the reduction of binary fuzzy relations and its applications this paper introduces the concept of t reduction of binary fuzzy relations and establishes its properties it presents an approach to providing the one valued restoration of any boundary transitive and t asymmetric binary fuzzy relation by its t reduction a theoretical basis for this approach is established showing that t reduction is the smallest fuzzy relation whose t transitive closure coincides with the given binary fuzzy relation and where an nxn fuzzy relation on a finite universe with cardinality n can be represented by n 1 values this approach opens up an original way for constructing the membership functions of certain binary fuzzy relations important examples for the application of this concept are given

on approximate reasoning with graded rules this contribution presents a comprehensive view on problems of approximate reasoning with imprecise knowledge in the form of a collection of fuzzy if then rules formalized by approximating formulas of a special type two alternatives that follow from the dual character of approximating formulas are developed in parallel the link to the theory of fuzzy control systems is also explained

sup t norm and inf residuum are one type of relational product unifying framework and consequences we present a simple framework which enables us to consider the well known sup t norm and inf residuum products of relations as two particular cases of a single more general type of product we present basic properties of the framework and consequences for the theory of fuzzy relations informally the paper implies that in many cases of fuzzy relational modeling such as in solving fuzzy relational equations there is no need to develop the methods for sup t norm and inf residuum products separately because these methods are just two particular instances of a single method

the cutting of compositions this paper investigates the cutability of the fuzzy relational compositions the sup t composition the triangular compositions of bandler and kohout and their improved versions the theoretical and practical importance of fuzzy cuts is illustrated by means of a careful selection of results from the fuzzy literature some new potential properties of fuzzy implications operators and triangular norms that play an important role in this cutability study are introduced and discussed extensively the paper concludes with a study of the fuzzy cutability of the sup t composition

intentional possibilistic approach to dealing with completeness vagueness and uncertainty abstract the paper outlines a possibilistic intentional framework suitable for dealing with incompleteness vagueness and ambiguity which captures their global as well as local aspects after discussing three levels of intentions in possibilistic modelling we present a formalization of an intentional possibilistic scheme for capturing the statics and dynamics of interaction of participants and observers of a modelling process we continue with a detailed discussion of perspectives in fuzzy logics formalization of perspectival models by means of fuzzy relational structures presents the foundations for discussion of the relationship of observation and participation the paper is concluded with exploratory examination of interrelationship between vagueness ambiguity incompleteness and inexactness within the framework of our possibilistic perspectival approach

optimal error estimates for semidiscrete galerkin approximations to equations of motion described by kelvin voigt viscoelastic fluid flow model in this paper a finite element galerkin method is applied to equations of motion arising in the kelvin voigt viscoelastic fluid flow model when the forcing function is in l l 2 some a priori estimates for the exact solution which are valid uniformly in time as t and even uniformly in the retardation time an 0 are derived it is shown that the semidiscrete method admits a global attractor further with the help of a priori bounds and sobolev stokes projection optimal error estimates for the velocity in l l 2 and l h 1 norms and for the pressure in l l 2 norm are established since the constants involved in error estimates have an exponential growth in time therefore in the last part of the article under certain uniqueness condition the error bounds are established which are valid uniformly in time finally some numerical experiments are conducted which confirm our theoretical findings

on a reflexivity preserving family of cardinality based fuzzy comparison measures in this paper we generalize and extend previous work on a triparametric family of cardinality based fuzzy similarity measures more specifically we i generalize this family to a broader family of fuzzy comparison measures ii summarize previously obtained results regarding this family and fill up a few gaps by adding some novel results iii compare our parametric family with an already existing one and iv elaborate on the ranking invariance between certain members of the presented triparametric family of cardinality based fuzzy comparison measures

inference of noisy nonlinear differential equation models for gene regulatory networks using genetic programming and kalman filtering a key issue in genomic signal processing is the inference of gene regulatory networks these are used both to understand the role of biological regulation in phenotypic determination and to derive therapeutic strategies for genetic based diseases in this paper gene regulatory networks are inferred via evolutionary modeling based on time series microarray measurements a nonlinear differential equation model is adopted it includes random noise parameters for intrinsic noise arising from stochasticity in transcription and translation and for external noise arising from factors such as the amount of rna polymerase levels of regulatory proteins and the effects of mrna and protein degradation an iterative algorithm is proposed for model identification genetic programming is applied to identify the structure of the model and kalman filtering is used to estimate the parameters in each iteration both standard and robust kalman filtering are considered the effectiveness of the proposed scheme is demonstrated by using synthetic data and by using microarray measurements pertaining to yeast protein synthesis

a frequentist view on cycle transitivity of reciprocal relations we establish a connection between two transitivity frameworks the transitivity of fuzzy relations based on a commutative quasi copula and the cycle transitivity of reciprocal relations w r t the dual quasi copula as an upper bound function loosely speaking it turns out that the latter can be characterized by imposing a lower bound on the relative frequency with which the former is fulfilled when applied to reciprocal relations we provide two compelling cases the 4 6 theorem expressing that the winning probability relation of a set of independent random variables is at least 66 66 product transitive and the 5 6 theorem expressing that the mutual rank probability relation associated with a given poset is at least 83 33 product transitive moreover these lower bounds turn out be rather conservative illustrating that from a frequentist point of view transitivity is abundant

optimal infinite horizon control for probabilistic boolean networks external control of a genetic regulatory network is used for the purpose of avoiding undesirable states such as those associated with disease heretofore intervention has focused on finite horizon control i e control over a small number of stages this paper considers the design of optimal infinite horizon control for context sensitive probabilistic boolean networks pbns it can also be applied to instantaneously random pbns the stationary policy obtained is independent of time and dependent on the current state this paper concentrates on discounted problems with bounded cost per stage and on average cost per stage problems these formulations are used to generate stationary policies for a pbn constructed from melanoma gene expression data the results show that the stationary policies obtained by the two different formulations are capable of shifting the probability mass of the stationary distribution from undesirable states to desirable ones

algorithms for computing the min transitive closure and associated partition tree of a symmetric fuzzy relation abstract based on a previously derived weight driven algorithm for the computation of the t transitive closure of an arbitrary binary fuzzy relation on a finite universe with t a triangular norm we establish three algorithms for the computation of the min transitive closure of a symmetric matrix with elements in 0 1 as a by product these algorithms enable to generate the partition tree associated with the min transitive closure of the given matrix in descending order of the cutting parameter

structure identification for gene regulatory networks via linearization and robust state estimation abstract inferring causal relationships among cellular components is one of the fundamental problems in understanding biological behaviours the well known extended kalman filter ekf has been proved to be a useful tool in simultaneously estimating both structure and actual gene expression levels of a gene regulatory network grn first order approximations however unavoidably result in modelling errors but the ekf based method does not take either unmodelled dynamics or parametric uncertainties into account which makes its estimation performances not very satisfactory to overcome these problems a sensitivity penalization based robust state estimator is adopted in this paper for revealing the structure of a grn based on the specific structure of the estimation problem it has been proved that under some weak conditions both the ekf based method and the method suggested in this paper provide a consistent estimate but the suggested method has a faster convergence speed compared with both the ekf and the unscented kalman filter ukf based methods simulation results and real data based estimations consistently show that both convergence speed and parametric estimation accuracy can be appreciably improved these lead to significant reductions in both false positive errors and false negative errors and may imply helpfulness of the suggested method in better understanding the structure and dynamics of actual grns

on the existence and construction of t transitive closures in this paper it is shown that any fuzzy relation r on an arbitrary universe x has a t transitive closure the triangular norm t involved is not subject to any conditions this existential result can be turned into an explicit expression in the case of a left continuous triangular norm t furthermore this continuity restriction vanishes in the case of a finite universe x in that case the explicit expression can also be operationalized

a priori error estimates of fully discrete finite element galerkin method for kelvin voigt viscoelastic fluid flow model abstract in this article a finite element galerkin method is applied to the kelvin voigt viscoelastic fluid model when its forcing function is in l l 2 some new a priori bounds for the velocity as well as for the pressure are derived which are independent of inverse powers of the retardation time the second order error estimate in l l 2 norm the first order error estimate in l h 0 1 norm for the velocity and the first order error estimate in l l 2 norm for the pressure for the semidiscrete method are derived which hold uniformly with respect to as 0 with the initial condition only in h 2 h 0 1 further under the smallness assumption on the data these error estimates are shown to be uniform in time as t for the complete discretization of the semidiscrete system a first order accurate backward euler method is applied and fully discrete error estimates are established finally numerical experiments are conducted to verify the theoretical results the results derived in this article are sharper than those derived earlier for finite element analysis of the kelvin voigt fluid model in the sense that the error estimates in this article hold true uniformly even as 0

a top down algorithm for generating the hasse tree of a fuzzy preorder closure this paper describes a new top down algorithm for the stepwise generation of the different levels or hasse diagrams of the hasse tree associated to the fuzzy preorder closure min transitive closure of a given reflexive binary fuzzy relation the algorithm is based upon a recently established weight driven method for computing the min transitive closure of a reflexive binary fuzzy relation the way in which this method gradually establishes the fuzzy preorder closure implies that for the generation of a specific level of the hasse tree the newly proposed algorithm does not require the complete computation of this closure

on the convergence of a time discretization scheme for the navier stokes equations version linearisee du schema d euler implicite utilisee pour l approximation des solutions des equations de navier stokes dans un domaine bidimensionnel

finite element approximation for the viscoelastic fluid motion problem in this article the variational formulation of the two dimensional viscoelastic fluid motion problem and its finite element approximation are considered an local error estimate for the velocity with h1 norm and the pressure with l2 norm is obtained and a uniform error estimate for the velocity and pressure with the above norms is provided if the given data satisfies the uniqueness condition

characterisation of main classes of fuzzy relations using fuzzy modal operators fuzzy modal operators express interactions between binary fuzzy relations and fuzzy sets each of these operators is determined by a binary fuzzy relation on a given universe and transforms one fuzzy set in this universe to another one in this paper we define several fuzzy modal operators and provide characterisations of main classes of binary fuzzy relations by means of these operators interpretation of these characterisations is presented in the context of fuzzy modal logics we show that these characterisations constitute the basis for determining characteristic axioms of particular classes of fuzzy modal logics

long time numerical solution of a parabolic equation with memory long time stability and convergence properties of two time discreti zation methods for an integro differential equation of parabolic type are studied the methods are based on the standard backward euler and second order back ward differencing methods the memory term is approximated by a quadrature rule with emphasis on such rules with reduced computational memory require ments discretization of the spatial partial differential operators by the finite element method is also considered

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

generalized morphisms a new tool for comparative evaluation of performance of fuzzy implications t norms and co norms in relational knowledge elicitation this paper describes a technique for identification of the knowledge structures implicit in scientific data and further develops the supporting methodology for analysis of fuzzy relational structures using the mathematical theory of fuzzy relations and generalized morphism initial knowledge elicited using repertory grid with semiotic descriptors is transformed to fuzzy relational knowledge structures which capture semantic relationships in a problem domain by means of bk product in analysis the fuzzy relational structures computed in the different sized domains or those computed by means of various fuzzy logics can be analyzed using generalized morphism namely global structure embedding or structure preserving as well as meta analysis of fuzzy relational structures the result of meta analysis is represented as the hasse diagram structure of meta relations by classifying equivalent classes of fuzzy relational structures the paper is concluded with a case study using real life engineering data from the aviation industry

on a linearized backward euler method for the equations of motion of oldroyd fluids of order one in this paper a linearized backward euler method is discussed for the equations of motion arising in the oldroyd model of viscoelastic fluids some new a priori bounds are obtained for the solution under realistically assumed conditions on the data further the exponential decay properties for the exact as well as the discrete solutions are established finally a priori error estimates in bh 1 and bl 2 norms are derived for the the discrete problem which are valid uniformly for all time t 0

closing reciprocal relations w r t stochastic transitivity we equip the set of reciprocal relations with an appropriate order relation capturing the bipolar semantics turning this set into a complete join semilattice we also introduce the notion of a compatible family of reciprocal relations and show that such a family has an infimum moreover we discuss the one to one correspondence between the set of 3 valued reciprocal relations and the set of complete crisp relations we refine the theorem of bandler and kohout concerning the existence of closures of elements of a poset w r t some given property rendering it applicable to the above join semilattice the paper is solely dedicated to the transitivity property although uniquely defined for 3 valued reciprocal relations general reciprocal relations can exhibit various types of transitivity we characterize the mappings g and h for which the corresponding g stochastic and h isostochastic transitive closure exists for any reciprocal relation in particular it follows that weak and strong stochastic transitive closures always exist while this is not the case for moderate stochastic transitivity moreover max isostochastic transitivity turns out to be the only practically relevant type of isostochastic transitivity finally we provide algorithms realizing each of these transitive closures

intrinsically bayesian robust kalman filter an innovation process approach in many contemporary engineering problems model uncertainty is inherent because accurate system identification is virtually impossible owing to system complexity or lack of data on account of availability time or cost the situation can be treated by assuming that the true model belongs to an uncertainty class of models in this context an intrinsically bayesian robust ibr filter is one that is optimal relative to the cost function in the classical sense and the prior distribution over the uncertainty class in the bayesian sense ibr filters have previously been found for both wiener and granulometric morphological filtering in this paper we derive the ibr kalman filter that performs optimally relative to an uncertainty class of state space models introducing the notion of bayesian innovation process and the bayesian orthogonality principle we show how the problem of designing an ibr kalman filter can be reduced to a recursive system similar to the classical kalman recursive equations except with effective counterparts such as the effective kalman gain matrix after deriving the recursive ibr kalman equations for discrete time we use the limiting method to obtain the ibr kalman bucy equations for continuous time finally we demonstrate the utility of the proposed framework for two real world problems sensor networks and gene regulatory network inference

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

error estimates with smooth and nonsmooth data for a finite element method for the cahn hilliard equation a finite element method for the cahn hilliard equation a semilinear parabolic equation of fourth order is analyzed both in a spatially semidiscrete case and in a completely discrete case based on the backward euler method error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data a detailed study of the regularity of the exact solution is included the analysis is based on local lipschitz conditions for the nonlinearity with respect to sobolev norms and the existence of a ljapunov functional for the exact and the discretized equations is essential a result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup upper semicontinuity with respect to the discretization parameters is obtained as a simple application of the nonsmooth data error estimate

computing a t transitive lower approximation or opening of a proximity relation since transitivity is quite often violated even by decision makers that accept transitivity in their preferences as a condition for consistency a standard approach to deal with intransitive preference elicitations is the search for a close enough transitive preference relation assuming that such a violation is mainly due to decision maker estimation errors in some way the more number of elicitations the more probable inconsistency is this is mostly the case within a fuzzy framework even when the number of alternatives or object to be classified is relatively small in this paper we propose a fast method to compute a t indistinguishability from a reflexive and symmetric fuzzy relation being t any left continuous t norm the computed approximation we propose will take o n 3 time complexity where n is the number of elements under consideration and is expected to produce a t transitive opening to the authors knowledge there are no other proposed algorithm that computes t transitive lower approximations or openings while preserving the reflexivity and symmetry properties

fuzzy multi criteria analysis of cutting techniques in a nuclear reactor dismantling project abstract three different cutting techniques employed for dismantling the thermal shield of the belgian br3 nuclear reactor are analyzed within the framework of classical multi criteria utility theory this utilitarian approach allows to incorporate into the decision analysis factors which are difficult to quantify in monetary terms such as the environmental impact of the cutting technique a well known drawback of this approach is that it imposes a linear ordering on the alternatives in real world situations it is common practice to deal with alternatives towards which the decision maker is indifferent or which are incomparable to the decision maker in order to allow for indifference and incomparability in the decision process the problem is redefined in terms of a newly developed fuzzy methodology this methodology leads to a fuzzy quasi order relation in the set of alternatives that can be analyzed at different a cuts producing classical quasi order relations a remarkable asset is that this technique is equally applicable for the analysis of the criteria involved in the decision process in conclusion this approach results in a better understanding of the role of both alternatives and criteria in this mcdm problem

relational compositions in fuzzy class theory we present a method for mass proofs of theorems of certain forms in a formal theory of fuzzy relations and classes the method is based on formal identification of fuzzy classes and inner truth values with certain fuzzy relations which allows transferring basic properties of sup t and inf r compositions to a family of more than 30 composition related operations including sup t and inf r images pre images cartesian products domains ranges resizes inclusion height plinth etc besides yielding a large number of theorems on fuzzy relations as simple corollaries of a few basic principles the method provides a systematization of the family of relational notions and generates a simple equational calculus for proving elementary identities between them thus trivializing a large part of the theory of fuzzy relations

l fuzzy interior systems the aim of this work is to introduce the concept of l fuzzy interior systems and l fuzzy interior operators we start by establishing a connection between l fuzzy interior systems and l fuzzy interior operators it is shown that an l fuzzy interior system is precisely the fuzzy system in opposition to the crisp system and an l fuzzy interior operator is a suitable interior operator that has a close relation to an l fuzzy interior system it is also shown that there is a galois correspondence between the category of l fuzzy interior system spaces and that of l fuzzy interior spaces

permutable fuzzy consequence and interior operators and their connection with fuzzy relations fuzzy operators are an essential tool in many fields and the operation of composition is often needed in general composition is not a commutative operation however it is very useful to have operators for which the order of composition does not affect the result in this paper we analyze when permutability appears that is when the order of application of the operators does not change the outcome we characterize permutability in the case of the composition of fuzzy consequence operators and the dual case of fuzzy interior operators we prove that for these cases permutability is completely connected to the preservation of the operator type we also study the particular case of fuzzy operators induced by fuzzy relations through zadeh s compositional rule and the inf composition for this cases we connect permutability of the fuzzy relations using the sup composition with permutability of the induced operators special attention is paid to the cases of operators induced by fuzzy preorders and similarities finally we use these results to relate the operator induced by the transitive closure of the composition of two reflexive fuzzy relations with the closure of the operator this composition induces

special properties closures and interiors of crisp and fuzzy relations this paper offers optimal definitions of the special properties which relations on a set may possess and which combine to define the classically important types of preorders both kinds of orders tolerances and equivalences the useful concept of local properties is introduced the definitions are shown to unify in a highly convenient way fuzzy and crisp considerations the concepts of closure and interior of a relation with respect to a property are defined necessary and sufficient conditions for the full existence of specific closures and interiors are given and formulae for all of them are presented and validated a widely important area of application of the closure and interior operators is discussed finally fast algorithms are given for the computer calculation of the more troublesome closures and a standardized procedure for the analysis of local orders is presented

algorithms for the computation of t transitive closures we present two weight driven algorithms for the computation of the t transitive closure of a symmetric binary fuzzy relation on a finite universe x with cardinality n or equivalently of a symmetric n spl times n matrix with elements in 0 1 with t a triangular norm the first algorithm is proven to be valid for any triangular norm t whereas the second algorithm is shown to be valid when t is either the minimum operator or an archimedean triangular norm furthermore we investigate how these algorithms can be appropriately adapted to generate the t transitive closure of nonsymmetric binary fuzzy relations or matrices as well

computing the lattice of all fixpoints of a fuzzy closure operator we present a fast bottom up algorithm to compute all fixpoints of a fuzzy closure operator in a finite set over a finite chain of truth degrees along with the partial order on the set of all fixpoints fuzzy closure operators appear in several areas of fuzzy logic and its applications including formal concept analysis fca that we use as a reference area of application in this paper several problems in fca such as computing all formal concepts from data with graded attributes or computing non redundant bases of all attribute dependencies can be reduced to the problem of computing fixpoints of particular fuzzy closure operators associated with the input data the development of a general algorithm that is applicable in particular to these problems is the ultimate purpose of this paper we present the algorithm its theoretical foundations and experimental evaluation

mi algebras the existing arithmetics of fuzzy numbers usually stemming from the cut arithmetic do not preserve some of the important properties of the standard arithmetics of real numbers although one cannot expect a generalization of standard arithmetic to perfectly preserve all its properties the most important properties should be preserved at least in a weakened form we present a novel framework for arithmetics of extensional fuzzy numbers that more or less preserves all the important algebraic properties of the arithmetic of real numbers and thus seems to be an important seed for further investigations on this topic the suggested approach to arithmetics of extensional fuzzy numbers is demonstrated by many examples and besides its algebraic properties it is also shown to carry some desirable practical properties the investigation leads to novel algebraic structures mi algebras mi monoids mi groups mi fields that abstract the discussed properties the main idea of these structures is based on a set of pseudoidentities that complements the only commonly used identity element in a monoid group structure these pseudoidentities are elements that generalize the identity like behavior and allow us to weaken the standard forms of algebraic properties that are highly desirable to preserve at least in a weakened form appropriate properties of suggested mi structures are introduced and demonstrated by a number of practical examples

weak and graded properties of fuzzy relations in the context of aggregation process we study graded properties a properties of fuzzy relations which are parameterized versions of properties of a fuzzy relation defined by l a zadeh namely we take into account fuzzy relations which are a reflexive a irreflexive a symmetric a antisymmetric a asymmetric a connected a transitive where a 0 1 we also pay our attention to the composed versions of these basic properties e g an a equivalence a orders we also consider the so called weak properties of fuzzy relations which are the weakest versions of the standard properties of fuzzy relations we take into account the same types of properties as in the case of the graded ones using functions of n variables we consider an aggregated fuzzy relation of given fuzzy relations we give conditions for functions to preserve graded and weak properties of fuzzy relations

context sensitive probabilistic boolean networks steady state properties reduction and steady state approximation context sensitive probabilistic boolean networks pbns have been recently introduced as a paradigm for modeling genetic regulatory networks and have served as the main model for the application of intervention methods including optimal control strategies to favorably effect system dynamics since it is believed that the steady state behavior of a context sensitive pbn is indicative of the phenotype it is important to study the alternation in the steady state probability distribution due to any variations in the formulations of the context sensitive pbns furthermore the huge computational complexity of the context sensitive pbn model necessitates generation of size reduction techniques and approximate methods for calculation of the steady state probability distribution of context sensitive pbns the goal of this paper is threefold i to study the effects of the various definitions of context sensitive pbns on the steady state probability distributions and the downstream control policy design ii to propose a reduction technique that maintains the steady state probability distribution and iii to provide an approximation method for calculating the steady state probability distribution of a context sensitive pbn

stability and convergence of second order time discrete projection method for the linearized oldroyd model abstract in this paper we consider the second order time discrete projection method for the linearized oldroyd model based on the time iterative discrete scheme by the projection method the original problem is decoupled into two linear subproblems and each subproblem can be solved easily unconditional stability and the corresponding convergence results of the numerical solutions are derived our main results are that the convergence orders in time for the velocity in l 2 norm is second order and for the pressure in h 1 norm is first order finally some numerical examples are provided to verify the performances of the developed numerical method

the stability of local properties of fuzzy relations under ordinal equivalence abstract we consider a notion of equivalence relation between fuzzy relations and its properties the relation is called ordinal equivalence since it partitions the family of fuzzy relations into equivalence classes by means of the order defined on elements of each relation we compare fuzzy relations by their membership values and as a consequence we propose two types of local properties of fuzzy relations new properties of fuzzy relations are more compatible with the ordinal equivalence relation namely they are stable in the equivalence classes moreover we study dependencies between these new properties of fuzzy relations finally notes on applications of the presented local properties and the ordinal equivalence relation are provided