A black-box program for the explicit calculation of Riemann matrices of arbitrary compact connected Riemann surfaces is presented. All such Riemann surfaces are represented as plane algebraic curves. These algebraic curves are allowed to have arbitrary singularities. The method of calculation of the Riemann matrix is essentially its deenition: we numerically integrate the holomorphic diierentia...

A new approximate Riemann solver for the equations of magnetohydrodynamics (MHD) with an isothermal equation of state is presented. The proposed method of solution draws on the recent work of Miyoshi and Kusano, in the context of adiabatic MHD, where an approximate solution to the Riemann problem is sought in terms of an average constant velocity and total pressure across the Riemann fan. This ...

We present an extension of the HLLC approximate Riemann solver by Toro, Spruce and Speares to the relativistic equations of fluid dynamics. The solver retains the simplicity of the original two-wave formulation proposed by Harten, Lax and van Leer (HLL) but it restores the missing contact wave in the solution of the Riemann problem. The resulting numerical scheme is computationally efficient, r...

1 Riemann Poisson manifolds were introduced by the author in [1] and studied in more details in [2]. Kähler-Riemann foliations form an interesting subset of the Riemannian foliations with remarkable properties ( see [3]). In this paper we will show that to give a regular Riemann Poisson structure on a manifold M is equivalent to to give a Kähler-Riemann foliation on M such that the leafwise sym...

We explore the structure of compact Riemann surfaces by studying their isometry groups. First we give two constructions due to Accola [1] showing that for all g ≥ 2, there are Riemann surfaces of genus g that admit isometry groups of at least some minimal size. Then we prove a theorem of Hurwitz giving an upper bound on the size of any isometry group acting on any Riemann surface of genus g ≥ 2...

Riemann hypothesis is proven by reducing the vanishing of Riemann Zeta to an orthogonality condition for the eigenfunctions of a nonHermitian operator having the zeros of Riemann Zeta as its eigenvalues. Eigenfunctions are analogous to the so called coherent states and in general not orthogonal to each other. The construction of the operator is inspired by the conviction that Riemann Zeta is as...

The MFCAV Riemann solver is practically used in many Lagrangian or ALE methods due to its merit of sharp shock profiles and rarefaction corners, though very often with numerical oscillations. By viewing it as a modification of the WWAM Riemann solver, we apply the MFCAV Riemann solver to the Lagrangian method recently developed by Maire. P. H et. al.. The numerical experiments show that the app...

Solutions to the Riemann-Hilbert problems with irregular singularities naturally associated to semisimple Frobenius manifold structures on Hurwitz spaces (moduli spaces of meromorphic functions on Riemann surfaces) are constructed. The solutions are given in terms of meromorphic bidifferentials defined on the underlying Riemann surface. The relationship between different classes of Frobenius ma...

Aut(H) = PGL2(R) = PSL2(R) = SL2(R)/{±I2}, where the plus sign denotes matrices with positive determinant. A fundamental result of Riemann states that every simply connected connected Riemann surface is isomorphic to C,P1(C) or H. In fact, any punctured Riemann surface, R (namely R ⊆ R with R a compact Riemann surface and R−R a finite set of points), which is hyperbolic, that is, 2− 2 · genus(R...

The aim of the paper is to study qualitative behavior of solutions to the equation du df(u) , . m + ~oT9(o)' where (x,t) £lx R+,u = u(x,t) € R. The main new feature with respect to previous works is that the flux function / may have finitely many inflection points, intervals in which it is afEne, and corner points. The function g is supposed to be zero at 0 and 1, and positive in between. We pr...