triangular and parametric approximations of fuzzy numbers inadvertences and corrections recent papers were dedicated to approximate fuzzy numbers by triangular trapezoidal or parametric fuzzy numbers with or without additional conditions unfortunately the result of approximation is not always a fuzzy number sometimes it is not a fuzzy set we point out the wrongs and inadvertences in some recent papers then we correct the results

a note on trapezoidal approximations of fuzzy numbers in this paper we propose some generalized and new properties of the trapezoidal approximations of fuzzy numbers first we claim that the trapezoidal approximation operator is linear with respect to the operations of fuzzy numbers second we show a new property concerning the centroid points of fuzzy numbers furthermore a comment for computing the trapezoidal approximations is provided finally some properties about the distance between a fuzzy number and its trapezoidal approximation are proved

note on trapezoidal approximation of fuzzy numbers we show by two examples that trapezoidal approximation of fuzzy number a in grzegorzewski mrowka fuzzy sets and systems 153 2005 115 135 are not always trapezoidal

trapezoidal approximations of fuzzy numbers the problem of the trapezoidal approximation of fuzzy numbers is discussed a set of criteria for approximation operators is formulated these constraints can be used for direct operator derivation a new nearest trapezoidal approximation operator preserving expected interval is suggested

on the centroids of fuzzy numbers in a paper by cheng a new approach for ranking fuzzy numbers by distance method fuzzy sets and systems 95 1998 307 317 a centroid based distance method was suggested for ranking fuzzy numbers both normal and non normal where the fuzzy numbers are compared and ranked in terms of their euclidean distances from their centroid points to the origin it is found that the centroid formulae provided by the above paper are incorrect and have led to some misapplications in this paper we present the correct centroid formulae for fuzzy numbers and justify them from the viewpoint of analytical geometry a numerical example demonstrates that cheng s formulae can significantly alter the result of the ranking procedure

conditioned weighted l r approximations of fuzzy numbers we compute the extended weighted l r approximation of a given fuzzy number by a method based on general results in hilbert spaces the weighted average euclidean distance being considered the metric properties of the extended weighted l r approximation of a fuzzy number are proved we elaborate on a general method to study the existence uniqueness and to calculate the l r approximations of fuzzy numbers under the preservation of some parameters we apply the results to find the weighted l r approximations preserving ambiguity and value and respectively width in the general and unimodal case

perturbation of homoclinics and subharmonics in duffing s equation abstract for lambda mu and g f small the equation x x 2x cubed g x lambda x mu cos t f t is considered near the separatrix s of the unperturbed equation lambda mu 0 0 for g f in a neighborhood of the zero functions a complete description is given of the bifurcation curves in lambda mu space to homoclinic points the perturbation of subharmonics outside s is also considered in a symmetric and nonsymmetric case

existence uniqueness and continuity of trapezoidal approximations of fuzzy numbers under a general condition abstract the main aim of this paper is to characterize the set of real parameters associated to a fuzzy number represented in a general form which include the most important characteristics with the following property for any given fuzzy number there exists at least a trapezoidal fuzzy number which preserves a fixed parameter the uniqueness of the nearest trapezoidal fuzzy number with this property is proved the average euclidean distance being considered as an important property each resulting trapezoidal approximation operator is continuous the main results are illustrated by examples

trapezoidal approximations of fuzzy numbers revisited fuzzy number approximation by trapezoidal fuzzy numbers which preserve expected interval is discussed the previously proposed approximation operator is improved so as to always produce a well formed trapezoidal fuzzy number

approximations by lr type fuzzy numbers recently many scholars investigated interval triangular and trapezoidal approximations of fuzzy numbers these publications can be grouped into two classes euclidean distance class and non euclidean distance class most approximations in euclidean distance class can be calculated by formulas but calculating approximations in the other class is more complicated furthermore approximations in euclidean distance class can be divided into two subclasses one is to study approximations of fuzzy numbers without constraints the other one is to study approximations preserving some attributes in this paper we use lr type fuzzy numbers to approximate fuzzy numbers the proposed approximations will generalize all recent approximations without constraints in euclidean class also an efficient formula is provided

homoclinic finger rings in rn abstract in this paper we investigate bifurcations of a degenerate homoclinic loop in r n we prove that a homoclinic finger ring an invariant manifold of a definite dimension textured with homoclinic orbits arises from the degenerate homoclinic orbit the size of the homoclinic finger ring is decided by not only its dimension of manifold but also its width for the rise of homoclinic finger rings of different dimensions we give conditions which are proved to form bifurcation manifolds in the parameter space we further estimate the width for the homoclinic finger ring and give a method to compute the bifurcation manifolds approximately

a new type of approximation for fuzzy intervals this paper suggests a new method to approximate a fuzzy interval u by a sequence of differentiable fuzzy intervals this new approximation method involves the construction of differentiable fuzzy intervals using the sup min convolution of fuzzy sets numerical examples and an algorithm for computational implementation of the method proposed are also given

approximation of fuzzy numbers using the convolution method in this study we consider the use of the convolution method for constructing approximations comprising fuzzy number sequences with useful properties for a general fuzzy number we show that this convolution method can generate differentiable approximations in finite steps for fuzzy numbers with finite non differentiable points in previous studies this convolution method was only used for constructing differentiable approximations of continuous fuzzy numbers the possible non differentiable points of which were the two endpoints of the 1 cut the construction of smoothers is a key step in the process for producing approximations we also show that if appropriate smoothers are selected then we can use the convolution method to provide approximations that are differentiable lipschitz and that also preserve the core

visibility graph analysis of economy policy uncertainty indices abstract uncertainty plays an important role in the global economy in this paper the economic policy uncertainty epu indices of the united states and china are selected as the proxy variable corresponding to the uncertainty of national economic policy by adopting the visibility graph algorithm the four economic policy uncertainty indices of the united states and china are mapped into complex networks and the topological properties of the corresponding networks are studied the hurst exponents of all the four indices are within 0 5 1 which implies that the economic policy uncertainty is persistent the degree distributions of the epu networks have power law tails and thus the networks are scale free the average clustering coefficients of the four epu networks are high and close to each other while these networks exhibit weak assortative mixing we also find that the epu network in united states based on daily data shows the small world feature since the average shortest path length increases logarithmically with the network size such that l n 0 626 ln n 0 405 our research highlights the possibility to study the epu from the view angle of complex networks

nearest interval approximation of a fuzzy number the problem of the interval approximation of fuzzy numbers is discussed a new interval approximation operator which is the best one with respect to a certain measure of distance between fuzzy numbers is suggested

trapezoidal approximations of fuzzy numbers preserving the expected interval algorithms and properties fuzzy number approximation by trapezoidal fuzzy numbers which preserves the expected interval is discussed algorithms for calculating the proper approximations are proposed and some properties of the approximation operators are discussed it is shown that an adequate approximation operator might be chosen through the comparisons of some characteristics of the fuzzy number like its ambiguity width its value and weighted expected value

subharmonic bifurcations in a perturbed nonlinear oscillation abstract in this paper we discuss subharmonic bifurcations inside the homoclinic orbits and outside for a general form of perturbed two dimensional hamiltonian systems with a pair of real parameters and a pair of function parameters g 1 f 1 by approximating periodic orbits to homoclinic orbits we determine bifurcation curves for in r 2 and bifurcation manifolds for g 1 f 1 in the corresponding space of functions furthermore we determine the codimensions of those bifurcation manifolds all results on subharmonic bifurcations are given with conditions on homoclinic orbits our results generalize hale and spezamiglio s in nonlinear anal 9 1985 181

metrics and orders in space of fuzzy numbers abstract two classes of metrics p q and p where 1 p 0 q 1 in space of fuzzy numbers are introduced then a method of ranking fuzzy numbers based on these metrics is proposed and investigated not only fuzzy numbers but also the left sided and right sided fuzzy numbers are considered applications in making decisions and statistics are suggested

existence of interval triangular and trapezoidal approximations of fuzzy numbers under a general condition abstract recently many scholars studied approximations of fuzzy numbers by specific fuzzy numbers under preservation of some operators in fact these approximations may not exist for some linear operators the purpose of this paper is to study necessary and sufficient conditions of linear operators which are preserved by interval triangular symmetric triangular trapezoidal or symmetric trapezoidal approximations of fuzzy numbers in addition an effective method for solving such problems is proposed

weighted trapezoidal and triangular approximations of fuzzy numbers in 2007 zeng and li proposed a weighted triangular approximation of a fuzzy number unfortunately this approximation may fail to be a fuzzy number in this paper we improve this approximation and propose a generalization by the name of weighted trapezoidal approximation their algorithms are also presented finally some examples and relevant properties are discussed

weighted trapezoidal approximation preserving cores of a fuzzy number recently various researchers have proved that approximations of fuzzy numbers may fail to be fuzzy numbers in this contribution we suggest a new weighted trapezoidal approximation of an arbitrary fuzzy number which preserves its cores we prove that the weighted trapezoidal approximation preserving core is always a fuzzy number we then discuss properties of the approximation strategy including translation invariance scale invariance identity and continuity the advantage is that the method presented here is simpler than other methods computationally

approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval the problem to find the nearest trapezoidal approximation of a fuzzy number with respect to a well known metric which preserves the expected interval of the fuzzy number is completely solved the previously proposed approximation operators are improved so as to always obtain a trapezoidal fuzzy number properties of this new trapezoidal approximation operator are studied

the mean value of a fuzzy number abstract the notion of expectation is defined for fuzzy numbers of intervals viewing them as consonant random sets this expectation is interval valued and remains additive in the sense of fuzzy interval addition as a by product some light is shed on the debate between probability and possibility theory and the problem of how to carry measure theoretic notions into the field of possibility theory

on a canonical representation of fuzzy numbers abstract fuzzy numbers and more generally linguistic values are approximate assessments given by experts and accepted by decision makers when obtaining more accurate values is impossible or unnecessary to simplify the task of representing and handling fuzzy numbers several authors have introduced real indices in order to capture the information contained in a fuzzy number in this paper we propose two parameters value and ambiguity for this purpose we use these parameters to obtain canonical representations and to deal with fuzzy numbers in decision making problems several examples illustrate these ideas

the expected value of a fuzzy number abstract the notions of expected interval and expected value of fuzzy number are introduced the expected interval is defined as the expected value of an interval random set generated by the fuzzy number and the expected value of this number is defined as the centre of the expected interval the linearity of the expected interval is studied

weighted semi trapezoidal approximations of fuzzy numbers recently many scholars investigated interval triangular and trapezoidal approximations of fuzzy numbers these publications can be grouped into two classes euclidean distance class and non euclidean distance class most approximations in euclidean distance class can be calculated by formulas but calculating approximations in the other class is more complicated in the present paper we study a special class of non linear approximations with respect to a weighted euclidean distance we call it weighted semi trapezoidal approximations the proposed approximations generalize all recent approximations in euclidean distance class first we embed fuzzy numbers into a hilbert space then compute weighted semi trapezoidal approximations by means of best approximations in a closed convex subset of the hilbert space finally we propose formulas of matrix type which is more clear than the previous contributions

on the interval approximation of a fuzzy number the notion of an approximation interval of a fuzzy number is introduced it is the interval which fulfills two conditions in the first its width is equal to the width of a fuzzy number being approximated in the second the hamming distance between this interval and the approximated number is minimal the introduced notion is compared with the notion of expected interval of a fuzzy number known in literature

approximation of fuzzy numbers by max product bernstein operators in this paper firstly we extend to an arbitrary compact interval the definition of the nonlinear bernstein operators of max product kind by proving that their order of uniform approximation is the same as in the particular case of the unit interval then similarly to the particular case of the unit interval we will prove that these operators preserve the quasi concavity too since these properties will help us to generate in a simple way fuzzy numbers of the same support it turns out that these results are very suitable in the approximation of fuzzy numbers moreover these operators approximate the non degenerate segment core with a good order of approximation in addition in the case when the fuzzy numbers are given in parametric form the bernstein max product operator generates a sequence of fuzzy numbers whose widths expected intervals ambiguities and values approximate with a convergent rate the width the expected interval the ambiguity and the value of the approximated fuzzy number finally we obtain a quantitative approximation with respect to a well known integral type metric for some subclasses of functions leading to applications to the approximation of some subclasses of fuzzy numbers

linearly independent homoclinic bifurcations parameterized by a small function abstract in this paper we discuss a small nonautonomous perturbation of an autonomous system on r n which has a homoclinic solution regarding the small perturbation as a parameter in an appropriate space of functions we discuss various situations of co existence of homoclinic orbits those conditions of various co existence actually define bifurcation manifolds in the space of functions for linearly independent homoclinic bifurcations