fuzzy non continuous mappings and fuzzy pre semi separation axioms the aim of the paper is mainly to introduce and study fuzzy pre semi open mapping fuzzy pre semi irresolute mapping fuzzy pre semi separation axioms and fuzzy pre semi connectedness in fuzzy topological spaces

cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

absorbing boundary in one dimensional anomalous transport in this paper we study the space time probability distribution q x t of a random walk subject to an absorbing boundary at the origin x 0 for motion controlled by l evy flights and l evy walks characterized by the exponent ensuremath gamma we find that the method of images usually applicable to brownian motion may break down for l evy processes we calculate the distribution q x t to be at xg0 after time tg0 having started at the origin assuming that the boundary is effective at time tg0 we show that q x t depends on the details of the underlying process q x t ensuremath sim mathit x ensuremath gamma 2 mathit t 1 1 ensuremath gamma 1 ensuremath le ensuremath gamma ensuremath le 2 for small x while total survival is independent of the spatial realization of motion and displays a universal behavior we also discuss the related smoluchowski boundary condition problem

paracompactness in l topological spaces in this paper a certain fuzzy paracompactness is presented by means of the concept of locally finite family of l fuzzy sets the concept of locally finite family can be used to characterize fuzzy countable compactness this fuzzy paracompactness is a natural generalization of lowen fuzzy compactness a fuzzy countably compact l set is fuzzy paracompact if and only if it is fuzzy compact a topological space x j is paracompact if and only if its induced space x j is fuzzy paracompact when ms l the fuzzy enhanced real line r l is fuzzy paracompact the product of a fuzzy compact set and a fuzzy paracompact set is fuzzy paracompact a fuzzy strongly t2 paracompact space is strongly s regular and strongly ws normal moreover a strongly s regular lindelof space is fuzzy paracompact

a note on lo fuzzy nearly compact spaces in this note we define lo fuzzy nc fuzzy sets and introduce the finite intersection property for fuzzy sets in the sense of lowen lo fip for short we give some equivalent characterizations of lo fuzzy nc spaces and lo fuzzy nc fuzzy sets by using lo fip regular open and regular closed fuzzy sets finally we prove that lo fuzzy nearly compactness is preserved under fuzzy almost continuous and almost open mapping

almost compact fuzzy sets in fuzzy topological spaces abstract the study of almost compact fuzzy topological spaces was initiated by concilio and gerla 4 which was followed by further investigations of the same in 11 the present paper aspires for an extension of the same concept to fuzzy sets in a fuzzy topological space we characterize almost compact fuzzy sets in terms of the concept of fuzzy filterbase some properties of such sets are established using the concepts of fuzzy ds closure and fuzzy ds cluster points of fuzzy filterbase

fuzzy nets and their application fuzzy net theory is enriched by the introduction of the fuzzy net of fuzzy sets the limsup liminf and limit of a fuzzy net of fuzzy sets are defined and their various properties are discussed alternative characterizations based on the notion of a fuzzy net are provided for several fuzzy topological concepts including open and closed fuzzy sets fuzzy continuity maps with closed fuzzy graph open fuzzy mapping etc thus the net theoretic approach is shown to be a promising tool in fuzzy topology

fµ strongly semiopen sets fµ strong semicontinuity and fµ strongly semiretracts throughout the literature if x d is an f ts and y x the induced f topological subspace y d y is defined so that d y n g y n d macho stadler and de prada vicente fuzzy sets and systems 58 1993 365 introduced a new concept of f topological subspaces which coincides with the usual definition in the case that m g y also they introduced the concepts of f m open sets and f m continuity in this paper using the previous concepts we introduce weaker forms of f m open sets and f m continuity the notion of an f retract was introduced by rodabough j math anal appl 79 1981 273 here we introduce the notion of a f m retract and weaker forms of it the basic properties of these notions are given many results are obtained

regression with censored data summary there are four regression techniques currently available for use with censored data which do not assume particular parametric families of survival distributions they are due to i cox 1972 ii miller 1976 iii buckley james 1979 and iv koul susarla van ryzin 1981 these four methods are described and their performances compared on the updated stanford heart transplant data conclusions on the usefulness of the four procedures are drawn

fractional laplacian in bounded domains the fractional laplacian operator ensuremath ensuremath ensuremath delta ensuremath alpha 2 appears in a wide class of physical systems including l evy flights and stochastic interfaces in this paper we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval the implementation of boundary conditions is justified by appealing to two physical models namely hopping particles and elastic springs the eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions some analytical results concerning the structure of the eigenvalue spectrum are also obtained

goodness of fit test for proportional subdistribution hazards model this paper concerns using modified weighted schoenfeld residuals to test the proportionality of subdistribution hazards for the fine gray model similar to the tests proposed by grambsch and therneau for independently censored data we develop a score test for the time varying coefficients based on the modified schoenfeld residuals derived assuming a certain form of non proportionality the methods perform well in simulations and a real data analysis of breast cancer data where the treatment effect exhibits non proportional hazards copyright 2013 john wiley sons ltd

nonparametric survival analysis with time dependent covariate effects a penalized partial likelihood approach techniques are developed for nonparametric analysis of data under a cox regression like model permitting time dependent covariate effects determined by a regression function beta 0 t estimators resulting from maximization of an appropriate penalized partial likelihood are shown to exist and a computational approach is outlined weak uniform consistency with a rate of convergence and pointwise asymptotic normality of the estimators are established under regularity conditions a consistent estimator of a common baseline hazard function is presented and used to construct a consistent estimator of the asymptotic variance of the estimator of the regression function extensions to multiple covariates general relative risk functions and time dependent covariates are discussed

on various weak fuzzy pairwise retracts the notion of a fuzzy retract was introduced by rodabaugh j math anal appl 79 1981 273 and this notion was extended to the fuzzy bitopological setting j fuzzy math 9 1 2001 207 in this paper we define the notions of fuzzy pairwise semiretracts fuzzy pairwise preretracts fuzzy pairwise strongly semiretracts fuzzy pairwise less strongly semiretracts and fuzzy pairwise semi preretracts we establish some properties of these concepts moreover the relations between these new concepts are discussed

fuzzy proximities and totally bounded fuzzy uniformities after the classical papers of zadeh 181 and chang 3 many concepts of general topology have been extended to fuzzy set theory in this paper we are interested to develop the study of fuzzy uniformities fuzzy proximities and of the connexions between such structures fuzzy uniformities have been introduced by lowen in 9 and with slight modifications in lo and by hutton in 5 the two approaches are quite different however the one proposed by hutton seems to suit much better to fuzzy set theory also because it allows to generalize a lot of results in the most proper way therefore in this paper we shall deal with hutton fuzzy uniformities the concept of fuzzy proximity used up to now see e g 7 8 15 1 is quite unsatisfactory indeed its fuzzyness is rather poor since these fuzzy proximities are in a canonical l 1 correspondence with the usual proximities see remark 2 6 moreover the open sets of the induced topologies are crisp and though every lowen fuzzy uniformity induces a fuzzy proximity this correspondence cannot work well since the two structures do not give the same fuzzy topology for these reasons we propose a new definition of fuzzy proximity it differs from the old one only in axiom p5 in short we could say that we privilege complementation with respect to intersection an agreement between the two operations being problematic in fuzzy set theory we point out that as a consequence a fuzzy set may be far from itself however this fact ought not to amaze anyone since it may happen just for pale sets which in a certain sense resemble the empty set nevertheless this modification enables us to associate a topology in a completely different way moreover every fuzzy uniformity induces a fuzzy proximity and vice versa we succeed to construct a fuzzy uniformity starting with a given fuzzy proximity and the induced topologies do not change at any step this implies that the topologies which admit a fuzzy proximity are exactly the completely regular ones moreover we prove that there exists a l l correspondence between fuzzy proximity spaces and a

on the eigenvalues of certain canonical higher order ordinary differential operators abstract we consider the operator of taking the 2pth derivative of a function with zero boundary conditions for the function and its first p 1 derivatives at two distinct points our main result provides an asymptotic formula for the eigenvalues and resolves a question on the appearance of certain regular numbers in the eigenvalue sequences for p 1 and p 3

properties of levy flights on an interval with absorbing boundaries we consider a l evy yer of orderthat starts from a point x0 on an interval 0 l with absorbing boundaries we 7nd a closed form expression for an arbitrary average quantity char acterizing the trajectory of the yer such as mean 7rst passage time average total path length probability to be absorbed by one of the boundaries using fractional di erential equations with a riesz kernel we 7nd exact analytical expressions for these quantities in the continuous limit we 7nd numerically the eigenfunctions and the eigenvalues of these equations we study how the results of monte carlo simulations of the l evy ights with di erent ight length distribu tions converge to the continuous approximations we show that if x0 is placed in the vicinity of absorbing boundaries the average total path length has a minimum near 1 corresponding to the cauchy distribution we discuss the relevance of these results to the problem of biological foraging and transmission of light through cloudy atmosphere c 2001 elsevier science b v all rights reserved

estimation of the survival function for gray s piecewise constant time varying coefficients model gray s extension of cox s proportional hazards ph model for right censored survival data allows for a departure from the ph assumption via introduction of time varying regression coefficients tvc using penalized splines gray s work focused on estimation inference and residual analyses but no estimator for the survival function has been proposed we derive a survival function estimator for one important member of the class of tvc models a piecewise constant time varying coefficients pc tvc model we also derive an estimate for the confidence limits of the survival function accuracy in estimating underlying survival times and survival quantiles is assessed for both cox s and gray s pc tvc model using a simulation study featuring scenarios violating the ph assumption finally an example of the estimated survival functions and the corresponding confidence limits derived from cox s ph and gray s pc tvc model respectively is presented for a liver transplant data set in the second part of the thesis we examine the effect of model misspecification for two classes of regression models for right censored survival data additive and multiplicative models for the conditional hazard rate a particular attention is given to data exhibiting time varying regression coefficients the class of multiplicative models is represented by cox ph model and gray s tvc model respectively and for additive models we use aalen s linear model both gray s tvc model and aalen s linear model incorporate time varying coefficients a simulation study is performed to cross analyze survival data which follows either a multiplicative or an additive model for the conditional hazard rate the effect of misspecifying the true model for the conditional hazard rate is assessed by looking at the power of the individual models to detect an existing effect bias and mean square error observed for each conditional model based estimator of survival we also show that aalen s model formulae is a first order taylor series approximation of that of gray s model which explains the comparably higher flexibility on part of the aalen s model as compared to the cox ph when the gray s tvc model for the data is misspecified

regularly open sets in fuzzy topological spaces abstract this paper is devoted to the study of the role of fuzzy regularly open sets we prove some properties of fuzzy almost continuous mappings and define fuzzy almost open mappings we prove that under a fuzzy almost continuous and fuzzy almost open map the inverse image of a fuzzy regularly open set is fuzzy regularly open further we define a new type of fuzzy separation axioms fuzzy almost separation axioms it is interesting that there are some deviations in the behaviour of these axioms as compared to those in general topology for example in a fuzzy almost t1 space not every fuzzy singleton is closed also a fuzzy space which is fuzzy almost t 2 1 2 as well as fuzzy almost t0 is fuzzy almost regular while in general topology we have to take an almost t2 space in place of almost t0 space

checking the cox model with cumulative sums of martingale based residuals summary this paper presents a new class of graphical and numerical methods for checking the adequacy of the cox regression model the procedures are derived from cumulative sums of martingale based residuals over follow up time and or covariate values the distributions of these stochastic processes under the assumed model can be approximated by zero mean gaussian processes each observed process can then be compared both visually and analytically with a number of simulated realizations from the approximate null distribution these comparisons enable the data analyst to assess objectively how unusual the observed residual patterns are special attention is given to checking the functional form of a covariate the form of the link function and the validity of the proportional hazards assumption an omnibus test consistent against any model misspecification is also studied the proposed techniques are illustrated with two real data sets

on fuzzy almost continuous functions abstract in this paper we redefine the notions of open fuzzy set closed fuzzy set adherent point of a fuzzy set and closure of a fuzzy set all the results pertaining to these notions see velicko 17 are extended to fuzzy setting and several related characterization of fuzzy almost continuous functions are obtained which are the counterparts of the corresponding results in general topological spaces 12 some more properties of fuzzy almost continuous functions are established subsequently a characaterization of normal fuzzy topological spaces is also obtained the notion of fuzzy almost quasi compact mapping is introduced and its characterizations and relationship with fuzzy almost continuous functions are discussed

on some weaker forms of fuzzy continuous and fuzzy open mappings on fuzzy topology spaces abstract the purpose of the present paper is to study some weaker forms of fuzzy continuous and fuzzy open maps between fuzzy topological spaces along with the introduction and investigations of a new class of non fuzzy continuous maps this latter type of mappings is seen to be independent of the other non continuous functions discussed here all these functions are characterized and certain properties concerning them are proved finally a comparative study as to their mutual dependence and interrelations under different conditions is taken up

semi open sets and semi continuity in fuzzy bitopological spaces abstract in this paper the concepts of fuzzy semi open sets and fuzzy semi continuous mappings due to azad 1981 have been generalized to fuzzy bitopological spaces and some of their properties are studied

goodness of fit tests for conditional models under censoring and truncation the problem of specification tests for conditional models is studied when the data are subject to left truncation and right censoring a general method is applied to derive tests for the polynomial regression the proportional hazards the additive risks and the proportional odds models bootstrap versions to approximate the critical values of the test are introduced and proved to work both from a theoretical viewpoint as well as in a small simulation study

on several types of degree of fuzzy compactness abstract sostak 1985 defined a fuzzy topology on a set x as a fuzzy subset of the family i x of fuzzy subsets of x satisfying some axioms the concept of the degree of compactness of a fuzzy set was first introduced by sostak 1985 in this paper we discuss the degrees of almost compactness near compactness countable compactness light compactness and strong compactness of a fuzzy set also we investigate the behavior of the degrees of compactness under several types of fuzzy continuous functions

efficient estimation for the cox model with varying coefficients summary a proportional hazards model with varying coefficients allows one to examine the extent to which covariates interact nonlinearly with an exposure variable a global partial likelihood method in contrast with the local partial likelihood method of fan et al 2006 is proposed for estimation of varying coefficient functions the proposed estimators are proved to be consistent and asymptotically normal semiparametric efficiency of the estimators is demonstrated in terms of their linear functionals evidence in support of the superiority of the method is presented in numerical studies and real examples

a compactness in bifuzzy topological spaces abstract compact fuzzy topological spaces were first introduced by chang 3 compact fuzzy topological spaces were introduced by gantner et al 4 abd el monsef et al 1 introduced the notion of almost compact fuzzy topological spaces mashbour et al 8 introduced the notion of bifuzzy topological spaces and defined the notion of compactness in those spaces in this paper we obtain some new results about compact bifuzzy topological spaces we introduce and study the notion of pairwise almost compact bifuzzy topological spaces

proportional hazards models with frailties and random effects we discuss some of the fundamental concepts underlying the development of frailty and random effects models in survival one of these fundamental concepts was the idea of a frailty model where each subject has his or her own disposition to failure their so called frailty additional to any effects we wish to quantify via regression although the concept of individual frailty can be of value when thinking about how data arise or when interpreting parameter estimates in the context of a fitted model we argue that the concept is of limited practical value individual random effects frailties whenever detected can be made to disappear by elementary model transformation in consequence unless we are to take some model form as unassailable beyond challenge and carved in stone and if we are to understand the term frailty as referring to individual random effects then frailty models have no value random effects models on the other hand in which groups of individuals share some common effect can be used to advantage even in this case however if we are prepared to sacrifice some efficiency we can avoid complex modelling by using the considerable power already provided by the stratified proportional hazards model stratified models and random effects models can both be seen to be particular cases of partially proportional hazards models a view that gives further insight the added structure of a random effects model viewed as a stratified proportional hazards model with some added distributional constraints will for group sizes of five or more provide no more than modest efficiency gains even when the additional assumptions are exactly true on the other hand for moderate to large numbers of very small groups of sizes two or three the study of twins being a well known example the efficiency gains of the random effects model can be far from negligible for such applications the case for using random effects models rather than the stratified model is strong this is especially so in view of the good robustness properties of random effects models nonetheless the simpler analysis based upon the stratified model remains valid albeit making a less efficient use of resources

group sequential tests with censored survival data adjusting for covariates summary group sequential tests for the analysis of survival data that compare two treatments and allow for the adjustment of other concomitant variables are discussed these methods allow early termination of a study if large treatment differences occur based on simulation results we compare the operating characteristics of these adjusted tests with the unadjusted sequentially computed log rank test

partially linear additive hazards regression with varying coefficients please see the supplementary materials for a correction to this article to explore the nonlinear interactions between some covariates and an exposure variable we propose the partially linear additive hazards model for survival data in a semiparametric setting we construct a local pseudoscore function to estimate the varying and constant coefficients and establish the asymptotic normality of the proposed estimators moreover we develop the weak convergence property for the local estimator of the baseline cumulative hazard function we conduct simulation studies to empirically examine the finite sample performance of the proposed methods and use real data from a breast cancer study for illustration

fuzzy preopen sets and fuzzy preseparation axioms abstract in this paper the concept of preopen sets due to a s mashhour m e abd el monsef and s n el debb 7 has been generalised to the fuzzy setting further fuzzy separation axioms have been introduced and investigated with the help of fuzzy preopen sets

fuzzy closed and fuzzy perfect mappings abstract it is shown that a fuzzy continuous map f is fuzzy perfect iff f i z is fuzzy closed for every fuzzy topological space z

erosion of regression effect in a survival study lack of persistence or erosion of the regression effect is an alternative to proportional hazards of particular interest in many medical applications such a departure from proportional hazards is often the most likely direction in which the model may be inadequate questions such as is the effect of treatment only transitory or to what extent does an initially measured prognostic variable maintain its impact frequently arise in the context of a simple changepoint model we propose a test of the null hypothesis of proportional hazards against the specific alternative of erosion of the regression effect the particular changepoint model used can be viewed as a first approximation to a more complex reality an approximation that enables us to avoid specifically modeling the functional form that any erosion might take practical guidelines for carrying out the test are provided the approach is illustrated in the context of a study on risk factors for breast cancer survival

on nearly compact and 0 rigid fuzzy sets in fuzzy topological spaces abstract this paper intends to initiate the study of nearly compact and rigid fuzzy sets in a fuzzy topological space these two classes of fuzzy sets are seen to be properly contained in the class of fuzzy compact sets and properly contain the class of fuzzy almost compact sets several characterizations and certain other properties of these two types of fuzzy sets are established by using the concepts of fuzzy closures and closures of fuzzy sets and fuzzy filterbases

fuzzy sets and functions on fuzzy spaces abstract we generally follow the terminology of azad j math anal appl 82 1981 14 32 and ming j math anal appl 76 1980 571 599 in addition to the fundamental concepts for fuzzy sets we emphasize the usefulness of the concepts of fuzzy point fuzzy elementhood for fuzzy sets a and b a new characterization is given for relations a b and a b this knowledge permits us to combine the two definitions of fuzzy point fuzzy elementhood in the third section some results are given concerning various special types of fuzzy sets in fuzzy topological spaces and the fuzzy semi regular fuzzy regular spaces defined by azad in the last section the definitions of h almost continuous w almost open functions urysohn space which are defined by hussain wilansky and noiri respectively are extended to fuzzy sets furthermore some results are obtained in the functions of the fuzzy topological spaces defined by azad and those are defined here

on separation axioms and ti fuzzy continuity abstract we introduce and generalise concepts of t continuous functions in a fuzzy setting and study them in connection with the fuzzy continuity and fuzzy separation axioms some fuzzy topological properties are also investigated under such functions

eigenproblems of large powers of the laplacian in bounded domains we present a method for computing the spectrum of large powers of the laplacian in a bounded domain restricting ourselves to the one and three dimensional cases since it does not seem possible to obtain information on the eigenvalues directly from the transcendental equation that gives the spectrum we introduce a wallis inspired method we obtain the expansion of the eigenfunction and the eigenvalues in power series where the inverse of the power at which the laplacian is raised plays the role of the small parameter we compare these eigenvalues to those obtained through a simple variational approach and remark that the latter offers an excellent approximation to the exact result

some remarks on lo rs fuzzy compactness abstract in this paper we define lo rs fuzzy compact fuzzy sets we study some properties of lo rs fuzzy compact spaces and lo rs fuzzy compact fuzzy sets we give some characterizations of these notions by fuzzy regularly open sets and fuzzy regularly closed sets we prove that lo rs fuzzy compactness is preserved under a fuzzy almost continuous and fuzzy almost open mapping

a comparison of some hausdorff notions in fuzzy topological spaces abstract several different notions of the hausdorff separation property for fuzzy topological spaces have appeared in the literature this paper is a study of the relationships between 12 of these concepts

a semiparametric transition model with latent traits for longitudinal multistate data we propose a general multistate transition model the model is developed for the analysis of repeated episodes of multiple states representing different health status transitions among multiple states are modeled jointly using multivariate latent traits with factor loadings different types of state transition are described by flexible transition specific nonparametric baseline intensities a state specific latent trait is used to capture individual tendency of the sojourn in the state that cannot be explained by covariates and to account for correlation among repeated sojourns in the same state within an individual correlation among sojourns across different states within an individual is accounted for by the correlation between the different latent traits the factor loadings for a latent trait accommodate the dependence of the transitions to different competing states from a same state we obtain the semiparametric maximum likelihood estimates through an expectation maximization em algorithm the method is illustrated by studying repeated transitions between independence and disability states of activities of daily living adl with death as an absorbing state in a longitudinal aging study the performance of the estimation procedure is assessed by simulation studies

separation axioms for fuzzy spaces abstract using the concepts of fuzzy points as introduced by wong 15 and of belonging to as introduced by r srivastava s n lal and a k srivastava 13 we introduce several notions of fuzzy separation axioms and examine their productivity relations between these axioms have been observed we also introduce four types of separated fuzzy sets and clarify some relations between them

fuzzy topology and fuzzy multifunctions the fundamental concept of a fuzzy set was first introduced by zadeh in his classical paper 141 since then work has been done by many authors in several directions which has resulted in the formation of a new mathematical field called fuzzy mathematics the theory of fuzzy topological spaces is a branch of such mathematics chang in 2 was the first to introduce the notion of fuzzy topology several others continued the work in this area for example see 12 131 we remark that general point set topology can be regarded as a special case of fuzzy topology where all membership functions are just characteristic functions in this paper we introduce several fuzzy topological concepts and examine their properties our basic tool in that process is the notion of the fuzzy neighborhood of a point introduced by warren in ill we also introduce the notion of fuzzy multifunctions which we strongly believe will be useful in the study of fuzzy games and general fuzzy optimization problems we study the continuity and measurability properties of these multifunctions also we introduce linear fuzzy multifunctions and study their properties in the light of the recent work of katsaras 5 katsaras and liu s and lowen 9 on fuzzy vector spaces

residuals for relative risk regression summary several possible definitions of residuals are given for relative risk regression with time varying covariates each such residual has a representation as an estimator of a stochastic integral with respect to the martingale arising from a subject s failure time counting process previously proposed residuals for individual study subjects and for specific time points are shown to be special cases of this definition as are previously derived regression diagnostics an illustration and various generalizations are also given able methods are required to detect various departures from modelling assumptions suitably defined residuals may play an important role in such identification however the nonparametric aspect of the model the possibility that modelled regression variables may be varying with follow up time and most importantly the usual presence of right censorship implies that specialized residual definitions are required the class of residuals considered here is most easily formulated using counting process notation for the failure time data let ni t yi t and zi t represent respectively for the ith subject the values of counting censoring and covariate processes at follow up time t i 1 n while ni u yi u zi u 0c u t specifies the corresponding histories for the ith subject prior to time t thus in a typical univariate failure time application ni with right continuous sample paths will take value zero prior to the time of failure on the ith subject and value one thereafter while yi with left continuous sample paths will take value one at times at which the ith subject is at risk for an observed failure and value zero otherwise the counting process ni can be uniquely decomposed so that for all t i

fuzzy subcontinuity inverse fuzzy subcontinuity and a new category of fuzzy topological spaces the notions of subcontinuity and inverse subcontinuity are extended to the fuzzy setting adopting fuzzy net theoretic approach various properties of fuzzy subcontinuous and inversely fuzzy subcontinuous mappings are studied a category of fuzzy topological spaces is constructed canonical example is constructed by using fuzzy translation of fuzzy numbers introduced by rodabaugh to show that fuzzy subcontinuity does not imply kfuzzy continuity in the process n compactness is characterized and fuzzy unit interval is shown to be n compact also a direct proof is provided showing that fuzzy addition of fuzzy numbers is fuzzy continuous

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

a note on k k azad s fuzzy hausdorffness concept abstract in this paper we investigate the fuzzy hausdorffness concept of azad 1 and note that it neither generalizes the usual hausdorffness nor is a good extension of usual hausdorffness a new concept of fuzzy hausdorffness called hausdorffness related to azad s concept is then proposed and we observe that it in addition to being free from the drawbacks in azad s concept also possesses several pleasing properties

a new approach to fuzzy topological spaces and fuzzy perfect mappings abstract a net theoretic approach is provided for defining a fuzzy topological space fuzzy topological spaces so defined form a category the main features and advantages of this category which is denoted by n here are discussed the notion of a perfect mapping is introduced and studied in the fuzzy setting it is found that in n a fts is n compact iff any mapping x x where x is any singleton is fuzzy perfect a fuzzy continuous mapping x y from an n compact fts x to a hausdorff fts y is fuzzy perfect in n however these results do not hold in c the chang s category in general it is found that the divergences in the study of fuzzy perfect mappings disappear if the study is carried out in n instead of c

fuzzy grills and a characterization of fuzzy proximity in his classical paper lo zadeh introduced the fundamental concept of fuzzy sets this idea was then applied to various other fields of mathematics including topology resulting in the study of fuzzy topology by chang i fuzzy uniformity by lowen 7 and fuzzy proximity by katsaras 5 thron is presented a new approach to poximity structures using the theory of grills an important concept introduced by choquet 3 the importance of grills in the proximity theory is noticed sufficiently in various other papers e g cf 2 4 also recently katsaras 6 introduced and studied filters ultrafilters and clusters in the new setup of fuzzy proximity on a set in the present paper we introduce fuzzy stacks and fuzzy grills and obtain a characterization of fuzzy proximity using fuzzy grills in section 2 fuzzy stacks and fuzzy grills and in section 3 on the lines of basic proximity or c proximity as described in 8 9 the concept of fuzzy basic proximity on a set are introduced also the fuzzy basic proximal neighborhood of a fuzzy set with respect to a fuzzy basic proximity is defined and some of the properties of sets of all fuzzy basic proximal neighborhoods of fuzzy sets are obtained finally in section 4 a necessary and sufficient condition for a fuzzy basic proximity to be a fuzzy proximity is obtained

fuzzy s continuous functions and semilocally connected fuzzy topological spaces abstract the concept of s continuous functions kohli 1978 is translated in fuzzy setting into two concepts fuzzy s continuous functions and fuzzy os continuous functions the concepts of semiconnected functions kohli 1978 and semilocally connected spaces kohli 1978 do the same interactions between those concepts in fuzzy setting and fuzzy continuity chang 1968 are discussed also characterizations of semilocally connected and semilocally o connected fuzzy topological spaces in terms of fuzzy s continuous functions and fuzzy os continuous functions respectively are obtained

functions and as closedness in fuzzy topological spaces abstract in this paper we discuss s closedness in fuzzy topological spaces and we give some characterizations of s closedness also we investigate the image and the inverse image of s closed spaces under some types of functions and we improve and generalize some results on s closed spaces in fuzzy setting

local partial likelihood estimation for lifetime data this paper considers a proportional hazards model which allows one to examine the extent to which covariates interact nonlinearly with an exposure variable for analysis of lifetime data a local partial likelihood technique is proposed to estimate nonlinear interactions asymptotic normality of the proposed estimator is established the baseline hazard function the bias and the variance of the local likelihood estimator are consistently estimated in addition a one step local partial likelihood estimator is presented to facilitate the computation of the proposed procedure and is demonstrated to be as efficient as the fully iterated local partial likelihood estimator furthermore a penalized local likelihood estimator is proposed to select important risk variables in the model numerical examples are used to illustrate the effectiveness of the proposed procedures

average time spent by levy flights and walks on an interval with absorbing boundaries we consider a levy flyer of order alpha that starts from a point x 0 on an interval o l with absorbing boundaries we find a closed form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed these two quantities are equivalent to the mean first passage times for levy flights and levy walks respectively using fractional differential equations with a riesz kernel we find exact analytical expressions for both quantities in the continuous limit we show that numerical solutions for the discrete levy processes converge to the continuous approximations in all cases except the case of alpha 2 and the cases of x 0 0 and x 0 l for alpha 2 when the second moment of the flight length distribution exists our result is replaced by known results of classical diffusion we show that if x 0 is placed in the vicinity of absorbing boundaries the average total length has a minimum at alpha 1 corresponding to the cauchy distribution we discuss the relevance of this result to the problem of foraging which has received recent attention in the statistical physics literature

relative efficiency of the log rank test within a multiplicative intensity model summary for large sample clinical trials with independent individuals randomly allocated to two treatment groups in which survival times follow a log linear multiplicative intensity model with treatment group as one covariate this paper calculates the asymptotic relative efficiency of the log rank test for treatment effect as compared with the optimal score test the method is to exhibit the failure hazard intensity not of proportional hazards form obtained by ignoring all covariates other than treatment group the efficiency formulae are illustrated in two examples and estimation from data of the loss of efficiency is illustrated for two clinical trial datasets

a graphical method for assessing goodness of fit in cox s proportional hazards model abstract suggested here is a simple graphical method for studying the goodness of fit in cox s regression model for survival data the method is easy to use as it does not require the estimation of alternative models and only involves quantities similar to those already appearing in the partial likelihood expression that is needed in the parameter estimation the rationale behind the graphs is very intuitive they make a direct comparison between observed and expected failure frequencies as estimated from the model in a correctly specified model one anticipates an approximate balance between such frequencies otherwise there will typically be groups of individuals for which the expected frequencies are systematically too high or too low to match with the data and this shows in the graphs introduced here in the concrete applications of the method the individuals are stratified in a way that depends on what aspect of the model is being checked against data there is always one graph for each stratum s

on fitting cox s regression model with time dependent coefficients summary this paper examines goodness of fit testing in the cox regression model with timevarying regression coefficients arjas 1988 approach is used to define test statistics of the kolmogorov smirnov and cramer von mises types their asymptotic limits are shown to be well known functionals of standard brownian motion leading to the construction of formal goodness of fit tests some numerical studies are also included to illustrate the performance of the tests for moderate sample sizes

directed family of fuzzy sets and fuzzy perfect maps the present paper deals with the perfect functions in fuzzy setting fuzzy filter bases of fuzzy sets are also studied vis a vis such functions finally certain fuzzy topological properties are taken up for their study with regard to fuzzy perfect functions

on fuzzy t convergences abstract in this paper we introduce and study the notions of fuzzy convergence and weakly convergence on a fuzzy topological space these notions can be considered as generalizations of the convergence defined in georgiou and papadopoulos 1998 also the forementioned convergence does not coincide with the one defined in chen and chenn 1997 the importance of these convergences relies on the fact that they behave equally well as the convergence defined in georgiou and papadopoulos 1998

fuzzy strongly semiopen sets and fuzzy strong semicontinuity abstract in this paper we first introduce and study the fuzzy strongly semiopen and fuzzy strongly semiclosed sets in fuzzy space then we introduce the strong semi interior and strong semiclosure of a fuzzy set and kuratowski s 14 sets theorem of general topology is generalized finally we introduce the fuzzy strongly semicontinuous fuzzy strongly semiopen and fuzzy strongly semiclosed mappings on fuzzy topological spaces and also establish some of their characteristic properties and discuss relations between fuzzy continuous fuzzy semicontinuous and fuzzy strongly semicontinuous mappings

on fuzzy compactness in this paper we characterize well known fuzzy compactness see 3 4 9 and 10 using the notion of fuzzy upper limit see 5 also we give and study new fuzzy compactness in fuzzy topological spaces

the spectrum of large powers of the laplacian in bounded domains we present exact results for the spectrum of the nth power of the laplacian in a bounded domain we begin with the one dimensional case and show that the whole spectrum can be obtained in the limit of large n we also show that it is a useful numerical approach valid for any n finally we discuss implications of this work and present its possible extensions for non integer n and for 3d laplacian problems