Four positive solutions for the semilinear elliptic equation: -Delta u+u=a(x)u^p+f(x) in {mathbb R}^N

We consider the existence of positive solutions of the following semilinear elliptic problem in : where , , , and . Under the conditions: 1° for all , 2° as , 3° there exist and such that 4°, we show that (*) has at least four positive solutions for sufficiently small but . Received December 11, 1998 / Accepted July 16, 1999 / Published online April 6, 2000     

Pluripolarity of sets with small Hausdorff measure

We show that any set E⊂C ⁿ , n≥ 2, with finite Hausdorff measure? is pluripolar. The result is sharp with respect to the measuring function. The new idea in the proof is to combine a construction from potential theory, related to the real variational integral , , with properties of the pluricomplex relative extremal function for the Bedford–Taylor capacity. Received: 20 May 1999     

The algebraic torus theorem

A structure theorem is proved for a finitely generated group with a finitely generated virtually polycyclic codimension one subgroup. Oblatum 12-II-1999 & 18-XI-1999?Published online: 21 February 2000     

A Sobolev mapping property of the Bergman kernel

We prove that if D is a pseudoconvex domain with Lipschitz boundary having an exhaustion function such that is plurisubharmonic, then the Bergman projection maps the Sobolev space boundedly to itself for any . Received March 10, 1999 / Published online May 8, 2000     

Comparing two methods of resolving homogeneous differential inclusions

Given a compact set we consider the differential inclusion We show how to use the main idea of the method of convex integration [ N], [G], [K] (to control convergence of the gradients of a sequence of approximate solutions by appropriate selection of the sequence) to obtain an optimal existence result. We compare this result with the ones available by the Baire category approach applied to the set of admissible functions with topology. A byproduct of our result is attainment in the minimization problems with integrands L having quasiaffine quasiconvexification that was, in fact, the reason of our interest to differential inclusions. This result can be considered as a first step towards characterization of those minimization problems which are solvable for all boundary data. This problem was solved in [S1] in the scalar case m=1. Received November 5, 1998 / Accepted July 17, 2000 / Published online December 8, 2000     

A new geometric construction of compact complex manifolds in any dimension

We consider holomorphic linear foliations of dimension m of (with ) fulfilling a so-called weak hyperbolicity condition and equip the projectivization of the leaf space (for the foliation restricted to an adequate open dense subset) with a structure of compact, complex manifold of dimension . We show that, except for the limit-case where we obtain any complex torus of any dimension, this construction gives non-symplectic manifolds, including the previous examples of Hopf, Calabi-Eckmann, Haefliger (linear case), Loeb-Nicolau (linear case) and López de Medrano-Verjovsky. We study some properties of these manifolds, that is to say meromorphic functions, holomorphic vector fields, forms and submanifolds. For each manifold, we construct an analytic space of deformations of dimension and show that, under some additional conditions, it is universal. Lastly, we give explicit examples of new compact, complex manifolds, in particular of connected sums of products of spheres and show the existence of a momentum-like map which classifies these manifolds, up to diffeomorphism. Received: 28 October 1998 / in final form: 7 September 1999     

A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities

From Wajnryb's presentation, we extract a simple presentation of the mapping class group of the genus g surface as a quotient of an Artin group by simple relations among the centers of sub-Artin groups. Topological meanings are given by using deformation of simple singularities. Received: 22 January 1998 / in final form: 16 February 1999     

Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval

Summary. A Laguerre-Galerkin method is proposed and analyzed for the Burgers equation and Benjamin-Bona-Mahony (BBM) equation on a semi-infinite interval. By reformulating these equations with suitable functional transforms, it is shown that the Laguerre-Galerkin approximations are convergent on a semi-infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre-Galerkin approximations to the transformed equations is developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Received October 6, 1997 / Revised version received July 22, 1999 / Published online June 21, 2000     

Explicit irreducible representations of the¶Iwahori–Hecke algebra of Type F4

A general method for computing irreducible representations of Weyl groups and Iwahori–Hecke algebras was introduced by the first author in [10]. In that paper the representations of the algebras of types A _n , B _n , D _n and G _n were computed and it is the purpose of this paper to extend these computations to F ₄. The main goal here is to compute irreducible representations of the Iwahori–Hecke algebra of type F ₄ by only using information in the character table of the Weyl group. Received: Received: 30 July 1998     

Certain classes of pluricomplex Green functions on ${\mathbb C}^n$

We consider (pluricomplex) Green functions defined on , with logarithmic poles in a finite set and with logarithmic growth at infinity. For certain sets, we describe all the corresponding Green functions. The set of these functions is large and it carries a certain algebraic structure. We also show that for some sets no such Green functions exist. Our results indicate the fact that the set of poles should have certain algebro-geometric properties in order for these Green functions to exist. Received November 24, 1998; in final form April 19, 1999 / Published online July 3, 2000     

Equivariant evaluation on free loop spaces

There is a natural evaluation map on the free loop space which sends a loop to its values at the kth roots of unity. This map is equivariant with respect to the action of the cyclic group on k elements . We study the induced map in -equivariant cohomology with mod two coefficients in the cases where for . Received: 17 February 2000; in final form: 28 February 2001 / Published online: 18 January 2002     

Scale functions on $p$-adic Lie groups

Let G be a p-adic Lie group. Then G is a locally compact, totally disconnected group, to which Willis [14] associates its scale function G : G→ℕ. We show that s can be computed on the Lie algebra level. The image of s consists of powers of p. If G is a linear algebraic group over ℚ _p , s(x)=s(h) is determined by the semisimple part h of x∈G. For every finite extension K of ℚ _p , the scale functions of G and H:=G(K) are related by s _H ∣ _G =s _G ^{[ K :ℚ} _p ^]. More generally, we clarify the relations between the scale function of a p-adic Lie group and the scale functions of its closed subgroups and Hausdorff quotients. Received: 20 February 1997; Revised version: 18 May 1998