On the global attractivity for a logistic equation with piecewise constant arguments

In this paper, we consider the following logistic equation with piecewise constant arguments:

     

Existence and global attractivity of positive periodic solutions for delay Lotka-Volterra competition patch systems with stocking

By means of Mawhin's continuation theorem of coincidence degree and Lyapunov functional, a set of easily verifiable criteria are established for the existence and global attractivity of positive periodic solutions for delay Lotka-Volterra competition patch system with stocking

     

Ab initio Tertiary Structure Prediction of Proteins

A daunting challenge in the area of computational biology has been to develop a method to theoretically predict the correct three-dimensional structure of a protein given its linear amino acid sequence. The ability to surmount this challenge, which is known as the protein folding problem, has tremendous implications. We introduce a novel ab initio approach for the protein folding problem. The accurate prediction of the three-dimensional structure of a protein relies on both the mathematical model used to mimic the protein system and the technique used to identify the correct structure. The models employed are based solely on first principles, as opposed to the myriad of techniques relying on information from statistical databases. The framework integrates our recently proposed methods for the prediction of secondary structural features including helices and strands, as well as -sheet and disulfide bridge formation. The final stage of the approach, which culminates in the tertiary structure prediction of a protein, utilizes search techniques grounded on the foundations of deterministic global optimization, powerful methods which can potentially guarantee the correct identification of a protein's structure. The performance of the approach is illustrated with bovine pancreatic trypsin inhibitor protein and the immunoglobulin binding domain of protein G.     

On Uniform Global Error Bounds for Convex Inequalities

This paper studies the existence of a uniform global error bound when a convex inequality g 0, where g is a closed proper convex function, is perturbed. The perturbation neighborhoods are defined by small arbitrary perturbations of the epigraph of its conjugate function. Under certain conditions, it is shown that for sufficiently small arbitrary perturbations the perturbed system is solvable and there exists a uniform global error bound if and only if g satisfies the Slater condition and the solution set is bounded or its recession function satisfies the Slater condition. The results are used to derive lower bounds on the distance to ill-posedness.     

On 2x2 conservation laws with large data

We deal with a strictly hyperbolic system of two conservation laws in one spatial dimension. One of the eigenvalues of the system is of Temple type (rarefaction and shock curves coincide), the other eigenvalue is only required to be genuinely nonlinear.We consider the initial value problem for data of the following kind: the total variation of the Temple component is bounded, possibly large, while the total variation of the other component is small. For such data we prove global existence, uniqueness and L⊃-Lipschitz continuous dependence of solutions.AMS Subject Classification: Primary 35L65; Secondary 35D05, 35L45.     

A Point Balance Algorithm for the Spherical Code Problem

The Spherical Code (SC) problem has many important applications in such fields as physics, molecular biology, signal transmission, chemistry, engineering and mathematics. This paper presents a bilevel optimization formulation of the SC problem. Based on this formulation, the concept of balanced spherical code is introduced and a new approach, the Point Balance Algorithm (PBA), is presented to search for a 1-balanced spherical code. Since an optimal solution of the SC problem (an extremal spherical code) must be a 1-balanced spherical code, PBA can be applied easily to search for an extremal spherical code. In addition, given a certain criterion, PBA can generate efficiently an approximate optimal spherical code on a sphere in the n-dimensional space ⁿ. Some implementation issues of PBA are discussed and putative global optimal solutions of the Fekete problem in 3, 4 and 5-dimensional space are also reported. Finally, an open question about the geometry of Fekete points on the unit sphere in the 3-dimensional space is posed.     

The asymptotic behavior of a class of nonlinear delay difference equations

The asymptotic behavior of difference equations of type 0, \end{equation*}">is studied, where and each are continuous real functions with decreasing and increasing. Results include sufficient conditions for permanence, oscillations and global attractivity.

     

Global Optimization: On Pathlengths in Min-Max Graphs

Let f be a smooth nondegenerate real valued function on a finite dimensional, compact and connected Riemannian manifold. The bipartite min-max graph is defined as follows. Its nodes are formed by the set of local minima and the set of local maxima. Two nodes (a local minimum and a local maximum) are connected in by means of an edge if some trajectory of the corresponding gradient flow connects them. Given a natural number k, we construct a function f such that the length of the shortest path in between two specific local minima exceeds k. The latter construction is independent of the underlying Riemannian metric.     

Adaptive extremal optimization by detrended fluctuation analysis

Global optimization is one of the key challenges in computational physics as several problems, e.g. protein structure prediction, the low-energy landscape of atomic clusters, detection of community structures in networks, or model-parameter fitting can be formulated as global optimization problems. Extremal optimization (EO) has become in recent years one particular, successful approach to the global optimization problem. As with almost all other global optimization approaches, EO is driven by an internal dynamics that depends crucially on one or more parameters. Recently, the existence of an optimal scheme for this internal parameter of EO was proven, so as to maximize the performance of the algorithm. However, this proof was not constructive, that is, one cannot use it to deduce the optimal parameter itself a priori. In this study we analyze the dynamics of EO for a test problem (spin glasses). Based on the results we propose an online measure of the performance of EO and a way to use this insight to reformulate the EO algorithm in order to construct optimal values of the internal parameter online without any input by the user. This approach will ultimately allow us to make EO parameter free and thus its application in general global optimization problems much more efficient.     

Global existence of weak solutions for the 3D chemotaxis-Navier–Stokes equations

In this paper, we consider the Cauchy problem for the three dimensional chemotaxis-Navier–Stokes equations. By exploring the new a priori estimates, we prove the global existence of weak solutions for the 3D chemotaxis-Navier–Stokes equations.