the special functions of fractional calculus as generalized fractional calculus operators of some basic functions abstract we propose a unified approach to the so called special functions of fractional calculus sfs of fc recently enjoying increasing interest from both theoretical mathematicians and applied scientists this is due to their role as solutions of fractional order differential and integral equations as the better mathematical models of phenomena of various physical engineering automatization chemical biological earth science economics etc nature our approach is based on the use of generalized fractional calculus gfc operators namely we show that all the wright generalized hypergeometric functions w ghf s q p z can be represented as generalized fractional integrals derivatives or differ integrals of three basic simpler functions as cos q p z exp z and 0 1 z reducible in particular to the elementary function z 1 z the beta distribution depending on whether p q p q or p q 1 and on the values of their indices and parameters in this way the sfs of fc can be separated into three classes with similar behaviour and also new integral and differential formulas can be derived useful for computational procedures

improved global exponential stability for delay difference equations with impulses abstract in this paper by means of constructing an impulsive delay difference inequality we study the global exponential stability for a class of delay difference equations with impulses a new criterion ensuring the global exponential stability of the equilibrium point is obtained which is less restrictive and conservative than that given in the earlier reference an illustrative example is given to demonstrate the effectiveness and advantage of the obtained result

approximate self similar solutions to a nonlinear diffusion equation with time fractional derivative abstract in this paper we consider a fractional nonlinear problem for anomalous diffusion the diffusion coefficient we use is of power type and hence the investigated problem generalizes the porous medium equation a generalization is made by introducing a fractional time derivative we look for self similar solutions for which the fractional setting introduces other than classical space time scaling the resulting similarity equations are of nonlinear integro differential type we approximate these equations by an expansion of the integral operator and by looking for solutions in a power function form our method can be easily adapted to solve various problems in self similar diffusion the approximations obtained give very good results in numerical analysis their simplicity allows for easy use in applications as our fitting with experimental data shows moreover our derivation justifies theoretically some previously used empirical models for anomalous diffusion

boundedness for impulsive delay differential equations and applications to population growth models abstract in this paper criteria on uniform ultimate boundedness are derived for impulsive delay differential equations using lyapunov functions and razumikhin techniques it should be noted that the boundedness criteria establish global existence of solutions as well as boundedness without assuming a priori that solutions can necessarily be continued to infinity those criteria are then used to obtain permanence results for population growth models some examples are discussed to illustrate the theorems

stability theorem for delay differential equations with impulses abstract in this paper we obtain sufficient conditions for the stability of the zero solution of the delay differential equation with impulses formula

on delay differential equations with impulses abstract sufficient conditions are obtained respectively for the asymptotic stability of the trivial solution of x t ax t j 1 bjx tj t tj t tj and for the existence of a nonoscillatory solution conditions are also obtained for all solutions to be oscillatory the asymptotic behaviour of an impulsively perturbed delay logistic equation is investigated as an extension to a nonlinear equation

an anomalous non self similar infiltration and fractional diffusion equation problems of anomalous infiltration in porous media are considered as follows from the analysis of experimental data modification of the infiltration equation is necessary a fractional diffusion equation with variable order of the time derivative operator for describing the liquid infiltration in porous media is proposed the physical meaning of this fractional equation is explained this equation provides good agreement with existing experimental data for both the subdiffusion and the superdiffusion the treatment of experimental data for the absorption of water in a fired clay brick and for water infiltration in cement mortar using this fractional equation of diffusion is presented various formulae which can be useful for applications have been developed

new razumikhin type theorems on the stability for impulsive functional differential systems in this paper we investigate the stability of the trivial solution for impulsive functional differential systems using several lyapunov functions including partial components coupled with the razumikhin technique and obtained some new razumikhin type theorems which avoid using the auxiliary function p under less restrictive conditions our results improve some of the earlier findings and are suitable for many applications some examples are given to illustrate the advantages of the theorems obtained

uniform boundedness and stability criteria in terms of two measures for impulsive integro differential equations1research supported by nserc canada 1 this paper establishes several criteria on uniform boundedness and stability in terms of two measures for impulsive integro differential equations by the method of lyapunov functions the interplay of two different measures and the discontinuities of solutions create additional difficulties in choosing suitable minimal set of functions along which the derivative of the lyapunov function is estimated consequently very complicated analysis is required in the proof of the theorems

fractional sturm liouville problem in this paper we define some fractional sturm liouville operators fslos and introduce two classes of fractional sturm liouville problems fslps namely regular and singular fslp the operators defined here are different from those defined in the literature in the sense that the operators defined here contain left and right riemann liouville and left and right caputo fractional derivatives for both classes we investigate the eigenvalue and eigenfunction properties of the fslos in the class of regular fslps we discuss two types of fslps as an operator for the class of singular fslps we introduce a fractional legendre equation fle and discuss its solution it is shown that the legendre polynomials resulting from an fle are the same as those obtained from the integer order legendre equation however the eigenvalues of the two equations differ using the legendre integral transform we demonstrate some applications of our results by solving two fractional differential equations one ordinary and the other partial it is our hope that this paper will initiate new research in the area of fslps and many of its variations

stability criteria of impulsive differential systems the notion of eventual stability has been recently discussed we extend this notion to impulsive systems of differential equations our technique depends on liapunov s direct method

a case cohort design for epidemiologic cohort studies and disease prevention trials summary suppose that a cohort of individuals is to be followed in order to relate failure rates to preceding covariate histories a design is proposed which involves covariate data only for cases experiencing failure and for members of a randomly selected subcohort odds ratio and relative risk estimation procedures are presented for such a case cohort design a small simulation study compares case cohort relative risk estimation procedures to full cohort and synthetic case control analyses relevance to epidemiologic cohort studies and disease prevention trials is discussed

a new approach to stability of impulsive functional differential equations in this work a new approach to stability theory of impulsive functional differential equations is proposed instead of putting all components of the state variable x in one liapunov function several functions of partial components of x which can be much easier constructed are used so that the conditions ensuring that stability are simpler and less restrictive also an example is given to illustrate the advantages of the obtained results

eigenvalue asymptotics for a fractional boundary value problem this letter presents a result concerning eigenvalue approximation of a boundary value problem with the caputo fractional derivative this approximation is derived by the use of the asymptotic for large x and l form of the exact solution the growth order of the eigenvalues is given and it is shown that their number is finite moreover a simple method of estimating the size of the spectrum is proposed the issue of a finite number of eigenvalues is a very peculiar and characteristic feature of differential equations with fractional order derivative the paper is concluded with a numerical verification that our approximations are very accurate this shows that the devised formulas can be readily used in applications of fractional boundary value problems

asymptotic integration of 1 α order fractional differential equations we establish the long time asymptotic formula of solutions to the 1 a order fractional differential equation 0 io t 1 ax a t x 0 t 0 under some simple restrictions on the functional coefficient a t where 0 io t 1 a is one of the fractional differential operators 0d t a x 0d t ax 0d t 1 ax and 0d t a tx x here 0d t a designates the riemann liouville derivative of order a 0 1 the asymptotic formula reads as b o 1 x s m a l l c x l a r g e as t for given b c r where x s m a l l and x l a r g e represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation 0 io t 1 ax 0 t 0

impulsive stabilization of high order hopfield type neural networks with time varying delays this paper studies the problems of global exponential stability for impulsive high order hopfield type neural networks nns with time varying delays by employing the lyapunov razumikhin technique some criteria ensuring global exponential stability are derived our results are then used to obtain some sufficient conditions under which some nns can be forced to converge by impulsive control numerical examples are also discussed to illustrate our results

exponential asymptotics of the mittag leffler function we consider the exponentially improved asymptotics of the mittag leffler function e a z and show how certain exponentially small terms make their appearance in the asymptotic expansion of e a z when z is large a new terminant function is introduced a generalized mittag leffler function e a b z and the stokes phenomenon in both arg z and the parameter a is examined

stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks in this paper we investigate the stochastic functional differential equations with infinite delay some sufficient conditions are derived to ensure the pth moment exponential stability and pth moment global asymptotic stability of stochastic functional differential equations with infinite delay by using razumikhin method and lyapunov functions based on the obtained results we further study the pth moment exponential stability of stochastic recurrent neural networks with unbounded distributed delays the result extends and improves the earlier publications two examples are given to illustrate the applicability of the obtained results

stability analysis of complex valued impulsive systems with time delay in this paper the global exponential stability of complex valued impulsive systems is addressed some new sufficient conditions are obtained to guarantee the global exponential stability by the lyapunov razumikhin theory which extend and improve most of recent results moreover the obtained razumikhin conditions are very simple and efficient to verify in real problems and helpful to investigate the stability of delayed neural networks and synchronization problems of chaotic systems under impulsive perturbation finally a numerical example is given to show the effectiveness of the obtained results

using em to obtain asymptotic variance covariance matrices the sem algorithm abstract the expectation maximization em algorithm is a popular and often remarkably simple method for maximum likelihood estimation in incomplete data problems one criticism of em in practice is that asymptotic variance covariance matrices for parameters e g standard errors are not automatic byproducts as they are when using some other methods such as newton raphson in this article we define and illustrate a procedure that obtains numerically stable asymptotic variance covariance matrices using only the code for computing the complete data variance covariance matrix the code for em itself and code for standard matrix operations the basic idea is to use the fact that the rate of convergence of em is governed by the fractions of missing information to find the increased variability due to missing information to add to the complete data variance covariance matrix we call this supplemented em algorithm the sem algorithm theory and particular examples reinforce the conclusion that the sem alg

razumikhin type theorems on exponential stability of impulsive infinite delay differential systems in this note we study the exponential stability of impulsive functional differential systems with infinite delays by using the razumikhin technique and lyapunov functions several razumikhin type theorems on exponential stability are obtained which shows that certain impulsive perturbations may make unstable systems exponentially stable some examples are discussed to illustrate our results

exponential stability of linear delay impulsive differential equations for an impulsive delay differential equation with bounded delay and bounded coefficients the following result is established if each solution is bounded on 0 together with its derivative for each bounded right hand side then the equation is exponentially stable an explicit stability theorem is presented

survival curve estimation with partial non random exposure information the objective of this paper is to estimate survival curves for two different exposure groups when the exposure group is not known for all observations and the data is subject to left truncation and right censoring the situation we consider is when the probability that the exposure group is missing may depend on whether the observation is censored or uncensored in which case the exposure is not missing at random the problem was motivated by a study of alzheimer s disease to estimate the distribution of ages at diagnosis for individuals with and without an apolipoprotein e4 allele the exposure group genotyping for this risk factor was incomplete and performed more frequently on the cases of alzheimer s disease the uncensored observations than the censored observations the survival curves are estimated in discrete time using an em algorithm a bootstrapping procedure is proposed that guarantees each bootstrap sample has the same proportion of observations with missing exposure a simulation is performed to evaluate the bias of the estimators and to investigate design and efficiency issues the methods are applied to the alzheimer s disease study

stability results for impulsive functional differential equations with infinite delays abstract this paper studies the stability problems for a class of impulsive functional differential equations with infinite delays of the form x t f t x t t x t k j k x t k k 1 2 by using the liapunov functions and razumikhin technique some new razumikhin type theorems on stability are obtained

exponential stability for impulsive delay differential equations by razumikhin method abstract in this paper we study exponential stability for impulsive delay differential equation of the form x t f t x t t t k x t i k t x t t t k k n by employing the razumikhin technique and lyapunov functions several exponential stability criteria are established some examples are also discussed to illustrate our results

monotonicity oscillations and stability of a solution to a nonlinear equation modelling the capillary rise abstract in this paper we analyse a singular second order nonlinear ode which models the capillary rise of a fluid inside a tubular column we prove global existence uniqueness and find several approximations along with the asymptotic behaviour of the solution moreover we are able to find a critical value of the nondimensional parameter for which the solution exhibits a transition in its behaviour from being monotone to oscillatory this is an analytical rigorous proof of the experimentally and numerically confirmed phenomenon

auxiliary covariate data in failure time regression summary we consider the problem of missing covariate data in the context of censored failure time relative risk regression auxiliary covariate data which are considered informative about the missing data but which are not explicitly part of the relative risk regression model may be available full covariate information is available for a validation set an estimated partial likelihood method is proposed for estimating relative risk parameters this method is an extension of the estimated likelihood regression analysis method for uncensored data pepe 1992 pepe fleming 1991 a key feature of the method is that it is nonparametric with respect to the association between the missing and observed including auxiliary covariate components asymptotic distribution theory is derived for the proposed estimated partial likelihood estimator in the case where the auxiliary or mismeasured covariates are categorical asymptotic efficiencies are calculated for exponential failure times using an exponential relative risk model the estimated partial likelihood estimator compares favourably with a fully parametric maximum likelihood analysis comparisons are also made with a standard partial likelihood analysis which ignores the incomplete observations important efficiency gains can be made with the estimated partial likelihood method small sample properties are investigated through simulation studies

logistic regression for two stage case control data summary samples of diseased cases and nondiseased controls are drawn at random from the population at risk after classification according to the exposure of interest subsamples of cases and controls are selected for purposes of covariable ascertainment a modification of the usual logistic regression analysis yields consistent estimates of covariable adjusted relative risks and their standard errors by balancing the numbers of exposed and nonexposed for whom covariable inforniation is ascertained within case and control samples some efficiency may be gained over the usual single stage design particularly when the exposure is rare and the relative risks associated with the covariables are large the procedure may be useful also when covariable information is missing for a large part of the sample

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

cox regression with incomplete covariate measurements abstract this article provides a general solution to the problem of missing covariate data under the cox regression model the estimating function for the vector of regression parameters is an approximation to the partial likelihood score function with full covariate measurements and reduces to the pseudolikelihood score function of self and prentice in the special setting of case cohort designs the resulting parameter estimator is consistent and asymptotically normal with a covariance matrix for which a simple and consistent estimator is provided extensive simulation studies show that the large sample approximations are adequate for practical use the proposed approach tends to be more efficient than the complete case analysis especially for large cohorts with infrequent failures for case cohort designs the new methodology offers a variance covariance estimator that is much easier to calculate than the existing ones and allows multiple subcohort augmentations to improve efficiency real data taken f

asymptotical stability of the exact solutions and the numerical solutions for a class of impulsive differential equations this paper is concerned with stability and asymptotical stability of a class of impulsive delay differential equations iddes stability and asymptotical stability of the system of iddes are studied by the properties of delay differential equations ddes without impulsive perturbations base on this idea numerical methods of iddes are constructed stability and asymptotical stability of numerical methods of iddes are also studied by the properties of numerical methods of ddes

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