null controllability of some systems of two parabolic equations with one control force in this paper we establish some exact controllability results for systems of two parabolic equations first we prove the existence of insensitizing controls for the l 2 norm of the gradient of solutions of linear heat equations then in the worst situation where null controllability for a system of two parabolic equations can hold we prove this result for some general couplings

exponential decay toward equilibrium via entropy methods for reaction diffusion equations abstract in this work we show how the entropy method enables to get in an elementary way and without linearization estimates of exponential decay towards equilibrium for solutions of reaction diffusion equations corresponding to a reversible reaction explicit rates of convergence combining the dissipative effects of diffusion and reaction are given

null controllability with constraints on the state for the reaction diffusion system this paper is addressed to study the null controllability with constraints on the state for the reaction diffusion system we transform the problem into an equivalent null controllability problem with constraint on the control using a carleman inequality adapted to the constraint we prove the null controllability with constraints on the state for the reaction diffusion system holds

a proposal for a defuzzification strategy by the concept of sensitivity analysis abstract a systematic procedure is proposed for defuzzifying fuzzy subsets and interval values employing the concept of sensitivity analysis with a kind of minmax principle such that the defuzzification minimizes the sensitivity rating the discrepancy between a would be most appropriate defuzzified value and the unknown ideal value of defuzzification in the possible worst case a criterion of defuzzification can be built into the sensitivity to give a rationale to the defuzzified value as examples defuzzification with respect to a criterion of distance and minimization problems of two types of functions with fuzzy parameters are discussed

statistical inference procedures for bivariate archimedean copulas abstract a bivariate distribution function h x y with marginals f x and g y is said to be generated by an archimedean copula if it can be expressed in the form h x y 1 f x g y for some convex decreasing function defined on 0 1 in such a way that 1 0 many well known systems of bivariate distributions belong to this class including those of gumbel ali mikhail haq thelot clayton frank and hougaard frailty models also fall under that general prescription this article examines the problem of selecting an archimedean copula providing a suitable representation of the dependence structure between two variates x and y in the light of a random sample x 1 y 1 x n y n the key to the estimation procedure is a one dimensional empirical distribution function that can be constructed whether the uniform representation of x and y is archimedean or not and independently of their marginals this semiparametric estimator based on a decomposition of kendall s tau statistic

optimal l p l q estimates for parabolic boundary value problems with inhomogeneous data in this paper we investigate vector valued parabolic initial boundary value problems mathcal a t x d mathcal b j t x d subject to general boundary conditions in domains g in mathbb r n with compact c 2m boundary the top order coefficients of mathcal a are assumed to be continuous we characterize optimal l p l q regularity for the solution of such problems in terms of the data we also prove that the normal ellipticity condition on mathcal a and the lopatinskii shapiro condition on mathcal a mathcal b 1 dots mathcal b m are necessary for these l p l q estimates as a byproduct of the techniques being introduced we obtain new trace and extension results for sobolev spaces of mixed order and a characterization of triebel lizorkin spaces by boundary data

boundary controllability of a cascade system coupling fourth and second order parabolic equations abstract a control system coupling fourth and second order parabolic equations is considered in this paper the main topic is the study of the control properties of this system when we only control the second order partial differential equation through a boundary condition depending on the choice of the diffusion coefficients we obtain positive and negative results for approximate and null controllability in particular we prove that for any given positive time t 0 we can find some diffusion coefficients such that the system is null controllable in time t if t t 0 and is not null controllable if t t 0

bivariate latent variable models for clustered discrete and continuous outcomes abstract we use the concept of a latent variable to derive the joint distribution of a continuous and a discrete outcome and then extend the model to allow for clustered data the model can be parameterized in a way that allows one to write the joint distribution as a product of a standard random effects model for the continuous variable and a correlated probit model for the discrete variable this factorization suggests a convenient approach to parameter estimation using quasi likelihood techniques our approach is motivated by the analysis of developmental toxicity experiments for which a number of discrete and continuous outcomes are measured on offspring clustered within litters fetal weight and malformation data illustrate the results

forecasting enrollments with fuzzy time series part ii abstract there have been a good many methods to forecast university enrollments in the literature however none of them could be applied when the historical data are linguistic values fuzzy time series is an effective tool to deal with such problems in this paper as an application of fuzzy time series in educational research the forecast of the enrollments of the university of alabama is carried out in so doing a fuzzy time series model is developed using historical data a complete procedure is proposed which includes fuzzifying the historical data developing a fuzzy time series model and calculating and interpreting the outputs to evaluate the forecasting model the robustness of the fuzzy time series model is tested advantages and problems of the forecasting method are also discussed

response models for mixed binary and quantitative variables summary a number of special representations are considered for the joint distribution of qualitative mostly binary and quantitative variables in addition to the conditional gaussian models and to conditional gaussian regression chain models some emphasis is placed on models derived from an underlying multivariate normal distribution and on models in which discrete probabilities are specified linearly in terms of unknown parameters the possibilities for choosing between the models empirically are examined as well as the testing of independence and conditional independence and the estimation of parameters often the testing of independence is exactly or nearly the same for a number of different models

minimal time for the null controllability of parabolic systems the effect of the condensation index of complex sequences let a d a be a diagonalizable generator of a c0 semigroup of contractions on a complex hilbert space x b l c y y being some suitable extrapolation space of x and u l2 0 t c under some assumptions on the sequence of eigenvalues k k 1 c of a d a we prove the existence of a minimal time t0 0 depending on bernstein s condensation index of and on b such that y ay bu is null controllable at any time t t0 and not null controllable for t t0 as a consequence we solve controllability problems of various systems of coupled parabolic equations in particular new results on the boundary controllability of one dimensional parabolic systems are derived these seem to be difficult to achieve using other classical tools

controllability of shadow reaction diffusion systems abstract we study the null controllability of linear shadow models for reaction diffusion systems arising as singular limits when the diffusivity of some of the components is very high this leads to a coupled system where one component solves a parabolic partial differential equation pde and the other one an ordinary differential equation ode we analyze these shadow systems from a controllability perspective and prove two types of results first by employing carleman inequalities and ode arguments we prove that the null controllability of the shadow model holds this result together with the effectiveness of the controls to control the original dynamics is illustrated by numerical simulations we also obtain a uniform carleman estimate for the reaction diffusion equations which allows to obtain the null control for the shadow system as a limit when the diffusivity tends to infinity in one of the equations

on the controllability of the lotka mckendrick model of population dynamics abstract in this work we consider the problem of exact controllability of the linear lotka mckendrick model of population dynamics we establish an observability inequality and show that any initial condition can be steered into any quasi steady state by a distributed control except for a small interval of ages near zero in contrast with that we observe that the boundary control is more efficient in the sense that it controls the entire population the biological implication of our conclusions is that a closed population is more efficiently controlled through birth control as opposed to migration and eradication in particular this supports the trap alter release method for controlling a population of stray cats as opposed to the trap and kill method

defuzzification filters and applications to power system stabilization problems defuzzification is a very important step in fuzzy systems applications there are a number of different defuzzification methods reported in the literature in this paper the concept of defuzzification filters in a control system setting is first discussed and a methodology for designing such filters considered as will be seen the design of such filters requires the knowledge of the plant model and its inverse a reference control signal is computed and then is used to generate the actual defuzzified control signal which will be applied to control the plant the application of the defuzzification filter is made by introducing the filter into a power system in which a neuro fuzzy self learning controller was applied to stabilize the system but success could not always be guaranteed with the defuzzification filter however the system is always stabilized simulation results are presented

some properties of defuzzification neural networks abstract fuzzy systems have gained more and more attention from researchers and practitioners of various fields in such systems the output represented by a fuzzy set sometimes needs to be transformed into a scalar value and this task is known as the defuzzification process several analytic methods have been proposed for this problem but lately the neural network approach has been used for this purpose when employed as defuzzifiers a neural network is called a defuzzification neural network in this paper some preliminary results on properties of such defuzzification networks will be reported

handling forecasting problems using fuzzy time series abstract in 6 9 song et al proposed fuzzy time series models to deal with forecasting problems in 10 sullivan and woodall reviewed the first order time invariant fuzzy time series model and the first order time variant model proposed by song and chissom 6 8 where the models are compared with each other and with a time invariant markov model using linguistic labels with probability distributions in this paper we propose a new method to forecast university enrollments where the historical enrollments of the university of alabama shown in 7 8 are used to illustrate the forecasting process the average forecasting errors and the time complexity of these methods are compared the proposed method is more efficient than the ones presented in 7 8 10 due to the fact that the proposed method simplifies the arithmetic operation process furthermore the average forecasting error of the proposed method is smaller than the ones presented in 2 7 8

a simple method of forecasting based on fuzzy time series in fuzzy time series forecasting various methods have been developed to establish the fuzzy relations on time series data having linguistic values for forecasting the future values however the major problem in fuzzy time series forecasting is the accuracy in the forecasted values the present paper proposes a new method of fuzzy time series forecasting based on difference parameters the proposed method is a simplified computational approach for the forecasting the method has been implemented on the historical enrollment data of university of alabama adapted by song and chissom and the forecasted values have been compared with the results of the existing methods to show is superiority further the proposed method has also been implemented on a real life problem of crop production forecast of wheat crop and the results have been compared with other methods

latent variable models for mixed discrete and continuous outcomes we propose a latent variable model for mixed discrete and continuous outcomes the model accommodates any mixture of outcomes from an exponential family and allows for arbitrary covariate effects as well as direct modelling of covariates on the latent variable an em algorithm is proposed for parameter estimation and estimates of the latent variables are produced as a by product of the analysis a generalized likelihood ratio test can be used to test the significance of covariates affecting the latent outcomes this method is applied to birth defects data where the outcomes of interest are continuous measures of size and binary indicators of minor physical anomalies infants who were exposed in utero to anticonvulsant medications are compared with controls

a robust method of forecasting based on fuzzy time series present study proposes an improved and versatile method of forecasting based on the concept fuzzy time series forecasting the developed model has been presented in a form of simple computational algorithms it utilizes various difference parameters being implemented on current state for forecasting the next state values to accommodate the possible vagueness in the data in a better way and making it a robust method the developed model has been implemented on the historical student enrollments data of university of alabama adapted by song and chissom and the obtained forecasted values have been compared with the existing methods to show its superiority the robustness of the model has also been tested in comparison the suitability of the developed model has also been examined in the crop production forecasting by implementing it on historical time series data of rice production of pantnagar farm india

correlations and copulas for decision and risk analysis the construction of a probabilistic model is a key step in most decision and risk analyses typically this is done by defining a joint distribution in terms of marginal and conditional distributions for the model s random variables we describe an alternative approach that uses a copula to construct joint distributions and pairwise correlations to incorporate dependence among the variables the approach is designed specifically to permit the use of an expert s subjective judgments of marginal distributions and correlations the copula that underlies the multivariate normal distribution provides the basis for modeling dependence but arbitrary marginals are allowed we discuss how correlations can be assessed using techniques that are familiar to decision analysts and we report the results of an empirical study of the accuracy of the assessment methods the approach is demonstrated in the context of a simple example including a study of the sensitivity of the results to the assessed correlations

global carleman inequalities for parabolic systems and applications to controllability this paper has been conceived as an overview on the controllability properties of some relevant linear and nonlinear parabolic systems specifically we deal with the null controllability and the exact controllability to the trajectories we try to explain the role played by the observability inequalities in this context and the need of global carleman estimates we also recall the main ideas used to overcome the difficulties motivated by nonlinearities first we considered the classical heat equation with dirichlet conditions and distributed controls then we analyze recent extensions to other linear and semilinear parabolic systems and or boundary controls finally we review the controllability properties for the stokes and navier stokes equations that are known to date in this context we have paid special attention to obtaining the necessary carleman estimates some open questions are mentioned throughout the paper we hope that this unified presentation will be useful for those researchers interested in the field

control of three heat equations coupled with two cubic nonlinearities we study the null controllability of three parabolic equations the control is acting only on one of the three equations the three equations are coupled by means of two cubic nonlinearities the linearized control system around 0 is not null controllable however using the cubic nonlinearities we prove the global null controllability of the control system the proof relies on the return method an algebraic solvability and smoothing properties of the parabolic equations

exact and approximate controllability of the age and space population dynamics structured model we prove exact and approximate controllability for a linear age dependent and spatially structured population dynamics problem the birth process is nonlocal and the demographic functions depends on age time and space our method relies on the derivation of a carleman inequality for the backward nonlocal adjoint system

interior degenerate singular parabolic equations in nondivergence form well posedness and carleman estimates abstract we consider non smooth general degenerate singular parabolic equations in non divergence form with degeneracy and singularity occurring in the interior of the spatial domain in presence of dirichlet or neumann boundary conditions in particular we consider well posedness of the problem and then we prove carleman estimates for the associated adjoint problem

fast and accurate center of gravity defuzzification of fuzzy system outputs defined on trapezoidal fuzzy partitions in this article three methods are presented to perform the center of gravity cog defuzzification method in the context of linguistic fuzzy models with t norm based inference one well known method the discretisation method and two new methods the slope based method and the modified transformation function method the methods are worked out for trapezoidal membership functions forming a fuzzy partition in the sense of ruspini experimental results show that the newly introduced methods exhibit excellent accuracy at an extremely low computational cost compared to the widely applied discretisation method

internal stabilizability for a reaction diffusion problem modeling a predator prey system abstract in this work we consider a 2 2 system of semilinear partial differential equations of parabolic type describing interactions between a prey population and a predator population featuring a holling type ii functional response to predation we address the question of stabilizing the predator population to zero upon using a suitable internal control supported on a small subdomain of the whole spatial domain and acting on predators we give necessary and sufficient conditions for this stabilizability result to hold

a copula based model for multivariate non normal longitudinal data analysis of a dose titration safety study on a new antidepressant a new model for multivariate non normal longitudinal data is proposed in a first step each longitudinal series of data corresponding to a given response is modelled separately using a copula to relate the marginal distributions of the response at each time of observation in a second step at each observation time the conditional on the past distributions of each response are related using another copula describing the relationship between the corresponding variables note that there is no need to consider the same family of distributions for these response variables the technique is illustrated in a dose titration safety study on a new antidepressant the haemodynamic effect on diastolic blood pressure systolic blood pressure and heart rate is studied these three responses are measured repeatedly over time on ten healthy volunteers during the dose escalation the available covariates are sex and the concentration of drug in the plasma at time of measurement

local null controllability of the three dimensional navier stokes system with a distributed control having two vanishing components in this paper we prove a local null controllability result for the three dimensional navier stokes equations on a smooth bounded domain of mathbb r 3 with null dirichlet boundary conditions the control is distributed in an arbitrarily small nonempty open subset and has two vanishing components lions and zuazua proved that the linearized system is not necessarily null controllable even if the control is distributed on the entire domain hence the standard linearization method fails we use the return method together with a new algebraic method inspired by the works of gromov and previous results by gueye

regression models for a bivariate discrete and continuous outcome with clustering abstract developmental toxicity studies of laboratory animals play a crucial role in the testing and regulation of chemicals and pharmaceutical compounds exposure to developmental toxicants typically causes a variety of adverse effects such as fetal malformations and reduced fetal weight at term in this article we discuss regression methods for jointly analyzing bivariate discrete and continuous outcomes that are motivated by the statistical problems that arise in analyzing data from developmental toxicity studies we focus on marginal regression models that is models in which the marginal expectation of the bivariate response vector is related to a set of covariates by some known link functions in these models the regression parameters for the marginal expectation are of primary scientific interest whereas the association between the binary and continuous response is considered to be a nuisance characteristic of the data we describe a likelihood based approach based on the general location mode

a degenerate population system carleman estimates and controllability abstract in this paper we consider a system in non divergence form which models the interaction between two different species u and v both of them depend on time on age and on space moreover the diffusion coefficients degenerate at the boundary of domain we study in particular null controllability of the system via observability inequalities and carleman estimates for the associated non homogeneous adjoint problem

null controllability of the stabilized kuramoto sivashinsky system with one distributed control this paper presents a control problem for a one dimensional nonlinear parabolic system which consists of a kuramoto sivashinsky korteweg de vries equation coupled to a heat equation we address the problem of controllability by means of a control supported in an interior open subset of the domain and acting on one equation only the local null controllability of the system is proved the proof is based on a carleman estimate for the linearized system around the origin a local inversion theorem is applied to get the result for the nonlinear system

maximum likelihood estimation for mixed continuous and categorical data with missing values summary maximum likelihood procedures for analysing mixed continuous and categorical data with missing values are presented the general location model of olkin tate 1961 and extensions introduced by krzanowski 1980 1982 form the basis for our methods maximum likelihood estimation with incomplete data is achieved by an application of the em algorithm dempster laird rubin 1977 special cases of the algorithm include orchard woodbury s 1972 algorithm for incomplete normal samples fuchs s 1982 algorithms for log linear modelling of partially classified contingency tables and day s 1969 algorithm for multivariate normal mixtures applications include a imputation of missing values b logistic regression and discriminant analysis with missing predictors and unclassified observations c linear regression with missing continuous and categorical predictors and d parametric cluster analysis with incomplete data methods are illustrated using data from the st louis risk research project some key word8 cluster analysis discriminant analysis em algorithm incomplete data linear regression logistic regression log linear model mixture model

assessment and propagation of model uncertainty in most examples of inference and prediction the expression of uncertainty about unknown quantities y on the basis of known quantities x is based on a model m that formalizes assumptions about how x and y are related m will typically have two parts structural assumptions s such as the form of the link function and the choice of error distribution in a generalized linear model and parameters whose meaning is specific to a given choice of s it is common in statistical theory and practice to acknowledge parametric uncertainty about given a particular assumed structure s it is less common to acknowledge structural uncertainty about s itself a widely used approach involves enlisting the aid of x to specify a plausible single best choice s for s and then proceeding as if s were known to be correct in general this approach fails to assess and propagate structural uncertainty fully and may lead to miscalibrated uncertainty assessments about y given x when miscalibration occurs it will often result in understatement of inferential or predictive uncertainty about y leading to inaccurate scientific summaries and overconfident decisions that do not incorporate sufficient hedging against uncertainty in this paper i discuss a bayesian approach to solving this problem that has long been available in principle but is only now becoming routinely feasible by virtue of recent computational advances and examine its implementation in examples that involve forecasting the price of oil and estimating the chance of catastrophic failure of the us space shuttle

adaptive learning defuzzification techniques and applications abstract in this paper adaptive learning defuzzification techniques are studied under the consideration of system performance indices by treating defuzzification processes as continuous mappings from space 0 1 n to the real line the concept of the optimal defuzzification mapping can be developed since all the continuous defuzzification mappings considered in this paper form a banach space approximation to the optimal mapping with some known functions can be expressed as the parametric optimization problem to find the optimal parameters adaptive learning of the optimal defuzzification mapping is investigated learning laws for the parameters in the defuzzification mapping are derived in one case numerical results indicate that the adaptive learning defuzzification method can give superior defuzzification results to some popular defuzzification methods

null controllability of a parabolic system with a cubic coupling term we consider a system of two parabolic equations with a forcing term present in one equation and a cubic coupling term in the other one we prove that the system is locally null controllable

likelihood inference in a correlated probit regression model correlated binary observations arise in a variety of applications for example in animal studies the term litter effect is used to describe the greater alikeness of responses within a litter as compared to that between litters at a given set of experimental conditions in a setting that motivated this work cells on individual atomic bomb survivors were scored as to the presence or absence of chromosomal aberrations the number of aberrant cells varied among subjects at a specified radiation exposure estimate age at exposure city of exposure and sex in a manner that substantially exceeds that consistent with a binomial error structure otake prentice 1984 this extra binomial variation or overdispersion reflects the fact that the binary response of cells from a given survivor tend to be more alike than are the responses of cells from distinct survivors having the same estimated radiation exposure level and other covariate values such overdispersion may result from individual differences in susceptibility to radiation damage from omitted covariates or most plausibly in this setting from substantial random errors in the estimated radiation exposure levels failure to acknowledge overdispersion may lead to serious underestimation of the standard errors associated with regression parameter estimates and to unduly precise inferences

controllability of a 4 4 quadratic reaction diffusion system we consider a 4 4 nonlinear reaction diffusion system posed on a smooth domain of r n n 1 with controls localized in some arbitrary nonempty open subset of the domain this system is a model for the evolution of concentrations in reversible chemical reactions we prove the local exact controllability to stationary constant solutions of the underlying reaction diffusion system for every n 1 in any time t 0 a specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics the proof is based on a linearization which uses return method and an adequate change of variables that creates crossed diffusion which will be used as coupling terms of second order the controllability properties of the linearized system are deduced from carleman estimates a kakutani s fixed point argument enables to go back to the nonlinear parabolic system then we prove a global controllability result in large time for 1 n 2 thanks to our local controllabillity result together with a known theorem on the asymptotics of the free nonlinear reaction diffusion system

ellipsoidally symmetric extensions of the general location model for mixed categorical and continuous data summary the general location model olkin tate 1961 krzanowski 1980 1982 little schluchter 1985 has categorical variables marginally distributed as a multinomial and continuous variables conditionally normally distributed with different means across cells defined by the categorical variables but a common covariance matrix across cells two extensions of the general location model are obtained the first replaces the common covariance matrix with different but proportional covariance matrices where the proportionality constants are to be estimated the second replaces the multivariate normal distributions of the first extension with multivariate t distributions where the degrees of freedom can also vary across cells and are to be estimated the t distribution is just one example of more general ellipsoidally symmetric distributions that can be used in place of the normal these extensions can provide more accurate fits to real data and can be viewed as tools for robust inference moreover the models can be very useful for multiple imputation of ignorable missing values maximum likelihood estimation using the aecm algorithm meng van dyk 1997 is presented as is a monotone data gibbs sampling scheme for drawing parameters and missing values from their posterior distributions to illustrate the techniques a numerical example is presented

on the issue of defuzzification and selection based on a fuzzy set abstract we are concerned with the problem of selecting a crisp element based on information provided by a fuzzy set a problem which manifests itself in the defuzzification step in fuzzy logic controllers we provide a unifying approach to this selection process among other characteristics this unification puts the defuzzification methods of mean of maxima and center of gravity in the same framework we show that this selection can be viewed as a three step operation transformation of the decision fuzzy set normalization to probability distribution selection based on the probability distribution a number of different procedures for selection are discussed

slide a simple adaptive defuzzification method we introduce a parameterized family of defuznfica tion operators called the semi linear defuzzification slide method this method is based upon a simple transformation of the fuzzy output set of the controller we suggest an algorithm for the learning of the parameter from a data set in an attempt to simplify the parameter learning we suggest a modified version of the slide method which results in a simple learning algorithm the development of the learning algorithm is based upon the use of the kalman filter n fuzzy logic control systems l 2 the defuzzification i step involves the selection of one value as the output of the controller more specifically starting with a fuzzy subset possibility distribution f over the output space x of the controller the defuzzification step uses this fuzzy subset to select a representative element x the two most often used methods of defuzzification found in the literature are the center of area coa and mean of maxima mom 3 6 methods we recall that the mom method takes as its defuzzified value the mean of the elements that attain the maximum membership grade in f the coa method takes as its defuzzified value d ci ui xi where ui f xi cj f xj in 7 and 8 we formulated a general defuzzification method via basic defuzzification distribution badd transformations the main idea of this badd approach is to transform the possibility distribution f into a probability distribution p and then the defuzzified value is the expected value of the probability distribution the process of transforming f into p was seen as a two step process the first step was to transform f into a new possibility distribution e and then to normalize e to obtain p the step of obtaining e from f involved the use of a badd transformation in which e xi f z where 6 e o co it was shown in 7 that the generalized badd method implies the coa and mom methods as special cases in particular the coa method is obtained for a 1 and the mom is obtained for 6 00 when s 0 we get a defuzzified value that is the unweighted mean of the output space d xi xi the badd transformation has the advantage of being based on a single parameter which allows us the potential of adaptively learning the best method of defuzzification for a given controller while the family of all defuzzified values that can be obtained by the generalized badd defuzzification method was parameterized an undesirable property of the method was the nonlinear

null controllability of some reaction diffusion systems with one control force this work is concerned with the null controllability of semilinear parabolic systems by a single control force acting on a subdomain

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

asymptotic behavior of solutions to chemical reaction diffusion systems abstract this paper concerns the study of the asymptotic behavior of solutions to reaction diffusion systems modeling multi components reversible chemistry with spatial diffusion by solution we understand any limit of adequate approximate solutions it is proved in any space dimension that as time tends to infinity the solution converges exponentially to the unique homogeneous stationary solution we adapt and extend to any number of components the entropy decay estimates which have been exploited for some particular 3 3 and 4 4 systems