valuations determined by quadratic transforms of a regular ring let r be a regular noetherian local ring and let v be the ring of a zero dimensional valuation v of the quotient field of r dominating r the aim of this paper is to characterize when v can be obtained from r by the sequence of successive quadratic transforms of r along v we also show some properties of such valuations in terms of the sequence

optimal experimental designs in the flow rate of particles this article focuses on analyzing the process of jam formation during the discharge by gravity of granular material stored in a two dimensional silo the aim of the article is two fold first optimal experimental designs are computed in which four approaches are considered d optimality a combination of d optimality and a cost gain function bayesian d optimality and sequential designing these results reveal that the efficiency of the design used by the experimenters can be improved dramatically a sensitivity analysis with respect to the most important parameter is also performed second estimation of the unknown parameters is done using least squares that is assuming normality and also via maximum likelihood assuming the exponential distribution simulations for the designs considered in this article show that the variance the mean squared error and the bias of the estimators using maximum likelihood are in most cases lower than those using least squares supplementary materials for this arti

parameter transformations for improved approximate confidence regions in nonlinear least squares in a previous paper it was shown that parameter effects nonlinearities of a nonlinear regression model experimental design parameterization combination can be quantified by means of a parameter effects curvature array a based on second derivatives of the model function in this paper the individual terms of a are interpreted and local compensation methods are suggested a method of computing the parameter effects array corresponding to a transformed set of parameters is given and we discuss how this result could be used to determine reparameterizations which have zero local parameter effects nonlinearity

applying experimental design and regression splines to high dimensional continuous state stochastic dynamic programming in stochastic dynamic programming sdp with continuous state and decision variables the future value function is computed at discrete points in the state spac e interpolation can be used to approximate the values of the future value function between these discrete points however for large dimensional problems the number of discrete points required to obtain a good approximation of the future value function can be prohibitively large statistical methods of experimental design and function estimation may be employed to overcome this curse of dimensionality in this paper we describe a method for estimating the future value function by multivariate adaptive regression splines mars fit over a discretization scheme based on orthogonal array oa experimental designs because orthogonal arrays only grow polynomially in the state space dimension our oa mars method is accurately able to solve higher dimensional sdp problems than previously possible to our knowledge the most efficient method published prior to this work employs tensor product cubic splines to approximate the future value function johnson et al 1993 the computational advantages of oa ma rs are demonstrated in comparisons with the method using tensor product cubic splines for applications of an inventory forecasting sdp with up to nine state variables computed on a small workstation in particular the storage of an adequate tensor product cubic spline for six dimensions exceeds the memory of our workstation and the run time for an accurate oa mars sdp solution would be at least an order of magnitude faster than using tensor product cubic splines for higher than six dimensions

curvature adjusted optimal design of sampling times for the inference of pharmacokinetic compartment models in pharmacokinetics compartment models are often used to describe the time course of blood concentration after the administration of a drug in this article we propose an optimal design criterion for precise estimation of parameters included in the compartment model and illustrate the non sequential design of sampling times of blood drug concentration data in individual pharmacokinetics the proposed optimal design criterion minimizes the determinant of the mean squared error matrix of the parameter estimator that is quadratically approximated by the curvature array therefore the proposed criterion considers the intrinsic and parameter effects nonlinearity underlying the compartment model and so is applicable in a pharmacokinetic experiment where the sample size of the blood drug concentration data is quite small

strong consistency and asymptotic efficiency for adaptive quantum estimation problems it is shown that for an adaptive quantum estimation scheme based on locally unbiased measurements the sequence of maximum likelihood estimators is strongly consistent and asymptotically efficient

identification and adaptive control of markov chains consider a countable state controlled markov chain whose transition probability is specified up to an unknown parameter alpha taking values in a compact metric space a to each alpha is associated a prespecified stationary control law zeta alpha the adaptive control law selects at each time t the control action zeta alpha t x t where x t is the state and alpha t is the maximum likelihood estimate of alpha the asymptotic behavior of this control scheme is investigated for the cases when the true parameter value alpha 0 does or does not belong to a and for the case when zeta is chosen to minimize an average cost criterion the analysis uses an appropriate extension of the notions of recurrence to nonstationary markov chains

the effect of incomplete knowledge of parameter values on single and multiple stage designs for logistic regression we consider the design of single and multiple stage dose response trials in which the probability of response is a logistic function of the dose knowledge of the parameters of the logistic at the time of planning the trial is represented by a gaussian prior distribution methods are presented for determining a design that approximately optimizes a measure of the accuracy of estimation averaged over the prior distribution changes in design due to uncertainty of parameter values are described as well as the changes in sample size required to produce a specified precision in multiple stage trials the initial stage is planned as if it were the only stage for succeeding stages the initial gaussian prior distribution is updated using outcomes at the previous stages at each such stage the design optimizes a chosen criterion averaged over the updated prior distribution the effectiveness of this methodology is evaluated by comparing the operating characteristics of two stage designs with those of single stage designs

a survey of some results in stochastic adaptive control some results in discrete time stochastic adaptive control are surveyed the survey divides itself into two parts bayesian and non bayesian adaptive control in the former area the problems of converting an incompletely observed system into a completely observed one multi armed bandit processes bayesian adaptive control of markov chains and bayesian adaptive control of linear systems are exposed and surveyed in the latter area non bayesian adaptive control of markov chains and the self tuning regulator are dealt with proofs are given where appropriate to illustrate the methods involved

optimal design for additive partially nonlinear models we develop optimal design theory for additive partially nonlinear regression models showing that bayesian and standardized maximin d optimal designs can be found as the products of the corresponding optimal designs in one dimension a sufficient condition under which analogous results hold for ds optimality is derived to accommodate situations in which only a subset of the model parameters is of interest to facilitate prediction of the response at unobserved locations we prove similar results for q optimality in the class of all product designs the usefulness of this approach is demonstrated through an application from the automotive industry where optimal designs for least squares regression splines are determined and compared with designs commonly used in practice

optimal designs for binary data abstract this article is concerned with the problem of selecting the values of the explanatory variable in logistic regression to obtain likelihood based confidence regions of minimum area one way of finding analytic solutions is by replacing the log likelihood by a quadratic surface around the maximum approximating in this way the likelihood regions by ellipses the optimal allocation for this local approximation is derived without unnecessary restrictions since the implementation of the optimal allocation requires accurate initial estimates a two stage procedure that uses the second stage to complement the first is recommended an alternative approach is to deal directly with the likelihood regions in this case the identification of the globally optimal allocation calls for numerical integration and optimization the results presented here attempt to strengthen those introduced by abdelbasit and plackett 1983 who derived optimal allocation for the local criterion under the restriction of symmet

a note on d optimal designs for partially nonlinear regression models hill 1980 states that a d optimal design for the estimation of a subset of the parameter vector in a partially nonlinear model depends on only those parameters that appear nonlinearly in the model in this note we demonstrate that hill s statement is incorrect and present a suffkient condition under which it is valid

inference and sequential design summary if an experiment is designed sequentially repeated sampling inference may not necessarily be made using distributional results that are valid for fixed designs a few simple illustrative examples and alternative approaches are given sometimes the sequential nature of design can be ignored asymptotically links are forged with inference for stochastic processes and missing data problems

confidence regions for parameter subsets in nonlinear regression on utilise la methode a deux niveaux proposee par hamilton watts et bates 1982 pour obtenir des approximations ameliorees des regions de confiance

nonlinear experiments optimal design and inference based on likelihood abstract nonlinear experiments involve response and regressors that are connected through a nonlinear regression type structure examples of nonlinear models include standard nonlinear regression logistic regression probit regression poisson regression gamma regression inverse gaussian regression and so on the fisher information associated with a nonlinear experiment is typically a complex nonlinear function of the unknown parameter of interest as a result we face an awkward situation designing an efficient experiment will require knowledge of the parameter but the purpose of the experiment is to generate data to yield parameter estimates our principal objective here is to investigate proper designing of nonlinear experiments that will let us construct efficient estimates of parameters we focus our attention on a very general nonlinear setup that includes many models commonly encountered in practice the experiments considered have two fundamental stages a static design in the initial stage

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

asymptotic properties of infinite directed unions of local quadratic transforms abstract let r m be a regular local ring of dimension at least 2 for each valuation domain birationally dominating r there is an associated sequence r n of local quadratic transforms of r we consider the case where this sequence r n n 0 is infinite and examine properties of the integrally closed local domain s n 0 r n in the case where s is not a valuation domain for this sequence there is an associated boundary valuation ring v n 0 i n v i where v i is the order valuation ring of r i there exists a unique minimal proper noetherian overring t of s t is the regular noetherian ufd obtained by localizing outside the maximal ideal of s and s v t in the present paper we define functions w and e where w is the asymptotic limit of the order valuations and e is the limit of the orders of transforms of principal ideals we describe v explicitly in terms of w and e and prove that v is either rank 1 or rank 2 we define an invariant associated to s that is either a positive real number or if is finite then s is archimedean and t is not local in this case the function w defines the rank 1 valuation overring w of v and w dominates s the rational dependence of over w t determines whether s is completely integrally closed and whether v has rank 1 we give examples where s is completely integrally closed if is infinite then s is non archimedean and t is local in this case the function e defines the rank 1 valuation overring e of v the valuation ring e is a dvr and e dominates t and in certain cases we prove that e is the order valuation ring of t

experimental design for binary data abstract models for binary data are usually such that the information matrix depends on the unknown parameters thus the standard criteria for optimality in regression experiments cannot be applied without modification methods of going around this difficulty include the use of initial point estimates sequential methods and bayesian analysis this article is mainly concerned with the robustness and the number of design points for methods involving initial estimates and for sequential methods in a small number of stages a final section discusses the criterion of constant information for models involving one or two parameters and summarizes recent results in this field

learning control of finite markov chains with unknown transition probabilities for a markovian decision problem in which the transition probabilities are unknown two learning algorithms are devised from the viewpoint of asymptotic optimality each time the algorithms select decisions to be used on the basis of not only the estimates of the unknown probabilities but also uncertainty of them it is shown that the algorithms are asymptotically optimal in the sense that the probability of selecting an optimal policy converges to unity

on goles universal machines a computational point of view this paper is a survey of e goles work on dynamical systems from the point of view of universality and computability the models considered by e goles and his collaborators are presented neural networks reaction diffusion automata chip firing games sand piles and artificial ants then we recall the corresponding universality results we also provide a classification of methods for proving universality and apply it to the aforementioned constructions

a quadratic design criterion for precise estimation in nonlinear regression models d optimal experimental designs for precise estimation in nonlinear regression models are obtained by minimizing the determinant of the approximate variance covariance matrix of the parameter estimates this determinant may not give a true indication of the volume of a joint inference region for the parameters however because of intrinsic and parameter effects nonlinearity in this article we investigate experimental designs that minimize a second order volume approximation unlike d optimal designs these designs depend on the noise and confidence levels and on the parameterization used and when used sequentially quadratic designs depend on the residuals from previous experiments and on the type of inference quadratic designs appear to be less sensitive to variations in initial parameter values used for design

the usefulness of optimum experimental designs optimum experimental designs were originally developed by kiefer mainly for response surface models this survey of recent developments emphasizes potential or actual usefulness for linear models the construction of exact designs particularly over irregular design regions is stressed as is the blocking of response surface designs other important areas include systematic designs that are robust against trend and designs for mixtures with irregular design regions several industrial examples are mentioned both d and c optimum designs are found for a non linear model of the economic response of cereal production to fertilizer level the c optimum design being for the conditions of maximum economic return locally optimum and bayesian designs are both described similar results for generalized linear models lead to designs for the ld 95 in a logistic model in which male and female subjects respond differently designs with structure in the variance suggest alternatives to the potentially wasteful product designs of taguchi designs for sequential clinical trials to include random balance are presented the last section outlines some applications including life testing and models in which time is a factor

a new family of optimal adaptive controllers for markov chains we consider the problem of adaptively controlling a markov chain with unknown transition probabilities a new family of adaptive controllers is exhibited which achieves a performance precisely equalling the optimal performance achievable if the transition probabilities i e the model or dynamics of the system were known instead hence the adaptive controllers presented here are truly optimal the performance of the system to be controlled is measured by the average of the costs incurred over an infinite operating time period these adaptive controllers can potentially be implemented on digital computers and used in the on line control of unknown systems

sequential experimental designs for generalized linear models we consider the problem of experimental design when the response is modeled by a generalized linear model glm and the experimental plan can be determined sequentially most previous research on this problem has been limited either to one factor binary response experiments or to augmenting the design when there are already sufficient data to compute parameter estimates we suggest a new procedure for the sequential choice of observations that offers five important advantages 1 it can be applied to multifactor experiments and is not limited to the one factor setting 2 it can be used with any glm not just binary responses 3 both fully sequential and group sequential settings are treated 4 it enables efficient design from the outset of the experiment and 5 the experimenter is not constrained to specify a single model and can use the prior to reflect uncertainty as to the link function and the form of the linear predictor our procedure is based on a d optimality criterion and on a bayesian

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

the derivative of x x and its uses the determinant of matrices of the form x x is used as the d optimal design criterion for parameter estimation in nonlinear regression and as the estimation criterion for multiresponse parameter estimation it is helpful to be able to calculate the gradient of this determinant so that sophisticated optimization methods can be used a method is given to efficiently compute the determinant using modern computational linear algebra and at the same time to provide the necessary information to compute the gradient the method is extended to cases in which a part of the x matrix is fixed examples from multiresponse parameter estimation and from experimental design are given

the computational complexity of sandpiles given an initial distribution of sand in an abelian sandpile what final state does it relax to after all possible avalanches have taken place in d 3 we show that this problem is p complete so that explicit simulation of the system is almost certainly necessary we also show that the problem of determining whether a sandpile state is recurrent is p complete in d 3 and briefly discuss the problem of constructing the identity in d 1 we give two algorithms for predicting the sandpile on a lattice of size n both faster than explicit simulation a serial one that runs in time mathcal o n log n and a parallel one that runs in time mathcal o log3n i e the class nc3 the latter is based on a more general problem we call additive ranked generability this leaves the two dimensional case as an interesting open problem

a new perspective on priors for generalized linear models abstract this article deals with specifications of informative prior distributions for generalized linear models our emphasis is on specifying distributions for selected points on the regression surface the prior distribution on regression coefficients is induced from this specification we believe that it is inherently easier to think about conditional means of observables given the regression variables than it is to think about model dependent regression coefficients previous use of conditional means priors seems to be restricted to logistic regression with one predictor variable and to normal theory regression we expand on the idea of conditional means priors and extend these to arbitrary generalized linear models we also consider data augmentation priors where the prior is of the same form as the likelihood we show that data augmentation priors are special cases of conditional means priors with current monte carlo methodology such as importance sampling and gibbs sampling our priors result in

accounting for intrinsic nonlinearity in nonlinear regression parameter inference regions joint confidence and likelihood regions for the parameters in nonlinear regression models can be defined using the geometric concepts of sample space and solution locus using a quadratic approximation to the solution locus instead of the usual linear approximation it is shown that these inference regions correspond to ellipsoids on the tangent plane at the least squares point accurate approximate inference regions can be obtained by mapping these ellipsoids into the parameter space and measures of the effect of intrinsic nonlinearity on inference can be based on the difference between the tangent plane ellipsoids and the sphere which would be obtained using a linear approximation

asymptotic inference from sequential design in a nonlinear situation summary it is shown that confidence regions for nonlinear parameters constructed by the repeated sampling principle are asymptotically valid for sequential designs in general linear models the related questions of consistency of parameter estimators and convergence of sequential design to an optimal design are answered positively an empirical finding of ford silvey 1980 is given a theoretical justification

sequential optimal design of neurophysiology experiments adaptively optimizing experiments has the potential to significantly reduce the number of trials needed to build parametric statistical models of neural systems however application of adaptive methods to neurophysiology has been limited by severe computational challenges since most neurons are high dimensional systems optimizing neurophysiology experiments requires computing high dimensional integrations and optimizations in real time here we present a fast algorithm for choosing the most informative stimulus by maximizing the mutual information between the data and the unknown parameters of a generalized linear model glm that we want to fit to the neuron s activity we rely on important log concavity and asymptotic normality properties of the posterior to facilitate the required computations our algorithm requires only low rank matrix manipulations and a two dimensional search to choose the optimal stimulus the average running time of these operations scales quadratically with the dimensionality of the glm making real time adaptive experimental design feasible even for high dimensional stimulus and parameter spaces for example we require roughly 10 milliseconds on a desktop computer to optimize a 100 dimensional stimulus despite using some approximations to make the algorithm efficient our algorithm asymptotically decreases the uncertainty about the model parameters at a rate equal to the maximum rate predicted by an asymptotic analysis simulation results show that picking stimuli by maximizing the mutual information can speed up convergence to the optimal values of the parameters by an order of magnitude compared to using random nonadaptive stimuli finally applying our design procedure to real neurophysiology experiments requires addressing the nonstationarities that we would expect to see in neural responses our algorithm can efficiently handle both fast adaptation due to spike history effects and slow nonsystematic drifts in a neuron s activity

multiobjective sampling design for parameter estimation and model discrimination in groundwater solute transport sampling design for site characterization studies of solute transport in porous media is formulated as a multiobjective problem optimal design of a sampling network is a sequential process in which the next phase of sampling is designed on the basis of all available physical knowledge of the system three objectives are considered model discrimination parameter estimation and cost minimization for the first two objectives physically based measures of the value of information obtained from a set of observations are specified in model discrimination value of information of an observation point is measured in terms of the difference in solute concentration predicted by hypothesized models of transport points of greatest difference in predictions can contribute the most information to the discriminatory power of a sampling design sensitivity of solute concentration to a change in a parameter contributes information on the relative variance of a parameter estimate inclusion of points in a sampling design with high sensitivities to parameters tends to reduce variance in parameter estimates cost minimization accounts for both the capital cost of well installation and the operating costs of collection and analysis of field samples sensitivities discrimination information and well installation and sampling costs are used to form coefficients in the multiobjective problem in which the decision variables are binary zero one each corresponding to the selection of an observation point in time and space the solution to the multiobjective problem is a noninferior set of designs to gain insight into effective design strategies a one dimensional solute transport problem is hypothesized then an approximation of the noninferior set is found by enumerating 120 designs and evaluating objective functions for each of the designs trade offs between pairs of objectives are demonstrated among the models the value of an objective function for a given design is shown to correspond to the ability of a design to actually meet an objective

designing experiments with respect to standardized optimality criteria we introduce a new class of standardized optimality criteria which depend on standardized covariances of the least squares estimators and provide an alternative to the commonly used criteria in design theory besides a nice statistical interpretation the new criteria satisfy an extremely useful invariance property which allows an easy calculation of optimal designs on many linearly transformed design spaces

crossing information in two dimensional sandpiles we prove that in a two dimensional sandpile automaton embedded in a regular infinite planar cellular space it is impossible to cross information if the bit of information is the presence or absence of an avalanche this proves that it is impossible to embed arbitrary logical circuits in a sandpile through quiescent configurations our result applies also for the non planar neighborhood of moore nevertheless we also show that it is possible to compute logical circuits with a two dimensional sandpile if a neighborhood of radius two is used in z2 crossing information becomes possible in that case and we conclude that for this neighborhood the sandpile is p complete and turing universal

a note on quadratic designs for nonlinear regression models summary hamilton watts 1985 state that their quadratic design criterion results in designs with only p points of support in this note we demonstrate how their design criterion may be used to obtain designs with p 1 design points thus allowing for a check of the adequacy of the assumed model

a d optimal design for estimation of parameters of an exponential linear growth curve of nanostructures we consider the problem of determining an optimal experimental design for estimation of parameters of a class of complex curves characterizing nanowire growth that is partially exponential and partially linear locally d optimal designs for some of the models belonging to this class are obtained by using a geometric approach further a bayesian sequential algorithm is proposed for obtaining d optimal designs for models with a closed form solution and for obtaining efficient designs in situations where theoretical results cannot be obtained the advantages of the proposed algorithm over traditional approaches adopted in recently reported nanoexperiments are demonstrated using monte carlo simulations the computer code implementing the sequential algorithm is available as supplementary materials

optimal adaptive controllers for unknown markov chains we consider the problem of adaptively controlling an unknown markov chain no prior information regarding the values of the transition probabilities is provided us except for a list of forbidden zero probability transitions which is usually obtained as a byproduct of the modeling process itself the goal is to design an adaptive controller to adequately control the unknown system when its performance is measured by the average cost incurred over a long operating time period our main result is the exhibition of a family of adaptive controllers which when applied to the unknown system will result in a performance precisely equal to the optimal performance attainable if the system i e the transition probabilities were known hence the adaptive controllers proposed here are truly optimal even when operating on an unknown system the results presented here extend similar results in 1 where we assume to be initially provided with a finite set of possible models one of which is guaranteed to be the true one this paper directly addresses those practical situations where a finite set of possible models with such a guarantee is hard to come by

an efficient design for model discrimination and parameter estimation in linear models summary we consider experimental designs in a regression set up where the unknown regression function belongs to a known family of nested linear models the objective of our design is to select the correct model from the family of nested models as well as to estimate efficiently the parameters associated with that model we show that our proposed design is able to choose the true model with probability tending to one as the number of trials grows to infinity we also establish that our selected design converges to the optimal design distribution for the true linear model ensuring asymptotic efficiency of least squares estimators of model parameters

ideal theory of infinite directed unions of local quadratic transforms abstract let r m be a regular local ring of dimension at least 2 associated to each valuation domain birationally dominating r there exists a unique sequence r n of local quadratic transforms of r along this valuation domain we consider the situation where the sequence r n n 0 is infinite and examine ideal theoretic properties of the integrally closed local domain s n 0 r n among the set of valuation overrings of r there exists a unique limit point v for the sequence of order valuation rings of the r n we prove the existence of a unique minimal proper noetherian overring t of s and establish the decomposition s t v if s is archimedean then the complete integral closure s of s has the form s w t where w is the rank 1 valuation overring of v

asymptotic properties of least squares estimates in stochastic regression models on etablit la consistance forte des estimations des moindres carres dans des modeles de regression stochastique sous l hypothese que le modele peut etre reparametrise de telle facon que les nouveaux vecteurs soient faiblement correles

directed unions of local quadratic transforms of a regular local ring abstract let r m be a d dimensional regular local domain with d 2 and let v be a valuation domain birationally dominating r such that the residue field of v is algebraic over r m let v be a valuation associated to v associated to r and v there exists an infinite directed family r n m n n 0 of d dimensional regular local rings dominated by v with r r 0 and r n 1 the local quadratic transform of r n along v let s n 0 r n abhyankar proves that s v if d 2 shannon observes that often s is properly contained in v if d 3 and granja gives necessary and sufficient conditions for s to be equal to v the directed family r n m n n 0 and the integral domain s n 0 r n may be defined without first prescribing a dominating valuation domain v if r n m n n 0 switches strongly infinitely often then s v is a rank one valuation domain and for nonzero elements f and g in m we have v f v g lim n ord r n f ord r n g if r n m n n 0 is a family of monomial local quadratic transforms we give necessary and sufficient conditions for r n m n n 0 to switch strongly infinitely often if these conditions hold then s v is a rank one valuation domain of rational rank d and v is a monomial valuation assume that v is rank one and birationally dominates s let s i 0 v m i granja martinez and rodriguez show that s implies s v we prove that s is finite if v has rational rank at least 2 in the case where v has maximal rational rank we give a sharp upper bound for s and show that s attains this bound if and only if the sequence switches strongly infinitely often

sequences of quadratic transforms of a noetherian local domain abstract let r be a noetherian local domain the aim of this paper is to characterize the sequences r i i 0 of successive quadratic transforms of r r 0 such that s i 0 r i is a valuation ring these characterizations involve ideals of s supported on the exceptional divisors along the sequence furthermore the characterizations are improved remarkably when the hilbert functions of the rings r i i 0 stabilize along the sequence we also give some examples to put our results in perspective