A significant number of large optimization problems exhibit structure known as partial separability, for example, least squares problems, where elemental functions are gathered into groups that are then squared. The sparsity of the Jacobian of a partially separable function can be exploited by computing the smaller Jacobians of the elemental functions and then assembling them into the full Jaco...

We introduce deformation theoretic methods for determining when a curve X in a nonhyperelliptic Jacobian JC will deform with JC to a non-Jacobian. We apply these methods to a particular class of curves in symmetric powers C of C where 3 e g−3. More precisely, given a pencil g1 d of degree d on C, let X be the curve parametrizing divisors of degree e in divisors of g1 d (see the paper for the pr...

We provide the first explicit construction of genus 2 curves over finite fields whose Jacobians are ordinary, have large prime-order subgroups, and have small embedding degree. Our algorithm works for arbitrary embedding degrees k and prime subgroup orders r. The resulting abelian surfaces are defined over prime fields Fq with q ≈ r. We also provide an algorithm for constructing genus 2 curves ...

– We review the definition and main properties of the distributional Jacobians, in particular for Sobolev maps u : R → R with values in the (k − 1)dimensional sphere S.

We state two recent results concerning the linearization of integrable systems on generalised Jacobians. Then we apply this to the (complexified) spherical pendulum.

where K is a subfield of even index at most 10 in a primitive cyclotomic field Q(ζp), or a subfield of index 2 in Q(ζpq) or Q(ζpα ). This result generalizes previous work of Brumer, Mestre, and Tautz-Top-Verberkmoes. Our curves are constructed as branched covers of the projective line, and the endomorphisms arise as quotients of double coset algebras of the Galois groups of these coverings. In ...

Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, Z[ζp] the ring of integers in the pth cyclotomic field, Cf,p : y p = f(x) the corresponding superelliptic curve and J(Cf,p) its jacobian. Assuming that either n = p + 1 or p does not divide ...

The algebraic–geometric approach is extended to study evolution equations associated with the energy-dependent Schrödinger operators having potentials with poles in the spectral parameter, in connection with Hamiltonian flows on nonlinear subvarieties of Jacobi varieties. The general approach is demonstrated by using new parametrizations for constructing quasi-periodic solutions of the shallow-...