Real Theta Characteristics And Automorphisms Of A Real Curve

Let X be a geometrically irreductble smooth projective cruve defined over R. of genus at least 2. that admits a nontrivial automorphism, sigma. Assume that X does not have any real points. Let tau be the antiholomorphic involution of the complexification lambda(C) of X. We show that if the action of sigma on the set S(X) of all real theta characteristics of X is trivial. then the order of sigma is even, say 2k and the automorphism tau o (sigma) over cap (lambda) of X-C has a fixed point, where (sigma) over cap is the automorphism of X x C-R defined by sigma We then show that there exists X with a real point and admitting a nontrivial automorphism sigma, such that the action of sigma on S(X) is trivial, while X/ not equal P-R(1) We also give an example of X with no real points and admitting a nontrivial automorphisim sigma such that the automorphism tau o (sigma) over cap (lambda) has a fixed point, the action of sigma on S(X) is trivial, and X/ not equal P-R(1)

A Numerical Scheme for Three-Dimensional Front Propagation and Control of Jordan Mode

As an example of a front propagation, we study the propagation of a three-dimensional nonlinear wavefront into a polytropic gas in a uniform state and at rest. The successive positions and geometry of the wavefront are obtained by solving the conservation form of equations of a weakly nonlinear ray theory. The proposed set of equations forms a weakly hyperbolic system of seven conservation laws with an additional vector constraint, each of whose components is a divergence-free condition. This constraint is an involution for the system of conservation laws, and it is termed a geometric solenoidal constraint. The analysis of a Cauchy problem for the linearized system shows that when this constraint is satisfied initially, the solution does not exhibit any Jordan mode. For the numerical simulation of the conservation laws we employ a high resolution central scheme. The second order accuracy of the scheme is achieved by using MUSCL-type reconstructions and Runge-Kutta time discretizations. A constrained transport-type technique is used to enforce the geometric solenoidal constraint. The results of several numerical experiments are presented, which confirm the efficiency and robustness of the proposed numerical method and the control of the Jordan mode.