in this paper, we study the boundedness and persistence of the solutions, the global stability of the unique positive equilibrium point and the rate of convergence of a solution that converges to the equilibrium $e=(bar{x}, bar{y})$ of the system of two difference equations of exponential form: begin{equation*} x_{n+1}=dfrac{a+e^{-(bx_n+cy_n)}}{d+bx_n+cy_n}, y_{n+1}=dfrac{a+e^{-(by_n+cx_n)}}{d+...

We study the asymptotic behavior of solutions to fully nonlinear second order parabolic equations including a generalized curvature flow equation which was introduced by Mullins in 1957 as a model of evaporationcondensation. We prove that, in the multi-dimensional half space, solutions of the problem with prescribed contact angle asymptotically converge to a selfsimilar solution of the associat...

The average concentration of a passive scalar advected from a point source by a multivariate normal velocity field is shown to deviate from a Gaussian profile. The flatness (kurtosis) is calculated using an asymptotic series expansion valid for velocity fields with short correlation times or weak space dependence. An explicit formula for the excess flatness at first order demonstrates maximum d...

In a recent paper, Lee and Neuhoff found an asymptotic formula for the distribution of the length of the errors produced by a vector quantizer with many quantization points. This distribution depends on the source probability density, the quantizer point density and the quantizer shape profile. (The latter characterizes the shapes of the quantization cells as a function of position.) The purpos...

We derive a profile-likelihood confidence interval and a score based confidence interval to estimate the population prevalences, test sensitivities, and test specificities of two conditionally independent diagnostic tests when no gold standard is available. We are motivated by a real-data example on the study of the properties for two fallible diagnostic tests for bovine immunodeficiency virus....

In this paper, we develop a new approach to deal with asymptotic behavior of the age-structured homogeneous epidemic systems. For homogeneous systems, there is no attracting nontrivial equilibrium, instead we have to examine existence and stability of persistent solutions. Assuming that the host population dynamics is described by the stable population model, we rewrite the basic system into a ...

We discuss some aspects of recent research as well as more general issues about motivation, useful methods and open problems in the field of cosmological singularities. In particular, we review some of the approaches to the general area and include discussions of the method of asymptotic splittings, singularity and completeness theorems and the use of the Bel-Robinson energy to prove completene...

In a one parameter family of dilaton gravity theories which allow the coupling of the dilaton to gravity and to a U(1) gauge field to differ, we have found the existence of everywhere regular spacetimes describing black holes hiding expanding universes inside their horizon.