principal poincare pontryagin function associated to polynomial perturbations of a product of d 1 straight lines abstract in this paper we study small polynomial perturbations of a hamiltonian vector field with hamiltonian f formed by a product of d 1 real linear functions in two variables we assume that the corresponding lines are in a general position in r 2 that is the lines are distinct non parallel no three of them have a common point and all critical values not corresponding to intersections of lines are distinct we prove in this paper that the principal poincare pontryagin function m k t associated to such a perturbation and to any family of ovals surrounding a singular point of center type belongs to the c t 1 t module generated by abelian integrals and some integrals i i j t with 1 i j d defined in the paper moreover i i j t are not abelian integrals they are iterated integrals of length two

the number of small amplitude limit cycles in arbitrary polynomial systems abstract in this paper we study the number of small amplitude limit cycles in arbitrary polynomial systems it is found that almost all the results for the number of small amplitude limit cycles are obtained by calculating lyapunov constants and determining the order of the corresponding hopf bifurcation it is well known that the difficulty in calculating the lyapunov constants increases with the increasing of the degree of polynomial systems so it is necessary and valuable for us to achieve some general results about the number of small amplitude limit cycles in arbitrary polynomial systems with degree m which is denoted by m m in this paper by applying the method developed by c christopher and n lloyd in 1995 and m han and j li in 2012 we first obtain the lower bounds for m 6 m 14 and then prove that m m m 2 if m 23 finally we obtain that m m grows as least as rapidly as 18 25 1 2 ln 2 m 2 2 ln m 2 for all large m it is proved by m han j li lower bounds for the hilbert number of polynomial systems j differential equations 252 2012 3278 3304 that the number of all limit cycles in arbitrary polynomial systems with degree m grows as least as rapidly as 1 2 ln 2 m 2 2 ln m 2

on the number of limit cycles near a homoclinic loop with a nilpotent singular point abstract in this article we study the expansion of the first melnikov function appearing by perturbing an integrable and reversible system with a homoclinic loop passing through a nilpotent singular point and obtain formulas for computing the first coefficients of the expansion based on these coefficients we obtain a lower bound for the maximal number of limit cycles near the homoclinic loop moreover as an application of our main results we consider a type of integrable and reversible polynomial systems obtaining at least 3 4 or 5 limit cycles respectively

on the fixed point set of nonexpansive order preserving maps for every pair x and y in c a point z in c is a fixed point for t provided tz z nonexpansive maps have been studied extensively for over two decades with the objectives motivated by applications of proving the existence of fixed points under appropriate conditions and finding iteration methods leading to fixed points in particular finding conditions for the convergence of the iterates themselves we refer the reader to the survey articles of bruck b2 and kirk ki most of the results of the general theory assume convexity conditions on the banach space x or some geometrical or topological conditions on t and its iterates thus the spaces l1 and l o are usually excluded from general studies investigations in loo and other c e spaces were mode by the second author and others e g is so crandall and tartar ct gave conditions under which a nonexpansive map t on l t or on l is also order preserving that is

simultaneity of centres in ℤ q equivariant systems we study the simultaneous existence of centres for two families of planar z q equivariant systems first we give a short review about z q equivariant systems next we present the necessary and sufficient conditions for the simultaneous existence of centres for a z 2 equivariant cubic system and for a z 2 equivariant quintic system

algebraic gauss manin systems and brieskorn modules we study the algebraic gauss manin system and the algebraic brieskorn module associated to a polynomial mapping with isolated singularities since the algebraic gauss manin system does not contain any information on the cohomology of singular fibers we first construct a non quasi coherent sheaf which gives the cohomology of every fiber then we study the algebraic brieskorn module and show that its position in the the algebraic gauss manin system is determined by a natural map to quotients of local analytic gauss manin systems and its pole part by the vanishing cycles at infinity comparing it with the deligne extension this implies for example a formula for the determinant of periods in the two dimensional case we can describe the global structure of the algebraic gauss manin system rather explicitly

robust estimates and tests for the one and two sample scale models summary the purpose of this paper is to investigate the scale problem and develop a class of robust techniques which are useful in estimation and testing the classical functional which has been used as a measure of variability of a distribution is the standard deviation its associated estimator the sample standard deviation is shown to have highly undesirable robustness properties another measure of variability the square root of the midvariance is proposed and investigated the midvariance is basically the asymptotic variance of an m estimator of location and is shown to behave like a winsorized variance the asymptotic distribution of the sample estimator of the midvariance is found and a small sample approximation of the distribution using the chi squared distribution is developed with the above machinery one sample and two sample tests of variability are constructed

center conditions rigidity of logarithmic differential equations abstract 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

robust estimators of scale finite sample performance in long tailed symmetric distributions abstract this article presents the results of a monte carlo study of the robustness of scale estimates in the presence of long tailed symmetric distributions the article examines the performance of several families of estimates in samples of size 20 from several distributions the family of a estimators finite sample approximations to the asymptotic variance of m estimators of location appears to be more robust than the sample standard deviation the median absolute deviation from the median mad trimmed standard deviations and m estimators of scale the most successful a estimator uses the biweight weighting function which is also the basis for high performance robust location estimates

robust estimation of the process standard deviation for control charts control charts are tools used to detect aberrant behavior in manufacturing processes the x chart plots subsample averages as a function of time and the r chart plots subsample ranges both of these charts rely on an estimate of the standard deviation of the process when it is operating correctly the estimate has traditionally been based on the average range of 2040 subgroups but this will produce an estimate that is biased high when outliers are present one standard solution is to construct a range chart for the original subgroups and estimate the standard deviation only from those subgroups within the control limits repeating the procedure as necessary proposals have also recently been made to use a trimmed mean of the subsample ranges with a fixed percentage of trimming as well as the trimmed mean of the subsample interquartile ranges this article presents a new approach to robust estimation of the process standard deviation the procedure first centers each subsample on its own median and then

the infinitesimal 16th hilbert problem in the quadratic case let h x y be a real cubic polynomial with four distinct critical values in a complex domain and let x h h y frac partial partial x h x frac partial partial y be the corresponding hamiltonian vector field we show that there is a neighborhood of x h in the space of all quadratic plane vector fields such that any x has at most two limit cycles

limit cycle bifurcations from a non degenerate center in this work we discuss the computational problems which appear in the computation of the poincare liapunov constants and the determination of their functionally independent number moreover we calculate the minimum number of bautin ideal generators which give the number of small limit cycles under certain hypothesis about the generators in particular we consider polynomial systems of the form x y pn x y y x qn x y where pn and qn are a homogeneous polynomial of degree n we use center bifurcation rather than multiple hopf bifurcations used a previous work 19 to estimate the cyclicity of a unique singular point of focus center type for n 4 5 6 7 and compare with the results given by the conjecture presented in 18

hopfs ergodic theorem for nonlinear operators 1 t f 1 n l i 0 are a e convergent in this paper we will give a generalization of hopfs theorem to a wide class of nonlinear operators in the nonlinear situation instead of being a contraction one usually assumes that t is nonexpansive or at least norm decreasing on l i or l the definition 1 can of course be literally generalized to nonlinear mappings and the partial sums can be also defined inductively by sof f and s 1 f f s tf from the point of view of the inductive definition it is equally well justified to define inductively sof f and s i f f t s f and we will adopt throughout this paper always the latter definition and define the 1 averages a f snf there are several reasons for doing this firstly for the

bifurcations of limit cycles in complex networks of hamiltonian systems and lloyd s conjecture abstract this paper is motivated by hilbert s sixteenth problem concerning the number and distribution of limit cycles in planar polynomial vector fields we propose an original method inspired by the complex systems framework in order to construct a family of hamiltonian systems admitting non degenerate centers located on the vertices of a planar network we introduce two novel polynomial perturbations adapted to those systems with the aim to produce a high number of limit cycles the first one is oriented along the gradient of the unperturbed system and the second one is again constructed with a complex systems approach using the melnikov integral we reach in a small number of particular cases the lower bound of cubical order for the hilbert number h n conjectured by lloyd in 1988

a cubic system with thirteen limit cycles abstract we construct a planar cubic system and demonstrate that it has at least 13 limit cycles the construction is essentially based on counting the number of zeros of some abelian integrals

on the number of critical periods for planar polynomial systems abstract in this paper we get some lower bounds for the number of critical periods of families of centers which are perturbations of the linear one we give a method which lets us prove that there are planar polynomial centers of degree l with at least 2 l 2 2 critical periods as well as study concrete families of potential reversible and lienard centers this last case is studied in more detail and we prove that the number of critical periods obtained with our approach does not increases with the order of the perturbation

polynomial systems a lower bound for the hilbert numbers let h n be the maximum possible number of limit cycles of systems dot x p x y dot y q x y where p and q are polynomials of degree at most n we are concerned with the rate of growth of h n as n increases it is known that h n geq kn 2 for some constant k in this paper we show that h n grows at least as rapidly as n 2 log n

robust statistical analysis of interlaboratory studies summary a common procedure in testing analytical methods is to send a portion of each of a number of samples to each of several laboratories the results of such a study are submitted to statistical analysis to determine the two important variance components in the problem replication error and laboratory bias outliers are relatively common in these data both among laboratory effects and among the residuals this paper presents a method of analysis for interlaboratory studies that is robust to the existence of outliers and long tailed distributions of random effects theoretical considerations as well as a monte carlo study are adduced as support for this new technique

on the number of critical periods for planar polynomial systems of arbitrary degree we construct a class of planar systems of arbitrary degree n having a reversible center at the origin and such that the number of critical periods on its period annulus grows quadratically with n as far as we know the previous results on this subject gave systems having linear growth

on the number of limit cycles bifurcating from a non global degenerated center abstract we give an upper bound for the number of zeros of an abelian integral this integral controls the number of limit cycles that bifurcate by a polynomial perturbation of arbitrary degree n from the periodic orbits of the integrable system 1 x d h 0 where h is the quasi homogeneous hamiltonian h x y x 2 k 2 k y 2 2 the tools used in our proofs are the argument principle applied to a suitable complex extension of the abelian integral and some techniques in real analysis

center conditions at infinity for abel differential equations an abel differential equation y p x y 2 q x y 3 is said to have a center at a set a a 1 a r of complex numbers if y a 1 y a 2 y a r for any solution y x with the initial value y a 1 small enough the polynomials p q are said to satisfy the polynomial composition condition on a if there exist polynomials p q and w such that p p and q q are representable as p x p w x q x q w x and w a 1 w a 2 w a r we show that for wide ranges of degrees of p and q restricted only by certain assumptions on the common divisors of these degrees the composition condition provides a very accurate approximation of the center one up to a finite number of configurations not accounted for to our best knowledge this is the first general i e not restricted to small degrees of p and q or to a very special form of these polynomials result in the center problem for abel equations as an important intermediate result we show that at infinity according to an appropriate projectivization of the parameter space the center conditions are given by a system of the moment equations of the form as a1 p k q 0 s 2 r k 0 1

bifurcations of limit cycles in equivariant quintic planar vector fields abstract in this paper we obtain 23 limit cycles for a z 3 equivariant near hamiltonian system of degree 5 which is the perturbation of a z 6 equivariant quintic hamiltonian system the configuration of these limit cycles is new and different from the configuration obtained by h s y chan k w chung and j li where the unperturbed system is a z 3 equivariant quintic hamiltonian system our unperturbed system is different from the unperturbed systems studied by y wu and m han the limit cycles are obtained by poincare pontryagin theorem and poincare bendixson theorem

effective metastability of halpern iterates in cat 0 spaces abstract this paper provides an effective uniform rate of metastability in the sense of tao on the strong convergence of halpern iterations of nonexpansive mappings in cat 0 spaces the extraction of this rate from an ineffective proof due to saejung is an instance of the general proof mining program which uses tools from mathematical logic to uncover hidden computational content from proofs this methodology is applied here for the first time to a proof that uses banach limits and hence makes a substantial reference to the axiom of choice

general center conditions and bifurcation of limit cycles for a quasi symmetric seventh degree system in this paper we study a class of quasi symmetric seventh degree system by making two appropriate transformations of system 3 and calculating general focal values carefully we obtain the conditions that the infinity and the elementary focus 12 0 become centers at the same time moreover 12 limit cycles including 6 small limit cycles from the elementary focus and 6 large limit cycles from the infinity can occur under a certain condition what is worth mentioning is that similar conclusions are less and our work is new in terms of research about quasi symmetric systems up till now

relative cohomology with respect to a lefschetz pencil let m be a complex projective manifold of dimension n 1 and f a meromorphic function on m obtained by a generic pencil of hyperplane sections of m the n th cohomology vector bundle of f0 f m r where r is the set of indeterminacy points of f is defined on the set of regular values of f0 and we have the usual gauss manin connection on it following brieskorn s methods in br we extend the n th cohomology vector bundle of f0 and the associated gauss manin connection to p1 by means of differential forms the new connection turns out to be meromorphic on the critical values of f0 we prove that the meromorphic global sections of the vector bundle with poles of arbitrary order at p1 is isomorphic to the brieskorn module of f in a natural way and so the brieskorn module in this case is a free c t module of rank n where c t is the ring of polynomials in t and n is the dimension of n th cohomology group of a regular fiber of f0

irreducible components of the space of holomorphic foliations of degree two in cp n n 3 in this paper we will prove that the space of holomorphic fo liations of codimension 1 and degree 2 in cp n n 3 has six irreducible components

on the number of limit cycles of a class of polynomial systems of lienard type abstract in this paper we study a planar system of the form x p k y y g m x e f n x y where p k y is a polynomial of degree k in y and f n x g m x are polynomials in x with degree of n and m respectively let h m n k denote the maximal number of limit cycles having odd multiplicities of this system we give some lower bounds of h m n k

lower bounds for the hilbert number of polynomial systems let h m denote the maximal number of limit cycles of polynomial systems of degree m it is called the hilbert number the main part of hilbert s 16th problem posed in 1900 is to find its value the problem is still open even for m 2 however there have been many interesting results on the lower bound of it for m 2 in this paper we give some new lower bounds of this number the results obtained in this paper improve all existing results for all m 7 based on some known results for m 3 4 5 6 in particular we obtain that h m grows at least as rapidly as 12ln2 m 2 2ln m 2 for all large m

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

asymptotic lower bounds on hilbert numbers using canard cycles abstract in this work we give an asymptotic lower bound for the hilbert number for real planar polynomial differential systems this lower bound equals up to leading order to the best existing one but the method we provide is new and involves slow fast systems the construction strongly relies on generalized lienard systems

higher order limit cycle bifurcations from non degenerate centers abstract the computational problems which appear in the computation of the poincare liapunov constants and the determination of their functionally independent number has led in recent works to consider only the lowest terms of these constants in this work we improve the results obtained in this direction for polynomials systems of the form x y p n x y y x q n x y where p n and q n are a homogeneous polynomial of degree n we use center bifurcation to estimate the cyclicity of a unique singular point of focus center type for different values of n and compare with the results given by the conjecture presented in 15

high breakdown point estimates of regression by means of the minimization of an efficient scale abstract a new class of robust estimates estimates is introduced the estimates have simultaneously the following properties a they are qualitatively robust b their breakdown point is 5 and c they are highly efficient for regression models with normal errors they are defined by minimizing a new scale estimate applied to the residuals asymptotically a estimate is equivalent to an m estimate with a function given by a weighted average of two functions one corresponding to a very robust estimate and the other to a highly efficient estimate the weights are adaptive and depend on the underlying error distribution we prove consistency and asymptotic normality and give a convergent iterative computing algorithm finally we compare the biases produced by gross error contamination in the estimates and optimal bounded influence estimates