with central chargeN

We study representations of the central extension of the Lie algebra of differential operators on the circle, the algebra. We obtain complete and specialized character formulas for a large class of representations, which we call primitive; these include all quasi-finite irreducible unitary representations. We show that any primitive representation with central chargeN has a canonical structure of an irreducible representation of the with the same central charge and that all irreducible representations of with central chargeN arise in this way. We also establish a duality between integral modules of and finite-dimensional irreducible modules ofgl _N , and conjecture their fusion rules.Supported by a Junior Fellowship from Harvard Society of Fellows and in part by NSF grant DMS-9205303.Supported in part by NSF grant DMS-9103792.     

Free energies of adsorption of amino acids, short linear peptides and 2,5-piperazinediones on silica from water as estimated from high-performance liquid-chromatographic retention data

Equilibrium constants (K) and free energies (— G) of adsorption of amino acids, short linear peptides and cyclic dipeptides (2,5-piperazinediones) on silica from neutral aqueous solutions were calculated from the retention values measured by means of high-performance liquid chromatography on a silica gel column. For most amino acids and linear peptides — G values were negative and K < 1, thus showing very weak adsorption. 2,5-Piperazinediones exhibited higher adsorbability (for most of them, — G > 0 and K > 1) as compared to related dipeptides. Influence of the structure of a-substituent on the adsorbability is analyzed. A linear dependence of — G on the number of aliphatic carbon atoms in a sorbate molecule was found for the series of aliphatic bifunctional amino acids, related dipeptides and 2,5-piperazinediones, as well as for the group from glycine to triglycyl glycine.     

Synthesis of sulfur-containing dehydrodesthiobiotin derivatives

Conclusions The reduction of the ethyl esters of- and'-oxodehydrodesthiobiotin with sodium borohydride, and subsequent treatment of the intermediate hydroxy esters with thioacetic acid, gave the ethyl esters of- and'- (aceitylmercapto)dehydrodesthiobiotin.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 11, pp. 2626–2627, November, 1973.We express our gratitude to V. V. Filippov for running the biological tests on (V), (VI) and (XI).     

A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms.     

Nanostructures on the Basis of Porous Alumina with Intercalated with Cholesteric Liquid Crystal

The introduction of liquid crystal materials into the porous material host will help us to get optically active nanocomposites. Nanocomposite based on aluminum oxide host with cholesteric liquid crystal is characterized by a shift of bandwidth wavelength minimum. In case of intercalation of cholesteric liquid crystals into the pores of aluminum oxide host the shift of transmittance minimum into the short wave region are observed. We analyzed the ways, which indicate the deformation of the pitch of the cholesteric helix and showed the role of capillary forces in the deformation of the pitch of cholesteric liquid crystal.     

Kinetics of atomic ordering in metal-doped graphene

Possible stably ordered substitutional structures based on a graphene-type crystal lattice are considered. A kinetic model of atomic ordering in metal-doped graphene with stoichiometric (1/8, 1/4, 1/2) and nonstoichiometric compositions is developed. Inasmuch as the intrasublattice and intersublattice ‘interchange’ (‘mixing’) energies are competitively different for graphene-based lattice, kinetic curves of the long-range order (LRO) parameters may be nonmonotonic for the structures described by two or three LRO parameters.     

Expanding the scope of atropisomeric monodentate P-donor ligands in asymmetric catalysis. Asymmetric allylic alkylation of 1,3-diphenylpropenyl-1-esters by Pd/BINEPINE catalysts

Monodentate binaphthophosphepines (BINEPINES) have been tested in the palladium-catalyzed asymmetric allylic alkylation of 1,3-diphenylallyl esters. Stereoselectivities of up to 92% ee have been achieved in the alkylation of the acetate ester with the C-nucleophile, generated in situ by the action of BSA (N,O-bis(trimethylsilyl)acetamide) on dimethylmalonate in the presence of a catalytic amount of sodium acetate.     

Accelerated Imaginary-time Evolution Methods for the Computation of Solitary Waves

Two accelerated imaginary-time evolution methods are proposed for the computation of solitary waves in arbitrary spatial dimensions. For the first method (with traditional power normalization), the convergence conditions as well as conditions for optimal accelerations are derived. In addition, it is shown that for nodeless solitary waves, this method converges if and only if the solitary wave is linearly stable. The second method is similar to the first method except that it uses a novel amplitude normalization. The performance of these methods is illustrated on various examples. It is found that while the first method is competitive with the Petviashvili method, the second method delivers much better performance than the first method and the Petviashvili method.