fuzzy n cell numbers and the differential of fuzzy n cell number value mappings in this paper we give the definition of a special kind of n dimension fuzzy numbers fuzzy n cell numbers discuss their operations and representation theorems define a complete metric on the fuzzy n cell number space and prove that the metric is equivalent to the supremum metric derived by the hausdorff metric between the level sets of the n dimension fuzzy numbers and obtain an embedding theorem of the fuzzy n cell number space isometrically embeds it into a concrete banach space we also consider the differential of the fuzzy mappings from an interval into the fuzzy n cell number space by using the embedding theorem

positive linear operators with equidistant nodes abstract in the present paper the approximation power of positive linear operators with equidistant nodes is investigated new pointwise estimates are given in terms of first and second order moduli of continuity showing that for positive operators having uniform jackson orders of approximation one may expect interpolation at the endopoints in particular the first solution to butzer s problem with equidistant nodes is given a negative result yields an explanation why bernstein operators and some of their modifications are optimal in a certain sense

estimation of a fuzzy regression model using fuzzy distances regression analysis is a powerful statistical tool that has many applications in different areas the problem of regression analysis under a fuzzy environment has been treated in the literature from different points of view and considering a variety of input output data crisp or fuzzy however we realize that in general most research papers have a conflict between the solution of the fuzzy regression problem using crisp distances minimizing a real error function and the interpretation of fuzzy data as possibility distributions the main aim of this paper is to develop a methodology to solve this problem introducing a fuzzy partial order and a family of fuzzy distance measures on the whole set of fuzzy numbers the new approach allows us to obtain linear and nonlinear models that reach the lowest fuzzy error the estimation process in general can be considered easier to apply in practice and it is not limited to triangular fuzzy numbers numerical examples are provided to illustrate the usefulness and applicability of these results and comparisons with existing methodologies show that the performance of the proposed solution is very satisfactory

quantitative korovkin type theorems on simultaneous approximation let k a b be a compact interval of the real axis and k c d be a subinterval of k by x c k r o we denote the banach space of realvalued and r times continuously differentiable functions on k equipped with norm given by ilghx max idjgl k here d is the j th differential operator o j r and ii l g denotes the cebygev norm in c k c moreover for ienw 0 the function ei n x xien is the i th monomial if hr 1 denotes the subspace of cr k spanned by e l then extending a well known result of shisha and mond 21 knoop and pottinger 8 proved

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

the theoretical fundamentals of learning theory based on fuzzy complex random samples statistical learning theory based on real valued random samples has been regarded as one of the influential developments for small samples statistical estimation and learning the key theorem of learning theory and the bounds on the rate of convergence of learning process are the most important theoretical fundamentals of the statistical learning theory in this paper we discuss a statistical learning theory based on fuzzy complex random samples firstly the definition of fuzzy complex numbers is introduced and the fuzzy complex random variables along with their numeric characteristic are investigated secondly we carry out further research focused on a special type of fuzzy complex number namely rectangular fuzzy complex number and establish some properties and develop important theorems we also prove the strong law of large numbers based on fuzzy complex random variables thirdly the definitions of the fuzzy complex expected risk functional the fuzzy complex empirical risk functional the fuzzy complex empirical risk minimization principle and the consistency are provided and discussed finally the key theorem of learning theory and the bounds on the rate of convergence of learning process based on fuzzy complex random samples are discussed

set valued choquet integrals revisited this paper performs a further investigation on the set valued choquet integral given by jang and kwon fuzzy sets and systems 112 2000 233 some incorrect properties are revised it is pointed out that the definition for convergence of sequence of sets or set valued functions is unreasonable by using the usual kuratowski s convergence some new convergence theorems for the set valued choquet integral are shown meanwhile some other properties are obtained

zero estimates on group varieties ii the aim of this article is to give an account of a class of estimates that have in recent years been found useful in the theory of transcendental numbers as our space is limited and the estimates are quite technical in nature wc can provide no hints of proofs and we prefer to illustrate the results with examples rather than supply full statements we also mention some of the applications one basic method in transcendence theory can be described in general terms as follows suppose it is desired to establish a certain proposition usually it will be possible to associate with the problem a collection of meromorphic functions ft fnm such a way that assuming the falsity of the proposition imposes strong arithmetical conditions on the functions these enable us to construct a polynomial p p x1 xn such that the function f p f1 fn is not identically zero but has a large number of zeroes by means of analytic interpolation arguments it can then be deduced that takes many small values and by using once more the arithmetical conditions we conclude that has even more zeroes than initially arranged but often merely knowing upper bounds for the degree of p leads to upper bounds for the number of zeroes of and if we are lucky these yield the contradiction that proves the original proposition in this article we shall limit attention to the last stage of this argument as it is here that the most interesting recent progress has taken place a simple example will suffice to illustrate the sort of estimates involved to show that w and e cannot both be algebraic for w 0 suppose they are then we end up with a non zero polynomial p p x y of degree at most l 1 in x and of degree at most jf 1 in y such that the function

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

algebraic properties and topological properties of the quotient space of fuzzy numbers based on mares equivalence relation abstract in this paper we obtain some algebraic properties and topological properties of the quotient space of fuzzy numbers with respect to the equivalence relation defined by mares every fuzzy number has only one mares core and equivalent fuzzy numbers have the same mares core in addition equivalence classes of fuzzy numbers can be only expressed as the sum of its mares core and the set of symmetric fuzzy numbers which shows the notable difference between the equivalence classes of fuzzy numbers and the cosets of the normal subgroup in a group based on these results we introduce a new concept of convergence under which the quotient space is complete as an application of the main results it is shown that if we identify every fuzzy number with the corresponding equivalence class there would be more differentiable fuzzy functions than what is found in the literature

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

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

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

uniform convergence of regularization methods for linear ill posed problems abstract after a general discussion about convergence and convergence rates for regularization methods in banach spaces we present a general method that can be used to modify regularization methods in l2 in such a way that uniform convergence which is often preferred in concrete applications to just l2 convergence is obtained we prove results about convergence rates in the uniform norm and discuss questions of parameter choice the theoretical results will be supported by numerical examples which indicate that although the results are asymptotic in character they are of some relevance also for actual computations

on the convergence of fuzzy sets three kind of convergences are defined on a space of fuzzy sets relationships between these convergences are studied some comments on the fixed point property are represented

periodic boundary value problems for impulsive fuzzy differential equations the peculiarity of the hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations in this work we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic linear differential problems is also provided

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

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

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

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

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

solving fuzzy complex system of linear equations abstract this paper proposes a new and simple method for solving general fuzzy complex system of linear equations in the original system the elements of unknown variable vector and right hand side vector are considered as complex fuzzy number initially the general system is solved by adding and subtracting the left and right bounds of the fuzzy complex unknown and right hand side fuzzy complex vector respectively subsequently above obtained solutions are used to get the final solution of the general fuzzy complex system of linear equations two theorems are stated and proved related to the proposed method a mathematical example and an application problem based on electrical circuit are solved using the present method results obtained are compared with the known results and are found to be in good agreement

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

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

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

a unified approach to asymptotic behaviors for the autoregressive model with fuzzy data we propose a unified estimator for the autoregressive model with fuzzy input output variables based on the least squares method the least squares estimation is investigated in presence of a unified r distance defined on the space of fuzzy numbers we investigate asymptotic properties of the unified estimator under some simple conditions as well as a generalization which reduces to the asymptotic properties under those distances when the distances are the special cases of r distance some simulation studies are included to compare the asymptotic properties of estimators formed under several distances being the special cases of r distance