Discrete infinite-dimensional type-K monotone dynamical systems and time-periodic reaction-diffusion systems

The asymptotic behavior of discrete type-K monotone dynamical systems and reaction-diffusion equations is investigated. The studying content includes the index theory for fixed points, permanence, global stability, convergence everywhere and coexistence. It is shown that the system has a globally asymptotically stable fixed point if every fixed point is locally asymptotically stable with respect to the face it belongs to and at this point the principal eigenvalue of the diagonal partial derivative about any component not belonging to the face is not one. A nice result presented is the sufficient and necessary conditions for the system to have a globally asymptotically stable positive fixed point. It can be used to establish the sufficient conditions for the system to persist uniformly and the convergent result for all orbits. Applications are made to time-periodic Lotka-Volterra systems with diffusion, and sufficient conditions for such systems to have a unique positive periodic solution attracting all positive initial value functions are given. For more general time-periodic type-K monotone reaction-diffusion systems with spatial homogeneity, a simple condition is given to guarantee the convergence of all positive solutions.     

Incomplete oblique projections for solving large inconsistent linear systems

The authors introduced in previously published papers acceleration schemes for Projected Aggregation Methods (PAM), aiming at solving consistent linear systems of equalities and inequalities. They have used the basic idea of forcing each iterate to belong to the aggregate hyperplane generated in the previous iteration. That scheme has been applied to a variety of projection algorithms for solving systems of linear equalities or inequalities, proving that the acceleration technique can be successfully used for consistent problems. The aim of this paper is to extend the applicability of those schemes to the inconsistent case, employing incomplete projections onto the set of solutions of the augmented system Ax − r = b. These extended algorithms converge to the least squares solution. For that purpose, oblique projections are used and, in particular, variable oblique incomplete projections are introduced. They are defined by means of matrices that penalize the norm of the residuals very strongly in the first iterations, decreasing their influence with the iteration counter in order to fulfill the convergence conditions. The theoretical properties of the new algorithms are analyzed, and numerical experiences are presented comparing their performance with several well-known projection methods. Dedicated to Clovis Gonzaga on the occassion of his 60th birthday.     

Improved stability criterion and output feedback control for discrete time-delay systems

This paper investigates the stability and stabilization for a class of linear systems with time-varying delay. We provide a new finite-sum inequality which is a powerful tool for stability analysis of time-delay systems. Applying the inequality, a new stability criterion is proposed in terms of linear matrix inequalities (LMIs). We also design a method for static output feedback (SOF) control problems which contains two parts. The first part is to find an initial values of the matrix variables. By utilizing the initial values, the condition for SOF control problems can be solved by an improved path-following method. Numerical examples demonstrate the effectiveness of the stability criterion and the SOF stabilization method.     

Descent methods for quasidifferentiable minimization

A descent method is given for minimizing a nondifferentiable function which can be locally approximated by pointwise minima of convex functions. At each iterate the algorithm finds several directions by solving several linear or quadratic programming subproblems. These directions are then used in an Armijo-like search for the next iterate. A feasible direction extension to inequality constrained minimization problems is also presented. The algorithms converge to points satisfying necessary optimality conditions which are sharper than the ones involved in convergence results for algorithms based on the Clarke subdifferential.This research was sponsored by Project 02.15.     

Permanence for non-autonomous food chain systems with delay

In this paper we first establish a new general criterion for the permanence of Kolmogorov-type systems of nonautonomous functional differential equations. Then, as applications of this criterion we study the permanence of a class of n-species general nonautonomous food chain systems with delay and new sufficient condition are established.     

On the finite-dimensional dynamical systems with limited competition

The asymptotic behavior of dynamical systems with limited competition is investigated. We study index theory for fixed points, permanence, global stability, convergence everywhere and coexistence. It is shown that the system has a globally asymptotically stable fixed point if every fixed point is hyperbolic and locally asymptotically stable relative to the face it belongs to. A nice result is the necessary and sufficient conditions for the system to have a globally asymptotically stable positive fixed point. It can be used to establish the sufficient conditions for the system to persist uniformly and the convergence result for all orbits. Applications are made to time-periodic ordinary differential equations and reaction-diffusion equations.

     

Weak uncertainty principle for fractals, graphs and metric measure spaces

We develop a new approach to formulate and prove the weak uncertainty inequality, which was recently introduced by Okoudjou and Strichartz. We assume either an appropriate measure growth condition with respect to the effective resistance metric, or, in the absence of such a metric, we assume the Poincaré inequality and reverse volume doubling property. We also consider the weak uncertainty inequality in the context of Nash-type inequalities. Our results can be applied to a wide variety of metric measure spaces, including graphs, fractals and manifolds.

     

High order starting iterates for implicit Runge-Kutta methods: an improvement for variable-step symplectic integrators

When dealing with implicit Runge–Kutta methods, the equationsdefining the stages are usually solved by iterative methods.The closer the first iterate is to the solution, the fewer iterationsare required. In this paper the author presents and analysesnew high order algorithms to compute such initial iterates.Numerical experiments are given to illustrate the performanceof the new procedures when combined with a variable-step symplecticintegrator.     

On permanence of all subsystems of competitive Lotka–Volterra systems with delays

In this paper, permanence for a class of competitive Lotka–Volterra systems is considered that have distributed delays and constant coefficients on interaction terms and have time dependent growth rate vectors with an asymptotic average. A computable necessary and sufficient condition is found for the permanence of all subsystems of the system and its small perturbations on the interaction matrix. This is a generalization from systems without delays to delayed systems of Ahmad and Lazer’s work on total permanence (S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Ann. Mat. 185 (2006) S47–S67). In addition to Ahmad and Lazer’s example showing that permanence does not imply total permanence, another example of permanent system is given having a non-permanent subsystem. As a particular case, a necessary and sufficient condition is given for all subsystems of the corresponding autonomous system to be permanent. As this condition does not rely on the delays, it actually shows the equivalence between such permanence of systems with delays and that of corresponding systems without delays. Moreover, this permanence property is still retained by systems as a small perturbation of the original system.     

Alternating Direction Method for Maximum Entropy Subject to Simple Constraint Sets

The problem of maximizing the entropy subject to simple constraint sets is reformulated as a structured variational inequality problem by introducing dual variables. A new iterative alternating direction method is then developed that generates alternatively the dual and primal iterates. For some existing maximum entropy problems in the literature, the new dual iterate can be derived from a simple projection and then the new primal iterate can be obtained via solving approximately n separate one-dimensional strong monotone equations. Therefore, the proposed method is very easy to carry out. Preliminary numerical results show that the method is applicable.     

Permanence for General Nonautonomous Impulsive Population Systems of Functional Differential Equations and Its Applications

In this paper, we investigate general impulsive nonautonomous population dynamical systems of functional differential equations. By utilizing the method of multiple Liapunov-like functionals to construct the permanence region, a general criterion on the permanence for the system is established. Furthermore, as applications of this general criterion, a class of impulsive nonautonomous n-species Lotka–Volterra competitive systems with delays and a class of impulsive nonautonomous 3-species Lotka–Volterra food chain systems with delays are discussed. Some new and useful sufficient conditions on the permanence for these systems are established.     

On permanence of Lotka–Volterra systems with delays and variable intrinsic growth rates

For competitive Lotka–Volterra systems, Ahmad and Lazer’s work [S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Annali di Matematica 185 (2006) S47–S67] on total permanence of systems without delays has been extended to delayed systems [Z. Hou, On permanence of all subsystems of competitive Lotka–Volterra systems with delays, Nonlinear Analysis: Real World Applications 11 (2010) 4285–4301]. In this paper, existence and boundedness of nonnegative solutions and permanence are considered for general Lotka–Volterra systems with delays including competitive, cooperative, predator–prey and mixed type systems. First, a condition is established for the existence and boundedness of solutions on a half line. Second, a necessary condition on the limits of the average growth rates is provided for permanence of all subsystems. Then the result for competitive systems is also proved for the general systems by using the same techniques. Just as for competitive systems, the eminent finding is that permanence of the system and all of its subsystems is completely irrelevant to the size and distribution of the delays.     

A NEW CONSTRAINTS IDENTIFICATION TECHNIQUE-BASED QP-FREE ALGORITHM FOR THE SOLUTION OF INEQUALITY CONSTRAINED MINIMIZATION PROBLEMS

In this paper, we propose a feasible QP-free method for solving nonlinear inequality constrained optimization problems. A new working set is proposed to estimate the active set. Specially, to determine the working set, the new method makes use of the multiplier information from the previous iteration, eliminating the need to compute a multiplier function. At each iteration, two or three reduced symmetric systems of linear equations with a common coefficient matrix involving only constraints in the working set are solved, and when the iterate is sufficiently close to a KKT point, only two of them are involved. Moreover, the new algorithm is proved to be globally convergent to a KKT point under mild conditions. Without assuming the strict complementarity, the convergence rate is superlinear under a condition weaker than the strong second-order sufficiency condition. Numerical experiments illustrate the efficiency of the algorithm.     

Shallow, deep and very deep cuts in the analytic center cutting plane method

In this paper we show that the cut does not need to go through the query point: it can be deep or shallow. The primal framework leads to a simple analysis of the potential variation, which shows that the inequality needed for convergence of the algorithm is in fact attained at the first iterate of the feasibility step. Received July 3, 1996 / Revised version received July 11, 1997 Published online August 18, 1998     

On the -condition

In [5], XI, S. Shelah formulated a condition on forcing notion ( -condition) which implies that the forcing it satisfies does not add reals. It was proved that, under some additional demands, this condition is preserved by revised countable support iterations. We are going to show that these demands can be weakened. A few examples of simple forcing notions that can iterate while preserving the-condition, and hence without adding reals, are presented.     

Permanent coexistence in chemostat models with delayed feedback control

Feedback control for chemostat model first appears in [P. De Leenher, H.L. Smith, Feedback control for chemostat models, J. Math. Biol. 46 (2003) 48–70] where the authors consider a dilution rate as a feedback control variable, keeping the input nutrient concentration to be constant. They showed that the coexistence of two organisms can be achieved under single nutrient source. In this paper, we improve the model in consideration of a time delay arising from controlling a dilution rate. It is shown that coexistence of two organisms can also be achieved in the improved model. The coexistence is mathematically insisted by solving permanence problem. We prove the problem by using the theory of “average Liapunov function”.     

Preconditioned Iterative Methods for Algebraic Systems from Multiplicative Half-Quadratic Regularization Image Restorations

<正>Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image.In this paper,we consider a class of convex and edge-preserving regularization functions,i.e.,multiplicative half-quadratic regularizations,and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations.At each Newton iterate,the preconditioned conjugate gradient method,incorporated with a constraint preconditioner,is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived,which can be used to estimate the convergence speed of the preconditioned conjugate gradient method.We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably well.     

An Inequality Involving Tight Closure and Parameter Ideals

An inequality is established involving colengths of the tightclosure of ideals of systems of parameters in local rings withsome mild conditions. As an application, a proof is given ofa result due to Goto and Nakamura (first conjectured by Watanabeand Yoshida), which states that the Hilbert–Samuel multiplicityof a parameter ideal is greater than or equal to the colengthof the tight closure of the ideal. The result is also furtherrefined. 2000 Mathematics Subject Classification 13D40, 13A35,13H15.     

Potential inequality

We prove a general inequality for kernels satisfying the maximum principle. This is then used to derive a sufficient condition for the kernel to define a continuous map of Lebesgue spaces. Exactly this condition happens to be necessary and sufficient for the validity of Hardy’s inequality with weights in one dimension. Some applications indicating the unifying nature of the potential inequality are given. A part of this paper was presented at the 865th AMS meeting, March 1991, Tampa, Florida.