applications of fixed point methods to discrete variational and quasi variational inequalities in this paper discrete analogues of variational inequalities v i and quasi variational inequalities q v i encountered in stochastic control and mathematical physics are discussed it is shown that those discrete v i s and q v i s can be written in the fixed point formx tx such that eithert or some power oft is a contraction this leads to globally convergent iterative methods for the solution of discrete v i s and q v i s which are very suitable for implementation on parallel computers with single instruction multiple data architecture particularly on massively parallel processors m p p s

time dependent roc curves for censored survival data and a diagnostic marker roc curves are a popular method for displaying sensitivity and specificity of a continuous diagnostic marker x for a binary disease variable d however many disease outcomes are time dependent d t and roc curves that vary as a function of time may be more appropriate a common example of a time dependent variable is vital status where d t 1 if a patient has died prior to time t and zero otherwise we propose summarizing the discrimination potential of a marker x measured at baseline t 0 by calculating roc curves for cumulative disease or death incidence by time t which we denote as roc t a typical complexity with survival data is that observations may be censored two roc curve estimators are proposed that can accommodate censored data a simple estimator is based on using the kaplan meier estimator for each possible subset x c however this estimator does not guarantee the necessary condition that sensitivity and specificity are monotone in x an alternative estimator that does guarantee monotonicity is based on a nearest neighbor estimator for the bivariate distribution function of x t where t represents survival time akritas m j 1994 annals of statistics 22 1299 1327 we present an example where roc t is used to compare a standard and a modified flow cytometry measurement for predicting survival after detection of breast cancer and an example where the roc t curve displays the impact of modifying eligibility criteria for sample size and power in hiv prevention trials

comparing the areas under two or more correlated receiver operating characteristic curves a nonparametric approach methods of evaluating and comparing the performance of diagnostic tests are of increasing importance as new tests are developed and marketed when a test is based on an observed variable that lies on a continuous or graded scale an assessment of the overall value of the test can be made through the use of a receiver operating characteristic roc curve the curve is constructed by varying the cutpoint used to determine which values of the observed variable will be considered abnormal and then plotting the resulting sensitivities against the corresponding false positive rates when two or more empirical curves are constructed based on tests performed on the same individuals statistical analysis on differences between curves must take into account the correlated nature of the data this paper presents a nonparametric approach to the analysis of areas under correlated roc curves by using the theory on generalized u statistics to generate an estimated covariance matrix

dynamic programming and maximum principle for discrete goursat systems abstract we obtain the dynamic programming equations for discrete goursat systems we prove that they are necessary and sufficient for optimality and we prove formulas for the exact or approximate solution of these equations using dynamic programming arguments we obtain semi dynamic programming equations and then we prove a maximum principle of pontryagin s type

dynamic programming approach to the optimal control of systems governed by goursat darboux equations the method of dynamic programming is extended to a class of two dimensional time problems the goursat darboux systems hyperbolic systems with initial boundary data on characteristics a formal derivation of a maximum principle of pontryagin s type is obtained by using the semi dynamic programming equations controlled goursat darboux systems have applications to the probiems of gas absorption and drying processes

triple i method of fuzzy reasoning abstract the theory of the triple i method with total inference rules of fuzzy reasoning is investigated by using zadeh s implication operator rz the computational formulae for both fuzzy modus ponens fmp and fuzzy modus tollens fmt are obtained the reversibility properties for fmp and fmt are analyzed and the reversibility criteria are given we also investigated the generalized problem of the triple i method and obtained the formulae for the triple i fmp and the triple i fmt

on solvability of functional equations and system of functional equations arising in dynamic programming the purpose of this paper is to study solvability of two classes of functional equations and a class of system of functional equations arising in dynamic programming of multistage decision processes by using fixed point theorems a few existence and uniqueness theorems of solutions and iterative approximation for solving these classes of functional equations are established under certain conditions some existence theorems of coincidence solutions for the class of system of functional equations are shown some examples are given to demonstrate the advantage of our results than existing ones in the literature

topological properties of prime filters in mtl algebras and fuzzy set representations for mtl algebras in the present paper some basic properties of prime filters in mtl algebras are studied by introducing some topological structures on the set of all prime filters and the set of all maximal filters respectively and by investigating the topological properties of them we conclude that the set of all prime filters is a compact t 0 topological space and the set of all maximal filters is a compact hausdorff topological space by studying the relation between valuations and filters in mtl algebras we prove that a mtl algebra is homomorphic to a subalgebra of a mtl cube over the set of all prime filters and illustrate that the stone representation theorem of boolean algebras is only a special case of fuzzy sets representation theorem of mtl algebras thus the well known stone representation theorem of boolean algebras is extended to mtl algebras

on the logic foundation of fuzzy reasoning abstract fuzzy reasoning and fuzzy formal deduction theory are two closely related subjects and each of them has been investigated by many researchers unfortunately it seems that the two branches have not been successfully connected the aim of this paper is to establish a so called quasi propositional deductive system syntactically as well as semantically and then set a logic foundation for developing the theory of fuzzy reasoning therefrom

optimal control of goursat darboux systems with discontinuous co state abstract we study an optimal control problem for goursat darboux systems over domains that are more general than rectangles with costs that include integral terms supported in the two dimensional domain as well as terms with singular one dimensional and zero dimensional support we prove that the co state satisfies a system of hamiltonian equations and a set of side conditions that involves jump discontinuities across a network of characteristic curves we present a method of solution of the hamiltonian equations and in the process of doing so we establish the unique solvability of these equations

solutions to two functional equations arising in dynamic programming this paper deals with the existence uniqueness and iterative approximation of solutions for two functional equations arising in dynamic programming of multistage decision processes the results presented in this paper extend improve and unify the results due to bellman bhakta and choudhury bhakta and mitra liu liu and ume and others

three and two valued łukasiewicz theories in the formal deductive system l nm logic in order to offer a logic foundation for fuzzy reasoning and to reduce the gap between fuzzy reasoning and artificial intelligence a new formal deductive system l was introduced in 1996 it was proved that the prominent nilpotent minimum logic is a logic system equivalent to l as is well known in classical propositional logic there is a one to one correspondence between deductively closed theories and topologically closed subsets of the stone space of all maximally consistent theories recently by means of the strong completeness of l we obtained the structural and topological characterizations of maximally consistent theories in this logic it was proved that the set of all maximally consistent theories over l with an induced topology is a cantor space in order to establish a similar connection between theories over l and closed subsets of this topological space to the classical case we introduce in this paper the notion of three valued lukasiewicz theory and the notion of two valued lukasiewicz boolean theory we prove that there is also a one to one correspondence between deductively closed three valued lukasiewicz theories and closed subsets of the space and boolean theories correspond to the closed subsets of the subspace of all maximally consistent boolean theories lastly we give an alternative proof for the structural characterization of the maximality of consistent theories over l by induction on the complexity of formulas which is independent of the strong completeness of l and also independent of the perfectness of the standard r 0 algebra

optimal control of volterra integral equations in two independent variables we prove an extremality condition for optimal control problems for systems of integral equations of volterra type with two independent variables these systems generalize both the hyperbolic control problems in two independent variables and the optimal control of ordinary i e with one independent variable volterra integral equations

simplification and independence of axioms of fuzzy logic systems imtl and nm monoidal t norm based logic mtl is the common formalization of all fuzzy logic of left continuous t norms and their residua this system has two important schematic extensions involutive monoidal t norm based logic imtl and nilpotent minimum logic nm in this paper we discuss simplification and independence of axioms of imtl and nm the shorter axioms of both logics and an independent axioms of the system imtl are given

characterizations of maximal consistent theories in the formal deductive system l nm logic and cantor space a maximal consistent theory is a maximal theory with respect to its consistency the present paper is divided into two parts the first one is devoted to characterize the maximality of a consistent theory in the formal deductive system l which is a logic system equivalent to the nilpotent minimum logic it is proved that each maximal consistent theory in this logic must be the deductive closure of a collection of simple compound formulas hence it follows that there is a one to one correspondence between the set of all maximal consistent theories and the set of evaluations e assigning to each propositional variable p its truth degree e p 0 12 1 the satisfiability theorem and compactness theorem of l are obtained the second part is to investigate the topological structure of the set of all maximal consistent theories over l and the results show that this topological space is a cantor space

the optimal control problem in the processes described by the goursat problem for a hyperbolic equation in variable exponent sobolev spaces with dominating mixed derivatives in this paper a necessary and sufficient condition such as the pontryagin s maximum principle for an optimal control problem with distributed parameters is given by a hyperbolic equation of the second order with l p x coefficients the results can be used in the theory of optimal processes for distribution pontryagin maximum principle for various controlled processes described by hyperbolic equations of second order with discontinuous coefficients in variable exponent sobolev spaces with dominant mixed derivatives

nearest neighbor estimation of a bivariate distribution under random censoring we consider the problem of estimating the bivariate distribution of the random vector x y when y may be subject to random censoring the censoring variable c is allowed to depend on x but it is assumed that y and c are conditionally independent given x x the estimate of the bivariate distribution is obtained by averaging estimates of the conditional distribution of y given x x over a range of values of x the weak convergence of the centered estimator multiplied by n 1 2 is obtained and a closed form expression for the covariance function of the limiting process is given it is shown that the proposed estimator is optimal in the beran sense this is similar to an optimality property the product limit estimator enjoys using the proposed estimator of the bivariate distribution an extension of the least squares estimator to censored data polynomial regression is obtained and its asymptotic normality established

formalized theory of general fuzzy reasoning first a rigorous mathematical theory for calculating the conclusion of general fuzzy reasoning is proposed second a formalized non fuzzy version of the theory of general fuzzy reasoning is founded in classical logics

a regression modelling framework for receiver operating characteristic curves in medical diagnostic testing summary receiver operating characteristic curves roc s are used to evaluate diagnostic tests when test results are not binary they describe the inherent capacity of the test for distinguishing between truly diseased and nondiseased subjects although methodology for estimating and for comparing roc s is well developed to date no general framework exists for evaluating covariate effects on roc s we formulate a general regression model which allows the effects of covariates on test accuracy to be succinctly summarised such covariates might include for example characteristics of the patient or test environment test type or severity of disease the regression models are shown to arise naturally from some classic models for continuous or ordinal test data regression parameters are fitted using an estimating equation approach the method is illustrated on data from a study of multiformat photographic images used for scintigraphy

on equivalent forms of fuzzy logic systems nm and imtl the nilpotent minima are important t norms in fuzzy logic and fuzzy reasoning two interesting formal systems nilpotent minimum logic nm and l based on nilpotent minima have been proposed and have been intensively studied in this paper their equivalence is pointed out moreover coincidence of two related algebraic structures nm and r0 algebras is also proved in addition we give an equivalent form of involutive monoidal t norm based logic imtl

a triangular norm based propositional fuzzy logic in 1997 wang introduced the formal system l for a type of fuzzy propositional calculus given by a particular t norm different from the three famous ones lukasiewicz godel product recently pei proved the completeness theorem for l with respect to w semantic by developing a theory of algebraic systems in this paper we aim to prove it by essentially metamathematical method furthermore we discussed originally the completeness for the schematic extension l n of l with respect to w n semantic

undefinability of min conjunction in mtl firstly a sufficient and necessary condition is given to decide whether the lattice operations of a wnm chain can be definable from its other operations secondly we show that it is indeed necessary to introduce the min conjunction into mtl as a primitive connective by constructing a left continuous t norm such that in the propositional calculus defined by this t norm the min conjunction and max disjunction are not definable from the strong conjunction implication and truth constant email protected

generalized consistency degrees of theories w r t formulas in several standard complete logic systems abstract the present paper carries out a deep analysis on the inconsistency of theories by means of deduction theorems completeness theorems and satisfiability degrees of formulas and introduces the concept of the degree of entailment of a contradiction from a theory in classical two valued logic system ukasiewicz fuzzy logic system godel fuzzy logic system product fuzzy logic system and the r 0 fuzzy logic system it is proved that in classical two valued logic system ukasiewicz fuzzy logic system and the r 0 fuzzy logic system respectively the concept of consistency degrees of theories defined by the divergence degrees of theories in x n zhou g j wang consistency degrees of theories in some systems of propositional fuzzy logic fuzzy sets and systems 152 2005 321 331 h j zhou g j wang a new theory index based on deduction theorems in several logic systems fuzzy sets and systems 157 2006 427 443 is reasonable and can accurately measure the consistency degrees of theories the concept of the degree of entailment of a contradiction from a theory is generalized by replacing the contradiction with a general formula and then the generalized consistency degrees of theories w r t general formulas in the above mentioned logic systems is established

maximum likelihood estimation of receiver operating characteristic roc curves from continuously distributed data we show that truth state runs in rank ordered data constitute a natural categorization of continuously distributed test results for maximum likelihood ml estimation of roc curves on this basis we develop two new algorithms for fitting binormal roc curves to continuously distributed data a true ml algorithm labroc4 and a quasi ml algorithm labroc5 that requires substantially less computation with large data sets simulation studies indicate that both algorithms produce reliable estimates of the binormal roc curve parameters a and b the roc area index az and the standard errors of those estimates

unified forms of triple i method first the present paper provides unified forms of triple i method for fuzzy modus ponens and fuzzy modus tollens of which diverse implication operators can be employed second it is clarified that in a sense zadeh s cri method for fuzzy modus ponens can be brought into line with the unified form of the triple i method lastly a unified form of a triple i method is established as well and a duality result concerning a triple i solutions of fuzzy modus ponens and fuzzy modus tollens is obtained

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

stone like representation theorems and three valued filters in r0 algebras nilpotent minimum algebras nilpotent minimum algebras nm algebras are algebraic counterpart of a formal deductive system where conjunction is modeled by the nilpotent minimum t norm a logic also independently introduced by guo jun wang in the mid 1990s such algebras are to this logic just what boolean algebras are to the classical propositional logic in this paper by introducing respectively the stone topology and a three valued fuzzy stone topology on the set of all maximal filters in an nm algebra we first establish two analogues for an nm algebra of the well known stone representation theorem for a boolean algebra which state that the boolean skeleton of an nm algebra is isomorphic to the algebra of all clopen subsets of its stone space and the three valued skeleton is isomorphic to the algebra of all clopen fuzzy subsets of its three valued fuzzy stone space respectively then we introduce the notions of boolean filter and of three valued filter in an nm algebra and finally we prove that three valued filters and closed subsets of the stone space of an nm algebra are in one to one correspondence and boolean filters uniquely correspond to closed subsets of the subspace consisting of all ultrafilters

a characterization of truth functions in the nilpotent minimum logic abstract by introducing a new family of partitions into the n cube 0 1 n the problem of characterizing truth tables of formulas in the nilpotent minimum logic is solved and their normal forms are presented so far only this kind of fuzzy truth functions have normal forms among all fuzzy propositional calculi which are based on left continuous but discontinuous t norm

smooth non parametric receiver operating characteristic roc curves for continuous diagnostic tests receiver operating characteristic roc curves have seen increasing use to assess the accuracy of diagnostic tests that yield continuous test results we can calculate a fully non parametric roc curve from the empirical false positive rate fpr and true positive rate tpr at each possible decision threshold but it may be quite jagged we can obtain a smooth roc curve by directly fitting a parametric model for example binormal or bilogistic to the actual test results but substantial lack of fit may result if the distributional assumptions are not valid a recently proposed alternative algorithm labroc4 is insensitive to such departures but is not fully flexible in that the form of the resulting roc curve is restricted to be binormal we propose a smooth non parametric roc curve derived from kernel density estimates of the two test result distributions we obtain pointwise standard errors for the estimated tpr at each fpr and use them to construct pointwise confidence intervals for the roc curve using a logit transformation we also obtain standard errors for the estimated tpr and fpr at given thresholds and use them to construct confidence rectangles in fpr tpr space that correspond to a specific threshold we adapt existing methods for the unsmoothed non parametric roc curve to obtain the area under the roc curve and its standard error we also give partial areas and corresponding standard errors we have created a fortran algorithm roc rol available upon request we compare roc curves and areas obtained by applying our methods and its competitors to two data sets one of which is fit well by parametric methods and labroc4 the other of which is not 1997 john wiley sons ltd