Using the Harish-Chandra D-module, we give a proof of Haiman’s theorem on the positivity of Macdonald polynomials. Ginzburg’s work on the connection between this D-module and the isospectral commuting variety is fundamental to this approach.

For a linear ODE with indeterminate coefficients, we explicitly exhibit a fundamental system of solutions in terms of the coefficients. We show that the generalized Wronskians of the fundamental system are given by an action of the Schur functions on the usual Wronskian, and thence enjoy Pieri’s and Giambelli’s formulae. As an outcome, we obtain a natural isomorphism between the free module gen...

A 165-fs all-fiber ring laser is demonstrated with a fundamental repetition rate of 235 MHz based on a 5.7-cm-long Er(3+)/Yb(3+) codoped phosphate glass fiber and a technique of nonlinear polarization evolution. In order to further enhance the fundamental repetition rate and compact the structure of the all-fiber laser, an optical integrated module is designed. By employing this novel optical m...

To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than t...

Let F be a field, V a 6-dimensional F-vector space and f a nondegenerate alternating bilinear form on V . We consider a 14-dimensional module for the symplectic group Sp(V, f) ∼= Sp(6,F) associated with (V, f), and classify the orbits on vectors. For characteristic distinct from 2, this module is irreducible and isomorphic to the Weyl module of Sp(V, f) for the fundamental weight λ3. If the cha...

in this paper we study the module contractibility ofbanach algebras and characterize them in terms the conceptssplitting and admissibility of short exact sequences. also weinvestigate module contractibility of banach algebras with theconcept of the module diagonal.