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

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

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

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

on primitivity of sets of matrices a nonnegative matrix a is called primitive if a k is positive for some integer k 0 a generalization by protasov and voynov 2012 of this concept to finite sets of matrices is as follows a set of matrices m a 1 a 2 a m is primitive if a i 1 a i 2 a i k is positive for some indices i 1 i 2 i k the concept of primitive sets of matrices comes up in a number of problems within the study of discrete time switched systems in this paper we analyze the computational complexity of deciding if a given set of matrices is primitive and we derive bounds on the length of the shortest positive product we show that while primitivity is algorithmically decidable unless p n p it is not possible to decide primitivity of a matrix set in polynomial time moreover we show that the length of the shortest positive sequence can be superpolynomial in the dimension of the matrices on the other hand defining p to be the set of matrices with no zero rows or columns we give a combinatorial proof of the protasov voynov characterization 2012 of primitivity for matrices in p which can be tested in polynomial time this latter observation is related to the well known 1964 conjecture of ern on synchronizing automata in fact any bound on the minimal length of a synchronizing word for synchronizing automata immediately translates into a bound on the length of the shortest positive product of a primitive set of matrices in p in particular any primitive set of n n matrices in p has a positive product of length o n 3

median value and median interval of a fuzzy number median value and median interval of a fuzzy number are investigated effects of concentration and dilation of a trapezoidal fuzzy number on the location of the median value and the median interval are studied relationship between the median interval and the mean interval is explored central value and central interval of a fuzzy number are proposed

a procedure for ordering fuzzy subsets of the unit interval abstract we introduce a function to help in the ordering of fuzzy subsets of the unit interval this function is the integral of the mean of the level sets associated with the fuzzy subsets various properties of this function are studied

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

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

discontinuity of the trapezoidal fuzzy number valued operators preserving core we prove that any trapezoidal fuzzy number valued operator preserving core is discontinuous with respect to any weighted metric on the space of fuzzy numbers as an application we obtain the discontinuity of the weighted trapezoidal approximation operator preserving core

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 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

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

set valued lyapunov functions for difference inclusions the paper relates set valued lyapunov functions to pointwise asymptotic stability in systems described by a difference inclusion pointwise asymptotic stability of a set is a property which requires that each point of the set be lyapunov stable and that every solution to the inclusion from a neighborhood of the set be convergent and have the limit in the set weak set valued lyapunov functions are shown via an argument resembling an invariance principle to imply this property strict set valued lyapunov functions are shown in the spirit of converse lyapunov results to always exist for closed sets that are pointwise asymptotically stable

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 nearest trapezoidal fuzzy number to a fuzzy quantity in this paper we introduce a fuzzy trapezoidal approximation using the metric distance between two fuzzy numbers an application of this approximation is also investigated

trapezoidal approximations of fuzzy numbers using quadratic programs abstract in this paper we will prove that the nearest trapezoidal approximation of fuzzy numbers with respect to weighted l 2 type metrics with or without additional constraints can be obtained via quadratic programs actually the approach is even more general based on so called finite polyhedral subsets of fuzzy numbers which include most of the important special classes of fuzzy numbers available in the literature in particular we will recapture the algorithm to compute the nearest weighted trapezoidal approximation of a fuzzy number by a method which we believe that has the potential to be extended to more complex approximation problems then we will improve the lipschitz constant of the trapezoidal approximation operator preserving the ambiguity to achieve this improved result we will exploit the fact that we have an analytical expression for this operator however note that the same result is obtained if this solution function is described by quadratic programs therefore for similar problems we still can obtain lipschitz constants for the approximation operator even if an analytical expression of this operator is not available

exponential cluster solutions to quantum transport equations connected graph methods are used to solve the transport equation arising in a semiclassical representation of the schrodinger propagator k t s x y for a spinless quantum system whose hamiltonian is a smooth position and momentum dependent perturbation of the laplacian in rd exponential cluster expansions for k are obtained these formally exact solutions to the time dependent schrodinger equation employ explicit graphically determined derivatives of the hamiltonian s symbol integrated along geodesics for the special case of particles interacting with external electromagnetic fields the propagator s gauge invariant derivative expansion coefficients are determined in closed form in terms of the lorentz force a tree graph formula for hamilton s principal function is extracted from this result

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

on conditions for convergence to consensus a new theorem on conditions for convergence to consensus of a multiagent time dependent time discrete dynamical system is presented the theorem is build up on the notion of averaging maps we compare this theorem to results by moreau 6 ieee transactions on automatic control vol 50 no 2 2005 about set valued lyapunov theory and convergence under switching communication topologies we give examples that point out differences of approaches including examples where moreau s theorem is not applicable but ours is further on we give examples that demonstrate that the theory of convergence to consensus is still not complete

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

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

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper