on number systems with finite degree of ambiguity abstract we show that it is decidable whether or not a given n recognizable set is representable by a number system having finite degree of ambiguity as a corollary we obtain an algorithm for computing the degree of ambiguity of a given set defined by a number system

a probabilistic approach to rank complex fuzzy numbers abstract techniques for ranking simple fuzzy numbers are abundant in the literature however we lack efficient methods for comparing complex fuzzy numbers that are induced by arithmetic operations in this paper a probabilistic approach is taken membership functions of fuzzy numbers are first converted into probability density functions the mellin transform is then introduced to compute the mean and the variance of a complex fuzzy number the fuzzy number with the higher mean is ranked higher if the means are equal the one with the smaller variance is judged higher rank two numerical examples and a fuzzy multiple attribute decision analysis are illustrated

linear numeration systems of order two abstract a numeration system is a sequence of integers such that any integer can be represented by means of the sequence using integers of bounded size we study numeration systems defined by linear recurrences of order two we give a necessary and sufficient condition on the system such that every integer has a canonical representation we show that this canonical representation can be computed from any representation by a rational function this rational function is the composition of two subsequential functions that are simply obtained from the system the addition of two integers represented in the system can be performed by a subsequential machine

fuzzy smoothness and sequences of fuzzy smooth functions abstract this paper is a continuation of the work of kalina tatra mountains math publ 14 1998 153 159 it further investigates fuzzy smoothness of fuzzy continuous but crisp functions finally sequences of fuzzy smooth functions and their sums are considered

bounded delay l codes abstract three natural notions of bounded delay for l codes are investigated all interrelations many of them rather surprising between ordinary codes of bounded delay and l codes of bounded delay of various types are established

on several definitions of the differential of a fuzzy mapping abstract several definitions of the differential of a fuzzy function are discussed in the restricted framework of fuzzy mappings from r to r

fuzzy conditional convergence and nearness relations this paper is concerned with fuzziness in basic mathematical structures such as sequences and series formalization of this fuzziness gives such concepts as conditional fuzzy limit conditional fuzzy convergence nearness limit and nearness convergence properties of these constructions and relations between them are obtained it is discussed how results and constructions of this paper are related to some principle problems of numerical computation

notes on fuzzy complex analysis several fuzzy complex analysis problems are discussed pointing out some errors in literature and some weaknesses of introduced concepts a modified approach to fuzzy complex analysis is proposed and its impact is discussed

approximate solutions existence and uniqueness of the cauchy problem of fuzzy differential equations abstract the cauchy problem of fuzzy differential equations is investigated by the use of the concept of h differentiability due to m l puri and d a ralescu j math anal appl 91 1983 552 558 the existence and uniqueness theorem is obtained for the solution of the cauchy problem x t f t x t x t 0 x 0 for the fuzzy valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in r n where the function f satisfies the generalized lipschitz condition

generalized lebesgue integrals of fuzzy complex valued functions in this paper on the basis of the theory of fuzzy complex numbers introduced by buckley we define fuzzy complex valued measurable functions fuzzy complex valued fuzzy measures and corresponding generalized lebesgue integrals of the fuzzy complex valued functions furthermore some basic properties and convergence theorems with respect to this kind of lebesgue integrals will be discussed

bases and ambiguity of number systems abstract the degree of ambiguity of a given number system can be effectively computed we show also that all degrees of ambiguity do exist both as regards number systems and the sets represented by them in contrast to language theory there are inherently unambiguous sets the paper contains also results concerning the possibility of changing the base of a number system

on the restudy of fuzzy complex analysis part 1 the sequence and series of fuzzy complex numbers and their convergences this paper studies the convergence of the sequence and series of fuzzy complex numbers we define the metric convergence and levelwise convergence of sequences on the metric space of fuzzy complex numbers and discuss the relationship between two convergences all this may be a foundation for researching fuzzy complex analysis

fuzzy complex analysis ii integration abstract this paper continues the author s research in fuzzy complex analysis where we now define and study the basic properties of a fuzzy contour integral for fuzzy mappings which map a rectifiable curve in the complex plane into fuzzy complex numbers

the cauchy problem for complex fuzzy differential equations abstract we discuss the existence of a solution to the cauchy problem for fuzzy differential equations that accommodates the notion of fuzzy sets defined by complex valued membership functions we first propose definitions of complex fuzzy sets and discuss entailed results which parallel those of regular fuzzy numbers we then give two existence results relevant to the cauchy problem for fuzzy differential equations in the case of bounded integral operators these results require either holder continuous or lipschitz continuous response functions

on the restudy of fuzzy complex analysis part ii the continuity and differentiation of fuzzy complex functions we consider functions mapping complex numbers into fuzzy complex numbers or mapping fuzzy complex numbers into complex numbers we study the continuity and differentiation of such functions

ancfis a neurofuzzy architecture employing complex fuzzy sets complex fuzzy sets cfss are an extension of type 1 fuzzy sets in which the membership of an object to the set is a value from the unit disc of the complex plane although there has been considerable progress made in determining the properties of cfss and complex fuzzy logic there has yet to be any practical application of this concept we present the adaptive neurocomplex fuzzy inferential system ancfis which is the first neurofuzzy system architecture to implement complex fuzzy rules and in particular the signature property of rule interference we have applied this neurofuzzy system to the domain of time series forecasting which is an important machine learning problem we find that ancfis performs well in one synthetic and five real world forecasting problems and is also very parsimonious experimental comparisons show that ancfis is comparable with existing approaches on our five datasets this work demonstrates the utility of complex fuzzy logic on real world problems

on fuzzy n cell numbers and n dimension fuzzy vectors fuzzy n cell numbers are a kind of special n dimension fuzzy numbers whose sets of cuts are all hyperrectangles n dimension fuzzy vectors are the cartesian products of n one dimension fuzzy numbers in this paper we show that the fuzzy n cell number space and the n dimension fuzzy vector space are identical and obtain the relations of the joint membership function and the edge membership functions of a fuzzy n cell number i e a n dimension fuzzy vector using fuzzy vector form representations of fuzzy n cell numbers we study the metric of two fuzzy n cell numbers the supremum of a set of fuzzy n cell numbers and operational rules of addition scalar multiplication and multiplication of fuzzy n cell numbers and get the fuzzy vector form characterizations of the differentiability measurability and integrability of fuzzy n cell number valued fuzzy mappings

some remarks for fuzzy complex analysis abstract this paper gives some results for fuzzy complex sets generalize and improve the corresponding work of fuzzy complex numbers in 1 we also introduce the derivative and integral of functions mapping complex numbers into the set of generalized fuzzy complex numbers and it is different to the definitions in 2 4 all this may be a foundation for researching fuzzy complex analysis

fuzzy time series and its models based upon the works by zadeh 1973 1975 in this paper a special dynamic process with linguistic values as its observations is defined and studied this dynamic process is called fuzzy time series to describe fuzzy time series fuzzy relational equations are employed as the models there are two classes of fuzzy time series time variant and time invariant the definition of fuzzy time series is given some properties of fuzzy time series are explored and procedures to develop fuzzy time series models are discussed finally a procedure for using fuzzy time series for foracasting is suggested

fuzzy complex numbers abstract this paper first defines fuzzy complex numbers and shows closure under the basic arithmetic operations it is shown that arithmetic of fuzzy complex numbers may be performed in terms of cuts next two special types of fuzzy complex numbers based on the forms z x iy and z re i for regular complex numbers are investigated introducing a metric on the space of fuzzy complex numbers then allows us to discuss continuity and differentiability of fuzzy complex functions

most odd degree hyperelliptic curves have only one rational point consider the smooth projective models c of curves y 2 f x with f x 2 z x monic and separable of degree 2g 1 we prove that for g 3 a positive fraction of these have only one rational point the point at innity we prove a lower bound on this fraction that tends to 1 as g 1 finally we show that c q can be algorithmically computed for such a fraction of the curves the method can be summarized as follows using p adic analysis and an idea of mccallum we develop a reformulation of chabauty s method that shows that certain computable conditions imply c q 1 on the other hand using further p adic analysis the theory of arithmetic surfaces a new result on torsion points on hyperelliptic curves and crucially the bhargava gross theorems on the average number and equidistribution of nonzero 2 selmer group elements we prove that these conditions are often satised for p 2

complex fuzzy logic a novel framework for logical reasoning termed complex fuzzy logic is presented in this paper complex fuzzy logic is a generalization of traditional fuzzy logic based on complex fuzzy sets in complex fuzzy logic inference rules are constructed and fired in a manner that closely parallels traditional fuzzy logic the novelty of complex fuzzy logic is that the sets used in the reasoning process are complex fuzzy sets characterized by complex valued membership functions the range of these membership functions is extended from the traditional fuzzy range of 0 1 to the unit circle in the complex plane thus providing a method for describing membership in a set in terms of a complex number several mathematical properties of complex fuzzy sets which serve as a basis for the derivation of complex fuzzy logic are reviewed in this paper these properties include basic set theoretic operations on complex fuzzy sets namely complex fuzzy union and intersection complex fuzzy relations and their composition and a novel form of set aggregation vector aggregation complex fuzzy logic is designed to maintain the advantages of traditional fuzzy logic while benefiting from the properties of complex numbers and complex fuzzy sets the introduction of complex valued grades of membership to the realm of fuzzy logic generates a framework with unique mathematical properties and considerable potential for further research and application

binary quartic forms having bounded invariants and the boundedness of the average rank of elliptic curves we prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants this extends to the quartic case the classical results of gauss and davenport in the quadratic and cubic cases respectively our techniques are quite general and may be applied to counting integral orbits in other representations of algebraic groups we use these counting results to prove that the average rank of elliptic curves over q when ordered by their heights is bounded in particular we show that when elliptic curves are ordered by height the mean size of the 2 selmer group is 3 this implies that the limsup of the average rank of elliptic curves is at most 1 5

easy multiplications i the realm of kleene s theorem abstract we study here the properties of a family of monoids which we call the rational monoids and which are monoids with a multiplication of low complexity a monoid is rational if its multiplication may be described by a rational function from a free monoid into itself the main results are that rational monoids like finite ones have the properties that green s relations d and f are equal and that kleene s theorem holds in rational monoids as in free ones every monoid described so far in which kleene s theorem holds is a rational monoid the closure of the family of rational monoids under rees quotient direct product and free product is then studied extensions of rational monoids will be considered in a forthcoming paper

fuzzy complex numbers representations operations and its analysis abstract as a key theoretical basis of fuzzy complex analysis a new representation of fuzzy complex numbers is proposed in this paper which unifies the exponential form trigonometric form and algebraic form of fuzzy complex numbers and some arithmetic operations are investigated the concepts of strong sum and strong difference are defined in order to simplify the addition and subtraction operations of complex fuzzy numbers and establish the calculus theory of fuzzy complex valued functions the metric between two fuzzy complex numbers the derivative differentiability and analyticity of fuzzy complex valued functions are defined and characterized the necessary and sufficient conditions of the analyticity of fuzzy complex valued functions are investigated furthermore the integral of fuzzy complex valued functions and cauchy integral theorem are given and discussed

a note on two point boundary value problems associated with non linear fuzzy differential equations in recent papers lakshmikantham et al and o regan et al assert that two point boundary value problem for a fuzzy differential equation is equivalent to an integral equation and the proofs of the main results in the above cited papers use this equivalence we prove by a counterexample that by using h differentiability or hukuhara differentiability this assertion does not hold moreover it is proved that a large class of two point boundary value problems have no solutions at all under h differentiability or hukuhara differentiability

notes on on the restudy of fuzzy complex analysis part i and part ii in this note we show by counterexamples that some results of qiu and colleagues on the restudy of fuzzy complex analysis part i the sequence and series of fuzzy complex numbers and their convergences fuzzy sets and systems 115 2000 445 450 on the restudy of fuzzy complex analysis part ii the continuity and differentiation of fuzzy complex functions fuzzy sets and systems 120 2001 517 521 concerning the convergence of the series of fuzzy complex numbers and the differentiation of fuzzy complex functions are incorrect and offer some of their modified versions we also show that the derivative introduced by them is inappropriate for the functions mapping complex numbers into fuzzy complex numbers

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

handling forecasting problems based on two factors high order fuzzy time series in our daily life people often use forecasting techniques to predict weather economy population growth stock etc however in the real world an event can be affected by many factors therefore if we consider more factors for prediction then we can get better forecasting results in recent years many researchers used fuzzy time series to handle prediction problems in this paper we present a new method to predict temperature and the taiwan futures exchange taifex based on the two factors high order fuzzy time series the proposed method constructs two factors high order fuzzy logical relationships based on the historical data to increase the forecasting accuracy rate the proposed method gets a higher forecasting accuracy rate than the existing methods

fuzzy limit theory of fuzzy complex numbers abstract we introduce three important concepts of fuzzy complex numbers fuzzy distance and fuzzy limit of fuzzy complex numbers and give some elementary properties of the fuzzy complex numbers and fuzzy distance and fuzzy limit of the fuzzy complex numbers we also discuss some important theorems of fuzzy complex numbers nested closed rectangles theorem cauchy s criterion for convergence accumulation principle etc

solving fuzzy equations a new solution concept abstract we have previously shown 2 that many fuzzy equations do not have solutions when the solution concept is based on the extension principle we therefore introduce two new solution procedures one based on the unified extension 6 and the other based on possibility theory after we solve the non fuzzy equation for the unknown variable we show for many types of equations that 1 the two new solutions are identical 2 the solution is either a real or generalized complex fuzzy number all uncertain parameters are modeled as real fuzzy numbers and 3 the previous solution based on the extension principle when it exists is a subset of the new solution in particular we show that the fuzzy quadratic equation with real fuzzy number coefficients always has a new solution

fuzzy complex analysis i differentiation abstract we consider functions mapping real numbers into the set of generalized complex fuzzy numbers c we define two derivatives of such functions which are generalizations of the derivative of real fuzzy mappings presented by dubois and prade 4 5 sufficient conditions on the function are given so that derivative will be a function mapping real numbers into c if the functional value is always a rectangular complex fuzzy number made up to lr real fuzzy numbers then we show that the derivative is easily computed from the derivatives of the lr fuzzy numbers other properties of the derivatives are presented

ellipsoids and ellipsoid shaped fuzzy sets as natural multi variate generalization of intervals and fuzzy numbers how to elicit them from users and how to use them in data processing in this paper we show that ellipsoids are natural multi variate generalization of intervals and ellipsoid shaped fuzzy sets are a natural generalization of fuzzy numbers we explain how to elicit them from users and how to use them in data processing

a method for multiple periodic factor prediction problems using complex fuzzy sets multiple periodic factor prediction mpfp problems exist widely in multisensor data fusion applications development of an effective prediction method should integrate information for multiple periodically changing factors because the uncertainty and periodicity coexist in the information used the prediction method should be able to handle them simultaneously in this study complex fuzzy sets are used to represent the information with uncertainty and periodicity a product sum aggregation operator psao is developed for a set of complex fuzzy sets which is used to integrate information with uncertainty and periodicity and a psao based prediction psaop method is then proposed to generate a solution of mpfp problems this study illustrates the details of the psaop method through two real applications in annual sunspot number prediction and bushfire danger rating prediction experiments indicate that the proposed psaop method effectively handles the uncertainty and periodicity in the information of multiple periodic factors simultaneously and can generate accurate predictions for mpfp problems

remarks on peano like theorems for fuzzy differential equations the validity of the classical peano theorem is noted for differential equations on the metric space en dp of normal fuzzy convex sets in rn where dp is the p th mean of the hausdorff distances between corresponding level sets this contrasts with differential equations on the space en d where d is the max metric which is not locally compact so more than just continuity is required to ensure the existence of solutions some additional properties differing from that given by kaleva are mentioned

numeration systems linear recurrences and regular sets a numeration system based on a strictly increasing sequence of positive integers u0 1 u1u2 expresses a non negative integer n as a sum n j 0 i ajuj in this case we say the string a i a i 1 a1 a0 is a representation for n

fuzzy continuous function and its properties abstract in this paper we introduce the concepts of fuzzy distance of fuzzy numbers fuzzy limit of fuzzy numbers fuzzy function and fuzzy limit of a fuzzy function and fuzzy continuous function on the set of fuzzy numbers and give some of their elementary properties we also discuss some important theorems of fuzzy numbers and the fuzzy continuous function on the m closed interval

comment on pythagorean and complex fuzzy set operations dick et al introduced a relation leq on pi i numbers and proved that pi i leq is a bounded distributive complete lattice in the present paper we point out that the relation leq is not a partial order and that pi i leq is not a lattice in addition we study some partial orders on pi i numbers