When subset-sums do not cover all the residues modulo p

Let . We prove that a subset of , where p is a prime number, with cardinality larger than such that its subset sums do not cover has an automorphic image which is rather concentrated; more precisely, there exists s prime to p such that

     

Geodesic currents and Teichmüller space

Consider a hyperbolic surface X of infinite area. The Liouville map assigns to any quasiconformal deformation of X a measure on the space of geodesics of the universal covering X? of X. We show that the Liouville map is a homeomorphism from the Teichmüller space onto its image, and that the image is closed and unbounded. The set of asymptotic rays to consists of all bounded measured laminations on X. Hence, the set of projective bounded measured laminations is a natural boundary for . The action of the quasiconformal mapping class group on continuously extends to this boundary for .     

Exponentiable monomorphisms in categories of domains

We give a characterization of exponentiable monomorphisms in the categories of ω-complete posets, of directed complete posets and of continuous directed complete posets as those monotone maps f that are convex and that lift an element (and then a queue) of any directed set (ω-chain in the case of ) whose supremum is in the image of f (Theorem 1.9). Using this characterization, we obtain that a monomorphism f:X→B in (, ) exponentiable in w.r.t. the Scott topology is exponentiable also in (, ). We prove that the converse is true in the category , but neither in , nor in .     

Caractérisation des feuilletages holomorphes singuliers, contenus dans des feuilletages Levi flat, sur les surfaces complexes compactes

Let be a holomorphic foliation with reduced singularities on a complex surface M and a real analytic codimension one foliation on M whose leaves contain the ones of . We show that a Levi flat group of diffeomorphisms of is resoluble and holomorphically conjugate to his normal form. We deduce, in one hand, that each singularity of  is conjugate to his normal form. In the other hand at each singularity m of , where is not defined, up a conjugacy, by the one form ω=xdy+ydx, one of the local invariant curves of , with non obvious holonomy, is contained in the set of singularities of . Moreover if M is a compact Stein variety we show, under some generic conditions, that has a 1-Liouvillian first integrating factor.     

Polynomial identities and noncommutative versal torsors

To any cleft Hopf Galois object, i.e., any algebra obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two “universal algebras” and . The algebra is obtained by twisting the multiplication of H with the most general two-cocycle σ formally cohomologous to α. The cocycle σ takes values in the field of rational functions on H. By construction, is a cleft H-Galois extension of a “big” commutative algebra . Any “form” of can be obtained from by a specialization of and vice versa. If the algebra is simple, then is an Azumaya algebra with center . The algebra is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of are satisfied. We construct an embedding of into ; this embedding maps the center of into when the algebra is simple. In this case, under an additional assumption, , thus turning into a central localization of . We completely work out these constructions in the case of the four-dimensional Sweedler algebra.     

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over we construct the associated spectral étale Lie groupoid over M such that is naturally isomorphic to G. Both these constructions are functorial, and is fully faithful left adjoint to . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid of an étale Lie groupoid G.     

Stability of singular Hopf bifurcations

A stability formula is given for the singular Hopf bifurcation arising in singularly perturbed systems of the form , in this paper. The derivation of the formula is based on a reduction technique and on an existing stability formula for Hopf bifurcation.     

On a relation between certain cohomological invariants

Let G be a group, the supremum of the projective lengths of the injective ZG-modules and the supremum of the injective lengths of the projective ZG-modules. The invariants and were studied in [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203-223] in connection with the existence of complete cohomological functors. If is finite then [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203-223] and , where is the generalized cohomological dimension of G [B.M. Ikenaga, Homological dimension and Farrell cohomology, J. Algebra 87 (1984) 422-457]. Note that if G is of finite virtual cohomological dimension. It has been conjectured in [O. Talelli, On groups of type Φ, Arch. Math. 89 (1) (2007) 24-32] that if is finite then G admits a finite dimensional model for , the classifying space for proper actions.We conjecture that for any group G and we prove the conjecture for duality groups, fundamental groups of graphs of finite groups and fundamental groups of certain finite graphs of groups of type .     

Classification of generalized symmetries of the Yang-Mills fields with a semi-simple structure group

A complete classification of generalized (or local) symmetries of the Yang-Mills equations on four dimensional Minkowski space with a semi-simple structure group is carried out. It is shown that any generalized symmetry, up to a generalized gauge symmetry, agrees with a first order symmetry on solutions of the Yang-Mills equations. Let be the decomposition of the Lie algebra of the structure group into simple ideals. First order symmetries for -valued Yang-Mills fields are found to consist of gauge symmetries, conformal symmetries for -valued Yang-Mills fields, 1?m?n, and their images under a complex structure of .     

q-Analog of the Möbius function and the cyclotomic identity associated to a profinite group

Let G be a profinite group and q an indeterminate. In this paper, we introduce and study a q-analog of the Möbius function and the cyclotomic identity arising from the lattice of open subgroups of G. When q is any integer, we show that they have close connections with the functors , , and introduced in [Y.-T. Oh, q-Deformation of Witt-Burnside rings, Math. Z. 257 (2007) 151-191]. In particular, we interpret the multiplicative property of the inverse of the table of marks and the Möbius function of G as a composition property of certain functors. Classification of , , and up to strict natural isomorphism as q varies over the set of integers and its application will be dealt with, too.     

Delooping the K-theory of exact categories

We generalize, from additive categories to exact categories, the concept of “Karoubi filtration” and the associated homotopy fibration in algebraic K-theory. As an application, we construct for any idempotent complete exact category an exact category such that .