We study several schemes of rst or second-order accuracy based on kinetic approximations to solve pressureless and isothermal gas dynamics equations. The pressureless gas system is weakly hyperbolic, giving rise to the formation of density concentrations known as delta-shocks. For the isothermal gas system, the innnite speed of expansion into vacuum leads to zero timestep in the Godunov method ...

We discuss the procedure for the exact solution of the Riemann problem in special relativistic MHD. We consider both initial states leading to a set of only three waves analogous to the ones in relativistic hydrodynamics, as well as generic initial states leading to the full set of seven MHD waves. Because of its generality, the solution presented here could serve as an important if not indispe...

Recommended by Ingo Witt We consider the Riemann problem for the Hopf equation with concave-convex flux functions. Applying the weak asymptotics method we construct a uniform in time description for the Cauchy data evolution and show that the use of this method implies automatically the appearance of the Oleinik E-condition. distributed under the Creative Commons Attribution License, which perm...

Phase transition problems in compressible media can be modelled by mixed hyperbolicelliptic systems of conservation laws. Within this approach phase boundaries are understood as shock waves that satisfy additional constraints, sometimes called kinetic relations. In recent years several tracking-type algorithms have been suggested for numerical approximation. Typically a core piece of these algo...

The theory of Riemann surfaces is a classical field of mathematics where geometry and analysis play equally important roles. The purpose of these notes is to present some basic facts of this theory to make this book more self contained. In particular we will deal with classical descriptions of Riemann surfaces, Abelian differentials, periods on Riemann surfaces, meromorphic functions, theta fun...

Riemann problems for the compressible Euler system in two space dimensions are complicated and difficult, but a viable alternative remains missing. We list merits of one-dimensional Riemann problems and compare them with those for the current two-dimensional Riemann problems, to illustrate their worthiness. We approach twodimensional Riemann problems via the methodology promoted by Andy Majda i...

Inspired by his vanishing results of tautological classes and by Harer’s computation of the virtual cohomological dimension of the mapping class group, Looijenga conjectured that the moduli space of smooth Riemann surfaces admits a stratification by affine subsets with a certain number of layers. Similarly, Roth and Vakil extended the conjecture to the moduli spaces of Riemann surfaces of compa...

This talk will come in four parts. First, I will introduce the Riemann hypothesis (RH) as it was first introduced, using the Riemann zeta function, and discuss briefly its connection to number theory and prime numbers. Second, I will make this connection explicit by discussion a famous problem equivalent to RH. Third, I will examine the generalised Riemann hypothesis (GRH) and, finally, will li...

We use discrete analogs of Riemann–Hilbert problem’s methods to derive the discrete Bessel kernel which describes the poissonized Plancherel measures for symmetric groups. To do this we define discrete analogs of a Riemann–Hilbert problem and of an integrable integral operator and show that computing the resolvent of a discrete integrable operator can be reduced to solving a corresponding discr...

(Generalized) Riemann Hypothesis (that all non-trivial zeros of the (Dirichlet L-function) zeta function have real part one-half) is arguably the most important unsolved problem in contemporary mathematics due to its deep relation to the fundamental building blocks of the integers, the primes. The proof of the Riemann hypothesis will immediately verify a slew of dependent theorems ([BRW], [SA])...