Center conditions II: Parametric and model center problems

We consider an Abel equation (*)y’=p(x)y ² +q(x)y ³ withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty ₀=y(0)≡y(1) for any solutiony(x) of (*). We introduce a parametric version of this condition: an equation (**)y’=p(x)y ² +εq(x)y ³ p, q as above, ℂ, is said to have a parametric center, if for any ε and for any solutiony(ε,x) of (**),y(ε,0)≡y(ε,1). We show that the parametric center condition implies vanishing of all the momentsm _k (1), wherem _k (x)=∫ ₀ ^{x pk} (t)q(t)(dt),P(x)=∫ ₀ ^x p(t)dt. We investigate the structure of zeroes ofm _k (x) and on this base prove in some special cases a composition conjecture, stated in [10], for a parametric center problem. The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.     

Center and isochronous center problems for quasi analytic systems

This paper considers the problems of determining center or focus and isochronous centers for the planar quasi-analytic systems. Two recursive formulas to determine the focal values and period constants are given. The convergence of first integral near the center is proved. Using the general results to quasi-quadratic systems, the problem of the isochronous center of the origin is completely solved. This research is partially supported by the National Natural Science Foundation of China (10671179 and 10771196) and the Natural Science Foundation of Yunnan Province (2005A0092M)     

Parametric nonlinear resonant Robin problems

We consider a nonlinear Robin problem driven by the p‐Laplacian. In the reaction we have the competing effects of two nonlinearities. One term is parametric, strictly $(p?1)$‐sublinear and the other one is $(p?1)$‐linear and resonant at any nonprincipal variational eigenvalue. Using variational tools from the critical theory (critical groups), we show that for all big values of the parameter λ the problem has at least five nontrivial smooth solutions.     

Center conditions: rigidity of logarithmic differential equations

In this paper, we prove that any degree d deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again. This is a generalization of Ilyashenko's result on Hamiltonian differential equations. The main tools are Picard-Lefschetz theory of a polynomial with complex coefficients in two variables, specially the Gusein-Zade/A'Campo's theorem on calculating the Dynkin diagram of the polynomial, and the action of Gauss-Manin connection on the so-called Brieskorn lattice/Petrov module of the polynomial. We will also generalize J.P. Francoise recursion formula and (∗) condition for a polynomial which is a product of lines in a general position. Some applications on the cyclicity of cycles and the Bautin ideals will be given.     

Parametric approximation problems arising in vector optimization

In this paper, a known scalarization result of vector optimization theory is reviewed and stated in a different form and a new short proof is presented. Moreover, it is shown how to apply this result to multi-objective optimization problems and to special problems in statistics and optimal control theory.The author is grateful to Professor H. Schellhaas and T. Staib for helpful discussions on this subject and to a referee for pointing out an error in an earlier version of this paper.     

Center and isochronous center at infinity for differential systems

In this article, the center conditions and isochronous center conditions at infinity for differential systems are investigated. We give a transformation by which infinity can be transferred into the origin. So we can study the properties of infinity with the methods of the origin. As an application of our method, we discuss the conditions of infinity to be a center and a isochronous center for a class of rational differential system. As far as we know, this is the first time that the isochronous center conditions of infinity are discussed.     

Parametric tail copula estimation and model testing

Parametric models for tail copulas are being used for modeling tail dependence and maximum likelihood estimation is employed to estimate unknown parameters. However, two important questions seem unanswered in the literature: (1) What is the asymptotic distribution of the MLE and (2) how does one test the parametric model? In this paper, we answer these two questions in the case of a single parameter for ease of illustration. A simulation study is provided to investigate the finite sample performance of the proposed estimator and test.     

Nonmonotone equilibrium problems: coercivity conditions and weak regularization

We consider a general equilibrium problem in a finite-dimensional space setting and propose a new coercivity condition for existence of solutions. We also show that it enables us to create a broad family of regularization methods with preserving well-definiteness and convergence of the iteration sequence without additional monotonicity assumptions. Some examples of applications are also given.     

Parametric versus nonparametric tolerance regions in detection problems

A major problem in statistical quality control is to detect a change in the distribution of independent sequentially observed random vectors. The case of a Gaussian pre-change distribution has been extensively analyzed. Here we are concerned with the non-normal multivariate case. In this setup it is natural to use tolerance regions as detection tools. These regions are defined in terms of density level sets, which can be estimated in a plug-in fashion. Under a normal mixture model we compare, through a simulation study, the performance of such a detection scheme for two density estimators: a (parametric) normal mixture and a (nonparametric) kernel estimator. The problem of the bandwidth choice for the latter is addressed. We also obtain a result concerning the convergence rates of the error probabilities under a general parametric model. Finally, a real data example is discussed.     

Absolute cyclicity, Lyapunov quantities and center conditions

In this paper we consider analytic vector fields X₀ having a non-degenerate center point e. We estimate the maximum number of small amplitude limit cycles, i.e., limit cycles that arise after small perturbations of X₀ from e. When the perturbation (Xλ) is fixed, this number is referred to as the cyclicity of Xλ at e for λ near 0. In this paper, we study the so-called absolute cyclicity; i.e., an upper bound for the cyclicity of any perturbation Xλ for which the set defined by the center conditions is a fixed linear variety. It is known that the zero-set of the Lyapunov quantities correspond to the center conditions (Caubergh and Dumortier (2004) [6]). If the ideal generated by the Lyapunov quantities is regular, then the absolute cyclicity is the dimension of this so-called Lyapunov ideal minus 1. Here we study the absolute cyclicity in case that the Lyapunov ideal is not regular.     

Lotka-Volterra's model and migrations: Breaking of the well-known center

This paper is devoted to the study of the effect of individual behavior on the Lotka-Volterra predation. We assume that the individuals have many activities in a day for example. Each population is subdivided into subpopulations corresponding to different activities. In order to be clear, I have chosen the case of two activities for each population. We assume that the activities change is faster than the other processes (reproduction, mortality, predation…). This means that we consider population in which the individuals change their activities many times in a day while the reproduction and the predation effects are sensible after about ten days, for example. We use the aggregation method developed in [1] to obtain the global dynamics. Indeed, we start with a micro-model governing the micro-variables, which are the subpopulation densities; the aggregation method permits us to obtain a simpler system governing the macro-variables, which are the global population densities. Furthermore, this method allows us to observe emergence of the dynamics. Indeed, the method implies that the dynamics of the micro-system is close to an invariant manifold after a short time. We show that the dynamics on this manifold is a perturbation of the well-known center of the Lotka-Volterra model. Finally, we prove that a weak change of behavior can lead to a subcritical Hopf bifurcation in the global dynamics.     

Embedding problems with local conditions

LetY→X be a connectedG-Galois cover of affine varieties in characteristicp, and supposeG=Γ/P for somep-groupP. We show that there is a connected Γ-Galois coverZ→X dominatingY→X, and thatZ→X can be chosen to have prescribed behavior over a given closed subset ofX. There are several versions of this result, depending on whether ramification is permitted, and whether adelic behavior is prescribed. The results are deduced from a general assertion about embedding problems, which is proven for profinite groups. Supported in part by NSF Grant DMS94-00836.     

Relative Pareto minimizers for multiobjective problems: existence and optimality conditions

In this paper we introduce and study enhanced notions of relative Pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we establish the existence of relative Pareto minimizers for general multiobjective problems under a refined version of the subdifferential Palais-Smale condition for set-valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers (as well as for conventional efficient and weak efficient counterparts) that are new in both finite-dimensional and infinite-dimensional settings. Our proofs are based on variational and extremal principles of variational analysis; in particular, on new versions of the Ekeland variational principle and the subdifferential variational principle for set-valued and single-valued mappings in infinite-dimensional spaces.     

Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions

We study an optimal design problem for the domain of an elliptic equation with Dirichlet boundary conditions. We introduce a relaxed formulation of the problem which always admits a solution, and we prove some necessary conditions for optimality both for the relaxed and for the original problem.     

Diffraction problems with impedance conditions

The two- or three-dimensional electromagnetic diffraction problem for a half-plane impedance or reactance sheet belongs to a class of elliptic transmission problems of mixed type. Sobolev spaces of order 1 and ±1/2 are naturally involved according to the energy norm and the trace theorem, respectively. This operator theoretic approach presents the equivalence to systems of Wiener-Hopf equations and their solution in the sense of a well-posed problem with respect to the spaces under consideration. Slightly different impedance numbers for the two banks of the screen lead to a perturbation problem. All results yield direct a priori estimates for the solutions.