The unbounded dead-end depth propertie is not a group invariant

Vegeu el resum a l'inici del document del fitxer adjunt

Inverse Problems for multiple invariant curves

Planar polynomial vector fields which admit invariant algebraic curves, Darboux integrating factors or Darboux first integrals are of special interest. In the present paper we solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors, under generic assumptions regarding the (multiple) invariant algebraic curves involved. In particular we prove, in this generic scenario, that the existence of a Darboux integrating factor implies Darboux integrability. Furthermore we construct examples where the genericity assumption does not hold and indicate that the situation is different for these.

Well-posedness and invariant measures for HJM models with deterministic volatility and Lévy noise

We give sufficient conditions for existence, uniqueness and ergodicity of invariant measures for Musiela's stochastic partial differential equation with deterministic volatility and a Hilbert space valued driving Lévy noise. Conditions for the absence of arbitrage and for the existence of mild solutions are also discussed.

Euclidean geometric invariant of framed knots in manifolds

We present an invariant of a three dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. An important feature of our work is that we are not using any nontrivial representation of the manifold fundamental group or knot group.

Inverse problems for invariant algebraic curves: explicit computations

Given an algebraic curve in the complex affine plane, we describe how to determine all planar polynomial vector fields which leave this curve invariant. If all (finite) singular points of the curve are nondegenerate, we give an explicit expression for these vector fields. In the general setting we provide an algorithmic approach, and as an alternative we discuss sigma processes.

Fast numerical algorithms for the computation of invariant tori in Hamiltonian systems

In this paper, we develop numerical algorithms that use small requirements of storage and operations for the computation of invariant tori in Hamiltonian systems (exact symplectic maps and Hamiltonian vector fields). The algorithms are based on the parameterization method and follow closely the proof of the KAM theorem given in [LGJV05] and [FLS07]. They essentially consist in solving a functional equation satisfied by the invariant tori by using a Newton method. Using some geometric identities, it is possible to perform a Newton step using little storage and few operations. In this paper we focus on the numerical issues of the algorithms (speed, storage and stability) and we refer to the mentioned papers for the rigorous results. We show how to compute efficiently both maximal invariant tori and whiskered tori, together with the associated invariant stable and unstable manifolds of whiskered tori. Moreover, we present fast algorithms for the iteration of the quasi-periodic cocycles and the computation of the invariant bundles, which is a preliminary step for the computation of invariant whiskered tori. Since quasi-periodic cocycles appear in other contexts, this section may be of independent interest. The numerical methods presented here allow to compute in a unified way primary and secondary invariant KAM tori. Secondary tori are invariant tori which can be contracted to a periodic orbit. We present some preliminary results that ensure that the methods are indeed implementable and fast. We postpone to a future paper optimized implementations and results on the breakdown of invariant tori.

Destruction of invariant curves in the restricted circular planar three body problem using comparison of action

"Vegeu el resum a l'inici del document del fitxer adjunt."

Generic super-exponential stability of invariant tori in Hamiltonian systems

In this article, we consider solutions starting close to some linearly stable invariant tori in an analytic Hamiltonian system and we prove results of stability for a super-exponentially long interval of time, under generic conditions. The proof combines classical Birkhoff normal forms and a new method to obtain generic Nekhoroshev estimates developed by the author and L. Niederman in another paper. We will mainly focus on the neighbourhood of elliptic fixed points, the other cases being completely similar.

Sobolev mappings: Lipschitz density is not an isometric invariant of the target

"Vegeu el resum a l'inici del document del fitxer adjunt."

The eta invariant and equivariant index of transversally elliptic operators

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a G-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.

The eta invariant and equivariant index of transversally elliptic operators

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a G-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.

Recovering the Elliott invariant from the Cuntz semigroup

Let A be a simple, separable C*-algebra of stable rank one. We prove that the Cuntz semigroup of C (T, A) is determined by its Murray-von Neumann semigroup of projections and a certain semigroup of lower semicontinuous functions (with values in the Cuntz semigroup of A). This result has two consequences. First, specializing to the case that A is simple, finite, separable and Z-stable, this yields a description of the Cuntz semigroup of C (T, A) in terms of the Elliott invariant of A. Second, suitably interpreted, it shows that the Elliott functor and the functor defined by the Cuntz semigroup of the tensor product with the algebra of continuous functions on the circle are naturally equivalent.

Human invariant V alpha 24-J alpha Q TCR supports the development of CD1d-dependent NK1.1+ and NK1.1- T cells in transgenic mice.

A sizable fraction of T cells expressing the NK cell marker NK1.1 (NKT cells) bear a very conserved TCR, characterized by homologous invariant (inv.) TCR V alpha 24-J alpha Q and V alpha 14-J alpha 18 rearrangements in humans and mice, respectively, and are thus defined as inv. NKT cells. Because human inv. NKT cells recognize mouse CD1d in vitro, we wondered whether a human inv. V alpha 24 TCR could be selected in vivo by mouse ligands presented by CD1d, thereby supporting the development of inv. NKT cells in mice. Therefore, we generated transgenic (Tg) mice expressing the human inv. V alpha 24-J alpha Q TCR chain in all T cells. The expression of the human inv. V alpha 24 TCR in TCR C alpha(-/-) mice indeed rescues the development of inv. NKT cells, which home preferentially to the liver and respond to the CD1d-restricted ligand alpha-galactosylceramide (alpha-GalCer). However, unlike inv. NKT cells from non-Tg mice, the majority of NKT cells in V alpha 24 Tg mice display a double-negative phenotype, as well as a significant increase in TCR V beta 7 and a corresponding decrease in TCR V beta 8.2 use. Despite the forced expression of the human CD1d-restricted TCR in C alpha(-/-) mice, staining with mCD1d-alpha-GalCer tetramers reveals that the absolute numbers of peripheral CD1d-dependent T lymphocytes increase at most by 2-fold. This increase is accounted for mainly by an increased fraction of NK1.1(-) T cells that bind CD1d-alpha-GalCer tetramers. These findings indicate that human inv. V alpha 24 TCR supports the development of CD1d-dependent lymphocytes in mice, and argue for a tight homeostatic control on the total number of inv. NKT cells. Thus, human inv. V alpha 24 TCR-expressing mice are a valuable model to study different aspects of the inv. NKT cell subset.

Cutting edge: influence of the TCR Vbeta domain on the selection of semi-invariant NKT cells by endogenous ligands.

Invariant Valpha14 (Valpha14i) NKT cells are a murine CD1d-dependent regulatory T cell subset characterized by a Valpha14-Jalpha18 rearrangement and expression of mostly Vbeta8.2 and Vbeta7. Whereas the TCR Vbeta domain influences the binding avidity of the Valpha14i TCR for CD1d-alpha-galactosylceramide complexes, with Vbeta8.2 conferring higher avidity binding than Vbeta7, a possible impact of the TCR Vbeta domain on Valpha14i NKT cell selection by endogenous ligands has not been studied. In this study, we show that thymic selection of Vbeta7(+), but not Vbeta8.2(+), Valpha14i NKT cells is favored in situations where endogenous ligand concentration or TCRalpha-chain avidity are suboptimal. Furthermore, thymic Vbeta7(+) Valpha14i NKT cells were preferentially selected in vitro in response to CD1d-dependent presentation of endogenous ligands or exogenously added self ligand isoglobotrihexosylceramide. Collectively, our data demonstrate that the TCR Vbeta domain influences the selection of Valpha14i NKT cells by endogenous ligands, presumably because Vbeta7 confers higher avidity binding.

Cutting edge: influence of the TCR V beta domain on the avidity of CD1d:alpha-galactosylceramide binding by invariant V alpha 14 NKT cells.

CD1d tetramers loaded with alpha-galactosylceramide (alpha-GalCer) bind selectively to mouse invariant Valpha14 (Valpha14i) NKT cells and their human counterparts. Whereas tetramer binding strictly depends on the expression of a Valpha14-Jalpha18 chain in murine NKT cells, the associated beta-chain (typically expressing Vbeta8.2 or Vbeta7) appears not to influence tetramer binding. In this study, we describe novel alpha-GalCer-loaded mouse and human CD1d-IgG1 dimers, which revealed an unexpected influence of the TCR-beta chain on the avidity of CD1d:alpha-GalCer binding. A subset of Valpha14i NKT cells clearly discriminated alpha-GalCer bound to mouse or human CD1d on the basis of avidity differences conferred by the Vbeta domain of the TCR-beta chain, with Vbeta8.2 conferring higher avidity binding than Vbeta7.