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

Special metrics and Triality

We investigate a new 8-dimensional Riemannian geometry defined by a generic closed and coclosed 3-form with stabiliser PSU(3), and which arises as a critical point of Hitchin's variational principle. We give a Riemannian characterisation of this structure in terms of invariant spinor-valued 1-forms, which are harmonic with respect to the twisted Dirac operator \Db on $\Delta\otimes\Lambda^1$. We establish various obstructions to the existence of topological reductions to PSU(3). For compact manifolds, we also give sufficient conditions for topological PSU(3)-structures that can be lifted to topological SU(3)-structures. We also construct the first known compact example of an integrable non-symmetric PSU(3)-structure. In the same vein, we give a new Riemannian characterisation for topological quaternionic K\"ahler structures which are defined by an $Sp(1)\cdot Sp(2)$-invariant self-dual 4-form. Again, we show that this form is closed if and only if the corresponding spinor-valued 1-form is harmonic for \Db and that these equivalent conditions produce constraints on the Ricci tensor.Comment: 41 pages. v2: typos corrected, presentation improved v3: final versio

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

On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem

We prove that the number of limit cycles generated by a small non-conservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing tangential Hilbert 16th problem. The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection (Gauss-Manin connection) with a quasiunipotent monodromy group.Comment: Final revisio

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

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