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 .     

Equivariant chain complexes, twisted homology and relative minimality of arrangements

We show that the π-equivariant chain complex (), , associated to a Morse-theoretic minimal CW-structure X on the complement of an arrangement , is independent of X. The same holds for all scalar extensions, , a field, where X is an arbitrary minimal CW-structure on a space M. When is a section of another arrangement , we show that the divisibility properties of the first Betti number of the Milnor fiber of  obstruct the homotopy realization of  as a subcomplex of a minimal structure on .If is aspherical and is a sufficiently generic section of , then may be described in terms of π, L and , for an arbitrary local system L; explicit computations may be done, when is fiber-type. In this case, explicit -presentations of arbitrary abelian scalar extensions of the first non-trivial higher homotopy group of , π_p(M), may also be obtained. For nonresonant abelian scalar extensions, the -rank of is combinatorially determined.     

Quasi sure analysis of local times of anticipating smooth semimartingales

Let be the anticipating smooth semimartingale and be its generalized local time. In this paper, we give some estimates about the quasi sure property of Xt and its quadratic variation process t〈X〉. We also study the fractional smoothness of and prove that the quadratic variation process of can be constructed as the quasi sure limit of the form , where is a sequence of subdivisions of [a,b], , i=0,1,…,n².     

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 .     

Perturbations of subalgebras of type II1 factors

In this paper we consider two von Neumann subalgebras and of a type II₁ factor . For a map φ on , we define

     

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

     

Independence of ? in Lafforgue's theorem

Let X be a smooth curve over a finite field of characteristic p, let ?≠p be a prime number, and let be an irreducible lisse -sheaf on X whose determinant is of finite order. By a theorem of L. Lafforgue, for each prime number ?′≠p, there exists an irreducible lisse -sheaf on X which is compatible with , in the sense that at every closed point x of X, the characteristic polynomials of Frobenius at x for and are equal. We prove an “independence of ?” assertion on the fields of definition of these irreducible ?′-adic sheaves : namely, that there exists a number field F such that for any prime number ?′≠p, the -sheaf above is defined over the completion of F at one of its ?′-adic places.     

Computation of blowing up centers

Given a birational projective morphism of quasi-projective varieties . We want to find the ideal sheaf over X such that the blowing up of X along corresponds to f. In this paper we approach the problem from two directions, solving two subcases. First we present a method that determines when f is the composition of blowing ups along known centers, then by another method, we compute directly from .     

Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation

Let Ω be a smooth bounded domain in , with N?5, a>0, α?0 and . We show that the exponent plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem

     

Congruences involving Bernoulli and Euler numbers

Let [x] be the integral part of x. Let p>5 be a prime. In the paper we mainly determine , , and in terms of Euler and Bernoulli numbers. For example, we have

     

Cap pairings and spectral sequences

Let be a small category. For an -diagram X and -diagrams A and B of pointed spaces, each pairing X∧A→B satisfying the projection formula induces a pairing . In this note we show that there is an induced pairing of homotopy spectral sequences compatible with abutments in the sense that