69 research outputs found
On the invariants of the quotients of the Jacobian of a curve of genus 2
The original publication is available at www.springerlink.comInternational audienceLet C be a curve of genus 2 that admits a non-hyperelliptic involution. We show that there are at most 2 isomorphism classes of elliptic curves that are quotients of degree 2 of the Jacobian of C. Our proof is constructive, and we present explicit formulae, classified according to the involutions of C, that give the minimal polynomial of the j-invariant of these curves in terms of the moduli of C. The coefficients of these minimal polynomials are given as rational functions of the moduli
Duality between integrable Stackel systems
For the Stackel family of the integrable systems a non-canonical
transformation of the time variable is considered. This transformation may be
associated to the ambiguity of the Abel map on the corresponding hyperelliptic
curve. For some Stackel's systems with two degrees of freedom the 2x2 Lax
representations and the dynamical r-matrix algebras are constructed. As an
examples the Henon-Heiles systems, integrable Holt potentials and the
integrable deformations of the Kepler problem are discussed in detail.Comment: LaTeX2e, 18 page
The classification of isotrivially fibred surfaces with p_g=q=2
An isotrivially fibred surface is a smooth projective surface endowed with a
morphism onto a curve such that all the smooth fibres are isomorphic to each
other. The first goal of this paper is to classify the isotrivially fibred
surfaces with completing and extending a result of Zucconi. As an
important byproduct, we provide new examples of minimal surfaces of general
type with and and a first example with .Comment: Main paper by M.Penegini. Appendix by S.Rollenske. 31 pages, 6
Figures. v2 changed group relations in Theorem 5.2, changes in Theorem 5.7,
new proof of Theorem 4.15, minor corrections of misprint
Thomae type formulae for singular Z_N curves
We give an elementary and rigorous proof of the Thomae type formula for
singular curves. To derive the Thomae formula we use the traditional
variational method which goes back to Riemann, Thomae and Fuchs.Comment: 22 page
Wannier functions for quasi-periodic finite-gap potentials
In this paper we consider Wannier functions of quasi-periodic g-gap () potentials and investigate their main properties. In particular, we discuss
the problem of averaging underlying the definition of Wannier functions for
both periodic and quasi-periodic potentials and express Bloch functions and
quasi-momenta in terms of hyperelliptic functions. Using this approach
we derive a power series expansion of the Wannier function for quasi-periodic
potentials valid at and an asymptotic expansion valid at large
distance. These functions are important for a number of applied problems
The Darboux point
A theory of global optimality based upon the Darboux-point concept is developed. A definition is proposed for the Darboux point, and the Darboux point is shown to exist on nonglobally optimal trajectories under relatively general conditions. A mutually exclusive classification of Darboux points is noted, and several properties are proved for one of these classes (the Type-1 Darboux point). Numerous examples are included to illustrate the Darboux-point definition and properties.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45205/1/10957_2004_Article_BF00932789.pd
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