Pseudo-Abelian integrals along Darboux cycles: A codimension one case

AbstractWe investigate a polynomial perturbation of an integrable, non-Hamiltonian system with first integral of Darboux type. In the paper [M. Bobieński, P. Mardešić, Pseudo-Abelian integrals along Darboux cycles, Proc. Lond. Math. Soc., in press] the generic case was studied. In the present paper we study a degenerate, codimension one case. We consider 1-parameter unfolding of a non-generic case. The main result of the paper is an analog of Varchenko–Kchovanskii theorem for pseudo-Abelian integrals

A 2-Surface Quantization of the Lorentzian Gravity

This is a contribution to the MG9 session QG1-a. A new quantum representation for the Lorentzian gravity is created from the Pullin vaccum by the operators assigned to 2-complexes. The representation uses the original, spinorial Ashtekar variables, the reality conditions are well posed and Thiemann's Hamiltonian is well defined. The results on the existence of a suitable Hilbert product are partial. They were derived in collaboration with Abhay Ashtekar.Comment: QG1-a session of MG

On the reduction of the degree of linear differential operators

Let L be a linear differential operator with coefficients in some differential field k of characteristic zero with algebraically closed field of constants. Let k^a be the algebraic closure of k. For a solution y, Ly=0, we determine the linear differential operator of minimal degree M and coefficients in k^a, such that My=0. This result is then applied to some Picard-Fuchs equations which appear in the study of perturbations of plane polynomial vector fields of Lotka-Volterra type

Background independent quantizations: the scalar field I

We are concerned with the issue of quantization of a scalar field in a diffeomorphism invariant manner. We apply the method used in Loop Quantum Gravity. It relies on the specific choice of scalar field variables referred to as the polymer variables. The quantization, in our formulation, amounts to introducing the `quantum' polymer *-star algebra and looking for positive linear functionals, called states. The assumed in our paper homeomorphism invariance allows to determine a complete class of the states. Except one, all of them are new. In this letter we outline the main steps and conclusions, and present the results: the GNS representations, characterization of those states which lead to essentially self adjoint momentum operators (unbounded), identification of the equivalence classes of the representations as well as of the irreducible ones. The algebra and topology of the problem, the derivation, all the technical details and more are contained in the paper-part II.Comment: 13 pages, minor corrections were made in the revised versio

Finite cyclicity of slow-fast Darboux systems with a two-saddle loop

International audienc

Pseudo-abelian integrals: Unfolding generic exponential case

AbstractWe consider functions of the form H0=P1a1⋯PkakeR/Q, with Pi, R, and Q∈R[x,y], which are (generalized Darboux) first integrals of the polynomial system MdlogH0=0. We assume that H0 defines a family γ(h)⊂H0−1(h) of real cycles in a region bounded by a polycycle.To each polynomial form η one can associate the pseudo-abelian integrals I(h) of M−1η along γ(h), which is the first order term of the displacement function of the orbits of MdH0+δη=0.We consider Darboux first integrals unfolding H0 (and its saddle-nodes) and pseudo-abelian integrals associated to these unfoldings. Under genericity assumptions we show the existence of a uniform local bound for the number of zeros of these pseudo-abelian integrals.The result is a part of a program to extend Varchenko–Khovanskii's theorem from abelian integrals to pseudo-abelian integrals and prove the existence of a bound for the number of their zeros in function of the degree of the polynomial system only