Free differential algebras: Their use in field theory and dual formulation

The gauging of free differential algebras (FDA's) produces gauge field theories containing antisymmetric tensors. The FDA's extend the Cartan-Maurer equations of ordinary Lie algebras by incorporating p-form potentials (p>1). We study here the algebra of FDA transformations. To every p-form in the FDA, we associate an extended Lie derivative l generating a corresponding gauge transformation. The field theory based on the FDA is invariant under these new transformations. This gives geometrical meaning to the antisymmetric tensors. The algebra of Lie derivatives is shown to close and provides the dual formulation of FDA's.     

A borel-weil-bott approach to representations of sl q (2, ℂ)

We use a quite concrete and simple realization of sl _q (2, 50.gif" alt="Copf" align="BASELINE" BORDER="0">) involving finite difference operators. We interpret them as derivations (in the noncommutative sense) on a suitable graded algebra, which gives rise to the noncommutative scheme ¹ II ^1* as the counterpart of the standard ¹ = Sl(2, 50.gif" alt="Copf" align="BASELINE" BORDER="0">)/B.     

On spectral properties ofq-oscillator operators

The property of self-adjointness of the operatorQ =a ₊ +a _- in three types ofq-oscillator algebras is considered. Spectral measures and generalized eigenfunctions ofQ are found in the cases when this operator is bounded. Generalized eigenvectors are expressed in terms ofq-Hermite polynomials. If the operatorQ is unbounded, then its closure is not self-adjoint. However, in this case, admits self-adjoint extensions. Deficiency subspaces are one-dimensional. These subspaces are explicitly found.     

Behavior of alternation points in best rational approximation

The behavior of the equioscillation points (alternants) for the error in best uniform approximation on [–1, 1] by rational functions of degreen is investigated. In general, the points of the alternants need not be dense in [–1, 1], even when approximation by rational functions of degree (m, n) is considered and asymptoticallym/n 50x72643374775/xxlarge8805.gif" alt="ge" align="MIDDLE" BORDER="0"> 1. We show, however, that if more thanO(logn) poles of the approximants stay at a positive distance from [–1, 1], then asymptotic denseness holds, at least for a subsequence. Furthermore, we obtain stronger distribution results when 50x72643374775/xxlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">n (0 < 50x72643374775/xxlarge955.gif" alt="lambda" align="BASELINE" BORDER="0"> 50x72643374775/xxlarge8804.gif" alt="le" align="MIDDLE" BORDER="0"> 1) poles stay away from [–1, 1]. In the special case when a Markoff function is approximated, the distribution of the equioscillation points is related to the asymptotics for the degree of approximation.The research of this author was supported, in part, by NSF grant DMS 920-3659.     

A bound for minimal graphs with a normal at infinity

It is shown that a minimal graph with a normal at infinity is in a-priori bounded vertical distance from its approximating halfcatenoid. This is used to show that the exterior contact angle problem is wellposed under natural geometric conditions on the domain, while the exterior Dirichlet problem can be solvable only for data which satisfy an oscillation bound.This paper was written under the support of the Deutsche Forschungsgemeinschaft while the author was visiting the department of mathematics at Stanford University.This article was processed by the author using the LaT_EX style filepljour1 from Springer-Verlag.     

The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms

Given a vector of real numbers=(₁,... _d ) ^d , the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d+1)×(d+1) convergent matrices {C⁽ⁿ⁾():n1} whose rows provide Diophantine approximations to . Such algorithms are specified by two mapsT:[0, 1] ^d [0, 1] ^d and A:[0,1] ^d GL(d+1,), which compute convergent matrices C⁽ⁿ⁾())...A(T())A(). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent is the upper bound of those values of for which there is some row of the convergent matrices such that for infinitely many values ofn that row of C⁽ⁿ⁾() has . The uniform approximation exponent is the upper bound of those values of such that for all sufficiently large values ofn and all rows of C⁽ⁿ⁾⁽⁾ one has . The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms and take constant values and on a set of Lebesgue measure one. It establishes the formula where are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A,d), whered is aT-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that holds for a large class of such algorithms. These results apply to thed-dimensional Jacobi-Perron algorithm and Selmer's algorithm. We show that; experimental evidence of Baldwin (1992) indicates (nonrigorously) that. We conjecture that holds for alld2.     

Hydrodynamic limit for a spin system on a multidimensional lattice

Summary The hydrodynamic limit for a Markov process of [0, 503/xxlarge8734.gif" alt="infin" align="MIDDLE" BORDER="0">)-valued spin fields on a periodic multidimensional lattice is studied. In the process a positive real number, called energy, is attached to each site of the lattice and each couple of adjacent sites exchange thier energy by random amounts at random times. The law of the exchange is such that the sum of the total energy is conserved, and that the process is reversible and of gradient type for the energy distribution. We show that under diffusion type scaling of space and time, the macroscopic energy distribution converges to a deterministic limit which is characterized by a non-linear diffusion equation 503/xxlarge8706.gif" alt="part" align="MIDDLE" BORDER="0">503/xxlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0">/503/xxlarge8706.gif" alt="part" align="MIDDLE" BORDER="0">t=2^–1503/xxlarge916.gif" alt="Delta" align="BASELINE" BORDER="0">P(503/xxlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0">), whereP is an increasing function which in a typical case equals const·503/xxlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0">².     

Theoretical study of structure and stability of fullerene derivative: C50O

A total of eight possible isomers of C₅₀O, an oxide of fullerene C₅₀ (D_5h), have been investigated by B3LYP/3‐21G calculations. The isomer, which has an annulene‐like structure with oxygen bridging across a [5,6] type C? C bond at the site between the pole and the equatorial belt, is found being the ground state of C₅₀O. Four isomers are relatively more stable and the energy differences between them are not large. This result indicates that more than one C₅₀O isomer will coexist once C₅₀O is synthesized. The relative stabilities of the C₅₀O isomers may be determined mainly by the strain release and by the formation of the cyclic phenylene substructure at the equatorial belt of the cage. The calculated nucleus independent chemical shifts (NICS) of the C₅₀O isomers will be useful because from them one can expect outstanding NMR properties that can lead to their identification and characterization. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

The inclusion complex of piromidic acid with dimethyl-β-cyclodextrin in aqueous solution and in the solid state

Inclusion complex formation of piromidic acid (PA) with dimethyl-50q46qm188524t7/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0">-cyclodextrin (DM-50q46qm188524t7/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0">-CD) in aqueous solution and in the solid state was confirmed by the solubility method, differential scanning calorimetry (DSC) and proton nuclear magnetic resonance (¹H-NMR) spectroscopy. The apparent stability constant,K _c , of the complex was estimated to be 244 M^–1. The stoichiometry of the complex was given as the ratio 1:2 of PA to DM-50q46qm188524t7/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0">-CD. The dissolution rate of the PA/DM-50q46qm188524t7/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0">-CD complex was much greater than that of intact PA.Presented at the Fourth International Symposium on Inclusion Phenomena and the Third International Symposium on Cyclodextrins, Lancaster, U.K., 20–25 July 1986.     

Determining PPARγ-ligand binding affinity using fluorescent assay with cis-parinaric acid as a probe

Upon the study of small-molecules binding to proteins, the traditional methods for calculating dissociation constants (Kd and Ki) have shortcomings in dealing with the single binding site models. In this paper, two equations have been derived to solve this problem. These two equations are independent of the total concentration or initial degree of saturation of receptor and the activity of the competitive molecule. Through nonlinear fitting against these two equations, Kd value of a probe can be obtained by binding assay, and Ki value of a ligand can be obtained by competitive assay. Moreover, only the total concentrations of receptor([R]t), ligand([L]t) and probe([P]t) are required for the data fitting. In this work, Ki values of some typical ligands of PPARγ were successfully determined by use of our equations, among which the Ki value of PPARγ-LY171883 was reported for the first time.