favored than an axial. The axial C-Cl and C-H bonds in these molecules are longer than the equatorial. The ³⁵ Cl NQR frequencies for the axial Cl atoms are lower than the equatorial. The ³⁵ Cl NQR frequency of the axial chlorine atom in 2-chloro-1-methylpiperidine is anomalously low. This is chiefly due to the high population density of its p σ-orbital and this is a result of the polarization of the C-Cl bond via the N atom unshared electron pair directly through the field. The effect of a similar unshared electron pair on the parameters of the C-Cl bond in the ClCH₂NH₂ molecule has been studied by the RHF/6-31(g) method for different angles of rotation of the ClCH₂ group around the C-N bond. __________ Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 7, pp. 1044–1052, July, 2005.     

钴的二苯甲酰甲烷配合物：晶体结构及其轴向置换反应

Cobalt(Ⅱ) can form complexes with Hdbm in different environments. Hdbm reacted with cobalt nitrate to give complex 1 [Co(dbm)₂·2H₂O]. When complex 1 reacted with pyridine， α-stilbazole or 4，4′-bipyridine respectively， complex 2 [Co(DBM)₂Py₂] (Py=pyridine)， 3 [Co(DBM)₂Sbz₂] (Sbz=α-stilbazole) or 4 [Co(DBM)₂BPy]_n was obtained in turn through metathetical reaction. The coordination modes are octahedral polyhedrons. In the crystal structures， the two dbms take the plane position and two other donor molecules take the axial position. CCDC： 196070 for complex 2; 186859 for complex 3.     

Influence of the Coordination of Donor Solvent Molecules to 1,4,7,10-Tetramethyl-1,4,7,10- tetraazacyclododecanenickel(II) and Analogous Nickel(II) Complexes on Their Steric Factors

Coordination equilibrium constants (K _NiS) of some donor solvent molecules to 1,4,7,10-tetramethyl-1,4,7,10-tetraazacyclododecanenickel(II) ([Ni(Me₄[12]aneN₄)]²⁺) were determined in nitrobenzene (a noncoordinating bulk solvent). The first (K _NiS1) and second stepwise coordination equilibrium constants (K _NiS2) for 1,4,7,10-tetraazacyclododecanenickel(II) ([Ni([12]aneN₄)]²⁺), 1,4,8,11-tetraazac yclotetradecane- nickel(II) ([Ni([14] aneN₄)]²⁺), 1,4,8,11-tetrathiacyclotetra-decanenickel(II) ([Ni([14]aneS₄)]²⁺) were also reinvestigated. The K _NiS values for [Ni(Me₄[12]aneN₄)]²⁺ were compared to those of [Ni([12]aneN₄)]²⁺, (1R,4S, 8R,11S)-1,4,8,11-tetramethyl-1,4,8,11-tetraazacyclotetradecanenickel(II) (R,S,R,S-[Ni(Me₄[14]aneN₄)]²⁺), R,R,S,S-[Ni(Me₄[14]aneN₄)]²⁺, [Ni([14]aneN₄)]²⁺, and [Ni([14]aneS₄)]²⁺. Coordination of pyridine (Py), N,N,N′,N′-tetramethylurea (TMU), and N,N-dimethylacetamide (DMA) to [Ni(Me₄[12]aneN₄)]²⁺ was observed, although these donor solvent molecules did not coordinate to R,S,R,S-[Ni(Me₄[14]aneN₄)]²⁺. The K _NiS values for Py, TMU, and DMA are 7.9, 2.8, and 9.0 dm³⋅mol⁻¹, respectively. Some hydrogen-bonding waters were coordinated to R,S,R,S-[Ni(Me₄[14]aneN₄)]²⁺, but such waters did not coordinate to [Ni(Me₄[12] aneN₄)]²⁺. Also, the K _NiS2 values were larger than the corresponding K _NiS1 values for [Ni([14]aneS₄)]²⁺. Furthermore, the K _NiS1 values for [Ni([12]aneN₄)]²⁺ were the largest among these nickel(II) complex cations. The K _NiS, K _NiS1, and K _NiS2 values are discussed in terms of properties of the donor solvents and steric strains of these nickel(II) complex cations.     

Complementary kinetic and thermodynamic resolution of a chiral biaryl axis

The condensation of 2′-formylbiphenyl-2-carboxylic acid 4 with (S)-valinol proceeds under kinetic control to give a major product, (4bR,7S,aS)-6,7-dihydro-7-isopropyldibenz[c,e]oxazolo[3,2-a]azepin-9(4bH)-one 6a (84%), in which the biaryl axis has the (S)-configuration. Heating 6a at 140°C with a catalytic amount of acid gives rise to an equilibrium dominated by the diastereoisomeric (4bS,7S)-lactam 6b (6a:6b ratio 27:73), in which the biaryl unit has the (R)-configuration. The structures of both lactams were established by X-ray crystallography; no other diastereoisomers were obtained.