37 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
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
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
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