non existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities abstract in this paper we study the limit cycles of the planar polynomial differential systems x a x y p n x y y x a y q n x y where p n and q n are homogeneous polynomials of degree n 2 and a r consider the functions p n cos sin cos q n cos sin sin q n cos sin cos p n cos sin sin 1 a 2 n 1 2 a first we prove that these differential systems have at most 1 limit cycle if there exists a linear combination of 1 and 2 with definite sign this result improves previous known results furthermore if 1 1 a 2 0 for some 1 2 0 we provide necessary and sufficient conditions for the non existence and the existence and uniqueness of the limit cycles of these differential systems when one of these mentioned limit cycles exists it is hyperbolic and surrounds the origin

upper bounds of limit cycles in abel differential equations with invariant curves abstract new criteria are established for upper bounds on the number of limit cycles of periodic abel differential equations having two periodic invariant curves one of them bounded the criteria are applied to obtain upper bounds of either zero or one limit cycle for planar differential systems

bifurcation of limit cycles and integrability of planar dynamical systems in complex form we describe a new algorithm for the calculation of liapunov quantities for two dimensional systems in terms of the complex form of the system it is more transparent conceptually and more efficient computationally we demonstrate the advantages of this approach by deriving the integrability conditions and information about the bifurcation of limit cycles for some cubic systems

slope estimation in the presence of informative right censoring modeling the number of observations as a geometric random variable summary a method is proposed for the estimation of rate of change from incomplete longitudinal data where the number of observations made for each subject is assumed to vary depending on the level of the response variable the proposed method involves a random slope model in which the number of observations is modeled as a geometric distribution with its mean dependent on the individual subject s rate of change the method adjusts for informative right censoring and provides estimates of the slopes of individual subjects as well as of the population under noninformative right censoring these estimators of the slopes are equivalent to bayes estimators fearn 1975 biometrika 62 89 100 the simulation study demonstrates that in cases where the censoring process is informative the proposed estimator is more efficient than either the unweighted or weighted estimator of slope the method is illustrated by the analysis of renal transplant data

semiparametric pseudo likelihoods in generalized linear models with nonignorable missing data we consider identifiability and estimation in a generalized linear model in which the response variable and some covariates have missing values and the missing data mechanism is nonignorable and unspecified we adopt a pseudo likelihood approach that makes use of an instrumental variable to help identifying unknown parameters in the presence of nonignorable missing data explicit conditions for the identifiability of parameters are given some asymptotic properties of the parameter estimators based on maximizing the pseudo likelihood are established explicit asymptotic covariance matrix and its estimator are also derived in some cases for the numerical maximization of the pseudo likelihood we develop a two step iteration algorithm that decomposes a nonconcave maximization problem into two problems of maximizing concave functions some simulation results and an application to a dataset from cotton factory workers are also presented

structural stability of planar homogeneous polynomial vector fields applications to critical points and to infinity lethmbe the space of planar homogeneous polynomial vector fields of degreemendowed with the coefficient topology we characterize the set mof the vector fields ofhmthat are structurally stable with respect to perturbations inhmand we determine the exact number of the topological equivalence classes in m the study of structurally stable homogeneous polynomial vector fields is very useful for understanding some interesting features of inhomogeneous vector fields thus by using this characterization we can do first an extension of the hartman grobman theorem which allows us to study the critical points of planar analytical vector fields whosek jets are zero for allk munder generic assumptions and second the study of the flows of the planar polynomial vector fields in a neighborhood of the infinity also under generic assumptions

abel like differential equations with no periodic solutions we present various criteria for the non existence of positive periodic solutions of generalized abel differential equations with periodic coefficients that can change sign as an application we obtain some families of planar vector fields without limit cycles

classification of the centers their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities abstract in this paper we classify the centers the cyclicity of its hopf bifurcation and their isochronicity for the polynomial differential systems in r 2 of arbitrary degree d 3 odd that in complex notation z x i y can be written as z i z z z d 3 2 a z 3 b z 2 z c z z 2 d z 3 where r and a b c d c if d 3 we obtain the well known class of all polynomial differential systems of the form a linear system with cubic homogeneous nonlinearities

periodic solutions for equation x a t x m b t x n c t x l with a t and b t changing signs in this paper we investigate the differential equation x s x t a t xm b t xn c t xl where a b c c 0 1 m n l and m n l z a solution x t with x 1 x 0 is called a periodic solution under some hypotheses which admit a t and b t without fixed sign we obtain the upper bound sometimes sharp for the number of isolated periodic solutions of the equation applying these results for the abel equation i e m 3 n 2 l 1 we get that if there exists 0 such that s t c t 0 resp s t a t b t 0 then the equation has at most 2 resp 4 non zero isolated periodic solutions furthermore suppose that a t t is a smooth curve which lies in r 0 0 1 with a 0 a 1 we obtain that if vector fields s x t 1 resp and c t x 1 are transverse to resp s x t 1 on in opposite directions then the number of non zero isolated periodic solutions of this abel equation is still no more than 2 resp 4 these conclusions generalize the known criteria about the abel equation which only refer to the cases with either a t or b t keeping sign finally as an application we study a kind of trigonometrical abel equation

missing covariates in generalized linear models when the missing data mechanism is non ignorable we propose a method for estimating parameters in generalized linear models with missing covariates and a non ignorable missing data mechanism we use a multinomial model for the missing data indicators and propose a joint distribution for them which can be written as a sequence of one dimensional conditional distributions with each one dimensional conditional distribution consisting of a logistic regression we allow the covariates to be either categorical or continuous the joint covariate distribution is also modelled via a sequence of one dimensional conditional distributions and the response variable is assumed to be completely observed we derive the e and m steps of the em algorithm with non ignorable missing covariate data for categorical covariates we derive a closed form expression for the e and m steps of the em algorithm for obtaining the maximum likelihood estimates mles for continuous covariates we use a monte carlo version of the em algorithm to obtain the mles via the gibbs sampler computational techniques for gibbs sampling are proposed and implemented the parametric form of the assumed missing data mechanism itself is not testable from the data and thus the non ignorable modelling considered here can be viewed as a sensitivity analysis concerning a more complicated model therefore although a model may have passed the tests for a certain missing data mechanism this does not mean that we have captured even approximately the correct missing data mechanism hence model checking for the missing data mechanism and sensitivity analyses play an important role in this problem and are discussed in detail several simulations are given to demonstrate the methodology in addition a real data set from a melanoma cancer clinical trial is presented to illustrate the methods proposed

maximum principles and bounds in a class of fourth order uniformly elliptic equations this paper is devoted to a class of fourth order uniformly elliptic equations posed by schaefer in 1987 we obtain maximum principles for certain functions which are defined on solutions of the elliptic equations the principles are then used to deduce some bounds on important quantities in the physical problems of interest

vector fields with homogeneous nonlinearities and many limit cycles abstract consider planar real polynomial differential equations of the form x l x x n x where x x y r 2 l is a 2 2 matrix and x n is a homogeneous vector field of degree n 1 most known results about these equations valid for infinitely many n deal with the case where the origin is a focus or a node and give either non existence of limit cycles or upper bounds of one or two limit cycles surrounding the origin in this paper we improve some of these results and moreover we show that for n 3 odd there are equations of this form having at least n 1 2 limit cycles surrounding the origin our results include cases where the origin is a focus a node a saddle or a nilpotent singularity we also discuss a mechanism for the bifurcation of limit cycles from infinity

a new chebyshev family with applications to abel equations we prove that a family of functions defined through some definite integrals forms an extended complete chebyshev system the key point of our proof consists of reducing the study of certain wronskians to the gram determinants of a suitable set of new functions our result is then applied to give upper bounds for the number of isolated periodic solutions of some perturbed abel equations

cubic systems and abel equations abstract we investigate cubic systems which can be transformed to an equation of abel form the conditions for the origin to be a centre and in particular an isochronous centre are obtained the maximum number of limit cycles which can bifurcate from a fine focus is determined and some information is obtained about the global phase portrait

models for nonresponse in sample surveys abstract the literature on the analysis of incomplete data using models is reviewed in the context of nonresponse in sample surveys the modeling approach provides a large body of methods for handling unit and item nonresponse some of which cannot be derived from the randomization theory of inference for surveys key concepts from the literature on incomplete data such as factorizations of the likelihood for special data patterns the em algorithm for general data patterns and ignorability of the response mechanism are discussed within the survey context model based procedures are related to common methods for handling nonresponse in surveys such as weighting or imputation of means within subclasses of the population

non ignorable missing covariates in generalized linear models we propose a likelihood method for estimating parameters in generalized linear models with missing covariates and a non ignorable missing data mechanism in this paper we focus on one missing covariate we use a logistic model for the probability that the covariate is missing and allow this probability to depend on the incomplete covariate we allow the covariates including the incomplete covariate to be either categorical or continuous we propose an em algorithm in this case for a missing categorical covariate we derive a closed form expression for the e and m steps of the em algorithm for obtaining the maximum likelihood estimates mles for a missing continuous covariate we use a monte carlo version of the em algorithm to obtain the mles via the gibbs sampler the methodology is illustrated using an example from a breast cancer clinical trial in which time to disease progression is the outcome and the incomplete covariate is a quality of life physical well being score taken after the start of therapy this score may be missing because the patients are sicker so this covariate could be non ignorably missing

a new uniqueness criterion for the number of periodic orbits of abel equations abstract a solution of the abel equation x a t x 3 b t x 2 such that x 0 x 1 is called a periodic orbit of the equation our main result proves that if there exist two real numbers a and b such that the function a a t b b t is not identically zero and does not change sign in 0 1 then the abel differential equation has at most one non zero periodic orbit furthermore when this periodic orbit exists it is hyperbolic this result extends the known criteria about the abel equation that only refer to the cases where either a t 0 or b t 0 does not change sign we apply this new criterion to study the number of periodic solutions of two simple cases of abel equations the one where the functions a t and b t are 1 periodic trigonometric polynomials of degree one and the case where these two functions are polynomials with three monomials finally we give an upper bound for the number of isolated periodic orbits of the general abel equation x a t x 3 b t x 2 c t x when a t b t and c t satisfy adequate conditions

a uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities abstract this paper is devoted to study the planar polynomial system x a x y p n x y y x a y q n x y where a r and p n q n are homogeneous polynomials of degree n 2 denote cos q n cos sin sin p n cos sin we prove that the system has at most 1 limit cycle surrounding the origin provided n 1 a 0 furthermore this upper bound is sharp this is maybe the first uniqueness criterion which only depends on a linear condition of for the limit cycles of this kind of systems we show by examples that in many cases the criterion is applicable while the classical ones are invalid the tool that we mainly use is a new estimate for the number of limit cycles of abel equation with coefficients of indefinite signs employing this tool we also obtain another geometric criterion which allows the system to possess at most 2 limit cycles surrounding the origin

existence of non trivial limit cycles in abel equations with symmetries abstract we study the periodic solutions of the generalized abel equation x a 1 a 1 t x n 1 a 2 a 2 t x n 2 a 3 a 3 t x n 3 where n 1 n 2 n 3 1 are distinct integers a 1 a 2 a 3 r and a 1 a 2 a 3 are 2 periodic analytic functions such that a 1 t sin t a 2 t cos t a 3 t sin t cos t are periodic positive even functions when n 3 n 1 n 3 n 2 0 we prove that the equation has no non trivial different from zero limit cycle for any value of the parameters a 1 a 2 a 3 when n 3 n 1 n 3 n 2 0 we obtain under additional conditions the existence of non trivial limit cycles in particular we obtain limit cycles not detected by abelian integrals

limit cycles of polynomial differential equations with quintic homogeneous nonlinearities abstract in this paper we mainly study the number of limit cycles which can bifurcate from the periodic orbits of the two centers x y y x x y 1 x 2 y 2 2 y x 1 x 2 y 2 2 when they are perturbed inside the class of all polynomial differential systems with quintic homogeneous nonlinearities we do this study using the averaging theory of first second and third orders

on a subharmonic functional of some even order elliptic problems abstract a procedure for finding a subharmonic functional of a solution for the problem lu n i 0 1 i c i i u 0 in the domain e which is either in a plane or on a line is presented the functional obtained is used to derive the maximum principles and uniqueness theorems for some higher order elliptic problems the procedure proposed is also applicable to nonlinear differential equations

algebraic and topological classification of the homogeneous cubic vector fields in the plane tsutomu date and masao iri in di gave an algebraic classification of systems i p x y i q x y where p and q are homogeneous polynomials of degree 2 for this they used the classification of the binary cubic forms and also the simultaneous classification of a linear binary form and a cubic binary form given by the algebraic invariant theory we begin by doing a similar study for systems p x y q x y where p and q are homogeneous polynomials of degree three i e cubic systems the classification s theorem of such systems is based on the classification of fourth order binary forms gurevich in gu did the classification of fourth order binary forms on the field of complex numbers since we did not find the classification on the real domain we adapt gurevich s proof to obtain it in section 1 we give some definitions and preliminary results while in section 2 we give the theorem of classification of fourth order binary forms on the real domain the method used in the proof caley s method let us obtain canonical forms of the fourth order binary forms and the algebraic characteristics section 3 is devoted to obtaining the algebraic classification of systems p x y j q x y where p and q are homogeneous polynomials of degree 3 given an arbitrary system x p q with p and q homogeneous polynomials of degree 3 we can know the equivalence class at which it belongs through the algebraic characteristics in section 4 we study the phase portraits of systems 1 p x y j q x y where p and q are homogeneous polynomials of degree n and p and q have no common factor such systems have been studied by j argemi in a here we give a shorter new proof of his results by using

periodic orbits in complex abel equations abstract this paper is devoted to prove two unexpected properties of the abel equation d z d t z 3 b t z 2 c t z where b and c are smooth 2 periodic complex valuated functions t r and z c the first one is that there is no upper bound for its number of isolated 2 periodic solutions in contrast recall that if the functions b and c are real valuated then the number of complex 2 periodic solutions is at most three the second property is that there are examples of the above equation with b and c being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of b and c this result is also in contrast with the characterization of the center variety for the examples of abel equations d z d t a t z 3 b t z 2 studied in the literature where the center variety is located in a finite number of connected components

alternative methods for cps income imputation abstract the u s bureau of the census imputes missing income items in the income supplement of the current population survey cps by a technique commonly known as the cps hot deck this article compares cps hot deck imputations of wages and salary amounts with alternatives based on regression models for the logarithm of wages and salary and for the wage rate comparisons are effected by comparing imputations with an internal revenue service irs wages and salary amount found by an exact match of cps data to irs records although limitations in the matching and in the comparison variable preclude a definitive conclusion we find that a the cps hot deck does not underestimate income aggregates to any serious extent b model based alternatives have slightly smaller mean absolute error than the hot deck when comparable data bases of respondents are used to carry out imputations and c multivariate models for imputing recipiency weeks and hours worked and earnings need to be developed to provide re

imputation of missing values when the probability of response depends on the variable being imputed abstract a method is developed for imputing missing values when the probability of response depends upon the variable being imputed the missing data problem is viewed as one of parameter estimation in a regression model with stochastic censoring of the dependent variable the prediction approach to imputation is used to solve this estimation problem wages and salaries are imputed to non respondents in the current population survey and the results are compared to the nonrespondents irs wage and salary data the stochastic censoring approach gives improved results relative to a prediction approach that ignores the response mechanism

limit cycles of abel equations of the first kind abstract consider the scalar differential equation x i 0 m a i t x n i where a i t are t periodic analytic functions and 1 n i n for any polynomial q x x n 0 i 1 m i x n i the equation can be written as x a 0 q x r t x let w be the wronskian of q and r with respect to x and q w the previous polynomials after removing multiplicity of roots and solutions of the differential equation we prove that if the vector field defined by the differential equation is transversal at every point of q x 0 or w t x 0 then the number of limit cycles isolated periodic solutions in the set of periodic solutions of the differential equation is at most 3 n 1