The quotient of the upper half plane by Γ1(N), acting by fractional linear transformations, is the complex manifold associated to an affine algebraic curve Y1(N)/C. This curve has a natural model Y1(N)/Q, which for N > 3 is a fine moduli scheme for elliptic curves with a point of exact order N . We will let X1(N) denote the smooth projective curve which contains Y1(N) as a dense Zariski open su...

We study the stability of the perfectly matched layer (PML) for symmetric second order hyperbolic partial differential equations on the upper half plane, with boundary conditions at y = 0. Using a mode analysis, we develop a technique to analyze the stability of the corresponding initial boundary value problem on the half plane. We apply our technique to the PML for the elastic wave equations s...

We construct a Ces\`{a}ro-type operator acting on Dirichlet space of the upper half plane using approach strongly continuous semigroups composition operators Banach spaces. then determine spectral and norm properties obtained operator.