on the representation of unity by binary cubic forms if f x y is a binary cubic form with integer coefficients such that f x 1 has at least two distinct complex roots then the equation f x y 1 possesses at most ten solutions in integers x and y nine if f has a nontrivial automorphism group if further f x y is reducible over z x y then this equation has at most 2 solutions unless f x y is equivalent under gl2 z action to either x x2 xy y2 or x x2 2y2 the proofs of these results rely upon the method of thue siegel as refined by evertse together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational diophantine approximation along the way we completely solve all thue equations f x y 1 for f cubic and irreducible of positive discriminant df 106 as corollaries we obtain bounds for the number of solutions to more general cubic thue equations of the form f x y m and to mordell s equation y2 x3 k where m and k are nonzero integers

the dynamics of spreading and immune strategies of sexually transmitted diseases on scale free network abstract we examine epidemic threshold and dynamics for sexually transmitted diseases stds spread using a multiple susceptible infected removed susceptible ode model on scale free networks we derive the threshold for the epidemic to be zero in infinite scale free network for a hard cut off scale free network we also prove the stability of disease free equilibrium and the persistence of stds infection the effects of two immunization schemes including proportional scheme and targeted vaccination are studied and compared we find that targeted strategy compare favorably to a proportional scheme in terms of effectiveness theory and simulations both prove that an appropriate condom using has prominent effect to control stds spread on scale free networks

global stability of a generalized epidemic model abstract the competitive exclusion principle is one of the most interesting and important phenomena in both theoretical epidemiology and biology we show that the equilibrium in which only the strain with the maximum basic reproductive number exists is globally asymptotically stable by using an average lyapunov function theorem and some dynamical system theory this result is anticipated by h j bremermann and h r thieme 1989 6 where they showed that the equilibrium is locally stable the global result has not been established previously

impact of imputation of missing values on classification error for discrete data numerous industrial and research databases include missing values it is not uncommon to encounter databases that have up to a half of the entries missing making it very difficult to mine them using data analysis methods that can work only with complete data a common way of dealing with this problem is to impute fill in the missing values this paper evaluates how the choice of different imputation methods affects the performance of classifiers that are subsequently used with the imputed data the experiments here focus on discrete data this paper studies the effect of missing data imputation using five single imputation methods a mean method a hot deck method a nai ve bayes method and the latter two methods with a recently proposed imputation framework and one multiple imputation method a polytomous regression based method on classification accuracy for six popular classifiers ripper c4 5 k nearest neighbor support vector machine with polynomial and rbf kernels and nai ve bayes on 15 datasets this experimental study shows that imputation with the tested methods on average improves classification accuracy when compared to classification without imputation although the results show that there is no universally best imputation method nai ve bayes imputation is shown to give the best results for the ripper classifier for datasets with high amount i e 40 and 50 of missing data polytomous regression imputation is shown to be the best for support vector machine classifier with polynomial kernel and the application of the imputation framework is shown to be superior for the support vector machine with rbf kernel and k nearest neighbor the analysis of the quality of the imputation with respect to varying amounts of missing data i e between 5 and 50 shows that all imputation methods except for the mean imputation improve classification error for data with more than 10 of missing data finally some classifiers such as c4 5 and nai ve bayes were found to be missing data resistant i e they can produce accurate classification in the presence of missing data while other classifiers such as k nearest neighbor svms and ripper benefit from the imputation

identifying variables responsible for data not missing at random when data are not missing at random nmar maximum likelihood ml procedure will not generate consistent parameter estimates unless the missing data mechanism is correctly modeled understanding nmar mechanism in a data set would allow one to better use the ml methodology a survey or questionnaire may contain many items certain items may be responsible for nmar values in other items the paper develops statistical procedures to identify the responsible items by comparing ml estimates mle statistics are developed to test whether the mles are changed when excluding items the items that cause a significant change of the mles are responsible for the nmar mechanism normal distribution is used for obtaining the mles a sandwich type covariance matrix is used to account for distribution violations the class of nonnormal distributions within which the procedure is valid is provided both saturated and structural models are considered effect sizes are also defined and studied the results indicate that more missing data in a sample does not necessarily imply more significant test statistics due to smaller effect sizes knowing the true population means and covariances or the parameter values in structural equation models may not make things easier either

computing all power integral bases of cubic fields applying baker s effective method and the reduction procedure of baker and davenport we present several lists of solutions of index form equations in totally real and complex cubic algebraic number fields these solutions yield all power integral bases of these fields 1 introduction let k be a cubic algebraic number field and denote by zk the ring of integers of k a power integral basis of k is an integral basis of the form 1 2 with some a e zk if there exists such an a then we say that k is monogenic since zk z e obviously if a has this property then it holds also for a k with any k e z from a practical point of view it is important to know whether there exists a power integral basis of k and if so what are the numerical values of a if the galois group of k is cyclic then the discriminant of k is a full square the problem of monogeneity in cyclic cubic fields was considered by m n gras 12 13 archinard 1 and dummit and kisilevsky 4 m n gras and archinard gave necessary and sufficient conditions for monogeneity and tested for several numerical examples whether or not the field is monogenic moreover dummit and kisilevsky proved that there exist infinitely many cyclic cubic fields with power integral bases for arbitrary algebraic number fields l it was proved by gyory 14 that up to obvious translations by elements of z there are only finitely many a e zl with zl z and he gave effective but rather large bounds for the sizes of these a in this paper we present a method which allows us to determine all possible values of a up to translation with rational integers such that 1 2 is a power integral

testing ignorable missingness in estimating equation approaches for longitudinal data summary we address the matter of determining whether or not missing data in longitudinal studies are ignorable with regard to quasilikelihood or estimating equations approaches this involves testing for whether or not the zero mean property of estimating equations holds true chen little 1999 proposed testing for significant differences among parameter estimators calculated from sample subsets with different patterns of missing data whereas we propose a more unified generalised score type test this avoids exhaustive estimation of parameters for each missing data pattern testing instead with a single quadratic score test statistic whether or not there is a common parameter under which the means of all the pattern specific estimating equations are zero comparisons are made for the two approaches with both simulations and real data examples

complete solutions to families of quartic thue equations using a method due to e thomas we prove that if lai 9 9 1027 then the diophantine equations 4 3 2 2 3 4 x ax y x y axy y and 4 3 2 2 3 4 x ax y 3x y axy y i1 have exactly twelve solutions namely x y 0 1 1 0 1 1 f 1 1 a 1 l fa and eight solutions x y o 1 i 0 i 1 i f1 respectively

large integral points on elliptic curves to ml friend dan shanks abstract we describe several methods which permit one to search for big integral points on certain elliptic curves i e for integral solutions x of certain diophantine equations of the form v2 x3 ax b a b e z in a large range ixi iy i b in time polynomial in loglogb we also give a number of individual examples and of parametric families of examples of specific elliptic curves having a relatively large integral point

a test of missing completely at random for generalised estimating equations with missing data we consider inference from generalised estimating equations when data are incomplete a test for missing completely at random is proposed to help decide whether or not we should adjust estimating equations to correct the possible bias introduced by a missing data mechanism that is not missing completely at random likelihood ratio tests have been introduced to test the missing completely at random hypothesis fuchs 1982 little 1988 for the estimating equation setting following the basic idea of little 1988 we propose a wald type test based on an information decomposition and recombination procedure which also provides an alternative method for estimating parameters one application of the test is to assess the adequacy of the marginal generalised estimating equation for longitudinal data with missing values simulations are done to evaluate its performance

percolation on correlated networks we reconsider the problem of percolation on an equilibrium random network with degree degree correlations between nearest neighboring vertices focusing on critical singularities at a percolation threshold we obtain criteria for degree degree correlations to be irrelevant for critical singularities we present examples of networks in which assortative and disassortative mixing leads to unusual percolation properties and new critical exponents

on the resolution of inhomogeneous norm form equations in two dominating variables applying baker s well known method and the reduction procedure described by baker and davenport we give a numerical algorithm for finding all solutions of inhomogeneous thue equations of type nk q x ay x 1 in the variables x y e z and a e zic with a max x y ll2 where k q a is a totally real cubic field

on the method of thue siegel the methods of thue and siegel based on explicit pade approximations to algebraic functions are used to examine diophantine approximations to roots of

a copula model for repeated measurements with non ignorable non monotone missing outcome a normal copula based selection model is proposed for continuous longitudinal data with a non ignorable non monotone missing data process the normal copula is used to combine the distribution of the outcome of interest and that of the missing data indicators given the covariates parameters in the model are estimated by a pseudo likelihood method we first use the gee with a logistic link to estimate the parameters associated with the marginal distribution of the missing data indicator given the covariates assuming that covariates are always observed then we estimate other parameters by inserting the estimates from the first step into the full likelihood function a simulation study is conducted to assess the robustness of the assumed model under different missing data processes the proposed method is then applied to one example from a community cohort study to demonstrate its capability to reduce bias

assessing the validity of weighted generalized estimating equations the inverse probability weighted generalized estimating equations approach robins et al 1994 robins et al 1995 effectively removes bias and provides valid statistical inference for regression parameter estimation in marginal models when longitudinal data contain missing values the validity of the weighted generalized estimating equations regarding consistent estimation depends on whether the underlying missing data process is properly modelled however there is little work available to examine whether or not this condition holds in this paper we propose a test constructed from two sets of estimating equations one set is known to be unbiased but the other set is not known we utilize the quadratic inference function qu et al 2000 method to assess their compatibility which is equivalent to testing for the validity of the weighted generalized estimating equations approach we conduct simulation studies to assess the performance of the proposed method the test procedure is illustrated through a real data example

on mordell s equation y2 k x3 a problem of stolarsky on determine l ensemble complet des solutions en nombres entiers de l equation de stolarsky y 2 999 x 3

a fixed point theorem for a general epidemic model abstract epidemic type models typically undergo a phase transition when the infection rate surpasses the epidemic threshold however for networks having degree degree correlations the epidemic threshold has never formally been defined and there is a shortage of rigorous mathematics describing epidemic phase transitions in the context of disease spreading on top of a complex network this paper provides a set of necessary and sufficient conditions for the occurrence of a persistent infected state as a proof of principle we demonstrate that the percolation and sis sir epidemic models on complex correlated networks satisfy the assumptions necessary for a single phase transition this paper attempts to highlight commonalities in a variety of different interacting particle systems

tests of homogeneity of means and covariance matrices for multivariate incomplete data existing test statistics for assessing whether incomplete data represent a missing completely at random sample from a single population are based on a normal likelihood rationale and effectively test for homogeneity of means and covariances across missing data patterns the likelihood approach cannot be implemented adequately if a pattern of missing data contains very few subjects a generalized least squares rationale is used to develop parallel tests that are expected to be more stable in small samples three factors were varied for a simulation number of variables percent missing completely at random and sample size one thousand data sets were simulated for each condition the generalized least squares test of homogeneity of means performed close to an ideal type i error rate for most of the conditions the generalized least squares test of homogeneity of covariance matrices and a combined test performed quite well also

a test of missing completely at random for multivariate data with missing values abstract a common concern when faced with multivariate data with missing values is whether the missing data are missing completely at random mcar that is whether missingness depends on the variables in the data set one way of assessing this is to compare the means of recorded values of each variable between groups defined by whether other variables in the data set are missing or not although informative this procedure yields potentially many correlated statistics for testing mcar resulting in multiple comparison problems this article proposes a single global test statistic for mcar that uses all of the available data the asymptotic null distribution is given and the small sample null distribution is derived for multivariate normal data with a monotone pattern of missing data the test reduces to a standard t test when the data are bivariate with missing data confined to a single variable a limited simulation study of empirical sizes for the test applied to normal and nonnormal data suggests th

a one parameter family of stationary solutions in the susceptible infected susceptible epidemic model abstract we study the uncorrelated susceptible infected susceptible sis model in epidemiology on top of a one parameter family of networks whose connectivity distribution ranges from scale free sf to exponential for each network the fraction of the population infected in the long term is a recursively defined hypergeometric function for highly contagious diseases with a high infection rate the fraction of the population infected is lower when the network is sf for less contagious diseases the fraction of the population infected is lower when the network is exponential this result points to an evolutionary advantage for a network being sf namely an sf network is more resistant to the spread of a deadly disease

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

an autoregressive linear mixed effects model for the analysis of longitudinal data which include dropouts and show profiles approaching asymptotes we are interested in longitudinal data of a continuous response that show profiles with an initial sharp change and approaching asymptotes for each patient and many patients drop out with a reason related to the response in this paper we focus on a model that assumes a dropout process is missing at random mar in this dropout process we can obtain consistent maximum likelihood estimators as long as both the mean and covariance structures are correctly specified however parsimonious covariance structures for the profiles approaching asymptotes are unclear an autoregressive linear mixed effects model can express the profile with random individual asymptotes we show that this model provides a new parsimonious covariance structure the covariance structure at steady state is compound symmetry and the other elements of the covariance depend on the measurement points in simulation studies the estimate of the asymptote is unbiased in mar dropouts but biased in non ignorable dropouts we also applied this model to actual schizophrenia trial data

projected partial likelihood and its application to longitudinal data summary an estimating equation which we call the projected partial score is introduced for longitudinal data analysis the estimating equation is obtained by projecting the partial likelihood score function onto the vector space spanned by a class of conditionally linear estimating equations we demonstrate that removing certain terms from the projection of the full likelihood score does not alter important inferential properties of the estimating equation and doing so is advantageous in handling missing data and time varying covariates within a prequential frame of reference it is shown that the estimating equation is optimal among the largest collection of estimating equations determined by the conditional moments furthermore the method possesses similar properties to generalised estimating equations in particular the correct conditional variance specification is necessary for efficiency but not for asymptotic consistency and distribution theory

a nonparametric test of missing completely at random for incomplete multivariate data missing data occur in many real world studies knowing the type of missing mechanisms is important for adopting appropriate statistical analysis procedure many statistical methods assume missing completely at random mcar due to its simplicity therefore it is necessary to test whether this assumption is satisfied before applying those procedures in the literature most of the procedures for testing mcar were developed under normality assumption which is sometimes difficult to justify in practice in this paper we propose a nonparametric test of mcar for incomplete multivariate data which does not require distributional assumptions the proposed test is carried out by comparing the distributions of the observed data across different missing pattern groups we prove that the proposed test is consistent against any distributional differences in the observed data simulation shows that the proposed procedure has the type i error well controlled at the nominal level for testing mcar and also has good power against a variety of non mcar alternatives

missing data mechanisms and homogeneity of means and variances covariances unless data are missing completely at random mcar proper methodology is crucial for the analysis of incomplete data consequently methods for effectively testing the mcar mechanism become important and procedures were developed via testing the homogeneity of means and variances covariances across the observed patterns e g kim bentler in psychometrika 67 609 624 2002 little in j am stat assoc 83 1198 1202 1988 the current article shows that the population counterparts of the sample means and covariances of a given pattern of the observed data depend on the underlying structure that generates the data and the normal distribution based maximum likelihood estimates for different patterns of the observed sample can converge to the same values even when data are missing at random or missing not at random although the values may not equal those of the underlying population distribution the results imply that statistics developed for testing the homogeneity of means and covariances cannot be safely used for testing the mcar mechanism even when the population distribution is multivariate normal

quasi likelihood regression models with missing covariates summary this paper presents methods to handle missing covariates when the quasi likelihood equations for the complete data are available our suggestion is to replace the functions of the missing data appearing in the quasi likelihood equation with their conditional means given the observed data or with unbiased predictors so that the resulting equation is unbiased we focus on two models one is a random effects model for count data where random effects are treated as missing covariates the other is the overdispersed binomial regression model with partially missing covariates we also investigate the efficiency of the proposed estimates relative to the maximum likelihood estimators

the ergodicity and extinction of stochastically perturbed sir and seir epidemic models with saturated incidence abstract in this paper we include stochastic perturbations into sir and seir epidemic models with saturated incidence and investigate their dynamics according to the basic reproduction number r 0 the long time behavior of the two stochastic systems is studied mainly we utilize stochastic lyapunov functions to show under some conditions the solution has the ergodic property as r 0 1 while exponential stability as r 0 1 at last we make simulations to conform our analytical results

simultaneous rational approximation to binomial functions we apply pade approximation techniques to deduce lower bounds for simultaneous rational approximation to one or more algebraic numbers in particular we strengthen work of osgood fel dman and rickert proving for example that max 2 p1 q 3 p2 q q 1 79155 for q q0 where the latter is an effective constant some of the diophantine consequences of such bounds will be discussed specifically in the direction of solving simultaneous pell s equations and norm form equations 0 introduction in 1964 baker 1 2 utilized the method of pade approximation to hypergeometric functions to obtain explicit improvements upon liouville s theorem on rational approximation to algebraic numbers by way of example he showed that 3 2 pq 10 6q 2 955 0 1 for all positive integers p and q and used such bounds to solve related diophantine equations chudnovsky 6 subsequently refined baker s results primarily through a detailed analysis of the arithmetical properties of certain pade approximants analogous to 0 1 he proved that 3 2 pq q 2 42971 0 2 for all integers p and q with q greater than some effectively computable constant q0 by working out the implicit constants in 0 2 easton 8 deduced 3 2 pq 6 6 10 6q 2 795 for positive integers p and q as well as related bounds for other cubic irrationalities similar results exist for simultaneous approximation to several algebraic numbers in particular baker 3 derived bounds of the form max 1 u m u pu q q 0 3 received by the editors june 30 1994 and in revised form january 31 1995 1991 mathematics subject classification primary 11j68 11j82 secondary 11d57

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

a mean score method for missing and auxiliary covariate data in regression models summary we consider regression analysis when incomplete or auxiliary covariate data are available for all study subjects and in addition for a subset called the validation sample true covariate data of interest have been ascertained the term auxiliary data refers to data not in the regression model but thought to be informative about the true missing covariate data of interest we discuss a method which is nonparametric with respect to the association between available and missing data allows missingness to depend on available response and covariate values and is applicable to both cohort and case control study designs the method previously proposed by flanders greenland 1991 and by zhao lipsitz 1992 is generalised and asymptotic theory is derived our expression for the asymptotic variance of the estimator provides intuition regarding performance of the method optimal sampling strategies for the validation set are also suggested by the asymptotic results