global bifurcation of limit cycles in a family of polynomial systems in this paper we study the number of limit cycles in a family of polynomial systems using bifurcation methods and melnikov functions we obtain the maximal number of limit cycles in global bifurcation

the number of limit cycles from a cubic center by the melnikov function of any order abstract in this paper we consider the system x y 1 x 2 p x y y x 1 x 2 q x y where p x y and q x y are arbitrary quadratic polynomials we study the maximum number of limit cycles bifurcating from the periodic orbits by using the melnikov function of any order we prove that the upper bound for the number of limit cycles is 3 and reached

bifurcation of critical periods from the reversible rigidly isochronous centers abstract in this paper we study bifurcation of critical periods by perturbing a nonlinear vector field containing m th degree homogeneous terms with a rigidly isochronous center at the origin first we give expressions of period bifurcation functions in the form of integrals and then study the number of critical periods for cases of m 1 2 3 respectively

nonparabolic dissipative systems modeling the flow of liquid crystals we study a simplified system which retains most of the interesting mathematical properties of the original ericksen leslie equations for the flow of liquid crystals this is a coupled nonparabolic dissipative dynamic system we derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions we also discuss uniqueness and some stability properties of the system 1995 john wiley sons inc

limit cycles appearing from the perturbation of a system with a multiple line of critical points abstract consider planar ordinary differential equations of the form x y c x y y x c x y where c x y is an algebraic curve we are interested in knowing whether the existence of multiple factors for c is important or not when we study the maximum number of zeros of the abelian integral m that controls the limit cycles that bifurcate from the period annulus of the origin when we perturb it with an arbitrary polynomial vector field with this aim we study in detail the case c x y 1 y m where m is a positive integer number and prove that m has essentially no impact on the number of zeros of m this result improves the known studies on m one of the key points of our approach is that we obtain a simple expression of m based on some successive reductions of the integrals appearing during the procedure

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

measure valued solutions to the ericksen leslie model equipped with the oseen frank energy abstract in this article we prove the existence of measure valued solutions to the ericksen leslie system equipped with the oseen frank energy we introduce a suitable concept of generalized gradient young measures for this system via a galerkin approximation we show the existence of weak solutions to a regularized system and attain measure valued solutions for vanishing regularization additionally it is shown that the measure valued solution fulfills an energy inequality

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

explicit characterization oflp young measures the full explicit description of young measures attainable by bounded sequences from the lebesgue spacelp rm was known so far for the casep only in the paper a suitable condition characterizing such measures is isolated also for the case 1 p a generalization of thelp young measures constructed recently by diperna and majda can be thus described also but in special cases only

the number of limit cycles in perturbations of polynomial systems with multiple circles of critical points abstract this paper investigates the problem for limit cycle bifurcations of system x y f x y e p x y y x f x y e q x y where f x y consists of multiple circles and p x y q x y are polynomials of degree n the upper bound for the maximal number of limit cycles emerging from the period annulus surrounding the origin is provided in terms of n and the involved multiplicities of circles by using the first order melnikov function furthermore hopf bifurcation for a cubic system of this type is discussed

finite element approximations of the ericksen leslie model for nematic liquid crystal flow the ericksen leslie model describes dynamics of low molar mass nematic liquid crystals where the spatiotemporal distribution of defects defining texture is represented by the director unit vector field bf d omega t rightarrow mathbb s 2 it consists of the navier stokes equations with an extra viscous stress tensor and a convective harmonic map heat flow equation to govern the dynamics of the director field two fully discrete finite element methods for a regularized system using the ginzburg landau regularization and for the limiting ericksen leslie model are proposed and well posedness and related discrete energy laws are established for the regularized model unconditional convergence of finite element solutions towards weak solutions of the continuum model as well as convergence towards measure valued solutions of the limiting ericksen leslie model are verified when the mesh parameters and the regularization parameter successively tend to zero computational experiments are also presented to show the importance of balancing numerical and regularization parameters to compare regularized and direct approaches and to show numerically the finite time formation annihilation and evolution of point defects

on the abelian integrals of quadratic reversible centers with orbits formed by genus one curves of higher degree abstract in this study we determine the number of zeros for abelian integrals in four cases of quadratic reversible centers of genus one based on the results of li et al we prove that the upper bound of the associated number of zeros for abelian integrals with orbits formed by quartics quintics sextics and eight degree curves under polynomial perturbations of arbitrary degree n depends linearly on n

limit cycle bifurcations in perturbations of planar piecewise smooth systems with multiply lines of critical points abstract this paper investigates the limit cycle bifurcation problem of planar piecewise polynomial systems whose unperturbed system has a center at the origin and may have invariant multiply lines by melnikov function method this paper provides the upper and lower bounds for the poincare cyclicity in particular the exact cyclicities of two specific cases are given using analyzing method one of which has been discussed in the paper 7

a bayesian semi parametric model for colorectal cancer incidences a bayesian semi parametric model is proposed to capture the interaction among demographic effects age and gender spatial effects county and temporal effects of colorectal cancer incidences simultaneously in particular an extension of multivariate conditionally autoregressive car processes to a partially informative gaussian demographic spatial temporal car dstcar process for a spatial temporal setting is proposed the precision matrix of the gaussian dstcar process is the kronecker product of several components the spatial component is modelled with a car prior a pth order intrinsic autoregressive prior iar p is implemented for the temporal component to estimate a smoothed and non parametric temporal trend the demographic component is modelled with a wishart prior data analysis shows significant spatial correlation only exists in the age group of 50 59 males and females in their 50s and 60s show fairly strong correlation the hypothesis testing based on bayes factor suggests that gender correlation cannot be ignored in this model

weak strong uniqueness for measure valued solutions we prove the weak strong uniqueness for measure valued solutions of the incompressible euler equations these were introduced by diperna and majda in their landmark paper commun math phys 108 4 667 689 1987 where in particular global existence to any l2 initial data was proven whether measure valued solutions agree with classical solutions if the latter exist has apparently remained open

well posedness of the ericksen leslie system we prove the local well posedness of the ericksen leslie system and the global well posedness for small initial data under a physical constraint condition on the leslie coefficients which ensures that the energy of the system is dissipated instead of the ginzburg landau approximation we construct an approximate system with the dissipated energy based on a new formulation of the system

global weak solution and blow up criterion of the general ericksen leslie system for nematic liquid crystal flows abstract in this paper we investigate the three dimensional general ericksen leslie e l system with ginzburg landau type approximation modeling nematic liquid crystal flows first by overcoming the difficulties from lack of maximum principle for the director equation and high order nonlinearities for the stress tensor we prove existence of global in time weak solutions under physically meaningful boundary conditions and suitable assumptions on the leslie coefficients which ensures that the total energy of the e l system is dissipated moreover for the e l system with periodic boundary conditions we prove the local well posedness of classical solutions under the so called parodi s relation and establish a blow up criterion in terms of the temporal integral of both the maximum norm of the curl of the velocity field and the maximum norm of the gradient of the liquid crystal director field

on the number of limit cycles of a perturbed cubic polynomial differential center abstract for e sufficiently small we consider the planar polynomial differential system x y y 2 a x 2 b x c e p x y y x y 2 a x 2 b x c e q x y where p x y and q x y are polynomials of degree n by using the averaging method of first order we bound the number of limit cycles that can bifurcate from period annulus of the unperturbed system

existence of solutions for the ericksen leslie system we study the general ericksen leslie system which describes the flow of liquid crystal materials the dissipation property of the system is established and is used to prove the global existence of weak solutions we also study the existence of classical solutions and the asymptotic stability of the solutions

characterization of generalized gradient young measures generated by sequences in w1 1 and bv generalized young measures as introduced by diperna and majda commun math phys 108 667 689 1987 provide a quantitative tool for studying the one point statistics of oscillation and concentration in sequences of functions in this work after developing a functional analytic framework for such measures including a compactness theorem and results on the generation of such young measures by l1 bounded sequences or even by sequences of bounded radon measures we turn to investigation of those young measures that are generated by bounded sequences of w1 1 gradients or bv derivatives we provide several techniques to manipulate such measures including shifting averaging and approximation by piecewise homogeneous young measures and then establish the main new result of this work the duality characterization of the set of bv or w1 1 gradient young measures in terms of jensen type inequalities for quasiconvex functions with linear growth at infinity this result is the natural generalization of the kinderlehrer pedregal theorem arch ration mech anal 115 329 365 1991 j geom anal 4 59 90 1994 for classical young measures to the w1 1 and bv case and contains its version for weakly converging sequences in w1 1 as a special case finally we give an application to a new lower semicontinuity theorem in bv

lower semicontinuity of surface energies using the theory of indicator measures a lower semicontinuity result for quasiconvex functions in w 1 1 and assuming only l 1 convergence is obtained

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75