In this paper, we study the asymptotics of the Krawtchouk polynomials $K^{N}_{n} {\left( {z;p,q} \right)}$ as the degree n becomes large. Asymptotic expansions are obtained when the ratio of the parameters $\frac{n}{N}$ tends to a limit c ∈ (0, 1) as n → ∞. The results are globally valid in one or two regions in the complex z-plane depending on the values of c and p; in particular, they are valid in regions containing the interval on which these polynomials are orthogonal. Our method is based on the Riemann-Hilbert approach introduced by Deift and Zhou.

The authors consider Maxwell's equations for an isomagnetic anisotropic and inhomogeneous medium in two dimensions, and discuss an inverse problem of determining the permittivity tensor ${\left( {\begin{array}{*{20}c} {{\varepsilon _{1} }} & {{\varepsilon _{2} }} \\ {{\varepsilon _{2} }} & {{\varepsilon _{3} }} \\ \end{array} } \right)}$ and the permeability μ in the constitutive relations from a finite number of lateral boundary measurements. Applying a Carleman estimate, the authors prove an estimate of the Lipschitz type for stability, provided that ε₁, ε₂, ε₃, μ satisfy some a priori conditions.

Let S be a Riemann surface with genus p and n punctures. Assume that 3p−3+n > 0 and n ≥ 1. Let a be a puncture of S and let $ \ifmmode\expandafter\tilde\else\expandafter\~\fi{S} = S \cup {\left\{ a \right\}}$. Then all mapping classes in the mapping class group Mod _S that fixes the puncture a can be projected to mapping classes of ${\text{Mod}}_{{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{S}}} $ under the forgetful map. In this paper the author studies the mapping classes in Mod _S that can be projected to a given hyperbolic mapping class in ${\text{Mod}}_{{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{S}}} $.

Under the assumption that the underlying measure is a non-negative Radon measure which only satisfies some growth condition, the authors prove that for a class of commutators with Lipschitz functions which include commutators generated by Calderón-Zygmund operators and Lipschitz functions as examples, their boundedness in Lebesgue spaces or the Hardy space H ¹(μ) is equivalent to some endpoint estimates satisfied by them. This result is new even when the underlying measure μ is the d-dimensional Lebesgue measure.

In this paper, the authors develop new global perturbation techniques for detecting the persistence of transversal homoclinic orbits in a more general nondegenerated system with action-angle variable. The unperturbed system is assumed to have saddlecenter type equilibrium whose stable and unstable manifolds intersect in one dimensional manifold, and does not have to be completely integrable or near-integrable. By constructing local coordinate systems near the unperturbed homoclinic orbit, the conditions of existence of transversal homoclinic orbit are obtained, and the existence of periodic orbits bifurcated from homoclinic orbit is also considered.

The authors consider the exact controllability of the vibrations of a thin shallow shell, of thickness 2ε with controls imposed on the lateral surface and at the top and bottom of the shell. Apart from proving the existence of exact controls, it is shown that the solutions of the three dimensional exact controllability problems converge, as the thickness of the shell goes to zero, to the solution of an exact controllability problem in two dimensions.

Consider a system where units have random magnitude entering according to a homogeneous or nonhomogeneous Poisson process, while in the system, a unit's magnitude may change with time. In this paper, the authors obtain some results for the limiting behavior of the sum process of all unit magnitudes present in the system at time t.