Isospectral Flow and Liouville-Arnold Integration in Loop Algebras

A number of examples of Hamiltonian systems that are integrable by classical means are cast within the framework of isospectral flows in loop algebras. These include: the Neumann oscillator, the cubically nonlinear Schr\"odinger systems and the sine-Gordon equation. Each system has an associated invariant spectral curve and may be integrated via the Liouville-Arnold technique. The linearizing map is the Abel map to the associated Jacobi variety, which is deduced through separation of variables in hyperellipsoidal coordinates. More generally, a family of moment maps is derived, identifying certain finite dimensional symplectic manifolds with rational coadjoint orbits of loop algebras. Integrable Hamiltonians are obtained by restriction of elements of the ring of spectral invariants to the image of these moment maps. The isospectral property follows from the Adler-Kostant-Symes theorem, and gives rise to invariant spectral curves. {\it Spectral Darboux coordinates} are introduced on rational coadjoint orbits, generalizing the hyperellipsoidal coordinates to higher rank cases. Applying the Liouville-Arnold integration technique, the Liouville generating function is expressed in completely separated form as an abelian integral, implying the Abel map linearization in the general case.Comment: 42 pages, 2 Figures, 1 Table. Lectures presented at the VIIIth Scheveningen Conference, held at Wassenaar, the Netherlands, Aug. 16-21, 199

Abelian Functions for Trigonal Curves of Genus Three

We develop the theory of generalized Weierstrass sigma- and \wp-functions defined on a trigonal curve of genus three. In particular we give a list of the associated partial differential equations satisfied by the \wp-functions, a proof that the coefficients of the power series expansion of the sigma-function are polynomials of moduli parameters, and the derivation of two addition formulae.Comment: 32 pages, no figures. Revised version has the a fuller description of the general (3,4) trigonal curve results, the first version described only the "Purely Trigonal" cas

Some remarks on the hyperelliptic moduli of genus 3

In 1967, Shioda \cite{Shi1} determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in \cite{Shi1} are incorrect. In this paper, we compute the correct equations among the invariants of the binary octavics and give necessary and sufficient conditions for two genus 3 hyperelliptic curves to be isomorphic over an algebraically closed field $k$, $\ch k \neq 2, 3, 5, 7$. For the first time, an explicit equation of the hyperelliptic moduli for genus 3 is computed in terms of absolute invariants.Comment: arXiv admin note: text overlap with arXiv:1209.044

Spectral Curves of Operators with Elliptic Coefficients

A computer-algebra aided method is carried out, for determining geometric objects associated to differential operators that satisfy the elliptic ansatz. This results in examples of Lamé curves with double reduction and in the explicit reduction of the theta function of a Halphen curve

Avaliação de diferentes métodos de colheita de embriões bovinos.

Edição dos resumos da 20. Reunião Anual da Sociedade Brasileira de Tecnologia de Embriões, Araxá, MG, agosto 2006

Extranuclear structural components that mediate dynamic chromosome movements in yeast meiosis

Telomere-led rapid chromosome movements or rapid prophase movements direct fundamental meiotic processes required for successful haploidization of the genome. Critical components of the machinery that generates rapid prophase movements are unknown, and the mechanism underlying rapid prophase movements remains poorly understood. We identified S. cerevisiae Mps2 as the outer nuclear membrane protein that connects the LINC complex with the cytoskeleton. We also demonstrate that the motor Myo2 works together with Mps2 to couple the telomeres to the actin cytoskeleton. Further, we show that Csm4 interacts with Mps2 and is required for perinuclear localization of Myo2, implicating Csm4 as a regulator of the Mps2-Myo2 interaction. We propose a model in which the newly identified functions of Mps2 and Myo2 cooperate with Csm4 to drive chromosome movements in meiotic prophase by coupling telomeres to the actin cytoskeleton.Fil: Lee, Chih Ying. Oklahoma Medical Research Foundation; Estados UnidosFil: Bisig, Carlos Gaston. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigaciones en Química Biológica de Córdoba. Universidad Nacional de Córdoba. Facultad de Ciencias Químicas. Centro de Investigaciones en Química Biológica de Córdoba; ArgentinaFil: Conrad, Michael M.. Oklahoma Medical Research Foundation; Estados UnidosFil: Ditamo, Yanina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigaciones en Química Biológica de Córdoba. Universidad Nacional de Córdoba. Facultad de Ciencias Químicas. Centro de Investigaciones en Química Biológica de Córdoba; ArgentinaFil: Previato de Almeida, Luciana. Oklahoma Medical Research Foundation; Estados UnidosFil: Dresser, Michael E.. Oklahoma Medical Research Foundation; Estados UnidosFil: Pezza, Roberto J.. Oklahoma Medical Research Foundation; Estados Unido

On Separation of Variables for Integrable Equations of Soliton Type

We propose a general scheme for separation of variables in the integrable Hamiltonian systems on orbits of the loop algebra $\mathfrak{sl}(2,\Complex)\times \mathcal{P}(\lambda,\lambda^{-1})$. In particular, we illustrate the scheme by application to modified Korteweg--de Vries (MKdV), sin(sinh)-Gordon, nonlinear Schr\"odinger, and Heisenberg magnetic equations.Comment: 22 page

Akns Hierarchy, Self-Similarity, String Equations and the Grassmannian

In this paper the Galilean, scaling and translational self--similarity conditions for the AKNS hierarchy are analysed geometrically in terms of the infinite dimensional Grassmannian. The string equations found recently by non--scaling limit analysis of the one--matrix model are shown to correspond to the Galilean self--similarity condition for this hierarchy. We describe, in terms of the initial data for the zero--curvature 1--form of the AKNS hierarchy, the moduli space of these self--similar solutions in the Sato Grassmannian. As a byproduct we characterize the points in the Segal--Wilson Grassmannian corresponding to the Sachs rational solutions of the AKNS equation and to the Nakamura--Hirota rational solutions of the NLS equation. An explicit 1--parameter family of Galilean self--similar solutions of the AKNS equation and the associated solution to the NLS equation is determined.Comment: 25 pages in AMS-LaTe