We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets K ⊂ H of a Hilbert space H by the metric entropy of the set K where the covering numbers N(K, ") of K by "-balls of H satisfy the Lorentz condition ∫ ∞ 0 ( log2N(K, ") )r/s d" <∞ for some fixed 0 < r, s ≤ ∞ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 <...

In the fundamental paper [11], J. Dixmier posed six problems for the Weyl algebra A1 over a field K of characteristic zero. Problem 3 was solved by Joseph and Stein [15] (using results of McConnel and Robson [20]), problem 5 was solved by the author in [7]. Using a (difficult) polarization theorem for the Weyl algebra A1 Joseph [15] solved problem 6. Problems 1, 2, and 4 are still open. Note th...

We study the inverse scattering problem for a Schrödinger operator related to static wave with variable velocity, using GLM (Gelfand–Levitan–Marchenko) integral equation. assume have noisy data, and we derive stability estimate error of solution equation by showing invertibility between suitable function spaces. To regularise problem, formulate variational total least squares show that, under c...

In this paper we establish existence and multiplicity results for a class of fully nonlinear elliptic equations of k-Hessian type with exponential nonlinearity. In particular, we characterize the precise dependence of the multiplicity of solutions with respect to both the space dimension and the value of k. The choice of exponential nonlinearity is motivated by the classical Liouville-Gelfand p...

This work is part of a project on weight bases for the irreducible representations of semisim-ple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of sln, the celebrated Gelfand-Tsetlin basis is the only such basis known. Using the setup of supporting graphs developed by Donnelly, we present a simple combina-t...

Using a technique based on the Yangian Gelfand-Zetlin algebra and the associated Yangian Gelfand-Zetlin bases we construct an orthogonal basis of eigenvectors in the Calogero-Sutherland Model with spin, and derive product-type formulas for norms of these eigenvectors. 1 e-mail: [email protected] 2 e-mail: [email protected]

In the study of simple modules over a simple complex Lie algebra, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this note, we define a family of categories which generalizes the BGG category. We classify the simple modules for some of these categories. As a consequence we show that these categories are semisimple....