We consider the zeros on the boundary ∂Ω of a Neumann eigenfunction φλj of a real analytic plane domain Ω. We prove that the number of its boundary zeros is O(λj) where −∆φλj = λ2jφλj . We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is O(λj). It follows that the number of nodal lines of φλj (components of the nodal set) which touch the b...

In this paper we analyse the stability of weak discrete stationary shocks. The difference approximation is conservative, dissipative and k-th order accurate (k = 1 or k = 3). In Sections 1-3, following the previous results of D. Michelson we adapt the multistep methods RungeKutta and Adams-Bashforth to solve systems of conservation laws. In section 4 we analyse the stability of the numerical me...

We derive the Euler equations from quantum dynamics for a class of fermionic many-body systems. We make two types of assumptions. The first type are physical assumptions on the solution of the Euler equations for the given initial data. The second type are a number of reasonable conjectures on the statistical mechanics and dynamics of the Fermion Hamiltonian. Department of Mathematics, Universi...