Volume 1 (2001), Number 4. Abstracts

S. Artemov, T. Yavorskaya (Sidon). On First Order Logic of Proofs [PDF]

The Logic of Proofs LP solved long standing Gödel's problem concerning his provability calculus (cf. [1]). It also opened new lines of research in proof theory, modal logic, typed programming languages, knowledge representation, etc. The propositional logic of proofs is decidable and admits a complete axiomatization. In this paper we show that the first order logic of proofs is not recursively axiomatizable.

Keywords. Logic of proofs, provability, recursive axiomatizability.

2000 Mathematics Subject Classification. 03F45 (primary), 03F30, 03F50.

J. Colliander, C. Kenig, G. Staffilani. On Solutions for the Kadomtsev—Petviashvili I Equation [PDF]

Oscillatory integral techniques are used to study the well-posedness of the KP-I equation for initial data that are small with respect to the norm of a weighted Sobolev space involving derivatives of total order no larger than 2.

Keywords. Kadomtsev—Petviashvili equation, initial value problem, well-posedness, oscillatory intergals.

2000 Mathematics Subject Classification. 35Q53, 35D25.

B. Fayad, A. Katok, A. Windsor. Mixed Spectrum Reparameterizations of Linear Flows on T² [PDF]

We prove the existence of mixed spectrum C^$\infty$ reparameterizations of any linear flow on T² with Liouville rotation number. For a restricted class of Liouville rotation numbers, we prove the existence of mixed spectrum real-analytic reparameterizations.

Keywords. Mixed spectrum, reparameterization, special flow, Liouville, cocycle.

2000 Mathematics Subject Classification. 37A20, 37A45, 37C05.

R. Fedorov. Lower Bounds for the Number of Orbital Topological Types of Planar Polynomial Vector Fields ``Modulo Limit Cycles'' [PDF]

We consider planar polynomial vector fields. We aim to find the (asymptotic) upper and lower bounds for the number of orbital topological equivalence classes for the fields of degree n. An evident obstacle for this is the second part of Hilbert's 16th problem. To circumvent this obstacle we introduce the notion of equivalence modulo limit cycles. Both upper and lower bounds can be obtained for this type of equivalence. In this paper we use the Viro gluing method to obtain the lower bound 2^cn2, where c > 0 is a constant.

Keywords. Planar polynomial vector field, structural stability, orbital topological equivalence, Viro gluing method

2000 Mathematics Subject Classification. Primary: 37C15; Secondary: 37E35.

A. Givental. Gromov—Witten Invariants and Quantization of Quadratic Hamiltonians [PDF]

We describe a formalism based on quantization of quadratic hamiltonians and symplectic actions of loop groups which provides a convenient home for most of the known general results and conjectures about Gromov—Witten invariants of compact symplectic manifolds and, more generally, Frobenius structures at higher genus. We state several results illustrating the formalism and its use. In particular, we establish Virasoro constraints for semisimple Frobenius structures and outline a proof of the Virasoro conjecture for Gromov—Witten invariants of complex projective spaces and other Fano toric manifolds. Details will be published elsewhere.

Keywords. Gromov—Witten invariants, Fock spaces, Frobenius structures, Virasoro constraints.

2000 Mathematics Subject Classification. 53D45, 14N35.

B. Gurevich, S. Katok. Arithmetic coding and entropy for the positive geodesic flow on the modular surface [PDF]

In this article we study geodesics on the modular surface by means of their arithmetic codes. Closed geodesics for which arithmetic and geometric codes coincide were identified in [9]. Here they are described as periodic orbits of a special flow over a topological Markov chain with countable alphabet, which we call the positive geodesic flow. We obtain an explicit formula for the ceiling function and two-sided estimates for the topological entropy of the positive geodesic flow, which turns out to be separated from one: the topological entropy of the geodesic flow on the modular surface.

Keywords. Geodesic flow, modular surface, Fuchsian group, entropy, topological entropy.

2000 Mathematics Subject Classification. 37D40, 37B40, 20H05.

Yu. Ilyashenko, A. Panov. Some Upper Estimates of the Number of Limit Cycles of Planar Vector Fields with Applications to Liénard Equations [PDF]

We estimate the number of limit cycles of planar vector fields through the size of the domain of the Poincaré map, the increment of this map, and the width of the complex domain to which the Poincaré map may be analytically extended. The estimate is based on the relationship between the growth and zeros of holomorphic functions [IYa], [I]. This estimate is then applied to getting the upper bound of the number of limit cycles of the Liénard equation $\dot x = y - F(x)$, $\dot y = -x$ through the (odd) power of the monic polynomial F and magnitudes of its coefficients.

Keywords. Limit cycles, Poincaré map, Liénard equation

2000 Mathematics Subject Classification. 34Cxx, 34Mxx.

N. Nadirashvili. An Application of Potential Analysis to Minimal Surfaces [PDF]

We study complete proper minimal immersions in a bounded domain in Euclidean space. We show that for certain domains there are no such immersions. The existence of such unproper immersions is known.

Keywords. Minimal surface, Liouville theorem, proper minimal immersion.

2000 Mathematics Subject Classification. 31, 58

O. Sheinman. Second Order Casimirs for the Affine Krichever—Novikov Algebras $\widehat{\mathfrak{gl}}_g,2$ and $\widehat{\mathfrak{sl}}_g,2$ [PDF]

The second order casimirs for the affine Krichever—Novikov algebras $\widehat{\mathfrak{gl}}_g,2$ and $\widehat{\mathfrak{sl}}_g,2$ are described. More general operators which we call semi-casimirs are introduced. It is proven that the semi-casimirs induce well-defined operators on conformal blocks and, for a certain moduli space of Riemann surfaces with two marked points and fixed jets of local coordinates, there is a natural projection of its tangent space onto the space of these operators.

Keywords. Infinite-dimensional Lie algebras, Riemann surfaces, current algebras, central extensions, highest weight representations, wedge representations, Casimir operators, moduli spaces, conformal blocks.

2000 Mathematics Subject Classification. 17B66, 17B67, 14H10, 14H15, 17B90, 30F30, 14H55, 81R10, 81T40.

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