Effective codescent morphisms, amalgamations and factorization systems

The paper deals with the question whether it is sufficient, when investigating the problem of the effectiveness of a descent morphism, to restrict the consideration only to the descent data (C,γ,ξ), where γ lies in a certain morphism class. The notion of a factorization system and the dual to the amalgamation property in the sense of Kiss, Marki, Pröhle and Tholen play the key role in our discussion.It is shown that a category inherits from a category the property that all descent morphisms are effective if either is regular and is a full coreflective, closed under pullbacks of certain epimorphisms, subcategory of or is regular, has coequalizers and there exists a topological functor . This implies that in the category of topological spaces, all regular monomorphisms are effective codescent morphisms (the result of Mantovani). The same is shown to be valid also for the categories of compact Hausdorff topological spaces, normal topological spaces, Banach spaces, (quasi-)uniform spaces, and (quasi-)proximity spaces. Moreover, the effectiveness of all codescent morphisms is established for the categories of Hausdorff topological spaces and (complete) metric spaces. The internal characterization of such morphisms p:B→E is given for the category of Hausdorff topological spaces, in the case of compact B and regular E.     

On the Mahler measure of resultants in small dimensions

We prove that sparse resultants having Mahler measure equal to zero are those whose Newton polytope has dimension one. We then compute the Mahler measure of resultants in dimension two, and examples in dimension three and four. Finally, we show that sparse resultants are tempered polynomials. This property suggests that their Mahler measure may lead to special values of -functions and polylogarithms.     

The inversion formula for automorphisms of the Weyl algebras and polynomial algebras

Let A_n be the nth Weyl algebra and P_m be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {A_n⊗P_m} is proved: an algebraAadmits a finite setδ₁,…,δ_sof commuting locally nilpotent derivations with generic kernels andiffA?A_n⊗P_mfor somenandmwith2n+m=s, and vice versa. The inversion formula for automorphisms of the algebra A_n⊗P_m (and for ) has been found (giving a new inversion formula even for polynomials). Recall that (see [H. Bass, E.H. Connell, D. Wright, The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (New Series) 7 (1982) 287-330]) given, then (the proof is algebro-geometric). We extend this result (using [non-holonomic] D-modules): given, then. Any automorphism is determined by its face polynomials [J.H. McKay, S.S.-S. Wang, On the inversion formula for two polynomials in two variables, J. Pure Appl. Algebra 52 (1988) 102-119], a similar result is proved for .One can amalgamate two old open problems (the Jacobian Conjecture and the Dixmier Problem, see [J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968) 209-242. [6]] problem 1) into a single question, (JD): is aK-algebra endomorphismσ:A_n⊗P_m→A_n⊗P_man algebra automorphism providedσ(P_m)⊆P_mand? (P_m=K[x₁,…,x_m]). It follows immediately from the inversion formula that this question has an affirmative answer iff both conjectures have (see below) [iff one of the conjectures has a positive answer (as follows from the recent papers [Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka J. Math. 42(2) (2005) 435-452. [10]] and [A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier Conjecture. ArXiv:math.RA/0512171. [5]])].     

The Stokes operator in weighted Lq-spaces II: weighted resolvent estimates and maximal Lp-regularity

In this paper we establish a general weighted L ^q -theory of the Stokes operator in the whole space, the half space and a bounded domain for general Muckenhoupt weights . We show weighted L ^q -estimates for the Stokes resolvent system in bounded domains for general Muckenhoupt weights. These weighted resolvent estimates imply not only that the Stokes operator generates a bounded analytic semigroup but even yield the maximal L ^p -regularity of in the respective weighted L ^q -spaces for arbitrary Muckenhoupt weights . This conclusion is archived by combining a recent characterisation of maximal L ^p -regularity by -bounded families due to Weis [Operator-valued Fourier multiplier theorems and maximal L _p -regularity. Preprint (1999)] with the fact that for L ^q -spaces -boundedness is implied by weighted estimates.     

Compact and subelliptic estimates for the $${\overline{\partial}}$$-Neumann operator on C2 pseudoconvex domains

In this paper, we will show that Catlin’s property (P) implies regularity for the -Neumann operator in W ¹ on C ² domains. We will also look at the subelliptic case and derive the natural generalization to C ^k domains when k ≥  2.     

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier–Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as t goes to the possible time of singularity T. For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, . For the Navier–Stokes equations the convergence of the velocity to the self-similar singularity is in L ^q (B(z,r)) for some , where the ball of radius r is shrinking toward a possible singularity point z at the order of as t approaches to T. In the convergence case with we present a simple alternative proof of the similar result in Hou and Li in arXiv-preprint, math.AP/0603126. This work was supported partially by KRF Grant(MOEHRD, Basic Research Promotion Fund) and the KOSEF Grant no. R01-2005-000-10077-0.     

Dégénérescence de séries d’Eisenstein hyperboliques

In this work, we deal with hyperbolic Eisenstein series and more particularly, given a degenerating family S _l (l ≥ 0) of Riemann Surfaces with their canonical hyperbolic metrics, we work out the degeneration of hyperbolic Eisenstein series associated to the pinching geodesics in S _l . Our principal results are theorems 2.2, 4.1 et 4.2.     

Machine scheduling with resource dependent processing times

We consider machine scheduling on unrelated parallel machines with the objective to minimize the schedule makespan. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a discrete renewable resource, e.g. workers. A given amount of that resource can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes the classical unrelated parallel machine scheduling problem by adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of an integer linear programming formulation for a relaxation of the problem, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham’s list scheduling, we show how to derive a 4-approximation algorithm. We also show how to tune our approach to yield a 3.75-approximation algorithm. This is achieved by applying the same rounding technique to a slightly modified linear programming relaxation, and by using a more sophisticated scheduling algorithm that is inspired by the harmonic algorithm for bin packing. We finally derive inapproximability results for two special cases, and discuss tightness of the integer linear programming relaxations.     

Cartan decomposition of the moment map

We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniques we establish the existence of a quotient and a version of Luna’s slice theorem as well as a version of the Hilbert–Mumford criterion. A global slice theorem is proved for proper actions. We give new proofs of results of Mostow on decompositions of groups and homogeneous spaces.First author partially supported by the Sonderforschungsbereich SFB/TR12 of the Deutsche Forschungsgemeinschaft and the DFG Schwerpunk program Globale Methoden in der komplexen Geometrie.Second author partially supported by NSA grant H98230–04–01–0070.     

Curvature estimates for graphs with prescribed mean curvature and flat normal bundle

We consider graphs with prescribed mean curvature and flat normal bundle. Using techniques of Schoen et al. (Acta Math 134:275–288, 1975) and Ecker and Huisken (Ann Inst H Poincaré Anal Non Linèaire 6:251–260, 1989), we derive the interior curvature estimate