Two timescales asymptotes of the weakly compressible Stokes system

We establish a two timescale asymptotics of the weakly compressible Stokes system which has dissipation of order $\epsilon >0$. For any $L^2$ initial data, over time scale of order 1, the solutions of the weakly compressible Stokes system converge strongly to those of the acoustic system as $\epsilon \to 0$. Over time scale of order $1/\epsilon$, the limit system is the incompressible Stokes system with the initial data projected on the incompressible mode. For the periodic domain, the convergence is weak due to the fast oscillation generated by acoustic waves. For the Navier-slip boundary condition with the reciprocal of slip length being square of the Knudsen number, the acoustic waves are damped by the viscous boundary layer, and consequently the strong convergence is justified.     

Graphs in which some and every maximum matching is uniquely restricted

A matching M in a graph G is uniquely restricted if there is no matching in G that is distinct from M but covers the same vertices as M. Solving a problem posed by Golumbic, Hirst, and Lewenstein, we characterize the graphs in which some maximum matching is uniquely restricted. Solving a problem posed by Levit and Mandrescu, we characterize the graphs in which every maximum matching is uniquely restricted. Both our characterizations lead to efficient recognition algorithms for the corresponding graphs.     

Supereulerian bipartite digraphs

A digraph D is supereulerian if D has a spanning closed ditrail. Bang‐Jensen and Thomassé conjectured that if the arc‐strong connectivity of a digraph D is not less than the independence number , then D is supereulerian. A digraph is bipartite if its underlying graph is bipartite. Let be the size of a maximum matching of D. We prove that if D is a bipartite digraph satisfying , then D is supereulerian. Consequently, every bipartite digraph D satisfying is supereulerian. The bound of our main result is best possible.     

Optimal insurance with belief heterogeneity and incentive compatibility

People may evaluate risk differently in the insurance market. Motivated by this, we examine an optimal insurance problem allowing the insured and the insurer to have heterogeneous beliefs about loss distribution. To reduce ex post moral hazard, we follow Huberman et al. (1983) to assume that alternative insurance contracts satisfy the principle of indemnity and the incentive-compatible constraint. Under the assumption that the insurance premium is calculated by the expected value principle, we establish a necessary and sufficient condition for an optimal insurance solution and provide a practical scheme to improve any suboptimal insurance strategy under an arbitrary form of belief heterogeneity. By virtue of this condition, we explore qualitative properties of optimal solutions, and derive optimal insurance contracts explicitly for some interesting forms of belief heterogeneity. As a byproduct of this investigation, we find that Theorem 3.6 of Young (1999) is not completely true.     

A higher order uniformly convergent method for singularly perturbed parabolic turning point problems

In this article, we study numerical approximation for a class of singularly perturbed parabolic (SPP) convection-diffusion turning point problems. The considered SPP problem exhibits a parabolic boundary layer in the neighborhood of one of the sides of the domain. Some a priori bounds are given on the exact solution and its derivatives, which are necessary for the error analysis. A numerical scheme comprising of implicit finite difference method for time discretization on a uniform mesh and a hybrid scheme for spatial discretization on a generalized Shishkin mesh is proposed. Then Richardson extrapolation method is applied to increase the order of convergence in time direction. The resulting scheme has second-order convergence up to a logarithmic factor in space and second-order convergence in time. Numerical experiments are conducted to demonstrate the theoretical results and the comparative study is done with the existing schemes in literature to show better accuracy of the proposed schemes.     

Hyperuniform and rigid stable matchings

We study a stable partial matching τ of the d‐dimensional lattice with a stationary determinantal point process Ψ on R^d with intensity α>1. For instance, Ψ might be a Poisson process. The matched points from Ψ form a stationary and ergodic (under lattice shifts) point process Ψ^τ with intensity 1 that very much resembles Ψ for α close to 1. On the other hand Ψ^τ is hyperuniform and number rigid, quite in contrast to a Poisson process. We deduce these properties by proving more general results for a stationary point process Ψ, whose so‐called matching flower (a stopping set determining the matching partner of a lattice point) has a certain subexponential tail behavior. For hyperuniformity, we also additionally need to assume some mixing condition on Ψ. Furthermore, if Ψ is a Poisson process then Ψ^τ has an exponentially decreasing truncated pair correlation function.     

A higher order difference method for singularly perturbed parabolic partial differential equations

This paper studies a higher order numerical method for the singularly perturbed parabolic convection-diffusion problems where the diffusion term is multiplied by a small perturbation parameter. In general, the solutions of these type of problems have a boundary layer. Here, we generate a spatial adaptive mesh based on the equidistribution of a positive monitor function. Implicit Euler method is used to discretize the time variable and an upwind scheme is considered in space direction. A higher order convergent solution with respect to space and time is obtained using the postprocessing based extrapolation approach. It is observed that the convergence is independent of perturbation parameter. This technique enhances the order of accuracy from first order uniform convergence to second order uniform convergence in space as well as in time. Comparative study with the existed meshes show the highly effective behavior of the present method.     

A priori bounds and well-posedness of a system associated with unsteady boundary layer flows

An integral equation with singularities is introduced to characterize unsteady laminar boundary layer flows and some properties of solutions of this integral equation are investigated. Utilizing these properties, a priori bounds are obtained for the skin friction function and the similarity stream function and the well-posedness of solutions is proved.     

Interior layers for an inhomogeneous Allen-Cahn equation

We consider the problem

     

Maximally resonant polygonal systems

A benzenoid system G is k-resonant if any set F of no more than k disjoint hexagons is a resonant pattern, i.e, G−F has a perfect matching. In 1990’s M. Zheng constructed the 3-resonant benzenoid systems and showed that they are maximally resonant, that is, they are k-resonant for all k≥1. Recently, the equivalence of 3-resonance and maximal resonance has been shown to be valid also for coronoid systems, carbon nanotubes, polyhexes in tori and Klein bottles, and fullerene graphs. So our main problem is to investigate the extent of graphs possessing this interesting property. In this paper, by replacing the above hexagons with even faces, we define k-resonance of graphs in surfaces, possibly with boundary, in a unified way. Some exceptions exist. For plane polygonal systems tessellated with polygons of even size at least six such that all inner vertices have the same degree three and the others have degree two or three, we show that such 3-resonant polygonal systems are indeed maximally resonant. They can be constructed by gluing and lapping operations on three types of basic graphs.