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Information × Registration Number 0225U002278, (0118U000646) , R & D reports Title Algebraic Curves, Integrable Systems and Applications to Modern Problems in Physics and Informatics popup.stage_title Алгебраїчні криві, інтегровані системи та застосування до сучасних проблем фізики та інформатики Head Repetsky Stanislav P., д.ф.-м.н. Registration Date 28-02-2025 Organization National University of Kyiv-Mohyla Academy popup.description1 popup.description2 Expanding the possibilities of analytical solution of differential equations, integrable and quasi-integral systems with a distinction between numerical and analytical solutions of applied problems in the study of disordered crystal structures. The project is aimed at the development of mathematical methods of theoretical physics related to soliton-type equations, in particular methods of algebraic geometry and the theory of Lie representations of algebras and groups, and the separation of numerical and analytical solutions to applied problems in the study of disordered crystal structures. Was obtained expressions for second kind integrals on non-hyperelliptic (n,s)-curves. Such a curve possesses a Weierstrass point at infinity which is a branch point where all sheets of the curve come together. The infinity serves as the basepoint for Abel's map, and the basepoint in the definition of the second kind integrals. We define second kind differentials as having a pole at the infinity, therefore the second kind integrals need to be regularized. Was obtained explicit expressions for genus 2 degenerate sigma-function in terms of genus 1 sigma-function and elementary functions as solutions of a system of linear partial differential equations satisfied by the sigma-function. A general Thomae formula was obtained, which is a development of second Thomae formula in hyperelliptic case. Namely, values at zero of the lowest non-vanishing derivatives of theta functions with singular characteristics of arbitrary multiplicity are expressed in terms of branch points and period matrix. The general Thomae formula gives first and second Thomae formulas as particular cases. A complete solution of Mumford’s second problem about representation of theta derivatives with rational characteristics in terms of theta constants with rational characteristics is found. Product Description popup.authors popup.nrat_date 2025-02-28 Close