Wave Solutions of Evolution Equations and Hamiltonian Flows on Nonlinear Subvarieties of Generalized Jacobians

The algebraic-geometric approach is extended to study solutions of N-component systems associated with the energy dependent Schrodinger operators having potentials with poles in the spectral parameter, in connection with Hamiltonian flows on nonlinear subvariaties of Jacobi varieties. The systems under study include the shallow water equation and Dym type equation. The classes of solutions are described in terms of theta-functions and their singular limits by using new parameterizations. A qualitative description of real valued solutions is provided

A CANONICAL FORM FOR A SYMPLECTIC INVOLUTION

We present a canonical form for a symplectic involution $S\in Sp(2g,\mathbb{Z})$, $S^2=1$; the construction is algorithmic. Application is made in the Riemann surface setting.Comment: 8 page

VEGF-165 serum levels and tyrosinase expression in melanoma patients: correlation with the clinical course.

Vascular endothelial growth factor (VEGF) is known to play a crucial role in the growth and metastatization of solid tumours. In cancer patients, high VEGF serum levels correlate with tumour status and prognosis, but to date few data have been reported concerning VEGF in melanoma patients. In the present study, immunoenzymatic and reverse transcription-polymerase chain reaction (RT-PCR) techniques were used to detect VEGF-165 serum levels and the presence of tyrosinase mRNA, respectively, in the peripheral blood of a cohort of 155 melanoma patients at different clinical stages (30 stage I, 40 stage II, 40 stage III and 45 stage IV; AJCC classification). Data were compared with both the extent of the disease and the clinical course. The aim was to assess the relationship between VEGF serum levels, the presence of detectable circulating melanoma cells and melanoma progression. A significant increase in VEGF serum levels was found in melanoma patients, in particular in those with metastatic disease; a higher incidence of relapses was found in stage I-III disease-free patients who showed an increase in VEGF during follow-up. VEGF serum levels were significantly higher in patients with detectable circulating melanoma cells than in those with negative tyrosinase mRNA expression. The finding of both an increase in VEGF and the presence of detectable melanoma cells during follow-up was associated with a relapse rate of 81%. The relapse rate was significantly lower when either of the two parameters were present separately. Multivariate analysis of both overall survival and time-to-progression selected baseline tyrosinase expression in peripheral blood but not VEGF serum levels as an independent prognostic factor