Invariant hypersurfaces for derivations in positive characteristic

Let A be an integral k-algebra of finite type over an algebraically closed field k of characteristic p>0. Given a collection D of k-derivations on A, that we interpret as algebraic vector fields on , we study the group spanned by the hypersurfaces V(f) of X invariant under D modulo the rational first integrals of D. We prove that this group is always a finite dimensional F_p-vector space, and we give an estimate for its dimension. This is to be related to the results of Jouanolou and others on the number of hypersurfaces invariant under a foliation of codimension 1. As a application, given a k-algebra B between A^p and A, we show that the kernel of the pull-back morphism is a finite F_p-vector space. In particular, if A is a UFD, then the Picard group of B is finite.     

The central simple modules of Artinian Gorenstein algebras

Let A be a standard graded Artinian K-algebra, with char K=0. We prove the following.

1.

A has the Weak Lefschetz Property (resp. Strong Lefschetz Property) if and only if has the Weak Lefschetz Property (resp. Strong Lefschetz Property) for some linear form z of A.

2.

If A is Gorenstein, then A has the Strong Lefschetz Property if and only if there exists a linear form z of A such that all central simple modules of (A,z) have the Strong Lefschetz Property.

As an application of these theorems, we give some new classes of Artinian complete intersections with the Strong Lefschetz Property.     

Invariants de classes : exemples de non-annulation en dimension supérieure

The so-called class-invariant homomorphism ψ measures the Galois module structure of torsors—under a finite flat group scheme G—which lie in the image of a coboundary map associated to an isogeny between (Néron models of) abelian varieties with kernel G. When the varieties are elliptic curves with semi-stable reduction and the order of G is coprime to 6, it is known that the homomorphism ψ vanishes on torsion points. In this paper, using Weil restrictions of elliptic curves, we give the construction, for any prime number p > 2, of an abelian variety A of dimension p endowed with an isogeny (with kernel μ _p ) whose coboundary map is surjective. In the case when A has rank zero and the p-part of the Picard group of the base is non-trivial, we obtain examples where ψ does not vanish on torsion points.

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Résumé  Le class-invariant homomorphism permet de mesurer la structure galoisienne des torseurs—sous un schéma en groupes fini et plat G—qui sont dans l’image du cobord associé à une isogénie, de noyau G, entre des (modèles de Néron de) variétés abéliennes. Quand les variétés sont des courbes elliptiques à réduction semi-stable et que l’ordre de G est premier à 6, on sait que cet homomorphisme s’annule sur les points de torsion. Dans cet article, en nous servant de restrictions de Weil de courbes elliptiques, nous construisons, pour tout nombre premier p > 2, une variété abélienne A de dimension p munie d’une isogénie (de noyau μ _p ) dont le cobord est surjectif. Si A est de rang nul, et si la p-partie du groupe de Picard de la base est non triviale, nous obtenons ainsi un exemple où le class-invariant homomorphism ne s’annule pas sur les points de torsion.

     

Multi-index Filtrations and Generalized Poincaré Series

A multi-index filtration on the ring of germs of functions can be described by its Poincaré series. We consider a finer invariant (or rather two invariants) of a multi-index filtration than the Poincaré series generalizing the last one. The construction is based on the fact that the Poincaré series can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring of functions. The generalization of the Poincaré series is defined as a similar integral with respect to the generalized Euler characteristic with values in the Grothendieck ring of varieties. For the filtration defined by orders of functions on the components of a plane curve singularity C and for the so called divisorial filtration for a modification of by a sequence of blowing-ups there are given formulae for this generalized Poincaré series in terms of an embedded resolution of the germ C or in terms of the modification respectively. The generalized Euler characteristic of the extended semigroup corresponding to the divisorial filtration is computed giving a curious “motivic version” of an A’Campo type formula. First two authors were partially supported by the grant MEC, PN I + D + i MTM2004-00958. Partially supported by the grants RFBR-04-01-00762, NSh-4719.2006.1 The author is thankful to the University of Valladolid for hospitality.     

Error expansion for the discretization of backward stochastic differential equations

We study the error induced by the time discretization of decoupled forward–backward stochastic differential equations (X,Y,Z)$(X,Y,Z)$. The forward component X$X$ is the solution of a Brownian stochastic differential equation and is approximated by a Euler scheme X^N$X^N$ with N$N$ time steps. The backward component is approximated by a backward scheme. Firstly, we prove that the errors (Y^N−Y,Z^N−Z)$(Y^N-Y,Z^N-Z)$ measured in the strong _Lp$L_p$-sense (p≥1$p\ge 1$) are of order N^−1/2$N^{-1/2}$ (this generalizes the results by Zhang [J. Zhang, A numerical scheme for BSDEs, The Annals of Applied Probability 14 (1) (2004) 459–488]). Secondly, an error expansion is derived: surprisingly, the first term is proportional to X^N−X$X^N-X$ while residual terms are of order N⁻¹$N^{-1}$.     

Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths

In this note we consider a chain of N$N$ oscillators, whose ends are in contact with two heat baths at different temperatures. Our main result is the exponential convergence to the unique invariant probability measure (the stationary state). We use the Lyapunov’s function technique of Rey-Bellet and coauthors [Luc Rey-Bellet, Statistical mechanics of anharmonic lattices, in: Advances in Differential Equations and Mathematical Physics (Birmingham, AL, 2002), in: Contemp. Math., vol. 327, Amer. Math. Soc., Providence, RI, 2003, pp. 283–298. MR MR1991548 (2005a:82068) [11]; Luc Rey-Bellet, Lawrence E. Thomas, Fluctuations of the entropy production in anharmonic chains, Ann. Henri Poincaré 3 (3) (2002) 483–502. MR MR1915300 (2003g:82060); Luc Rey-Bellet, Lawrence E. Thomas, Exponential convergence to non-equilibrium stationary states in classical statistical mechanics, Comm. Math. Phys. 225 (2) (2002) 305–329. MR MR1889227 (2003f:82052); Luc Rey-Bellet, Lawrence E. Thomas, Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators, Comm. Math. Phys. 215 (1) (2000) 1–24. MR MR1799873 (2001k:82061) [12]; Jean-Pierre Eckmann, Claude-Alain Pillet, Luc Rey-Bellet, Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Comm. Math. Phys. 201 (3) (1999) 657–697. MR MR1685893 (2000d:82025); Jean-Pierre Eckmann, Claude-Alain Pillet, Luc Rey-Bellet, Entropy production in nonlinear, thermally driven Hamiltonian systems, J. Statist. Phys. 95 (1–2) (1999) 305–331. MR MR1705589 (2000h:82075)], with different model of heat baths, and adapt these techniques to two new case recently considered in the literature by Bernardin and Olla [Cédric Bernardin, Stefano Olla, Fourier’s law for a microscopic model of heat conduction, J. Statist. Phys. 121 (3–4) (2005) 271–289. MR MR2185330] and Lefevere and Schenkel [R. Lefevere, A. Schenkel, Normal heat conductivity in a strongly pinned chain of anharmonic oscillators, J. Stat. Mech. Theory Exp. 2006 (02) (2006) L02001].     

Tail expansions for the distribution of the maximum of a random walk with negative drift and regularly varying increments

Let F$F$ be a distribution function with negative mean and regularly varying right tail. Under a mild smoothness condition we derive higher order asymptotic expansions for the tail distribution of the maxima of the random walk generated by F$F$. The expansion is based on an expansion for the right Wiener–Hopf factor which we derive first. An application to ruin probabilities is developed.     

Invariant Measures for Univalent Self-Maps of the Unit Disc

We develop a translation-type model for univalent self-maps φ of the unit disc having an interior fixed-point and use the model to classify the φ-invariant measures on . We are particularly interested in maps which can be embedded in continuous semigroups of holomorphic self-maps of . Received: February 2, 2007. Revised: June 18, 2007. Accepted: July 4, 2007.     

A Spectrally Minimal Realization Formula for H^{\infty}({\mathbb{D}})

In this article we prove a representation theorem for functions, such that the realization formula is spectrally minimal in the following sense: the spectrum of the main operator in the realization intersects the unit circle precisely at those points where the given function has no holomorphic extension. We also extend this result to operator-valued H ^∞ functions. Received: March 12, 2007. Revised: April 12, 2007. Accepted: May 7, 2007.