Volume 6 (2006), Number 2. Abstracts

E. Bannai, M. Koike, M. Shinohara, and M. Tagami. Spherical Designs Attached to Extremal Lattices and the Modulo p Property of Fourier Coefficients of Extremal Modular Forms [PDF]

A theorem of Venkov says that each nontrivial shell of an extremal even unimodular lattice in Rn with 24|n is a spherical 11-design. It is a difficult open question whether there exists any 12-design among them. In the first part of this paper, we consider the following problem: When do all shells of an even unimodular lattice become 12-designs? We show that this does not happen in many cases, though there are also many cases yet to be answered. In the second part of this paper, we study the modulo p property of the Fourier coefficients of the extremal modular forms f = ∑ i ≥ 0aiqi (where q = eiτ) of weight k with k even. We are interested in determining, for each pair consisting of k and a prime p, which of the following three (exclusive) cases holds: (1) p|ai for all i ≥ 1; (2) p|ai for all i ≥ 1 with p \not|i, and there exists at least one j ≥ 1 with p \not|aj; (3) there exists at least one j ≥ 1 with p \not|j such that p \not|aj. We first prove that case (1) holds if and only if (p−1)|k. Then we obtain several conditions which guarantee that case (2) holds. Finally, we propose a conjecture that may characterize situations in which case (2) holds.

Keywords. Spherical design, extremal lattice, extremal modular form, Assmus–Mattson theorem, Lehmer conjecture

2000 Mathematics Subject Classification. Primary: 05Exx; Secondary: 05B05, 11E12, 11F11, 11F30, 11F33

S. Bautista and C. Morales. Existence of Periodic Orbits for Singular-Hyperbolic Sets [PDF]

It is well known that on every compact 3-manifold there is a C1 flow displaying a singular-hyperbolic isolated set which has no periodic orbits. By contrast, in this paper we prove that every singular-hyperbolic attracting set of a C1 flow on a compact 3-manifold has a periodic orbit.

Keywords. Singular-hyperbolic set, attracting set, periodic orbit

2000 Mathematics Subject Classification. Primary: 37D30; Secondary: 37D45

S. Gindikin. The Horospherical Cauchy–Radon Transform on Compact Symmetric Spaces [PDF]

Harmonic analysis on noncompact Riemannian symmetric spaces is in a sense equivalent to the theory of the horospherical transform. There are no horospheres on compact symmetric spaces, but we define a complex version of the horospherical transform, which plays a similar role for harmonic analysis on them.

Keywords. Symmetric space, horospherical transform, spherical Fourier transform, Cauchy–Radon transform, inversion formula, Plancherel formula

2000 Mathematics Subject Classification. 14M17, 22E46, 44A15

R. Ismagilov, M. Losik, and P. Michor. A 2-Cocycle on a Symplectomorphism Group [PDF]

For a symplectic manifold (M,ω) with exact symplectic form, we construct a 2-cocycle on the symplectomorphism group and indicate cases in which this cocycle is not trivial.

Keywords. Group extension, symplectomorphism

2000 Mathematics Subject Classification. 58D05, 20J06, 22E65

D. Khmelev and M. Yampolsky. The Rigidity Problem for Analytic Critical Circle Maps [PDF]

It is shown that if f and g are any two analytic critical circle mappings with the same irrational rotation number, then the conjugacy that maps the critical point of f to that of g has regularity C1+α at the critical point, with a universal value of α > 0. As a consequence, a new proof of the hyperbolicity of the full renormalization horseshoe of critical circle maps is given.

Keywords. Critical circle mapping, rigidity, renormalization

2000 Mathematics Subject Classification. 37E10

V. Malyshev. One-Dimensional Mechanical Networks and Crystals [PDF]

We propose a rigorous model of one dimensional crystal or of a biological mechanical network. We prove the thermal expansion and Hooke's law for this model.

Keywords. Gibbs distribution, biological mechanical network, crystal, thermal expansion, Hooke's law

2000 Mathematics Subject Classification. 60Kxx, 82C99

D. Schleicher and M. Stoll. An Introduction to Conway's Games and Numbers [PDF]

This note attempts to furnish John H. Conway's combinatorial game theory with an introduction that is easily accessible and yet mathematically precise and self-contained and which provides complete statements and proofs for some of the folklore in the subject.

Conway's theory is a fascinating and rich theory based on a simple and intuitive recursive definition of games, which yields a very rich algebraic structure. Games form an abelian GROUP in a very natural way. A certain subgroup of games, called numbers, is a FIELD that contains both the real numbers and the ordinal numbers. Conway's theory is deeply satisfying from a theoretical point of view, and at the same time it has useful applications to specific games such as Go.

Keywords. Conway game, surreal number, combinatorial game theory

2000 Mathematics Subject Classification. 91-02, 91A05, 91A46, 91A70

D. Timashev. Equivariant Symplectic Geometry of Cotangent Bundles, II [PDF]

We examine the structure of the cotangent bundle T*X of an algebraic variety X acted on by a reductive group G from the viewpoint of equivariant symplectic geometry. In particular, we construct an equivariant symplectic covering of T*X by the cotangent bundle of a certain variety of horospheres in X, and integrate the invariant collective motion on T*X. These results are based on a “local structure theorem” describing the action of a certain parabolic in G on an open subset of X, which is interesting by itself.

Keywords. Cotangent bundle, moment map, horosphere, symplectic covering, cross-section, invariant collective motion, flat

2000 Mathematics Subject Classification. Primary: 14L30; Secondary: 53D05, 53D20

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