characterization of a monodromic singular point of a planar vector field the newton diagram and the lowest degree quasi homogeneous terms of an analytic planar vector field allow us to determine whether an isolated singular point of the vector field is monodromic or has a characteristic trajectory

polynomial lienard equations near infinity abstract we study the behaviour near infinity of the generalized lienard equations x y y m k 0 a k x k y n k 0 b k x k providing a complete classification using poincare compactification as well as poincare lyapunov compactification we show that all the necessary information is contained in a m and b n except for the so called center focus problem occuring in case m 2 n 1 with m n odd and where the behaviour also depends on the value of the other coefficients a i and b j we also indicate how to take care about obtaining uniform information near infinity

monodromy of a class of analytic generalized nilpotent systems through their newton diagram newton diagram of a planar vector field allows to determine whether a singular point of an analytic system is a monodromic singular point we solve the monodromy problem for the nilpotent systems and we apply our method to a wide family of systems with a degenerate singular point so called generalized nilpotent cubic systems

integrability of centers perturbed by quasi homogeneous polynomials abstract the objective of this work is to give sufficient and necessary conditions for integrability of differential equation systems in the plane of the form y2p 1 x2 x y x2q 1 y2 x y wherex2 x y andy2 x y are p q quasi homogeneous polynomials of degree 2pq q 1 and 2pq p 1 respectively

estimating smooth monotone functions many situations call for a smooth strictly monotone function f of arbitrary flexibility the family of functions defined by the differential equation d 2 f w df where w is an unconstrained coefficient function comprises the strictly monotone twice differentiable functions the solution to this equation is f c 0 c 1 d 1 exp d 1 w where c 0 and c 1 are arbitrary constants and d 1 is the partial integration operator a basis for expanding w is suggested that permits explicit integration in the expression of f in fitting data it is also useful to regularize f by penalizing the integral of w 2 since this is a measure of the relative curvature in f applications are discussed to monotone nonparametric regression to the transformation of the dependent variable in non linear regression and to density estimation

growth at infinity and index of polynomial maps abstract let f k n k n be a polynomial map such that f 1 0 is compact where k r or c then we give a condition implying that there is a uniform bound for the ojasiewicz exponent at infinity in certain deformations of f this fact gives a result about the invariance of the global index of f

canards in r3 abstract we give a geometric analysis of canard solutions in three dimensional singularly perturbed systems with a folded two dimensional critical manifold by analysing the reduced flow we obtain singular canard solutions passing through a singularity on the fold curve we classify these singularities called canard points as folded saddles folded nodes and folded saddle nodes we prove the existence of canard solutions in the case of the folded saddle we show the existence of canards in the folded node case provided a generic non resonance condition is satisfied and in a subcase of the folded saddle node the proof is based on the blow up method

non parametric estimation of the odds ratios for continuous exposures using generalized additive models with an unknown link function the generalized additive model gam is a powerful and widely used tool that allows researchers to fit non parametrically the effect of continuous predictors on a transformation of the mean response variable such a transformation is given by a so called link function and in gams this link function is assumed to be known nevertheless if an incorrect choice is made for the link the resulting gam is misspecified and the results obtained may be misleading in this paper we propose a modified version of the local scoring algorithm that allows for the non parametric estimation of the link function by using local linear kernel smoothers to better understand the effect that each covariate produces on the outcome results are expressed in terms of the non parametric odds ratio or curves bootstrap techniques were used to correct the bias in the or estimation and to construct point wise confidence intervals a simulation study was carried out to assess the behaviour of the resulting estimates the proposed methodology was illustrated using data from the aids register of galicia nw spain with a view to assessing the effect of the cd4 lymphocyte count on the probability of being aids diagnosed via tuberculosis tb this application shows how the link s flexibility makes it possible to obtain or curve estimates that are less sensitive to the presence of outliers and unusual values that are often present in the extremes of the covariate distributions

the index of singularities of vector fields and finite jets abstract we describe when the index of a singularity of a smooth vector field is determined by a finite jet at the singularity we also give some criteria to determine some terms from the formal series expansion of the vector field at the singularity which determine the index

an upper bound of the index of an equilibrium point in the plane abstract we give an upper bound of the index of an isolated equilibrium point of a c 1 vector field in the plane the vector field is decomposed in gradient and hamiltonian components this decomposition is related with the loewner vector field associated to this decomposition we consider the set where the gradient and hamiltonian components are linearly dependent the number of branches of starting at the equilibrium point determines the upper bound of the index

global phase portraits and bifurcation diagrams for hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the y axis abstract this paper studies the global dynamics for hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the y axis by linear changes of variables it can be written as four systems one of them has been characterized in llibre 2018 in this paper the remaining three systems are investigated firstly the normal forms of these systems are given by an algebraic classification of their infinite singular points then we classify the global phase portraits of these systems having canonical forms on the poincare disk moreover we obtain the bifurcation diagrams for the corresponding global phase portraits combining this work with llibre 2018 we provide the complete classification of global phase portraits for hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the y axis

geometric treatments and a common mechanism in finite time singularities for autonomous odes abstract geometric treatments of blow up solutions for autonomous ordinary differential equations and their blow up rates are concerned our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces and dynamics on their center stable manifolds in particular we show that dynamics on center stable manifolds of invariant sets at infinity with appropriate time scale desingularizations as well as blowing up of singularities characterize dynamics of blow up solutions as well as their rigorous blow up rates similarities for characterizing finite time extinction and asymptotic behavior of compacton traveling waves to blow up solutions are also shown

monotone b spline smoothing abstract estimation of growth curves or item response curves often involves monotone data smoothing methods that have been studied in the literature tend to be either less flexible or more difficult to compute when constraints such as monotonicity are incorporated built on the ideas of koenker ng and portnoy and ramsay we propose monotone b spline smoothing based on l 1 optimization this method inherits the desirable properties of spline approximations and the computational efficiency of linear programs the constrained fit is similar to the unconstrained estimate in terms of computational complexity and asymptotic rate of convergence through applications to some real and simulated data we show that the method is useful in a variety of applications the basic ideas utilized in monotone smoothing can be useful in some other constrained function estimation problems

limit cycles for planar semi quasi homogeneous polynomial vector fields abstract this paper is concerned with the limit cycles for planar semi quasi homogeneous polynomial systems we give some explicit criteria for the nonexistence and existence of periodic orbits a lower bound is given for the maximum number of limit cycles of such a system the cyclicity and center problems are studied for some subfamilies of semi quasi homogeneous polynomial systems our results generalize those obtained for polynomial semi homogeneous systems

quasi homogeneous normal forms in this paper we extend the concepts of the normal form theory for vector fields that are expanded in quasi homogeneous components of a fixed type these expansions have been used by some authors in the analysis of the determinacy of a given singularity also the use of reparametrizations in the time are considered namely beyond the use of c conjugation to determine normal forms we present a method useful to determine how much a vector field can be simplified by using c equivalence the results obtained are applied in the case of the bogdanov takens singularity firstly using c conjugation and later showing the improvements provided by the c equivalence

topological equivalence of a plane vector field with its principal part defined through newton polyhedra abstract we introduce a blowing up of singularities of vector fields associated with newton polyhedra in the space of the exponents by means of which we prove a generalization of the hartman grobman theorem namely a plane vector field possessing characteristic orbits is locally topologically equivalent with its principal part under suitable non degeneracy hypotheses under stronger hypotheses a similar equivalence result is proven between a plane vector field and a single quasihomogeneous component of its principal part the case of second order equations is also studied

analytic integrability for some degenerate planar vector fields abstract in this paper we study the analytic integrability of degenerate vector fields of the form y 3 2 a x 3 y x 5 3 a x 2 y 2 around the origin for these vector fields it is proved that integrability does not imply formal orbital equivalence to the hamiltonian leading part moreover it is shown the existence of a system in this class which has a center but is neither analytically integrable nor formal orbital reversible

monodromy center focus and integrability problems for quasi homogeneous polynomial systems abstract this paper deals with planar quasi homogeneous polynomial vector fields and addresses three major questions the monodromy the center focus and the integrability problems we characterize the monodromic planar quasi homogeneous polynomial vector fields and we give a condition to distinguish between a center and a focus in this case also we provide conditions which characterize the integrability of quasi homogeneous polynomial systems under non resonance conditions the results obtained allow us to analyse two monodromic planar systems with degenerate linear part one of them with nilpotent linearization and another one with null linear part

an approach to mel nikov theory in celestial mechanics using a completely analytic procedure based on a suitable extension of a classical method we discuss an approach to the poincare mel nikov theory which can be conveniently applied also to the case of nonhyperbolic critical points and even if the critical point is located at the infinity in this paper we concentrate our attention on the latter case and precisely on problems described by kepler type potentials in one or two degrees of freedom in the presence of general time dependent perturbations we show that the appearance of chaos possibly including arnol d diffusion can be proved quite easily and in a direct way without resorting to singular coordinate transformations such as the mcgehee or blowing up transformations natural examples are provided by the classical gylden problem originally proposed in celestial mechanics but also of interest in different fields and by the general three body problem in classical mechanics

the center problem for a family of systems of differential equations having a nilpotent singular point abstract we study the analytic system of differential equations in the plane x y t i 0 f q p 2 i s where p q n p q s n 1 p q 0 n n and f i p i q i t are quasi homogeneous vector fields of type t p q and degree i with f q p y 0 t and q q p 2 s 1 0 0 the origin of this system is a nilpotent and monodromic isolated singular point we prove for this system the existence of a lyapunov function and we solve theoretically the center problem for such system finally as an application of the theoretical procedure we characterize the centers of several subfamilies

the monotone smoothing of scatterplots we consider the problem of summarizing a scatterplot with a smooth monotone curve a solution that combines local averaging and isotonic regression is proposed and we demonstrate its use with two examples in the second example the procedure is applied in a regression setting to some data from box and cox 1964 and it is shown how this new procedure generalizes box and cox s well known family of transformations in the same example the bootstrap is applied to obtain a measure of the variability of the procedure

topological equivalence of planar vector fields and their generalised principal part abstract let x r2 be the space of c planar vector fields we consider the space v x r2 of vector fields with an isolated singularity and a fixed newton diagram we define the generalised principal part x of the vector field x v and give the nondegeneracy condition on x using the newton diagram we prove that x v is locally topologically equivalent to its minimal generalised principal part x if x is nondegenerate and x is not a monodromic vector field in the proof we use the normal form method and the blowing up method

tests for monotonicity of a regression mean with guaranteed level in this paper a nonparametric procedure for testing for monotonicity of a regression mean with guaranteed level is proposed the procedure is based on signs of differences of observations from the response variable the test is calibrated against the most difficult null hypothesis when the regression function is constant and produces an exact test in this context in general the test is conservative the power of the test is good and comparable with that of other nonparametric tests it is shown that the testing procedure has asymptotic power 1 against certain local alternatives the method is also robust against heavy tailed error distributions and even maintains good power when the errors are for example cauchy distributed a simulation study is provided to demonstrate finite sample behaviour of the testing procedure

a new algorithm for determining the monodromy of a planar differential system abstract we give a new algorithmic criterium that determines wether an isolated degenerate singular point of a system of differential equations on the plane is monodromic this criterium involves the conservative and dissipative parts associated to the edges and vertices of the newton diagram of the vector field