Volume 11 (2011), Number 3. Abstracts M. Chaperon. Generalised Hopf Bifurcations: a Birth Lemma [PDF] We state and prove a “birth lemma” generalising the Poincaré–Andronov–Hopf and Sacker–Naimark bifurcation theorems. It implies the birth of many topologically different compact invariant manifolds in generic families of dynamics depending on at least two parameters. Keywords. Arnold, Thom, bifurcation, Hopf, Sacker–Naimark, normally hyperbolic, moment-angle manifolds, catastrophe, coupling. 2000 Mathematics Subject Classification. 34C23, 34K18, 37G05, 37G10, 37G15, 37G35, 37G40. D. Davis, D. Fuchs and S. Tabachnikov. Periodic Trajectories in the Regular Pentagon [PDF] We consider periodic billiard trajectories in a regular pentagon. It is known that the trajectory is periodic if and only if the tangent of the angle formed by the trajectory and the side of the pentagon belongs to (sin 36°)ℚ[ √5]. Moreover, for every such direction, the lengths of the trajectories, both geometric and combinatorial, take precisely two values. In this paper, we provide a full computation of these lengths as well as a full description of the corresponding symbolic orbits. We also formulate results and conjectures regarding the billiards in other regular polygons. Keywords. Periodic billiard trajectories, regular pentagon, Veech alternative, closed geodesics, regular dodecahedron. 2000 Mathematics Subject Classification. Primary: 37E35; Secondary: 37E05, 37E15. W. Ebeling and S. Gusein-Zade. Monodromy of Dual Invertible Polynomials [PDF] A generalization of Arnold's strange duality to invertible polynomials in three variables by the first author and A. Takahashi includes the following relation. For some invertible polynomials f the Saito dual of the reduced monodromy zeta function of f coincides with a formal “root” of the reduced monodromy zeta function of its Berglund–Hübsch transpose f^{T}. Here we give a geometric interpretation of “roots” of the monodromy zeta function and generalize the above relation to all non-degenerate invertible polynomials in three variables and to some polynomials in an arbitrary number of variables in a form including “roots” of the monodromy zeta functions both of f and f^{T}. Keywords. Invertible polynomials, monodromy, zeta functions, Saito duality. 2000 Mathematics Subject Classification. 32S05, 32S40, 14J33. A. Eremenko and A. Gabrielov. Singular Perturbation of Polynomial Potentials with Applications to PT-Symmetric Families [PDF] We discuss eigenvalue problems of the form −w′′ + Pw = λw with complex polynomial potential P(z) = tz^{d} + …, where t is a parameter, with zero boundary conditions at infinity on two rays in the complex plane. In the first part of the paper we give sufficient conditions for continuity of the spectrum at t = 0. In the second part we apply these results to the study of topology and geometry of the real spectral loci of PT-symmetric families with P of degree 3 and 4, and prove several related results on the location of zeros of their eigenfunctions. Keywords. Singular perturbation, Schrödinger operator, eigenvalue, spectral determinant, PT-symmetry. 2000 Mathematics Subject Classification. 34M35, 35J10. A. Felikson and S. Natanzon. Labeled Double Pants Decompositions [PDF] A double pants decomposition of a 2-dimensional surface is a collection of two pants decomposition of this surface introduced by the authors. There are two natural operations acting on double pants decompositions: flips and handle-twists. It is shown by the authors that the groupoid generated by flips and handle-twists acts transitively on admissible double pants decompositions, where the class of admissible decompositions has a natural topological and combinatorial description. In this paper, we label the curves of double pants decompositions and show that for all but one surfaces the same groupoid acts transitively on all labeled admissible double pants decompositions. The only exclusion is a sphere with two handles, where the groupoid has 15 orbits. Keywords. Pants decomposition, Mapping class group. 2000 Mathematics Subject Classification. 57M50. Yu. Ilyashenko and V. Moldavskis. Total Rigidity of Generic Quadratic Vector Fields [PDF] We consider a class of foliations on the complex projective plane that are determined by a quadratic vector field in a fixed affine neighborhood. Such foliations, as a rule, have an invariant line at infinity. Two foliations with singularities on ℂP^{2} are topologically equivalent provided that there exists a homeomorphism of the projective plane onto itself that preserves orientation both on the and in ℂP^{2} and brings the leaves of the first foliation to that of the second one. We prove that a generic foliation of this class may be topologically equivalent to but a finite number of foliations of the same class, modulo affine equivalence. This property is called total rigidity. A recent result of Lins Neto implies that the finite number above does not exceed 240. This is the first of the two closely related papers. It deals with the rigidity properties of quadratic foliations, whilst the second one studies the foliations of higher degree. Keywords. Foliations, topological equivalence, rigidity. 2000 Mathematics Subject Classification. 37F75. A. Neishtadt, A. Vasiliev, and A. Artemyev. Resonance-Induced Surfatron Acceleration of a Relativistic Particle [PDF] We study motion of a relativistic charged particle in a plane slow electromagnetic wave and background uniform magnetic field. The wave propagates normally to the background field. The motion of the particle can be described by a Hamiltonian system with two degrees of freedom. Parameters of the problem are such that in this system one can identify slow and fast variables: three variables are changing slowly and one angular variable (the phase of the wave) is rotating fast everywhere except for a neighborhood of a certain surface in the space of the slow variables called a resonant surface. Far from the resonant surface dynamics of the slow variables may be approximately described by the averaging method. In the process of evolution of the slow variables the particle approaches this surface and may be captured into resonance with the wave. Capture into this resonance results in acceleration of the particle along the wave front (surfatron acceleration). We study the phenomenon of capture and show that a captured particle never leaves the resonance and its energy infinitely grows. Passage through the resonant surface without capture leads to scattering at the resonance, i.e., a small phase-sensitive deviation of actual motion from the motion predicted by the averaging method. We find that repeated scatterings result in diffusive growth of the particle energy. The considered problem is a representative of a wide class of problems concerning passages through resonances in nonlinear systems with fast rotating phases. Estimates of accuracy of the averaging method in this class of problems were for the first time obtained by V.I. Arnold. Keywords. Adiabatic invariants, passage through resonance, surfatron acceleration. 2000 Mathematics Subject Classification. 34E10, 34D10, 37N05. M. Passare, J. M. Rojas, and B. Shapiro. New Multiplier Sequences via Discriminant Amoebae [PDF] In their classic 1914 paper, Pólya and Schur introduced and characterized two types of linear operators acting diagonally on the monomial basis of ℝ[x], sending real-rooted polynomials (resp. polynomials with all nonzero roots of the same sign) to real-rooted polynomials. Motivated by fundamental properties of amoebae and discriminants discovered by Gelfand, Kapranov, and Zelevinsky, we introduce two new natural classes of polynomials and describe diagonal operators preserving these new classes. A pleasant circumstance in our description is that these classes have a simple explicit description, one of them coinciding with the class of log-concave sequences. Keywords. Multiplier sequence, discriminant, amoeba, chamber. 2000 Mathematics Subject Classification. Primary: 12D10, Secondary: 32H99. R. Rimányi, V. Schechtman, and A. Varchenko. Conformal Blocks and Equivariant Cohomology [PDF] In this paper we show that the conformal blocks constructed in the previous article by the first and the third author may be described as certain integrals in equivariant cohomology. When the bundles of conformal blocks have rank one, this construction may be compared with the old integral formulas of the second and the third author. The proportionality coefficients are some Selberg type integrals, which are computed. Finally, a geometric construction of the tensor products of vector representations of the Lie algebra gl(m) is proposed. Keywords. Wess–Zumino–Witten model, Knizhnik–Zamolodchikov equations, equivariant cohomology, Selberg integrals, Kac–Moody Lie algebras. 2000 Mathematics Subject Classification. 81T40, 55N91, 17B67. V. Sedykh. On the Topology of Cooriented Wave Fronts in Spaces of Small Dimensions [PDF] We consider Legendre singularities with respect to Legendre equivalence preserving a coorientation of the contact structure. In this case, we calculate the adjacency indices of multisingularities of generic Legendre mappings to smooth manifolds of the dimension n ≤ 6. As a corollary, we find new coexistence conditions on singularities of wave fronts. Namely, we find all linear relations with real coefficients between the Euler characteristics of manifolds of singularities of any generic compact cooriented wave front in any n-dimensional space. Keywords. Legendre mappings, wave fronts, (multi)singularities, adjacency index, Euler characteristic. 2000 Mathematics Subject Classification. 57R45, 58K30. D. Siersma and M. Tibăr. Betti Bounds of Polynomials [PDF] We initiate a classification of polynomials f: ℂ^{n} → ℂ of degree d having the top Betti number of the general fibre close to the maximum. We find a range in which the polynomial must have isolated singularities and another range where it may have at most one line singularity of general Morse transversal type. Our method uses deformations into particular pencils with non-isolated singularities. Keywords. Deformation of hypersurfaces and polynomials, Betti numbers, classification, general fibres, singularities at infinity, boundary singularities. 2000 Mathematics Subject Classification. 2S30, 58K60, 55R55, 32S50. V. Vassiliev. Topological Complexity and Schwarz Genus of General Real Polynomial Equation [PDF] We prove that the minimal number of branchings of arithmetic algorithms of approximate solution of the general real polynomial equation x^{d} + a_{1}x^{d−1} + ⋯ + a_{d−1}x + a_{d} = 0 of odd degree d grows to infinity at least as log_{2} d. The same estimate is true for the ε-genus of the real algebraic function associated with this equation, i.e., for the minimal number of open sets covering the space ℝ^{d} of such polynomials in such a way that on any of these sets there exists a continuous function whose value at any point (a_{1}, …, a_{d}) is approximately (up to some sufficiently small ε > 0) equal to one of real roots of the corresponding equation. Keywords. Complexity, cross-section, Schwarz genus, ramified covering, 13th Hilbert problem, real polynomial. 2000 Mathematics Subject Classification. Primary: 55R80, 12Y05; Secondary: 55S40, 68W30. |
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