We study traveling wave solutions of an equation for surface waves of moderate amplitude arising as a shallow water approximation of the Euler equations for inviscid, incompressible and homogeneous fluids. We obtain solitary waves of elevation and depression, including a family of solitary waves with compact support, where the amplitude may increase or decrease with respect to the wave speed. Our approach is based on techniques from dynamical systems and relies on a reformulation of the evolution equation as an autonomous Hamiltonian system which facilitates an explicit expression for bounded orbits in the phase plane to establish existence of the corresponding periodic and solitary traveling wave solutions.

This paper proves that contractive ordinary differential equation systems remain contractive when diffusion is added. Thus, diffusive instabilities, in the sense of the Turing phenomenon, cannot arise for such systems. An important biochemical system is shown to satisfy the required conditions.

The existence of weak solutions to the continuous coagulation equation with multiple fragmentation is shown for a class of unbounded coagulation and fragmentation kernels, the fragmentation kernel having possibly a singularity at the origin. This result extends previous ones where either boundedness of the coagulation kernel or no singularity at the origin for the fragmentation kernel was assumed.

A hybrid evolutionary model is used to propose a hierarchical homology of protein sequences to identify protein functions systematically. The proposed model offers considerable potentials, considering the inconsistency of existing methods for predicting novel proteins. Because some novel proteins might align without meaningful conserved domains, maximizing the score of sequence alignment is not the best criterion for predicting protein functions. This work presents a decision model that can minimize the cost of making a decision for predicting protein functions using the hierarchical homologies. Particularly, the model has three characteristics: (i) it is a hybrid evolutionary model with multiple fitness functions that uses genetic programming to predict protein functions on a distantly related protein family, (ii) it incorporates modified robust point matching to accurately compare all feature points using the moment invariant and thin-plate spline theorems, and (iii) the hierarchical homologies holding up a novel protein sequence in the form of a causal tree can effectively demonstrate the relationship between proteins. This work describes the comparisons of nucleocapsid proteins from the putative polyprotein SARS virus and other coronaviruses in other hosts using the model.

Given finite disjoint sets { }, = 1, …, in Euclidean n-space, a general problem with numerous applications is to find simple nontrivial functions () which separate the sets { } in the sense that () ≤ () for all ⊂ and ≠ = 1, …, This can always be done (e.g., with the piecewise linear function obtained by the Voronoi Partition defined for the points in [Formula: see text]). However, typically one seeks linear functions () if possible, in which case we say the sets { } are piecewise linear separable. If the sets are separable in a linear sense, there are generally many such functions that separate, in which case we seek a 'best' (in some sense) separator that is referred as a robust separator. If the sets are not separable in a linear sense, we seek a function which comes as close as possible to separating, according to some criterion.

Given a pair of finite, disjoint sets and in , a fundamental problem with numerous applications is to find a simple function () defined over which separates the sets in the sense that () > 0 for all ∈ and () < 0 for all ∈ . This can always be done (e.g., with the piecewise linear function defined by the Voronoi partition implied by the points in ⋃ ). However typically one seeks a linear (or possibly a quadratic) function if possible, in which case we say that and are linearly (quadratically) separable. If and are separable in a linear or quadratic sense, there are generally many such functions which separate. In this case we seek a 'robust' separator, one that is best in a sense to be defined. When and are not separable in a linear or quadratic sense we seek a function which comes as close as possible to separating, according to some well defined criterion. In this paper we examine the optimization problems associated with the set separation problem, characterize them (convex or non-convex) and suggest algorithms for their solutions.

Given a pair of finite disjoint sets and in Euclidean -space, a fundamental problem with numerous applications is to efficiently determine a hyperplane (, ) which separates these sets when they are separable, or 'nearly' separates them when they are not. We seek a hyperplane that separates them in the sense that a measure of the Euclidean distance between the separating hyperplane and of the points is as large as possible. This is done by 'weighting' points relative to ∪ according to their distance to (, ), with the closer points getting a higher weight, but still taking into account the points distant from (, ). The negative exponential is chosen for that purpose. In this paper we examine the optimization problem associated with this set separation problem and characterize it (convex or non-convex).