resolution of a system of fuzzy polynomial equations using the grobner basis the occurrence of imprecision in the real world is inevitable due to some unexpected situations the imprecision is often involved in any engineering design process the imprecision and uncertainty are often interpreted as fuzziness fuzzy systems have an essential role in the uncertainty modelling which can formulate the uncertainty in the actual environment in this paper a new approach is proposed to solve a system of fuzzy polynomial equations based on the grobner basis in this approach first the h cut of a system of fuzzy polynomial equations is computed and a parametric form for the fuzzy system with respect to the parameter of h is obtained then a grobner basis is computed for the ideal generated by the h cuts of the system with respect to the lexicographical order using faugere s algorithm i e f 4 algorithm the grobner basis of the system has an upper triangular structure therefore the system can be solved using the forward substitution hence all the solutions of the system of fuzzy polynomial equations can easily be obtained finally the proposed approach is compared with the current numerical methods some theorems together with some numerical examples and applications are presented to show the efficiency of our method with respect to the other methods

algorithms for solving fuzzy relational equations in a probabilistic setting abstract we discuss methods suitable for finding approximate solutions to fuzzy relational equations defined in finite universe of discourse and taking the form of either a system of equations x k r y k k 1 2 k with r being unknown fuzzy relation or a single equation x r y with x being looked for the key idea of the proposed algorithms is to incorporate some statistics concerning the solution of a single equation and utilize it in the final process of approximation of the unknown fuzzy relation set numerical considerations support the established algorithms

on the minimal solutions of max min fuzzy relational equations in this paper the minimal solutions of max min fuzzy relational equations are investigated a sufficient and necessary condition for discriminating whether a given solution is minimal or not is shown furthermore we propose a new algorithm for computing all minimal solutions less than or equal to a given one

coverage probability bias objective bayes and the likelihood principle summary we review objective bayes procedures based on both parametric and predictive coverage probability bias and explore the extent to which such procedures contravene the likelihood principle in the case of a scalar parameter the discussion encompasses choice of objective priors objective posterior probability statements and objective predictive probability statements we conclude with some remarks concerning the future development and implementation of objective priors based on small coverage probability bias

the inverse problem for distributive lattices abstract the inverse problem givena in vl n b in vl m andb in vl m find all a in vl n such that a a b b where l is distributive lattice denotes max min composition vl n denotes the set of all vector with n member on l and x a1 a2 an b1 b2 bn ai bj n m is investigated the necessary and sufficient condition for the existence of a solution is shown and an analytical solution is given

single observation unbiased priors this paper studies a class of default priors which we call single observation unbiased priors soup a prior for a parameter is a soup if the corresponding posterior mean of the parameter based on a single observation is an unbiased estimator of the parameter we prove that under mild regularity conditions a default prior for a convolution parameter is noninformative in the sense of yielding a posterior inference invariant under amalgamation only if it is a soup therefore when amalgamation invariance is desirable as in our motivating example of performing imputation for census undercount soup is the only possible coherent noninformative prior for bayesian predictions that will be utilized under aggregation the use of soup also mutually calibrates bayesian and frequentist inferences for aggregates of convolution parameters across many small areas we describe approaches that identify soups in many common models in particular a constructive duality method that identifies soups in pairs of distribution families we introduce o completeness a necessary and sufficient condition for a prior distribution to be uniquely characterized by the corresponding posterior mean uniqueness of the soup is determined by the o completeness of the dual family o completeness of a natural exponential family is implied by its completeness hence the diaconis ylvisaker characterization of the conjugate prior for natural exponential families via linear posterior expectation is a direct consequence of the completeness of such families for most of the examples we have examined the inverse of the variance function is the soup for the mean parameter of the corresponding family suggesting that hartigan s results on asymptotic unbiasedness can be generalized to some families with discrete parameters we also discuss a possible extension of berger s result on the inadmissibility of unbiased estimators as the nonexistence of soup can be a first step in establishing such inadmissibility

statistically grounded logic operators in fuzzy sets abstract in this study we are concerned with a category of logic operators for fuzzy sets that is inherently associated with underlying statistical properties of membership grades they operate on these constructs are referred to as statistically grounded logic operators namely statistically grounded ors sors for short and statistically grounded ands sands for short operators in essence they arise as a solution to the optimization problem of the following form arg min m 0 1 k 1 n w x k x k m where xk k 1 2 n are the corresponding membership grades to be combined the weight function w 0 1 0 1 captures the nature of the logic and and or aggregation respectively we show that the weight function could be induced by t norms and t conorms the weight functions could be also implied by some statistical characteristics of data membership grades the choice of the t norm t conorm depends upon the predefined form of the logic operators to be developed we demonstrate that sands and sors offer an efficient operational framework for constructing fuzzy rough sets lead to the increased sensitivity of computing possibility and necessity measures bring a new insight into fuzzy relational equations and deliver an interpretation vehicle for fuzzy clustering that is provided in the form of dependency analysis

probability matching priors for non regular cases we study probability matching priors in non regular cases both one parameter and multi parameter cases are considered the resulting priors are shown to satisfy certain differential equations several examples are discussed it is observed that the reference prior in the sense of bernardo is in many cases also probability matching as a necessary tool for finding such priors asymptotic expansion of posterior distributions for multi parameter non regular cases is also obtained

on asymptotic normality of limiting density functions with bayesian implications walker 1969 presented a straightforward approach to asymptotic normality of posterior distributions based on independent and identically distributed i i d observations dawid 1970 extended this work by providing a weaker set of conditions to cover some distributions whose ranges depend on the parameters of interest more recently heyde and johnstone 1979 simplified walker s conditions for stochastic processes in this paper we introduce three basic normality conditions for limiting density functions in the bayesian context our approach is data determined and is nonprobabilistic in nature nevertheless the results thus obtained can be easily converted to probabilistic ones when compared with those given by other investigators it is shown that our conditions as a whole have a greater generality and flexibility than those previously assumed either by dawid or heyde and johnstone in particular for the i i d case the commonly assumed wald type uniform boundedness condition for tail behaviours is shown to be unnecessary for asymptotic normality section 3 3 the advantage of our approach is further exemplified by a simple proof of the asymptotic normality of distributions conjugate to the exponential family

on the invariance of noninformative priors various noninformative priors have been proposed in the literature this article explores the invariance or lack thereof of some of these noninformative priors including the reference prior of berger and bernardo the reverse reference prior of j k ghosh and the probabilitymatching prior of peers and stein under reparameterization berger and bernardo s m group ordered reference prior is shown to remain invariant under a special type of reparameterization the reverse reference prior of j k ghosh is shown not to remain invariant under reparameterization however the probability matching prior is shown to remain invariant under any reparameterization also for spherically symmetric distributions certain noninformative priors are derived using the principle of group invariance

likelihood estimation and inference in a class of nonregular econometric models we study inference in structural models with a jump in the conditional density where location and size of the jump are described by regression curves two prominent examples are auction models where the bid density jumps from zero to a positive value at the lowest cost and equilibrium job search models where the wage density jumps from one positive level to another at the reservation wage general inference in such models remained a long standing unresolved problem primarily due to nonregularities and computational difficulties caused by discontinuous likelihood functions this paper develops likelihood based estimation and inference methods for these models focusing on optimal bayes and maximum likelihood procedures we derive convergence rates and distribution theory and develop bayes and wald inference we show that bayes estimators and confidence intervals are attractive both theoretically and computationally and that bayes confidence intervals based on posterior quantiles provide a valid large sample inference method

frequentist validity of posterior quantiles for a two parameter exponential family summary berger bernardo s 1989 reference priors and tibshirani 1989 priors are derived for distributions in bar lev reiser s 1982 two parameter exponential family when either the location or the scale parameter is of interest the reference prior is shown to be a tibshirani prior furthermore conditions under which a tibshirani prior is a higherorder matching prior are investigated when both the parameters are of interest it is also shown that the reference prior agrees with the prior matching the posterior and frequentist expansions based on the signed square root of log likelihood ratio the normal inverse gaussian and gamma distributions are used to illustrate the results

on priors providing frequentist validity of bayesian inference for multiple parametric functions summary we characterise priors which match up to o n 1 the posterior joint cumulative distribution function of multiple parametric functions with the corresponding frequentist cumulative distribution function this work extends and unifies the work of ghosh mukerjee 1993 and datta ghosh 1995a on the topic of probability matching priors a set of necessary and sufficient conditions is obtained for the above characterisation some of these conditions depend only on the parametric functions and not on the prior examples are given where the joint probability matching is possible and where it is not possible

infinite fuzzy relational equations on a complete brouwerian lattice abstract let i be a finite and j an infinite index sets a a ij i j and b b i i i a given matrix and a column vector with elements from a complete brouwerian lattice in this paper the problem of a minimal solution of a fuzzy relational equation a x b where is the sup inf composite operation is investigated

on the existence of minimal solutions for fuzzy equations with tolerances abstract we investigate the existence of minimal elements of solution sets for fuzzy equation problems with tolerances where the cardinality of the basic spaces of fuzzy inputs and outputs of an object need not be finite it turns out the semi continuity of applied membership functions plays a central role crisp models with fuzzy parameters are considered as well

solvability and unique solvability of max min fuzzy equations abstract the problem of solvability and the problem of unique solvability of a fuzzy relation equation in an arbitrary max min algebra are considered and corresponding necessary and sufficient conditions are presented the results allow to solve both problems by an o mnp algorithm where m n p are the dimensions of the corresponding relations in the equation the existence of the greatest and of the least solution is also considered

resolution of fuzzy relational equations method algorithm and software with applications analytical methods and algorithms for inverse problem resolution of fuzzy linear systems of equations in some bl algebras godel algebra in case of max min and min max compositions and goguen algebra in case of max product composition are presented algorithms with software realization for solving fuzzy linear systems of equations are proposed applications include fuzzy optimization with fuzzy linear systems of equation constraint fuzzy machines and covering problem

some properties of infinite fuzzy relational equations on complete brouwerian lattices this paper deals with fuzzy relational equations email protected x b where both a a j j j and b are known b has an irredundant continuous join decomposition x x j j j t is unknown represents sup inf composition j is an infinite index set on complete brouwerian lattices at first a necessary and sufficient condition for existence of the attainable solutions the unattainable solutions respectively and some properties of the attainable solutions the unattainable solutions respectively are formulated then the solution set of the fuzzy relational equations is described

some remarks on noninformative priors abstract this article focuses primarily on a comparison between the reference priors of berger and bernardo and the reverse reference priors suggested by j k ghosh sufficient conditions are given that provide agreement between the two classes of priors several examples are given showing the agreement or disagreement between the two in addition these priors are compared under a criterion that requires the frequentist coverage probability of the posterior region of a real valued parametric function to match a nominal level with a remainder of o n 1 where n denotes the sample size the latter priors first introduced by welch and peers are obtained by solving a differential equation due to peers finally in the presence of several parameters of interest a general class of priors that satisfies the matching criterion separately for each parameter is constructed and examples are given to illustrate how reference or reverse reference priors fit within this class of priors

minimal join decompositions and their applications to fuzzy relation equations over complete brouwerian lattices this paper deals with the existence of minimal solutions of fuzzy relation equations over complete brouwerian lattices from the viewpoint of join decompositions first the concept of a minimal join decomposition is introduced then the existence conditions and properties of such decomposition are discussed as their applications the conditions for the existence of minimal solutions of fuzzy relation equations over complete brouwerian lattices and complete lattices are given respectively

entire functions on banach spaces with a separable dual abstract let e and f be complex banach spaces we show that if e has a separable dual then every holomorphic function from e into f which is bounded on weakly compact sets is bounded on bounded sets

entire functions on c0 abstract it is shown that every holomorphic function on c 0 which is bounded on weakly compact sets is bounded on bounded sets

some properties of sup min fuzzy relational equations on infinite domains abstract this paper deals with a sup min fuzzy relational equation on infinite domains some properties of its solution set will be obtained in particular a necessary and sufficient condition describing the attainable solution resp the unattainable solution is formulated in the end the solution set will be given under some conditions

finitary solvability conditions for systems of fuzzy relation equations in this paper we consider the problem of solving systems of fuzzy relation equations two types of these systems with different compositions are considered and processed simultaneously a new sufficient condition and new solvability criteria are proposed for systems of both types all these conditions are finitary and thus can be easily verified the sufficient condition and criteria of solvability characterize a relationship between a skeleton nxn matrix n is a number of equations and a vector of the right hand side

solution sets of finite fuzzy relation equations with sup inf composition over bounded brouwerian lattices this paper considers the resolution of finite fuzzy relation equations with sup inf composition over a bounded brouwerian lattice the solution sets of finite fuzzy relation equations on a bounded brouwerian lattice are described in a similar way as those of linear spaces of n dimensional vectors in linear algebra

optimization of linear problems subjected to the intersection of two fuzzy relational inequalities defined by dubois prade family of t norms abstract in this paper optimization of a linear objective function with fuzzy relational inequality constraints is investigated in doing so the feasible region is formed by the intersection of two inequality fuzzy systems and dubois prade family of t norms which are considered as fuzzy composition the most well known continuous t norms are archimedean such as frank yager hamacher sugeno weber and schweizer sklar family an interesting family of t norms that is not archimedean has been introduced by dubois and prade in this paper the resolution of the feasible region of the problem is initialy investigated when it is defined with max duboise prade composition a necessary and sufficient condition along with three other necessary conditions are derived for determining the feasibility of the problem moreover two procedures have also been presented with the aim of simplifying the current linear problems a method is proposed to generate random feasible max dubois prade fuzzy relational inequalities and an algorithm is accordingly presented to solve the problem finally an example is described to illustrate this algorithm

weakly continuous mappings on banach spaces abstract it is shown that every n homogeneous continuous polynomial on a banach space e which is weakly continuous on the unit ball of e is weakly uniformly continuous on the unit ball of e applications of the result to spaces of polynomials and holomorphic mappings on e are given

simple modifications for signed roots of likelihood ratio statistics summary in the context of inference about a scalar parameter in the presence of nuisance parameters some simple modifications for the signed root of the log likelihood ratio statistic r are developed that reduce the order of error in the standard normal approximation to the distribution of r from o n 1 2 to o n 1 barndorff nielsen has introduced a variable u such that the error in the standard normal approximation to the distribution of r r r log u r is of order o n 3 2 but calculation of urequires the specification of an exact or approximate ancillary statistic a this paper proposes an alternative variable to u denoted by t that is available without knowledge of a and satisfies t u op n 1 in general thus the standard normal approximation to the distribution of r r log t r has error of order o n 1 and it can be used to construct approximate confidence limits having coverage error of order o n 1 in certain cases however tand uare identical the derivation of tinvolves the bayesian approach to constructing confidence limits considered by welch and peers and peers similar modifications for the signed root of the conditional likelihood ratio statistic are also developed and these modifications are seen to be useful when a large number of nuisance parameters are present several examples are presented including inference for natural parameters in exponential models and inference about location scale models with type ii censoring in each case the relationship between tand u is discussed numerical examples are also given including inference for regression models inference about the means of log normal distributions and inference for exponential lifetime models with type i censoring where barndorff nielsen s variable u is not available

noninformative priors for one parameter of many summary we consider the problem of constructing a prior that is noninformative for a single parameter in the presence of nuisance parameters our approach is to require that the resulting marginal posterior intervals have accurate frequentist coverage stein 1985 derived nonrigorously a sufficient condition for such a prior through the use of orthogonal parameters we give a general form for the class of priors satisfying stein s condition the priors are proportional to the square root of the information element for the parameter of interest times an arbitrary function of the nuisance parameters this is in contrast to jeffreys 1946 invariant prior for the overall parameter which is proportional to the square root of the determinant of the information matrix several examples are given and comparisons are made to the reference priors of bernardo 1979

resolution of matrix equations over arbitrary brouwerian lattices this paper studies the problem of solving a matrix equation over an arbitrary not necessarily complete brouwerian lattice a criterion for the solvability and a method for finding all solutions of the equation are obtained over an arbitrary self dual brouwerian lattice an equivalent condition for the unique solution a necessary and sufficient condition for the minimal solution and a procedure for constructing minimal solutions less than or equal to any given solution of the equation are presented it is shown that the results of the paper can be applied to the determination of generalized inverses of a matrix over an arbitrary brouwerian lattice

method of solution to fuzzy equations in a complete brouwerian lattice abstract up to now how to solve a fuzzy relation equation in a complete brouwerian lattice is still an open problem as di nola et al point out to this problem the key problem is whether there exists a minimal element in the solution set when a fuzzy relation equation is solvable in this paper we first show that there is a minimal element in the solution set of a fuzzy relation equation a x b where a a 1 a 2 a n and b are known and x x 1 x 2 x n t is unknown when its solution set is nonempty and b has an irredundant finite join decomposition further we give the method to solve a x b in a complete brouwerian lattice under the same conditions finally a method to solve a more general fuzzy relation equation in a complete brouwerian lattice when its solution set is nonempty is also given under similar conditions

inverse problem in fuzzy relational equations abstract the paper deals with the problem of handling fuzzy relational equations of the form x r y in the case of an empty solution set with respect to x x r y discussing the essence of the structure of the fuzzy relation r with respect to provided y algorithms have been proposed which enable us to modify the original equation i e the fuzzy set y in a way leading to its genuine solution or at least an approximate one it is explained how these modifications relax a set of constraints needed to be satisfied for any solvable equation a grade of the modification of the original problem is expressed in terms of distortion of the fuzzy set y which forms a cornerstone of the proposed procedures numerical examples are provided some aspects related to the area of diagnostic models in which these equations are studied are also treated

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

on pre solution matrices of fuzzy relation equations over complete brouwerian lattices abstract each solution of a system of fuzzy relation equations over a complete brouwerian lattice can be decomposed into a matrix which we call pre solution matrix every pre solution matrix can induce at least one solution in this paper we discuss connections between solutions and pre solution matrices of a system of fuzzy relation equations over a complete brouwerian lattice then we explore properties of pre solution matrices and characterize lower pre solution matrices moreover we discuss constructions of pre solution matrices

some properties of infinite fuzzy relational equations with sup inf composition this paper deals with infinite fuzzy relational equations with sup inf composition first some properties of them are investigated then some necessary and sufficient conditions for the existence of attainable solutions resp unattainable solutions and partially attainable ones are given if the solution set is nonempty finally the set of attainable solutions is described

some properties of minimal solutions for a fuzzy relation equation abstract some properties of the solution set and minimal solutions of a fuzzy relation equation are considered in this paper we show the necessary and sufficient condition for existence of a minimal solution of a finite fuzzy relation equation defined on infinite index sets

on fuzzy relational equations and the covering problem previous studies have shown that fuzzy relational equations fres based on either the max continuous archimedean t norm or the max arithmetic mean composition can be transformed into the covering problem which is an np hard problem exploiting the properties common to the continuous archimedean t norm and the arithmetic mean this study proposes a generalization of them as the u norm enabling fres that are based on the max continuous u norm composition also to be transformed into the covering problem this study also proposes a procedure for transforming the covering problem into max product fres consequently max continuous u norm fres can be solved by extending any procedure for solving either the covering problem or max product fres

processing in relational structures fuzzy relational equations abstract the paper provides an overview of methodology for processing fuzzy information in relational structures completed in terms of fuzzy relational equations the origin and the central role of relational calculus in general and fuzzy sets in particular is explained with special emphasis paid to its role and representation capabilities it is pointed out that fuzzy relational equations play a significant role as a platform for a uniform development of techniques in fuzzy sets many of the problems and frameworks developed in fuzzy sets so far can be easily translated into the language of fuzzy relational equations which immediately takes a significant advantage of their solid well developed formalisms techniques and clarity of exposition we will study the role of approximate solutions to fuzzy relational equations as a convenient tool to handle probabilistic type of uncertainty moreover those solutions can easily explain the origin and generate a way in which fuzzy sets of higher generality such as for instance type 2 sets interval valued sets can be algorithmically determined finally we study the role of equations in diverse fields of applications making use of an ample framework of general systems theory

factorization of holomorphic mappings in infinite dimensions this note is concerned with a generalization of the first identity to the non linear setting of holomorphic i e complex analytic functions on e in fact our interest in this generalization arose from a study made by the second author 15 of a suitable generalization of the second identity to the holomorphic setting however our results and methods have little in common with 15 in somewhat simplistic terms our results provide further evidence that in many situations one is justified in regarding the space h e of holomorphic functions on e as the holomorphic dual of e which shares many properties with the standard linear dual of e see for example 6 a brief outline of this paper follows after which we recall some basic results about holomorphic mappings between infinite dimensional locally convex spaces in section 2 we investigate when a holomorphic function f u c on a balanced open subset u of e admits a factorization through u a characterization of factorizable functions is given in terms of the derivative o f f our basic technique is

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

analytic approximations of uniformly continuous functions in real banach spaces abstract it is shown that if a separable real banach space x admits a separating analytic function with an additional condition property k concerning uniform behaviour of radii of convergence then every uniformly continuous mapping on x into any real banach space y can be approximated by analytic operators in particular the result applies to c0

resolution of finite fuzzy relation equations abstract general schemes for solving fuzzy relation equations with finite sets are derived in this paper the relationship between the results derived here and those of previous publications regarding this subject is also discussed

weakly continuous holomorphic functions on banach spaces with a shrinking and unconditional basis let e and f be complex banach spaces and let u be an open ball in e we show that if e has a shrinking and unconditional basis then every holomorphic function f u f that is weakly continuous on u bounded sets is weakly uniformly continuous on u bounded sets

on matrix equations in a class of complete and completely distributive lattices matrix equations of the form ax b and ya b are considered in a class of complete and completely distributive lattices where the multiplication operation of matrices is defined either by the max min or by the min max way necessary and sufficient conditions for the solvability of the equations are obtained and for the solvable case the entire solution set and the largest solution are determined by using relative pseudo or relative lower pseudocomplements in lattices the results are seen to be a generalization of 4 5 6 and in a sense also an extension of 3 as an application the g inverse and thierrin vagner inverses of matrices are considered

on solving relational equations in brouwerian lattices abstract the problems of finding generic and minimal solutions of a fuzzy relational equations in brouwerian lattices are considered

conditions for the existence of the least solution and minimal solutions to fuzzy relation equations over complete brouwerian lattices this paper discusses properties and existence conditions of minimal solutions to sup inf fuzzy relation equations over complete brouwerian lattices the relationship between minimal solutions to a system of fuzzy relation equations and minimal solutions to a fuzzy relation equation in the system is illustrated by two examples necessary and sufficient conditions for the existence of the least solution and minimal solutions to a system of fuzzy relation equations are given

bayesian intervals with good frequentist behaviour in the presence of nuisance parameters given a random sample from a distribution with density function that depends on an unknown parameter 1 p we are concerned with the problem of setting confidence intervals for a particular component of it say 1 treating the remaining components as a nuisance parameter adopting an objective bayesian approach we show that the bayes intervals with a certain conditional prior density on the parameter of interest 1 are confidence intervals as well having nearly the correct frequency of coverage the frequentist performance of the proposed intervals is tested in a simulation study for gamma mean and shape parameters

estimating a product of means bayesian analysis with reference priors abstract suppose that we observe x n 1 and independently y n 1 and are concerned with inference mainly estimation and confidence statements about the product of means this problem arises most obviously in situations of determining area based on measurements of length and width it also arises in other practical contexts however for instance in gypsy moth studies the hatching rate of larvae per unit area can be estimated as the product of the mean of egg masses per unit area times the mean number of larvae hatching per egg mass approximately independent samples can be obtained for each mean see southwood 1978 noninformative prior bayesian approaches to the problem are considered in particular the reference prior approach of bernardo 1979 an appropriate reference prior for the problem is developed and relatively easily implementable formulas for posterior moments e g the posterior mean and variance and credible sets are derived comparisons with alternative noninf

system of fuzzy relation equations as a continuous model of if then rules we claim that a fuzzy relation which models a system of fuzzy if then rules has the property of continuity only if the underlying system of fuzzy relation equations is solvable in view of this connection it is important to consider the problem of solvability of a system of fuzzy relation equations with respect to an unknown fuzzy relation this problem is analyzed in the present paper we propose several new necessary and sufficient criteria for the so called mamdani relation to be a solution to the system we prove that in the general case these same criteria are sufficient but not always necessary for the solvability of the system a new easy to check criterion is proposed for the solvability of the system with special fuzzy parameters

a modified pso algorithm for linear optimization problem subject to the generalized fuzzy relational inequalities with fuzzy constraints fri fc in this paper optimization of a linear objective function subject to a generalized fuzzy relational inequalities is investigated in which an arbitrary continuous t norm is considered as fuzzy composition and fuzzy inequality replaces ordinary inequality in the constraints unlike most optimization algorithms in fuzzy relational inequalities with fuzzy constraints fri fc we find a near feasible solution having a better objective value than those resulting from the resolution of the similar problems with crisp ordinary inequality constraints such solutions are called super optima in this paper subsequently an algorithm is proposed to find a super optimum with pre specified desirable infeasibility for this purpose we firstly study some structural properties of the fri fc problem and present a new formulization that is independent of t norms used in the constraints of the problem this new formulation converts the primary problem into an equivalent problem with simple constraints without considering any penalty parameters however it is proved that the transformed equivalent problem enables our algorithm to distinguish unfavorable points and generate a sequence of solutions converging to a super optimum under a certain sufficient condition finally a modified pso is presented in which the ability of pso to solve unconstrained problems with continuous domain the structure of the transformed problem and some fuzzy structural modifications are combined and form an efficient algorithm to solve the generalized fri fc problems the modified pso algorithm has been applied to the generalized fri fc problem defined with ten well known continuous t norms additionally an idea of the fri fc problems as outer approximators for fri problems with ordinary inequalities is also investigated

on priors providing frequentist validity for bayesian inference summary we derive the differential equation that a prior must satisfy if the posterior probability of a one sided credibility interval for a parametric function and its frequentist probability agree up to o n 1 this equation turns out to be identical with stein s equation for a slightly different problem for which also our method provides a rigorous justification our method is different in details from stein s but similar in spirit to dawid 1991 and bickel ghosh 1990 some examples are provided