We introduce a colored coalescent process which recovers random colored genealogical trees. Here a colored genealogical tree has its vertices colored black or white. Moving backward along the colored genealogical tree, the color of vertices may change only when two vertice coalesce. The rule that governs the change of color involves a parameter $x$. When $x=1/2$, the colored coalescent process can be derived from a variant of the Wright-Fisher model for a haploid population in population genetics. Explicit computations of the expectation and the cumulative distribution function of the coalescent time are carried out. For example, our calculation shows that when $x=1/2$, for a sample of $n$ colored individuals, the expected time for the colored coalescent process to reach a black MRAC or a white MRAC, respectively, is $3-2/n$. On the other hand, the expected time for the colored coalescent process to reach a MRAC, either black or white, is $2-2/n$, which is the same as that for the standard Kingman coalescent process.

The foundations of weak turbulence theory is explored through its application to the (alpha) Fermi-Pasta-Ulam (FPU) model, a simple weakly nonlinear dispersive system. A direct application of the standard kinetic equations would miss interesting dynamics of the energy transfer process starting from a large-scale excitation. This failure is traced to an enforcement of the exact resonance condition, whereas mathematically the resonance should be broadened due to the energy transfer happening on large but finite time scales. By allowing for the broadened resonance, a modified three-wave kinetic equation is derived for the FPU model. This kinetic equation produces some correct scaling predictions about the statistical dynamics of the FPU model, but does not model accurately the detailed evolution of the energy spectrum. The reason for the failure seems not to be one of the previously clarified reasons for breakdown in the weak turbulence theory.

We consider a family of scalar delay differential equations $x'(t)=f(t,x_t)$, with a nonlinearity $f$ satisfying a negative feedback condition combined with a boundedness condition. We present a global stability criterion for this family, which in particular unifies the celebrated 3/2-conditions given for the Yorke and the Wright type equations. We illustrate our results with some applications.

The Alpha version of the Fermi-Pasta-Ulam problem is revisited through direct numerical simulations and an application of weak turbulence theory. The energy spectrum, initialized with a large scale excitation, is traced through a series of distinct qualitative phases en route to eventual equipartition. Weak turbulence theory is applied in an attempt to provide and effective quantitative description of the evolution of the energy spectrum. Some scaling predictions are well-confirmed byt the numerical simulations.

Summary We study a flow of fresh and salt water in a two di mensional axially symmetric coastal aquifer with a well on the central axis. Th e flow is governed by a nonlinear Darcy's law. We also show the behaviour of the solu tion when the out flow of salt water at well goes to 0.

We consider a class of singular systems of Lane-Emden type  

A semilinear elliptic equation with generalized cubic nonlinearity is stud- ied. Global bifurcation diagrams and the existence of multiple solutions are obtained and in certain cases, exact multiplicity is proved.

Surrogate optimization is a computational procedure that uses a sequence of approximations of the objective function to predict an optimum. Even in the case when the derivative information of the objective function is not available, the surro- gate functions can still be constructed only based on the function values. Moreover, the optimization method can be extended for better ways to compute numerical so- lutions of certain nonlinear differential equations. In this paper, we will present a direct convergence analysis of the surrogate optimization process in one dimensional fashion via polynomial interpolations of objective function values. Numerical exper- iments will be given to illustrate the effectiveness of the algorithms developed. An application in multi-agent cooperative search problem will be also presented. 1. Introduction. Without loss of generality, we may assume that all unconstrained optimization problems to be studied in this paper have the form min x∈Rn f (x),