Schur Function Identities Arising From the Basic Representation of {A^{(2)}_{2}}

A Lie theoretic interpretation is given for some formulas of Schur functions and Schur Q-functions. Two realizations of the basic representation of the Lie algebra \({A^{(2)}_2}\) are considered; one is on the fermionic Fock space and the other is on the bosonic polynomial space. Via the boson–fermion correspondence, simple relations of the vacuum expectation values of fermions turn out to be algebraic relations of Schur functions.     

Two-dimensional correlation gel permeation chromatography (2D GPC) study of the CH<Subscript>3</Subscript>SO<Subscript>3</Subscript>H-catalyzed polymerization of triethoxysilyl-terminated polystyrene. Molecular weight effect on the aggregate–aggregate interactions

The two-step polymerization process of two well-defined polymeric silane coupling agents, triethoxysilyl-terminated polystyrene with molecular weights equal to 2400 [TESi-PS (2400)] and 8000 [TESi-PS (8000)], catalyzed by 0.1 mol/kg CH₃SO₃H, was traced as a function of reaction time using gel-permeation chromatography (GPC). Two sets of GPC traces, collected during the condensation, were then converted to two-dimensional (2D) correlation spectra by using generalized 2D correlation theory. The 2D correlation spectra elucidated details of the aggregate–aggregate correlations [in particular, the difference between the correlations of TESi-PS (2400) and TESi-PS (8000)], thus demonstrating the effect of aggregation on the polymerization.     

Two-dimensional gel permeation chromatography (2-D GPC) correlation studies of the aggregate–aggregate interactions in acid-catalyzed triethoxysilyl-terminated polystyrene systems: weak catalysis by HNO<Subscript>3</Subscript>

Time-resolved gel-permeation chromatography (GPC) profiles were measured over a long period of time (t=0–768 h) for the HNO₃-catalyzed reaction mixture of a well-defined polymeric silane coupling agent, triethoxysilyl-terminated polystyrene. The results showed that HNO₃ catalyzes the rate of polymerization, but only to a small extent. Two-dimensional (2-D) GPC correlation spectra were calculated from time-resolved GPC profiles and were used to examine the mechanism of polymerization. The resolution enhancement obtained by 2-D correlation provided ample evidence for existence of four monomeric components. It has been demonstrated that dynamic variation in the population of these components occurs over a long reaction time, reflecting the mechanism of a slow polymerization process.     

Polynomial τ-Functions of the NLS-Toda Hierarchy and the Virasoro Singular Vectors

A family of polynomial -functions for the NLS-Toda hierarchy is constructed. The hierarchy is associated with the homogeneous vertex operator representation of the affine algebra of type A ₁ ⁽¹⁾. These -functions are given explicitly in terms of Schur functions that correspond to rectangular Young diagrams. It is shown that an arbitrary polynomial -function which is an eigenvector of d, the degree operator of , is contained in the family. By the construction, any -function in the family becomes a Virasoro singular vector. This consideration gives rise to a simple proof of known results on the Fock representation of the Virasoro algebra with c = 1.     

Folded structures of L-leucylglycine oligopeptides and their aggregational behavior in aqueous solution: Raman scattering spectra and proton NMR spin-lattice relaxation studies

The aggregational behavior of three L-leucylglycine oligopeptides (residue numbers of glycine are 3, 4, and 5) in aqueous solution was investigated by the use of Raman scattering and 1H NMR spin-lattice relaxation methods. The results indicate that their oligopeptides take up a folded structure to form dimeric aggregates above their critical aggregation concentration. The application of one-dimensional aggregate theory to these systems provides the following prediction. Elongation up to 6 glycine residues makes it possible to form dimeric aggregates, but further elongation (up to 7 glycine residues) makes the aggregates very unstable, and up to 8 or 9 glycine residues makes the formation of dimeric aggregates very difficult. The one-dimensional aggregate theory may be used to predict the existence of peptide aggregates through intermolecular hydrogen bonding.     

Two-dimensional gel permeation chromatography (2D GPC) correlation studies of the aggregate-aggregate interactions in acid-catalyzed triethoxysilyl-terminated polystyrene systems. The effect of specific catalysts on growth process

A set of time resolved gel-permeation chromatography (GPC) profiles, for the reaction mixture of a well-defined polymeric silane coupling agent, triethoxysilyl-terminated polystyrene with molecular weight equal to 8000 (TESi-PS (8000)), catalyzed by HCl (0.1 mol/kg) in tetrahydrofuran (THF), was measured over a long time scale (1–768 h). The GPC profiles were then converted to two-dimensional (2D) correlation spectra. The 2D GPC correlation spectra were compared with those for the CH₃SO₃H (0.1 mol/kg)-catalyzed TESi-PS (8000)-THF system. It has been demonstrated that predominant production of less-reactive oligomers in the HCl-catalyzed system hinders further growth of the oligomer, while formation of reactive oligomers leads to further growth of polymeric precursors in the CH₃SO₃H-catalyzed system.     

Mixed expansion formula for the rectangular Schur functions and the affine Lie algebra

Formulas are obtained that express the Schur S-functions indexed by Young diagrams of rectangular shape as linear combinations of “mixed” products of Schur's S- and Q-functions. The proof is achieved by using representations of the affine Lie algebra of type . A realization of the basic representation that is of “”-type plays the central role.     

Concentration-resolved 2D correlation gel permeation chromatography study of aggregate-aggregate interactions in the polymerization products of triethoxysilyl-terminated polystyrene silane-coupling agent

Concentration-resolved 2D correlation gel permeation chromatography (GPC) has been used to examine the intricate details of the HCl-catalyzed polymerization of a polymeric silane-coupling agent (SCA), triethoxysilyl-terminated polystyrene (TESiPS). The concentration-resolved 2D correlation GPC maps directly reflect the marked difference in the aggregate-aggregate interactions of dilute and concentrated monomeric units, which govern the differences in the polymerized products. There is an optimum concentration of SCA for the enhancement of interfacial strength and subsequent polymerization. Thus, the concentration-resolved 2D correlation GPC technique can be used as a powerful tool for elucidation of aggregate-aggregate interactions and reaction mechanisms in a surface- or interface-enhanced reaction system. It has been shown that the yield value of polymerization products can be improved to a marked extent by choosing a high initial monomer concentration, due to the increase in the production of oligomers. Multiple reaction processes are promoted by the self-assembly of the monomeric and oligomeric components.     

Compound Basis Arising from the Basic {A^{(1)}_{1}}-Module

Given a conditionally completely positive map on a unital *-algebra , we find an interesting connection between the second Hochschild cohomology of with coefficients in the bimodule of adjointable maps, where M is the GNS bimodule of , and the possibility of constructing a quantum random walk [in the sense of (Attal et al. in Ann Henri Poincar 7(1):59–104, 2006; Lindsay and Parthasarathy in Sankhya Ser A 50(2):151–170, 1988; Sahu in Quantum stochastic Dilation of a class of Quantum dynamical Semigroups and Quantum random walks. Indian Statistical Institute, 2005; Sinha in Banach Center Publ 73:377–390, 2006)] corresponding to . D. Goswami was supported by a project funded by the Indian National Academy of Sciences. L. Sahu had research support from the National Board of Higher Mathematics, DAE (India) is gratefully acknowledged.     

Pfaffian identities and Virasoro operators

Letters in Mathematical Physics - A formula for Schur Q-functions is presented which describes the action of the Virasoro operators. For a strict partition $$\lambda =(\lambda _1,\lambda _2,\ldots...