Volume 25, Issue 5

Abstract. We prove some global Morrey regularity results for almost minimizers of functionals of the form This regularity is valid up to the boundary, provided the boundary data is sufficiently regular. The main assumption on f is that for each and , the function behaves asymptotically like , where h is an N-function with comparable to . We provide a couple of applications of this result: an application to a broad class of PDEs, and a result which, for a large class of functionals, provides a minimizing sequence with uniform regularity.

Abstract. In the present paper we develop the splitting-up scheme (so-called method of fractional steps) for the investigation of existence problem for a class of nonlinear hyperbolic equations containing some nonlinear terms which do not satisfy the Lipschitz condition. Through a careful blending of the numerical scheme and deep compactness results of both analytic and probabilistic nature we establish the existence of a weak probabilistic solution for the problem. Our work is the stochastic counterpart of some important results of Roger Temam obtained in the late sixties of the last century in his works on the development of the splitting-up method for deterministic evolution problems.

Abstract. We deal with some pcf (possible cofinality theory) investigations mostly motivated by questions in abelian group theory. We concentrate on applications to test problems but we expect the combinatorics will have reasonably wide applications. The main test problem is the “trivial dual conjecture” which says that there is a quite free abelian group with trivial dual. The “quite free” stands for “-free” for a suitable cardinal , the first open case is . We almost always answer it positively, that is, prove the existence of -free abelian groups with trivial dual, i.e., with no non-trivial homomorphisms to the integers. Combinatorially, we prove that “almost always” there are which are quite free and have a relevant black box. The qualification “almost always” means except when we have strong restrictions on cardinal arithmetic, in fact restrictions which hold “everywhere”. The nicest combinatorial result is probably the so-called “Black Box Trichotomy Theorem” proved in ZFC. Also we may replace abelian groups by R -modules. Part of our motivation (in dealing with modules) is that in some sense the improvement over earlier results becomes clearer in this context.

Abstract. According to the Birch and Swinnerton-Dyer conjectures, if is an abelian variety, then its L -function must capture a substantial part of the properties of  A . The smallest number field where A has all its endomorphisms defined must also play a role. This article deals with the relationship between these two objects in the specific case of modular abelian varieties associated to weight 2 newforms for the group Γ 1 ( N ). Specifically, our goal is to relate , with the order at of Euler products restricted to primes that split completely in . This is attained when a power of A f is isogenous over to the Weil restriction of the building block of  A f . We give separated formulae for the CM and non-CM cases.

Abstract. Given a mapping class f of an oriented surface and a lagrangian λ in the first homology of , we define an integer . We use to describe a universal central extension of the mapping class group of as an index-four subgroup of the extension constructed from the Maslov index of triples of lagrangian subspaces in the homology of the surface. We give two descriptions of this subgroup. One is topological using surgery, the other is homological and builds on work of Turaev and work of Walker. Some applications to TQFT are discussed. They are based on the fact that our construction allows one to precisely describe how the phase factors that arise in the skein theory approach to TQFT-representations of the mapping class group depend on the choice of a lagrangian on the surface.

Abstract. Erratum to http://dx.doi.org/10.1515/form.2011.073.