Lenard scheme for two dimensional periodic Volterra chain

We prove that for compatible weakly nonlocal Hamiltonian and symplectic operators, hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating Nijenhuis operators are weakly nonlocal or not. We construct a recursion operator of the two dimensional periodic Volterra chain from its Lax representation and prove that it is a Nijenhuis operator. Furthermore we show this system is a (generalised) bi-Hamiltonian system. Rather surprisingly, the product of its weakly nonlocal Hamiltonian and symplectic operators gives rise to the square of the recursion operator.Comment: Submit to Journal of Mathematical Physic

Integrable Systems in n-dimensional Riemannian Geometry

In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an (n)-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary operator. This gives us a natural connection between finite dimensional geometry, infinite dimensional geometry and integrable systems. Moreover one finds a Lax pair in (\orth{n+1}) with the vector modified Korteweg-De Vries equation (vmKDV) \vk{t}= \vk{xxx}+\fr32 ||\vk{}||^2 \vk{x} as integrability condition. We indicate that other integrable vector evolution equations can be found by using a different Ansatz on the form of the Lax pair. We obtain these results by using the {\em natural} or {\em parallel} frame and we show how this can be gauged by a generalized Hasimoto transformation to the (usual) {\em Fren{\^e}t} frame. If one chooses the curvature to be zero, as is usual in the context of integrable systems, then one loses information unless one works in the natural frame

Genome-wide scan on plasma triglyceride and high density lipoprotein cholesterol levels, accounting for the effects of correlated quantitative phenotypes

BACKGROUND: Plasma triglyceride and high density lipoprotein cholesterol levels are inversely correlated and both are genetically related. Two correlated traits may be influenced both by shared and unshared genes. The power to detect unshared trait-specific genes may be increased by incorporating correlated traits as covariates. The power to localize the shared genes may be improved by bivariate analysis. Univariate genome scans were carried out on triglyceride (high density lipoprotein cholesterol) with and without using high density lipoprotein cholesterol (triglyceride) as a covariate, and bivariate linkage analysis on triglyceride and high density lipoprotein cholesterol using the 330 Framingham pedigrees of the Genetic Analysis Workshop 13 data. The results of five genome scans were compared to determine the chromosomal regions which may harbor the genes influencing variation specific to triglycerides, specific to high density lipoprotein cholesterol, or the covariation of both triglyceride and high density lipoprotein cholesterol. RESULTS: The results of our five genome scans identified some chromosomal regions with possible quantitative trait loci (QTL) that may specifically influence one trait, such as the regions on chromosome 1 (at 1 cM near marker 280we5), on high density lipoprotein cholesterol, or control the covariation of both traits, such as the regions on chromosome 7 (at 169 cM near marker GATA30D09), chromosome 12 (at 3 cM near marker GATA4H03), chromosome 20 (at 49 cM near marker GATA29F06), chromosome 2 (at 146 cM near marker GATA8H05), and chromosome 6 (at 148 cM near marker GATA184A08) on triglyceride and high density lipoprotein cholesterol. The one on chromosome 6 had a LOD score of 3.1 with the bivariate linkage analysis. CONCLUSION: There is strong evidence for a QTL on chromosome 6 near marker GATA184A08 appearing to influence the variation of high density lipoprotein cholesterol and triglycerides in the Framingham population