a note on fuzzy relational equations with min implication composition the fuzzy relational equations with min implication composition are considered where the implication operation involved is the residuum with respect to the minimum operation the solution set of such a system of equations whenever nonempty can be characterized by a minimum solution and finitely many maximal solutions it is shown that a system of fuzzy relational equations with min implication composition can be reformulated in polynomial time into a system of 0 1 mixed integer linear inequalities and consequently the structure of its solution set can be revealed in a succinct manner besides it is shown that the determination of all maximal solutions is polynomially reducible to the transversal hypergraph generation problem of which some practically well performed algorithms are available

some kinds of view the fuzzy filters of bl algebras the concepts of view the mathml source fuzzy implicative positive implicative and fantastic filters of bl algebras are introduced and some related properties are investigated some characterizations of these generalized fuzzy filters are derived in particular we describe the relationships among ordinary fuzzy implicative positive implicative and fantastic filters view the mathml source fuzzy implicative positive implicative and fantastic filters and view the mathml source fuzzy implicative positive implicative and fantastic filters of bl algebras finally we prove that a fuzzy set f of a bl algebra l is an view the mathml source fuzzy implicative filter of l if and only if it is both an view the mathml source fuzzy positive implicative filter and an view the mathml source fuzzy fantastic filter

fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max min composition in this paper we study the new linear objective function optimization with respect to the fuzzy relational inequalities defined by max min composition in which fuzzy inequality replaces ordinary inequality in the constraints fuzzy inequalities enable us to attain the optimal points that are better solutions than those resulting from the resolution of the similar problems with ordinary inequality constraints denoted by x o i in this paper we have presented an algorithm to generate such optimal solutions also two examples are given to illustrate the algorithm presented and some real application of this work

error estimation of perturbations under cri the analysis of stability and robustness of fuzzy reasoning is an important issue in areas like intelligent systems and fuzzy control an interesting aspect is to what extent the perturbation of input in a fuzzy reasoning scheme causes the oscillation of the output in particular when the error limits restrictions of the input values are given what the error limits of the output values are in this correspondence we estimate the upper and lower bounds of the output error affected by the perturbation parameters of the input and obtain the limits of the output values when the input values range over some interval in many fuzzy reasoning schemes under compositional rule of fuzzy inference cri

indecomposable higher chow cycles on jacobians we construct some natural indecomposable elements of ch g j c 1 with trivial regulator and in particular prove that ch g ind j c 1 mb q is uncountable for c a generic curve or a generic hyperelliptic curve of genus g geq 3

interpreting and extracting fuzzy decision rules from fuzzy information systems and their inference information systems which contain only crisp data precise and unique attribute values for all objects have been widely investigated due to the fact that in realworld applications imprecise data are abundant uncertainty is inherent in real information systems in this paper information systems are called fuzzy information systems and formalized by objects attributes f in which f is a fuzzy set and expresses some uncertainty between an object and its attribute values to interpret and extract fuzzy decision rules from fuzzy information systems the meta theory based on modal logic proposed by resconi et al is modified the modified meta theory not only expresses uncertainty between objects and their attributes but also uncertainty in the process of recognizing fuzzy information systems in addition according to perception computing proposed by zadeh granules of fuzzy information systems can be represented by fuzzy decision rules so that fuzzy inference methods can be used to obtain the decision attribute of a new object finally a novel way of combining evidences based on the modified meta theory is introduced which extends the concept of combining evidences based on dempster shafer theory

on the local convergence of a fifth order iterative method in banach spaces a new predictor corrector iterative procedure that combines newton s method as predictor scheme and a fifth order iterative method as a corrector is designed for solving nonlinear equations in banach spaces we analyze the local order of convergence and the regions of accessibility of the new method comparing it with newton s method both theoretical and numerically

efficient high order methods based on golden ratio for nonlinear systems abstract we derive new iterative methods with order of convergence four or higher for solving nonlinear systems by composing iteratively golden ratio methods with a modified newton s method we use different efficiency indices in order to compare the new methods with other ones and present several numerical tests which confirm the theoretical results

resolution of fuzzy relation equations i based on boolean type implications the problem of solving fuzzy relation equations i based on boolean type implications is studied in the present paper decomposition of fuzzy relation equations i based on boolean type implications is presented in a finite case the solution existence of fuzzy relation equations i based on boolean type implications is studied and for nice boolean type implications some new solvability criteria based upon the notion of solution matrices are discussed the complete solution set of fuzzy relation equation 1 based on boolean type implication can be determined by the maximal solution set of this equation which is finite an effective method to solve fuzzy relation equations i based on boolean type implications is proposed

approximation theorem of the fuzzy transform in fuzzy reasoning and its application to the scheduling problem in this paper the theoretical foundation of fuzzy reasoning is analyzed and the idea that the fuzzy transform given in the fuzzy reasoning method should be continuous with respect to a certain fuzzy distance is proposed also the fuzzy transforms given in the two fuzzy reasoning methods the mamdani method and the iii method are proved to be continuous based on the continuity of the fuzzy transform the approximation theorem of the continuous fuzzy number transform is proven then on the basis of the approximation theorem a simple fuzzy number transform is constructed in order to implement the fuzzy reasoning based on multiple rules at last the fuzzy reasoning based on multiple rules implemented by a simple fuzzy number transform is applied to machine scheduling problems and numerical computational results of different scale scheduling problems with the objective of minimizing the total number of tardy jobs show that it is more effective than usual heuristics based on rules and in the computational time it has the obvious advantage over the reasoning by fuzzy rules directly

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

some types of generalized fuzzy filters of bl algebras the aim of this paper is to introduce the notions of interval valued q fuzzy boolean implicative positive implicative and fantastic filters in bl algebras and to investigate some related properties some characterizations of these generalized fuzzy filters are derived finally the relations among these generalized fuzzy filters are discussed it is proved that an interval valued q fuzzy filter in bl algebras is an interval valued q fuzzy boolean implicative filter if and only if it is both an interval valued q fuzzy positive implicative filter and an interval valued q fuzzy fantastic filter

normal functions picard fuchs equations and elliptic fibrations onk3 surfaces using gauss manin derivatives of generalized normal functions we arrive at results on the non triviality of the transcendental regulator for km of a very general projective algebraic manifold our strongest results are for the transcendental regulator for k1 of a very general k3 surface and its self product we also construct an explicit family of k1 cycles on h e8 e8 polarized k3 surfaces and show they are indecomposable by a direct evaluation of the real regulator critical use is made of natural elliptic fibrations hypersurface normal forms and an explicit parametrization by modular functions

basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations several optimal eighth order methods to obtain simple roots are analyzed the methods are based on two step fourth order optimal methods and a third step of modified newton the modification is performed by taking an interpolating polynomial to replace either f z n or f z n in six of the eight methods we have used a hermite interpolating polynomial the other two schemes use inverse interpolation we discovered that the eighth order methods based on jarratt s optimal fourth order methods perform well and those based on king s or kung traub s methods do not in all cases tested the replacement of f z by hermite interpolation is better than the replacement of the derivative f z

ball convergence comparison between three iterative methods in banach space under hypothese only on the first derivative we present a convergence ball comparison between three iterative methods for approximating a locally unique solution of a nonlinear equation in a banach space setting the convergence ball and error estimates are given for these methods under hypotheses only on the first frechet derivative in contrast to earlier studies such as adomian 1994 1 babajee et al 2008 13 cordero and torregrosa 2007 17 cordero et al 18 darvishi and barati 2007 19 using hypotheses reaching up to the fourth frechet derivative although only the first derivative appears in these methods this way we expand the applicability of these methods numerical examples are also presented in this study

an algorithm for solving fuzzy relation equations with max t composition operator this paper studies the problem of solving a max t composite finite fuzzy relation equation where t is a special class of pseudo t norms if the equation is solvable then its set of feasible solutions is determined by the greatest solution and a finite number of minimal solutions some necessary conditions are presented for the minimal solutions in terms of the maximum solution and zero value under these conditions some minimal solutions of the system can be obtained easily some procedures are also proposed in order to simplify the original system the simplified system is then decomposed if possible into several subsystems with smaller dimensions which are very easy to solve furthermore a method is presented to solve each subsystem by combining the method and those procedures an efficient algorithm is proposed to obtain the set of feasible solutions of the original system two examples are also given to illustrate the algorithm

a new tool to study real dynamics in this paper the author presents a graphical tool that allows to study the real dynamics of iterative methods whose iterations depends on one parameter in an easy and compact way this tool gives the information as previous tools such as feigenbaum diagrams and lyapunov exponents for every initial point the convergence plane can be used inter alia to find the elements of a family that have good convergence properties to see how the basins of attraction changes along the elements of the family to study two point methods such as secant method or even to study two parameter families of iterative methods to show the applicability of the tool an example of the dynamics of the damped newton s method applied to a cubic polynomial is presented in this paper

dynamical aspects of some convex acceleration methods as purely iterative algorithm for newton s maps in this paper we define purely iterative algorithm for newton s maps which is a slight modification of the concept of purely iterative algorithm due to smale for this we use a characterization of rational maps which arise from newton s method applied to complex polynomials we prove the scaling theorem for purely iterative algorithm for newton s map then we focus our study in dynamical aspects of three root finding iterative methods viewed as a purely iterative algorithm for newton s map whittaker s iterative method the super halley iterative method and a modification of the latter we give a characterization of the attracting fixed points which correspond to the roots of a polynomial also numerical examples are included in order to show how to use the characterization of fixed points finally we give a description of the parameter spaces of the methods under study applied to a one parameter family of generic cubic polynomials

a note on resolution of fuzzy relation equations i based on boolean type implications this paper shows by examples that lemma 2 1 lemma 3 1 corollary 3 1 corollary 3 2 theorem 3 1 corollary 3 3 theorem 3 3 and lemma 4 1 in y luo resolution of fuzzy relation equation i based on boolean type implications comput math appl 52 2006 421 428 about resolution of fuzzy relation equations based on boolean type implications are false we also point out that the solution set of example 5 1 is not right and we give the right solution set of example 5 1

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

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

approximation theory of fuzzy systems based upon genuine many valued implications siso cases it is proved that the single input and single output siso fuzzy systems based upon genuine many valued implications are universal approximators it is shown theoretically that fuzzy control systems based upon genuine many valued implications are equivalent to those based upon t norm implications the general approach to construct fuzzy systems is given it is also shown that defuzzifier based upon center of areas is not appropriate to the fuzzy systems based upon genuine many valued implications

variable universe stable adaptive fuzzy control of a nonlinear system abstract a kind of stable adaptive fuzzy control of a nonlinear system is implemented based on the variable universe method proposed first in 1 first of all the basic structure of variable universe adaptive fuzzy controllers is briefly introduced then the contraction expansion factor which is a key tool of the variable universe method is defined by means of the integral regulation idea and then a kind of adaptive fuzzy controller is designed by using such a contraction expansion factor the simulation on the first order nonlinear system is done and as a result its simulation effect is quite good in comparison with the corresponding result in 2 3 second it is proved that the variable universe adaptive fuzzy control is asymptotically stable by use of liapunov theory the simulation on a second order nonlinear system shows that its simulation effect is also quite good in comparison with the corresponding result in 2 besides a useful tool called symbolic factor is proposed which may be of universal significance it can greatly reduce the setting time and enhance the robustness of the system

linear fractional multi objective optimization problems subject to fuzzy relational equations with a continuous archimedean triangular norm in this paper linear fractional multi objective optimization problems subject to a system of fuzzy relational equations fres using the max archimedean triangular norm composition are considered first some theorems and results are presented to simplify systems of fuzzy relational equations next the feasible set of the optimization problem is reduced then the linear fractional multi objective optimization problem is converted to a linear one using nykowski and zolkiewski s approach furthermore the efficient solutions are obtained by applying the improved e constraint method in addition we design an algorithm to generate consistent systems of fuzzy relational equations randomly using the covering problem finally the proposed method is effectively tested by solving a randomly generated consistent test problem

a family of higher order iterations free from second derivative for nonlinear equations in r abstract the aim of this article is to develop a family of higher order iterations free from second derivative for solving nonlinear equations in r their theoretical computational and dynamical aspects are fully investigated and theorems are established to provide their order of convergence and asymptotic error constant it is observed that the family includes sixth order methods and for a particular case its eighth order can be achieved in this family methods use three functions and one first derivative evaluations the family of methods can be shown to be optimal by using kung traub conjecture a number of numerical examples are worked out to demonstrate the applicability of these methods the improved results are obtained in comparison to some of the existing robust methods on the considered test examples local convergence analysis and dynamical study of the proposed family of methods are also carried out

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

the generalized truth degree of quantitative logic in the logic system l n n valued nm logic system the concept of the truth degree of a formula is the crucial tool and the building block in quantitative logic from which the concept of logic metric in quantitative logic is derived logic metric takes an important role in quantitative logic related to which are other concepts in quantitative logic such as divergence consistency etc in the present paper having combined the theory of generalized tautologies with the theory of truth degrees in quantitative logic we have proposed the theory of s c truth degrees of formulas related to theory c in the logic system l n n valued nm logic system and discussed some of its properties with the help of the properties of s c truth degrees t c a email protected c a b email protected c b we have obtained the c logic metric on the set f s of formulas in the propositional logic system l n n valued nm logic system by the work of this paper we can generalize the theory of quantitative logic in all round way and establish an approximate reasoning s framework related to theory c in the logic system l n n valued nm logic system

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

fractal basins of attraction associated with a damped newton s method an intriguing and unexpected result for students learning numerical analysis is that newton s method applied to the simple polynomial z 3 1 0 in the complex plane leads to intricately interwoven basins of attraction of the roots as an example of an interesting open question that may help to stimulate student interest in numerical analysis we investigate the question of whether a damping method which is designed to increase the likelihood of convergence for newton s method modifies the fractal structure of the basin boundaries the overlap of the frontiers of numerical analysis and nonlinear dynamics provides many other problems that can help to make numerical analysis courses interesting

fuzzy reasoning based on a new fuzzy rough set and its application to scheduling problems in this paper we study the fuzzy reasoning based on a new fuzzy rough set first we define a broad family of new lower and upper approximation operators of fuzzy sets between different universes using a set of axioms then based on the approximation operators above we propose the fuzzy reasoning based on the new fuzzy rough set by means of the above fuzzy reasoning based on the new fuzzy rough set for a given premise we can obtain the fuzzy reasoning consequence expressed by the fuzzy interval constructed by the above two approximations of fuzzy sets furthermore through the defuzzification of the lower and upper approximations we can get the corresponding two values constructing the interval used as the fuzzy reasoning consequence after defuzzification then from the above interval a suitable value can be selected as the final reasoning consequence so that some special constraints are satisfied as possibly at last we apply the fuzzy reasoning based on the new fuzzy rough set to the scheduling problems and numerical computational results show that the fuzzy reasoning based on the new fuzzy rough set is more suitable for the scheduling problems compared with the fuzzy reasoning based on the cri method and the iii method

the general α decomposition problem of fuzzy relations abstract this paper deals with a general decomposition problem of fuzzy relations which can be stated as follows given a fuzzy relation r f x y determine two fuzzy relations q f x z and t f z y such that r q t where x resp y is a finite set firstly we point out that every fuzzy relation r is always generally decomposable and give an algorithm to construct q and t with r q t for a given r secondly we show that the general content r with r min z r q t q f x z t f z y is equal to the chromatic number of the simple graph f r generated by r therefore finding an exact algorithm for calculating r is an np complete problem

soft bl algebras based on fuzzy sets by means of soft sets and q soft sets some characterizations of implicative positive implicative and fantastic filteristic soft bl algebras are investigated finally we prove that a soft set is an implicative filteristic soft bl algebra if and only if it is both a positive implicative filteristic soft bl algebra and a fantastic filteristic soft bl algebra

necessary and sufficient conditions for the existence of solution of generalized fuzzy relation equations a x b abstract in 2013 li and jin studied a particular type of fuzzy relational equations on finite sets where the introduced min bi implication composition is based on ukasiewicz equivalence in this paper such fuzzy relation equations are studied on a more general level namely complete residuated lattice valued fuzzy relation equations of type y y a x y x y b x are analyzed and the existence of solutions s is studied first a necessary condition for the existence of solution is established then conditions for lower and upper limits of solutions are given and finally sufficient conditions for the existence of the smallest and largest solutions respectively are characterized if such general or global solutions do not exist there might still be partial or point wise solutions this is a novel way to study fuzzy relation equations such point wise solutions are studied on ukasiewicz product and godel t norm based residuated lattices on the real unit interval

fuzzy reasoning as a control problem different from the dominant view of treating fuzzy reasoning as generalization of classical logical inference in this paper fuzzy reasoning is treated as a control problem a new fuzzy reasoning method is proposed that employs an explicit feedback mechanism to improve the robustness of fuzzy reasoning the fuzzy rule base given a priori serves as a controlled object and the fuzzy reasoning method serves as the corresponding controller the fuzzy rule base and the fuzzy reasoning method constitute a control system that may be open loop or closed loop depending on the underlying reasoning goals constraints the fuzzy rule base the fuzzy reasoning method and the corresponding reasoning goals constraints define the three distinct ingredients of fuzzy reasoning while various existing fuzzy reasoning methods are essentially a static mapping from the universe of single fuzzy premises to the universe of single fuzzy consequences the new fuzzy reasoning method maps sequences of fuzzy premises to sequences of fuzzy consequences and is a function of the underlying reasoning goals constraints the monte carlo simulation shows that the new fuzzy reasoning method is much more robust than the optimal fuzzy reasoning method proposed in our previous work the explicit feedback mechanism embedded in the fuzzy reasoning method does significantly improve the robustness of fuzzy reasoning which is concerned with the effects of perturbations associated with given fuzzy rule bases and or fuzzy premises on fuzzy consequences the work presented in this paper sets a new starting point for various principles of feedback control and optimization to be applied in fuzzy reasoning or logical inference and to explore new forms of reasoning including robust reasoning and adaptive reasoning it can be also expected that the new fuzzy reasoning method presented in this paper can be used for modeling and control of complex systems and for decision making under complex environments

decomposition and resolution of min implication fuzzy relation equations based on s implications the problem of solving min implication fuzzy relation equations based on s implications is studied in this paper decomposition of min implication fuzzy relation equations is given in a finite case the solution existence problem of min implication fuzzy relation equations is discussed and for ns implications a new solvability criterion based upon solution matrices introduced in this paper is studied it is shown that for each solution r of a min implication fuzzy relation equation there exists a maximal solution r of this equation such that r is greater than or equal to r whenever the solution set of this equation is nonempty the complete solution set of min implication fuzzy relation equation can be determined by the maximal solution set of this equation which is finite an effective method to solve min implication fuzzy relation equations based on s implication is proposed

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

the extensions l n of the formal system l n and their completeness in this paper the method of well combined semantics and syntax proposed by pavelka is applied to the research of the propositional calculus formal system l the part of constant values are taken as formulas formulas are fuzzified in two manners semantically and syntactically and inferring processes are fuzzified a sequence of new extensions l n of the system l is proposed and the completeness of l n is proved

fuzzy implicative and boolean filters of r 0 algebras the notions of fuzzy filters fuzzy implicative filters and fuzzy boolean filters are introduced in r0 algebras respectively some of their properties are investigated the relations between fuzzy implicative filters and fuzzy boolean filters are discussed the characterizations of fuzzy implicative filters are studied and the following is proved to be equivalent 1 a fuzzy filter f is implicative 2 f is a fuzzy boolean filter 3 x y z l f x z f x z y f y z 4 x y l f x y f x y y 5 x l n 1 f nx f x

robustness of fuzzy reasoning via logically equivalence measure abstract in this paper we discuss robustness of fuzzy reasoning after proposing the definition of perturbation of fuzzy sets based on some logic oriented equivalence measure we present robustness results for various fuzzy logic connectives fuzzy implication operators inference rules and fuzzy reasoning machines and discuss the relations between the robustness of fuzzy reasoning and that of fuzzy conjunction and implication operators the robustness results are presented in terms of equalities of fuzzy sets based on some logic oriented equivalence measure and the maximum of which ensures the corresponding equality holds is derived

perturbation of fuzzy sets and fuzzy reasoning based on normalized minkowski distances in this paper we extend the concept of the perturbation of fuzzy sets based on normalized minkowski distances and present some new conclusions on perturbation raised by various operations of fuzzy sets these operations are induced by triangular norms and conorms furthermore we discuss the perturbation of fuzzy reasoning

non fuzzy versions of fuzzy reasoning in classical logics abstract fuzzy reasoning features fuzzy logic in its narrow sense fln and fuzzy modus ponens fmp is the core of fuzzy reasoning as is well known fmp is far more general and complicated than the conventional rule of modus ponens mp and takes the latter as special case under certain conditions this paper proves that there exist non fuzzy versions of fmp in classical 2 valued logic as well as in conventional multivalued systems and reveals the roots of these seemingly surprising results

weak convergence conditions for the newton s method in banach space using general majorizing sequences using the majorizing principle we develop sufficient convergence conditions for the semilocal convergence of newton s method in a banach space setting the involved majorizing sequences are very general and as special cases include most relevant sequences from the published literature our convergence conditions can be weaker and our error analysis can be finer than before

unified full implication algorithms of fuzzy reasoning this paper discusses the full implication inference of fuzzy reasoning for all residuated implications induced by left continuous t norms unified a triple i algorithms are constructed to generalize the known results as the corollaries of the main results of this paper some special algorithms can be easily derived based on four important residuated implications these algorithms would be beneficial to applications of fuzzy reasoning based on properties of residuated implications the proofs of the many conclusions are greatly simplified

complete solution sets of inf interval valued fuzzy relation equations fuzzy relation equations play an important role in fuzzy set theory interval valued fuzzy set theory is an extension of fuzzy theory in which a closed subinterval of the unit interval is assigned with membership degree therefore it is very significant to study interval valued fuzzy relation equations from both the theoretical and practical viewpoints in this paper the solution sets of interval valued fuzzy relation equations with inf composition is investigated where is interval valued r s or ql implication necessary and sufficient conditions such that there exist solutions for these equations are first shown some sufficient conditions for existence of maximal solutions for these equations are represented and then it is shown that the complete solution sets of inf interval valued fuzzy relation equations can be determined by their maximal solutions finally the solution sets of linear interval valued fuzzy relation equations are described by a method similar to that in linear algebra

graphical representations for the homogeneous bivariate newton s method in this paper we propose a new and effective strategy to apply newton s method to the problem of finding the intersections of two real algebraic curves that is the roots of a pair of real bivariate polynomials the use of adequate homogeneous coordinates and the extension of the domain where the iteration function is defined allow us to avoid some numerical difficulties such as divisions by values close to zero in fact we consider an iteration map defined on a real augmented projective plane so we obtain a global description of the basins of attraction of the fixed points associated to the intersection of the curves as an application of our techniques we can plot the basins of attraction of the roots in the following geometric models hemisphere hemicube mobius band square and disk we can also give local graphical representations on any rectangle of the plane

fuzzy reasoning method by optimizing the similarity of truth tables this paper presents a new fuzzy reasoning method by optimizing the similarity of truth tables os method for short its basic idea is to find a fuzzy set such that the truth tables generated by the antecedent rule and the consequent rule are as similar as possible based on this idea the principle of os method and the fuzzy reasoning with os method are given and discussed and then the os methods with certain similarity measure and several fuzzy implications are investigated finally numerical examples are analyzed to compare the proposed method with compositional rule of inference cri method

the r0 type fuzzy logic metric space and an algorithm for solving fuzzy modus ponens fuzzy modus ponens briefly fmp is the most fundamental form of fuzzy reasoning and has been extensively discussed by diverse researchers the aim of the present paper is to propose a formalized form of fmp called generalized modus ponens in the fuzzy logic system l and solve it in l and then provide its numerical version as a new algorithm for solving fmp as a preparation some related questions such as what is a fuzzy logic metric space why is r 0 implication operator selected and what kind of merits does the fuzzy pseudo metric logic space f s r possess etc are analyzed

the gauss seidelization of iterative methods for solving nonlinear equations in the complex plane abstract in this paper we introduce a process we have called gauss seidelization for solving nonlinear equations we have used this name because the process is inspired by the well known gauss seidel method to numerically solve a system of linear equations together with some convergence results we present several numerical experiments in order to emphasize how the gauss seidelization process influences on the dynamical behavior of an iterative method for solving nonlinear equations

robustness of fuzzy reasoning and spl delta equalities of fuzzy sets fuzzy reasoning methods or approximate reasoning methods are extensively used in intelligent systems and fuzzy control in this paper the author discusses how errors in premises affect conclusions in fuzzy reasoning that is he discusses the robustness of fuzzy reasoning after reviewing his previous work 1996 he presents robustness results for various implication operators and inference rules all the robustness results are formulated in terms of spl delta equalities of fuzzy sets two fuzzy sets are said to be spl delta equal if they are equal to an extent of spl delta

a fuzzy logic for an ordinal sum t norm among the class of residuated fuzzy logics a few of them have been shown to have standard completeness both for propositional and predicate calculus like godel nm and monoidal t norm based logic systems in this paper a new residuated logic nmg which aims at capturing the tautologies of a class of ordinal sum t norms and their residua is introduced and its standard completeness both for propositional calculus and for predicate calculus are proved

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

the quintuple implication principle of fuzzy reasoning we propose the quintuple implication principle qip for fuzzy reasoning we applies qip to solve fmp and fmt for residual implications the mamdani type fuzzy reasoning is logical interpreted from the viewpoint of qip the qip is applied to multiple fuzzy rules fuzzy modus ponens fmp and fuzzy modus tollens fmt are two fundamental patterns of approximate reasoning suppose a and b are fuzzy predicates and if a then b is a fuzzy rule approximate reasoning often requires to derive an approximation b of b from a given approximation a of a or vice versa to solve these problems zadeh introduces the well known compositional rule of inference cri which models fuzzy rule by implication and computes b a resp by composing a b resp with a b wang argues that the use of the compositional operation is logically not sufficiently justified and proposes the triple implication principle tip instead both cri and tip do not explicitly use the closeness of a and a or that of b and b in the process of calculating the consequence which makes the thus computed approximation sometimes useless or misleading in this paper we propose the quintuple implication principle qip for fuzzy reasoning which characterizes the approximation b of b a of a resp as the formula which is best supported by a b a a and a a b b b and b resp based upon monoidal t norm logic mtl this paper applies qip to solve fmp and fmt for four important implications most importantly we show that qip when using godel implication computes exactly the same approximation as mamdani type fuzzy inference does this is surprising as mamdani interprets fuzzy rules in terms of the minimum operation while cri tip and qip all interpret fuzzy rules in terms of implication

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

some properties of fuzzy reasoning in propositional fuzzy logic systems in order to analyze the logical foundation of fuzzy reasoning this paper first introduces the concept of generalized roots of theories in lukasiewicz propositional fuzzy logic luk godel propositional fuzzy logic god product propositional fuzzy logic p and nilpotent minimum logic nm the r 0 propositional fuzzy logic l next it is proved that all consequences of a theory c named d c are completely determined by its generalized root whenever c has a generalized root moreover it is proved that every finite theory c has a generalized root which can be expressed by a specific formula finally we demonstrate the existence of a non fuzzy version of fuzzy modus ponens fmp in luk god p and nm l and we provide its numerical version as a new algorithm for solving fmp