This paper is a sequel of [ER]. Specifically, letW be a Coxeter group, generated by si, i ∈ I. Then, following [ER], one can define a new deformation A+ = A+(W ) of the group algebra Z[W+] of the group W+ of even elements in W . This deformation is an algebra over the ring R = Z[t±1 ijk] = Z[T] of regular functions on a certain torus T of deformation parameters. The main result of [ER] implies ...

We show that Bowen’s equation, which characterises the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In particular, we consider an arbitrary subset Z of a compact metric space and require only that the lower Lyapunov exponents be positive on Z, together with a tempered c...

Motivated by physical models and the so-called Crocco equation, we study the controllability properties of a class of degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold for this problem in general. First, we prove that we can drive the solution to rest at time T in a suitable subset of the space domain (regional null controllability). However, ...

This paper considers some typical optimal control problems for a class of strongly nonlinear parabolic systems. After some necessary preparation, it is shown that the family of admissible trajectories is a weakly closed and weakly sequentially compact subset of a reflexive Banach space and that the set of attainable states at any given time is a weakly compact subset of a Hilbert space. Using t...

Let S (respectively, S 0) be a nite subset of a compact connected Riemann surface X (respectively, X 0) of genus at least two. Let M (respectively, M 0) denote a moduli space of parabolic stable bundles of rank two over X (respectively, X 0) with xed determinant of degree one, having a nontrivial quasi-parabolic structure over each point of S (respectively, S 0), and of parabolic degree less th...

Let k be an algebraically closed field of characteristic exponent p ≥ 1. Let G be a connected reductive algebraic group over k and let g be the Lie algebra of G. Note that G acts on G and on g by the adjoint action. Let UG be the variety of unipotent elements of G. Let Ng be the variety of nilpotent elements of g. In [L2], we have proposed a definition of a partition of UG into smooth locally c...

In this paper we continue the study of character sheaves on a reductive group. To each subset of the set of simple reflections in the Weyl group we associate an algebra of the same kind as an Iwahori Hecke algebra with unequal parameters in terms of parabolic character sheaves. We also prove a Mackey type formula for character sheaves. We define a duality operation for character sheaves. We als...

where Ω is a bounded and open subset in R, with a smooth boundary ∂Ω. Problems of form (P) arise in flow in porous media, nonlinear heat equation with absorption, chemical isothermic reactions and unidirectional Non-Newtonian fluids, see [4] . Therefore, it is important to obtain information about the existence of the global attractor and its regularity. In recent years, numbers of works have c...

Let g be a finite-dimensional semisimple Lie algebra over C, and let U be the qanalogue of its universal enveloping algebra defined by Drinfel’d [3] and Jimbo [5]. According to [7, 3.5.6, 6.2.3 & 6.3.4], for each dominant weight λ in the weight lattice of g there is an irreducible, finite-dimensional highest weight U -module V (λ) with highest weight λ. Kashiwara [6] and Lusztig [7, 14.4.12] ha...