kms states and conformal measures from a non constant holomorphic map on a connected riemann surface we construct an etale second countable locally compact hausdorff groupoid whose associated groupoid c algebra admits a one parameter group of automorphisms with the property that its kms states correspond to conformal measures in the sense of sullivan in this way certain quadratic polynomials give rise to quantum statistical models with a phase transition arising from spontaneous symmetry breaking

collet eckmann and holder we prove that collet eckmann condition for rational functions which requires exponential expansion only along the critical orbits yields the holder regularity of fatou components this implies geometric regularity of julia sets with non hyperbolic and critically recurrent dynamics in particular polynomial collet eckmann julia sets are locally connected if connected and their hausdorff dimension is strictly less than 2 the same is true for rational collet eckmann julia sets with at least one non empty fully invariant fatou component

type 2 fuzzistics for nonsymmetric interval type 2 fuzzy sets forward problems interval type 2 fuzzy sets it2 fs play a central role in fuzzy sets as models for words and in engineering applications of t2 fss these fuzzy sets are characterized by their footprints of uncertainty fou which in turn are characterized by their boundaries upper and lower membership functions mf the centroid of an it2 fs which is an it1 fs provides a measure of the uncertainty in the it2 fs the main purpose of this paper is to quantify the centroid of a non symmetric it2 fs with respect to geometric properties of its fou this is very important because interval data collected from subjects about words suggests that the fous of most words are non symmetrical using the results in this paper it is possible to formulate and solve forward problems i e to go from parametric non symmetric it2 fs models to data with associated uncertainty bounds we provide some solutions to such problems for non symmetrical triangular trapezoidal gaussian and shoulder fous

defuzzification of the discretised generalised type 2 fuzzy set experimental evaluation the work reported in this paper addresses the challenge of the efficient and accurate defuzzification of discretised generalised type 2 fuzzy sets as created by the inference stage of a mamdani fuzzy inferencing system the exhaustive method of defuzzification for type 2 fuzzy sets is extremely slow owing to its enormous computational complexity several approximate methods have been devised in response to this defuzzification bottleneck in this paper we begin by surveying the main alternative strategies for defuzzifying a generalised type 2 fuzzy set 1 vertical slice centroid type reduction 2 the sampling method 3 the elite sampling method and 4 the a planes method we then evaluate the different methods experimentally for accuracy and efficiency for accuracy the exhaustive method is used as the standard the test results are analysed statistically by means of the wilcoxon nonparametric test and the elite sampling method shown to be the most accurate in regards to efficiency vertical slice centroid type reduction is demonstrated to be the fastest technique

centroid of triangular and gaussian type 2 fuzzy sets abstract in this paper we present exact computation procedures for the centroid of type 2 fuzzy sets with either triangular or gaussian secondary membership functions we assume the standard formulation of the centroid based on the sup min extension principle the centroid of a triangular type 2 fuzzy set can be considered as a subsequent analytical extension of the km type reduction procedure dedicated to interval type 2 sets the centroid of a gaussian type 2 fuzzy set is recursively expressed by an independent procedure

granular neural networks with evolutionary interval learning to deal with different membership functions of the same linguistic term a new interval reasoning method using new granular sets is proposed based on yin yang methodology to make interval valued granular reasoning efficiently and optimize interval membership functions based on training data effectively a granular neural network gnn with a new high speed evolutionary interval learning is designed simulation results in nonlinear function approximation and bioinformatics have shown that the gnn with the evolutionary interval learning is able to extract interval valued granular rules effectively and efficiently from training data by using the new evolutionary interval learning algorithm

non uniform hyperbolicity in complex dynamics we say that a rational function f satisfies the summability condition with exponent if for every critical point c which belongs to the julia set j there exists a positive integer nc so that sum n 1 infty f n prime f n c c alpha infty and f has no parabolic periodic cycles let max be the maximal multiplicity of the critical points

mpeg vbr video traffic modeling and classification using fuzzy technique we present an approach for mpeg variable bit rate vbr video modeling and classification using fuzzy techniques we demonstrate that a type 2 fuzzy membership function i e a gaussian mf with uncertain variance is most appropriate to model the log value of i p b frame sizes in mpeg vbr video the fuzzy c means fcm method is used to obtain the mean and standard deviation std of t p b frame sizes when the frame category is unknown we propose to use type 2 fuzzy logic classifiers flcs to classify video traffic using compressed data five fuzzy classifiers and a bayesian classifier are designed for video traffic classification and the fuzzy classifiers are compared against the bayesian classifier simulation results show that a type 2 fuzzy classifier in which the input is modeled as a type 2 fuzzy set and antecedent membership functions are modeled as type 2 fuzzy sets performs the best of the five classifiers when the testing video product is not included in the training products and a steepest descent algorithm is used to tune its parameters

rational rotation numbers for maps of the circle we consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms that is maps like x to x t frac c 2 pi sin 2 pi x bmod 1 withc 1 we prove that the set of parameter values corresponding to irrational rotation numbers has lebesgue measure 0 in other words the intervals on which frequency locking occurs fill up the set of full measure

min and max operators for fuzzy intervals and their potential use in aggregation operators this paper aims at expressing min and max operations when triangular fuzzy intervals are taken as inputs that is when zadeh s extension principle is considered first presented approach consists in representing fuzzy intervals by means of their profiles in this context a computation algorithm can be easily derived for implementing the min and max operations another methodology based on interval relations is then proposed for determining a general analytical expression of the min and max operations the potential use of these expressions in the framework of uncertain aggregation operators is illustrated with the two additive choquet integral

designing fuzzy sets with the use of the parametric principle of justifiable granularity this study is concerned with a design of membership functions of fuzzy sets the membership functions are formed in such a way that they are experimentally justifiable and exhibit a sound semantics these two requirements are articulated through the principle of justifiable granularity the parametric version of the principle is discussed in detail we show linkages with type 2 fuzzy sets which are constructed on a basis of type 1 fuzzy sets several experimental studies are reported which illustrate a behavior of the introduced method

on km algorithms for solving type 2 fuzzy set problems computing the centroid and performing type reduction for type 2 fuzzy sets and systems are operations that must be taken into consideration karnik mendel km algorithms are the standard ways to do these operations however because these algorithms are iterative much research has been conducted during the past decade about centroid and type reduction computations this tutorial paper focuses on the research that has been conducted to 1 improve the km algorithms 2 understand the km algorithms leading to further improved algorithms 3 eliminate the need for km algorithms 4 use the km algorithms to solve other nonfuzzy logic system problems and 5 use or not use km algorithms for general type 2 fuzzy sets and fuzzy logic systems

holder estimates on cr manifolds with a diagonalizable levi form a for all e 0 the operator cl maps lip s po into lip s 2 m e po here lip s p denotes the space of square integrable forms of degree p q with 1 g q otheoperators o l o l q and0 map lip s p into lip s l m e po c for all e o the operators 8b8zo 8z8ho 1 q a aa cl 8 8 8 q 8 and 8 0 ab map lip s p into lip s p hebre aiag web asm acts on forms of type p q with 1 q 0 e zf f is a square integrable function on a with f e lip p then s f e lip s e p for all e 0 where s is the orthogonal projection on square integrable cr functions

noncommutative algebras for hyperbolic diffeomorphisms the gibbs states of classical equilibrium statistical mechanics can be extended to states on non commutative algebras satisfying the kubo martin schwinger boundary condition this way of looking at gibbs states is applied here to the study of differentiable dynamical systems when some strong or weak hyperbolicity conditions are satisfied

model reduction of discrete time interval type 2 t s fuzzy systems this paper addresses the model reduction problem of discrete time interval type 2 it2 takagi sugeno t s fuzzy systems which represent the discrete time nonlinear systems subject to uncertainty with the use of it2 fuzzy sets the uncertainty of the discrete time nonlinear system can be captured by the lower and upper membership functions for a given high order discrete time it2 t s fuzzy system the purpose is to find a lower dimensional system to approximate the original system to achieve the approximation performance an mathcal h infty norm is used to suppress the error between the original system and its simplified system by introducing a membership functions dependent technique and applying a convex linearization method a membership functions dependent condition which takes the information of membership functions into account is obtained to reduce the dimensions of system matrices and the number of fuzzy rules of the system all the obtained theorems are represented as in the form of linear matrix inequalities finally simulation results are demonstrated to show the effectiveness of the derived results

reduced interval type 2 neural fuzzy system using weighted bound set boundary operation for computation speedup and chip implementation this paper proposes a reduced interval type 2 neural fuzzy system using weighted bound set boundaries rit2nfs wb for the simplification of type reduction operations the objective of this simplification is to reduce the system training time in software implementation and chip size in hardware implementation especially when the number of rules is large the antecedent part in the rit2nfs wb uses interval type 2 fuzzy sets it2fss and the consequent part can be of the takagi sugeno kang tsk or mamdani type the rit2nfs wb is built through an online structure and parameter learning to improve model accuracy in addition the interpretability of the rit2nfs wb is improved by considering distributions of the it2fss in input variables a distinguishability oriented cost function is used in parameter learning to generate distinguishable it2fss and improve semantics based interpretability for highly overlapped it2fss they are merged to reduce the number of it2fss and improve complexity based interpretability the software implemented tsk type rit2nfs wb is hardware implemented on a field programmable gate array chip to accelerate the chip execution speed the chip utilizes not only the parallel execution properties of fuzzy rules and bound set boundaries but the pipeline technique as well in particular the flexibility of the chip is considered so that no redesign of the circuits is required when the rit2nfs wb is applied to different problems the characteristics of the software and hardware implemented rit2nfs wb are verified through various examples and comparisons with various type 1 and interval type 2 fuzzy models

groupoids associated with endomorphisms to a compact hausdorff space which covers itself we associate an r discrete locally compact hausdorff groupoid its c algebra carries an action of the circle allowing it to be regarded as a crossed product by an endomorphism and as a generalization of the cuntz algebra op we consider examples related to coverings of the circle and of a heisenberg 3 manifold given a compact hausdorff space x and a continuous surjective map a x x we want to associate to the pair x a a c algebra c x a which in the case a is a homeomorphism coincides with the well known crossed product c x x z in this paper we consider the case of a covering a x x and c x a will be defined via a groupoid that generalizes the construction of the cuntz groupoid op considered by renault in re in that case x is a unilateral sequence space and a is the unilateral shift we start with some definitions and notation that make sense in greater generality definition given a set x and a map a x x let i x n y e xx zx x i 3k 1 0 n i k akx aly with the set of composable pairs i 2 x n y w m z e jx j w y the multiplication and inversion are given by x n y y m z x m n z and x n y 1 y n x the range and source maps are r x n y x 0 x s x n y y 0 y with these operations if becomes a groupoid with the unit space identified with x the equivalence relation associated to if will be denoted by r where r x y e x x x i 3k 1 0 akx ly and the isotropy group bundle is i x n x exxzxx ik 1 0 n l k akx alx received by the editors march 25 1993 and in revised form january 21 1994 originally communicated to the proceedings of the ams by palle e t jorgensen 1991 mathematics subject classification primary 46l05 46l55 46l80 research supported in part by the national science foundation 1995 american mathematical society 0002 9947 95 1 00 25 per page

quasiconformal homeomorphisms and dynamics i solution of the fatou julia problem on wandering domains on demontre un resultat dynamique sur l iteration d une application analytique complexe sur la sphere de riemann

prescribing zeros of functions in the nevanlinna class on weakly pseudo convex domains in let d be a bounded weakly pseudo convex domain in c2 of uniform strict type for any positive divisor m of d with finite area there exists a holomorphic function f in the nevanlinna class such that m is the zero set of f the proof is to study the solutions of a with l ad boundary values 2 let d be a bounded weakly pseudo convex domain in c in this paper we study the poincare lelong equation 1 iaou a ond where a is a d closed 1 1 form in d in particular we are interested in solutions u such that the boundary values of u are in l od this question arises when one tries to prescribe zeros of functions in the nevanlinna class in d it is also well known that the solutions of 1 are closely related to the solutions of the cauchy riemann equation 2 ou f ond where f is a 8 closed 0 1 from in d when d is strongly pseudoconvex in cn n 2 henkin 6 and skoda 18 have studied equations 1 and 2 independently and obtained precise estimates of the solutions u on the boundary from this they were able to completely characterize the zero sets of the functions in the nevanlinna class on strongly pseudo convex domains they showed that a zero divisor is the zero set of a function in the nevanlinna class if and only if the zero divisor satisfies the blaschke condition see definitions 1 3 1 4 in i this generalized the classical blaschke theorem in c to strongly pseudo convex domains in cn an excellent account of the henkin skoda theorem on the unit ball in cn can be found in chapter 17 of rudin s book 16 see also harvey polking 5 when d is only weakly pseudo convex such results are unknown except for the special case when d is a complex ellipsoid see bonami charpentier 2 in this case one can use dilation to study the solutions of equations 1 and 2 from the known solutions for the ball in this paper we solve 1 and 2 with boundary values in ls od when d is a received by the editors january 30 1988 1980 mathematics subject classification 1985 revision primary 32a10 32a20 research was partially supported by nsf grant dms 87 96300 d 1989 american mathematical society 0002 9947 89 1 00 25 per page

a new indirect approach to the type 2 fuzzy systems modeling and design this paper proposes a new method for designing interval type 2 fuzzy logic systems it2 flss considering two issues first quality of clustering the output space and secondly approximating the output of the it2 fls based on a new output processing method based on these two issues we present a new cluster validity index capable of being used for type 1 fuzzy c means fcms interval type 2 fcm it2 fcm and possibilistic c means pcms clustering algorithms this validity index is highly efficient in determining clusters with the least similarity between them and the highest similarity between data vectors in each cluster then a new definition for uncertainty bounds is presented in order to eliminate the type reduction process in it2 flss and to increase accuracy of the existing uncertainty bounds in the literature finally effectiveness of the proposed approaches compared to several well known existing methods has been investigated computational results have verified accuracy and effectiveness of the proposed method

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

simplified interval type 2 fuzzy logic systems type reduction tr followed by defuzzification is commonly used in interval type 2 fuzzy logic systems it2 flss because of the iterative nature of tr it may be a computational bottleneck for the real time applications of an it2 fls this has led to many direct approaches to defuzzification that bypass tr the simplest of which is the nie tan direct defuzzification method nt method this paper provides some theoretical analyses of the nt method that answer the question why is the nt method good to use this paper also provides a direct relationship between tr followed by defuzzification using km algorithms and the nt method it also provides an improved nt method numerical examples illustrate our theoretical results and suggest that the nt method is a very good way to simplify an interval type 2 fuzzy set

a fast geometric method for defuzzification of type 2 fuzzy sets generalized type 2 fuzzy logic systems cannot currently be used for practical problems because the amount of computation required to defuzzify a generalized type 2 fuzzy set is too large this paper presents a new method for defuzzifing a type 2 fuzzy set the new much faster technique is based on geometric representations and operations the results of a real world example contained in this paper show this new approach to be over 200 000 times faster than type reduction we present a new method for assessing the accuracy of the membership function of a type 2 fuzzy set this method is used to show that the new representation used by the defuzzifier is not detrimental to the accuracy of the set we also discuss the differences between the new approach and type reduction identifying the origin of this massive improvement in execution speed

simplified type 2 fuzzy sliding controller for wing rock system wing rock is a highly nonlinear phenomenon in which aircrafts with slender delta wings undergo limit cycle roll oscillations at high angles of attack a simplified type 2 fuzzy sliding controller is designed for suppressing wing rock phenomena and tracking the desired trajectories to reduce the computational complexity of a type reducer the end points of a type reduced set are approximated by the outputs of two standard fuzzy sliding mechanisms in the proposed simplified type 2 fuzzy sliding controller furthermore the sliding modes of the fuzzy sliding control system are guaranteed simulation results are included to show the effectiveness of the proposed simplified type 2 fuzzy sliding controller

equalization of nonlinear time varying channels using type 2 fuzzy adaptive filters presents a kind of adaptive filter type 2 fuzzy adaptive filter faf one that is realized using an unnormalized type 2 takagi sugeno kang tsk fuzzy logic system fls we apply this filter to equalization of a nonlinear time varying channel and demonstrate that it can implement the bayesian equalizer for such a channel has a simple structure and provides fast inference a clustering method is used to adaptively design the parameters of the faf two structures are used for the equalizer transversal equalizer te and decision feedback equalizer dfe a decision tree structure is used to implement the decision feedback equalizer in which each leaf of the tree is a type 2 faf this dfe vastly reduces computational complexity as compared to a te simulation results show that equalizers based on type 2 fafs perform much better than nearest neighbor classifiers nnc or equalizers based on type 1 fafs

self similarity of siegel disks and hausdorff dimension of julia sets let f z ez z where is an irrational number of bounded type according to siegel f is linearizable on a disk containing the origin in this paper we show the hausdorff dimension of the julia set j f is strictly less than two and if is a quadratic irrational such as the golden mean then the siegel disk for f is self similar about the critical point in the latter case we also show the rescaled first return maps converge exponentially fast to a system of commuting branched coverings of the complex plane

uncertainty bounds and their use in the design of interval type 2 fuzzy logic systems we derive inner and outer bound sets for the type reduced set of an interval type 2 fuzzy logic system fls based on a new mathematical interpretation of the karnik mendel iterative procedure for computing the type reduced set the bound sets can not only provide estimates about the uncertainty contained in the output of an interval type 2 fls but can also be used to design an interval type 2 fls we demonstrate by means of a simulation experiment that the resulting system can operate without type reduction and can achieve similar performance to one that uses type reduction therefore our new design method based on the bound sets can relieve the computation burden of an interval type 2 fls during its operation which makes an interval type 2 fls useful for real time applications

the lyapunov exponent of holomorphic maps we prove that for any polynomial map with a single critical point its lower lyapunov exponent at the critical value is negative if and only if the map has an attracting cycle similar statement holds for the exponential maps and some other complex dynamical systems we prove further that for the unicritical polynomials with positive area julia sets almost every point of the julia set has zero lyapunov exponent part of this statement generalizes as follows every point with positive upper lyapunov exponent in the julia set of an arbitrary polynomial is not a lebegue density point

on zero varieties of holomorphic functions in hardy spaces abstract in contrast to the famous henkin skoda theorem concerning the zero varieties of holomorphic functions in the nevanlinna class on the open unit ball bn in c n n 2 it is proved in this article that for any nonnegative increasing convex function t defined on r there exists g o b n satisfying s n g 1 d such that there is no f hp bn 0 z f z g here ng 1 denotes the integrated zero counting function associated with the slice function g this means that the zero sets of holomorphic functions belonging to the hardy spaces hp bn 0

optimal holder and lp estimates for b on boundaries of convex domains of finite type we prove optimal holder and lp estimates for solutions of the tangential cauchy riemann equation on the boundary of a bounded convex domain of finite type in cn using the integral kernel method

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

type 2 fuzzy markov random fields and their application to handwritten chinese character recognition in this paper we integrate type 2 t2 fuzzy sets with markov random fields mrfs referred to as t2 fmrfs which may handle both fuzziness and randomness in the structural pattern representation on the one hand the t2 membership function mf has a 3 d structure in which the primary mf describes randomness and the secondary mf evaluates the fuzziness of the primary mf on the other hand mrfs can represent patterns statistical structurally in terms of neighborhood system and clique potentials and thus have been widely applied to image analysis and computer vision in the proposed t2 fmrfs we define the same neighborhood system as that in classical mrfs to describe uncertain structural information in patterns we derive the fuzzy likelihood clique potentials from t2 fuzzy gaussian mixture models the fuzzy prior clique potentials are penalties for the mismatched structures based on prior knowledge because chinese characters have hierarchical structures we use t2 fmrfs to model character structures in the handwritten chinese character recognition system the overall recognition rate is 99 07 which confirms the effectiveness of the proposed method

the reduction of interval type 2 lr fuzzy sets type reduction of interval type 2 it2 fuzzy sets is essential in conducting the type 2 fuzzy sets expressed with the resolution forms of it2 fuzzy sets several type reduction methods such as km ekm and centroid flow have been proposed these methods are relatively easy to implement but still computation intensive because they need to invoke an iterative switching point finding procedure this study derives a theorem and proposes a heuristic algorithm which can fast and accurately identify the minimum and maximum switching points of a piecewise smooth it2 fuzzy set it also demonstrates that it is easy to derive the close form expressions of the switching points of a piecewise smooth it2 fuzzy set if both of its upper and lower membership functions can be parameterized as lr fuzzy sets which are defined in this paper then the type reduction of piecewise smooth it2 fuzzy sets can be simplified to solve the close form expressions of their switching points in terms of lr parameters experiments on it2 fuzzy sets with various piecewise smooth membership functions including linear gaussian and hybrid shaped ones were made the results showed that the proposed type reduction method can obtain solutions which accurately approximate to the desired switching points with much lower computational overhead than the karnik mondel km and enhanced km ekm methods

hierarchical collapsing method for direct defuzzification of general type 2 fuzzy sets abstract karnik mendel km and enhanced km ekm algorithms are proved to be efficient approaches when dealing with general type 2 fuzzy sets represented by their planes on the other hand collapsing method as a type reduction algorithm for interval type 2 fuzzy sets has shown to be an effectual type reduction method for interval type 2 fuzzy sets in this paper we aim at developing mathematical formulations based on collapsing procedure and plane decomposition in order to utilize geometric properties of general type 2 fuzzy sets for preparing a direct defuzzification procedure the proposed methods are fast and robust extensive simulations have shown that the proposed algorithm outperforms most of the existing approaches in this field and its results are closer to the exact optimal defuzzified values

equation de lewy resolubilite globale de l equation partial_b u f sur la frontiere de domaines faiblement pseudo convexes de mathbf c 2 mathrm ou mathbf c n

on the continuity of type 1 and interval type 2 fuzzy logic systems this paper studies the continuity of the input output mappings of fuzzy logic systems flss including both type 1 t1 and interval type 2 it2 flss we show that a t1 fls being an universal approximator is equivalent to saying that a t1 fls has a continuous input output mapping we also derive the condition under which a t1 fls is discontinuous for it2 flss we consider six type reduction and defuzzification methods the karnik mendel method the uncertainty bound method the wu tan method the nie tan method the du ying method and the begian melek mendel method and derive the conditions under which continuous and discontinuous input output mappings can be obtained guidelines for designing continuous it2 flss are also given this paper is to date the most comprehensive study on the continuity of flss our results will be very useful in the selection of the parameters of the membership functions to achieve a desired continuity e g for most traditional modeling and control applications or discontinuity e g for hybrid and switched systems modeling and control

ranking type 2 fuzzy numbers type 2 fuzzy sets are a generalization of the ordinary fuzzy sets in which each type 2 fuzzy set is characterized by a fuzzy membership function in this paper we consider the problem of ranking a set of type 2 fuzzy numbers we adopt a statistical viewpoint and interpret each type 2 fuzzy number as an ensemble of ordinary fuzzy numbers this enables us to define a type 2 fuzzy rank and a type 2 rank uncertainty for each intuitionistic fuzzy number we show the reasonableness of the results obtained by examining several test cases

enhanced karnik mendel algorithms the karnik mendel km algorithms are iterative procedures widely used in fuzzy logic theory they are known to converge monotonically and superexponentially fast however several usually two to six iterations are still needed before convergence occurs methods to reduce their computational cost are proposed in this paper extensive simulations show that on average the enhanced km algorithms can save about two iterations which corresponds to more than a 39 reduction in computation time an additional at least 23 computational cost can be saved if no sorting of the inputs is needed

encoding words into normal interval type 2 fuzzy sets hm approach this paper focuses on an approach called the hm approach hma to determine for the first time a normal interval type 2 fuzzy set model for a word that uses interval data about a word that are collected either from a group of subjects or from one subject the hma has two parts 1 data part which is the same as the data part of the enhanced interval approach eia 44 and 2 fuzzy set part which is very different from the second part of the eia the most notable difference being that in the hma the common overlap of subject data intervals is interpreted to indicate agreement by all of the subjects for that overlap and therefore a membership grade of 1 is assigned to the common overlap another difference between the hma and eia is the way in which data intervals are collectively classified into either a left shoulder interior or right shoulder footprint of uncertainty the hma does this more simply than does the eia and requires fewer probability assumptions about the intervals than does the eia

counterexamples to analytic hypoellipticity for domains of finite type consider the hypersurface z1 z2 q z2 p zi cc c2 where p r is a subharmonic nonharmonic polynomial such a surface is pseudoconvex more precisely is the boundary of a pseudoconvex domain and of finite type it is the boundary of a model domain f z2 p zi so called because such domains provide good approximations to general pseudoconvex domains of finite type in c2 see 37 11 a nonvanishing antiholomorphic tangent vector field is a oaz 1 2i ap oaz i o a29z as coordinates for the surface we use c x jr 3 z t z t ip z the vector field pulls back to l o oz i op 9z a3 t equip c x jr with lebesgue measure and consider the orthogonal projection from l2 onto its intersection with the kernel of l the intersection of this kernel with the schwartz space may be shown to have infinite dimension and it is well known that l fails to be c hypoelliptic however c hypoellipticity does hold in a modified form see 30 if lu e c in an open set and if in addition u lv for some v e l2 where l is the formal adjoint of l then u e c there our main purpose is to point out that the analogous assertion for real analytic hypoellipticity is false if p z rz m m an even integer 4 specifically for p a general homogeneous subharmonic nonharmonic polynomial let s z t w s be the szeg6 kernel that is the distribution kernel associated to the operator defined by the orthogonal projection of l2 c x jr with respect to lebesgue measure onto the kernel of l we shall see in section 5 that this projection maps co to c further s is smooth off the diagonal by 37 define the distribution

a direct approach for determining the switch points in the karnik mendel algorithm the karnik mendel algorithm is used to compute the centroid of interval type 2 fuzzy sets determining the switch points needed for the lower and upper bounds of the centroid through an iterative process it is commonly acknowledged that there is no closed form solution for determining such switch points many enhanced algorithms have been proposed to improve the computational efficiency of the karnik mendel algorithm however all of these algorithms are still based on iterative procedures in this paper a direct approach based on derivatives for determining the switch points without multiple iterations has been proposed together with mathematical proof that these switch points are correctly determining the lower and upper bounds of the centroid experimental simulations show that the direct approach obtains the same switch points but is more computationally efficient than any of the existing iterative algorithms thus we propose that this algorithm should be used in any application of interval type 2 fuzzy sets in which the centroid is required

improving the speed of center of sets type reduction in interval type 2 fuzzy systems by eliminating the need for sorting in the deployment of interval type 2 fuzzy systems one of the most important steps is the type reduction the commonly used center of sets type reducer requires the solution of two nonlinear constrained optimization problems frequently used approaches to solve them are the karnik mendel algorithms and their variants however these algorithms suffer from the need for sorting which is known to be computationally very expensive using the reformulations proposed in this paper for center of sets type reducer it is possible to eliminate the need for sorting this makes interval type 2 fuzzy systems more appropriate for cost sensitive real time applications extensive simulations are presented to illustrate the faster convergence speed of the proposed method over six other enhanced variants of the karnik mendel algorithm as applied to center of sets type reduction of interval type 2 fuzzy systems

interval type 2 mutual subsethood fuzzy neural inference system it2msfunis this paper presents an interval type 2 mutual subsethood fuzzy neural inference system it2msfunis a mutual subsethood measure between two interval type 2 fuzzy sets it2 fs has been derived and has been used in determining the similarity between the it2 fs inputs and it2 fs antecedents the consequent weights are taken to be interval sets the inputs to the system are fuzzified into it2 fss with gaussian primary membership function having fixed center and uncertain variance aggregation of type 2 mutual subsethood based activation spreads is performed using product operator the output is obtained using simplified type reduction followed by defuzzification the system learns using memetic procedure involving differential evolution for global search and gradient descent for local exploitation in solution space the mathematical modeling and empirical studies of it2msfunis bring forth its efficacy in problems pertaining to function approximation time series prediction control and classification comparisons with other type 1 and type 2 neuro fuzzy systems verify that it2msfunis compares excellently with other models with a performance better than most of them both in terms of total number of trainable parameters and result accuracy empirical studies indicate the intelligent decision making capability of the proposed model the main contribution of this paper lies in the identification of mutual subsethood to find out the correlation between it2 fss and to find out its applicability in diverse application domains the improved performance of the proposed method can be attributed to the better contrast handling capacity of mutual subsethood method and uncertainty handling capacity of it2 fss the integration of mutual subsethood with interval type 2 fuzzy logic puts forth a novel model with various merits as demonstrated amply with the help of well known problems reported in the literature

on answering the question where do i start in order to solve a new problem involving interval type 2 fuzzy sets this paper which is tutorial in nature demonstrates how the embedded sets representation theorem rt for a general type 2 fuzzy set t2 fs when specialized to an interval i t2 fs can be used as the starting point to solve many diverse problems that involve it2 fss the problems considered are set theoretic operations centroid uncertainty measures similarity inference engine computations for mamdani it2 fuzzy logic systems linguistic weighted average person membership function approach to type 2 fuzzistics and interval approach to type 2 fuzzistics each solution obtained from the rt is a structural solution but is not a practical computational solution however the latter are always found from the former it is this author s recommendation that one should use the rt as a starting point whenever solving a new problem involving it2 fss because it has had such great success in solving so many such problems in the past and it answers the question where do i start in order to solve a new problem involving it2 fss

regranulation a granular algorithm enabling communication between granular worlds abstract in this paper we describe a granular algorithm for translating information between two granular worlds represented as fuzzy rulebases these granular worlds are defined on the same universe of discourse but employ different granulations of this universe in order to translate information from one granular world to the other we must regranulate the information so that it matches the information granularity of the target world this is accomplished through the use of a first order interpolation algorithm implemented using linguistic arithmetic a set of elementary granular computing operations we first demonstrate this algorithm by studying the common fuzzy pd rulebase at several different granularities and conclude that the 3 3 granulation may be too coarse for this objective we then examine the question of what the natural granularity of a system might be this is studied through a 10 fold cross validation experiment involving three different granulations of the same underlying mapping for the problem under consideration we find that a 7 7 granulation appears to be the minimum necessary precision

super exponential convergence of the karnik mendel algorithms for computing the centroid of an interval type 2 fuzzy set computing the centroid of an interval t2 fs is an important operation in a type 2 fuzzy logic system where it is called type reduction but it is also a potentially time consuming operation the karnik mendel km iterative algorithms are widely used for doing this in this paper we prove that these algorithms converge monotonically and super exponentially fast both properties are highly desirable for iterative algorithms and explain why in practice the km algorithms have been observed to converge very fast thereby making them very practical to use

toward general type 2 fuzzy logic systems based on zslices higher order fuzzy logic systems flss such as interval type 2 flss have been shown to be very well suited to deal with the high levels of uncertainties present in the majority of real world applications general type 2 flss are expected to further extend this capability however the immense computational complexities associated with general type 2 flss have until recently prevented their application to real world control problems this paper aims to address this problem by the introduction of a complete representation framework which is referred to as zslices based general type 2 fuzzy systems the proposed approach will lead to a significant reduction in both the complexity and the computational requirements for general type 2 flss while it offers the capability to represent complex general type 2 fuzzy sets as a proof of concept application we have implemented a zslices based general type 2 fls for a two wheeled mobile robot which operates in a real world outdoor environment we have evaluated the computational performance of the zslices based general type 2 fls which is suitable for multiprocessor execution finally we have compared the performance of the zslices based general type 2 fls against type 1 and interval type 2 flss and a series of results is presented which is related to the different levels of uncertainty handled by the different types of flss

alpha plane representation for type 2 fuzzy sets theory and applications this paper 1 reviews the alpha plane representation of a type 2 fuzzy set t2 fs which is a representation that is comparable to the alpha cut representation of a type 1 fs t1 fs and is useful for both theoretical and computational studies of and for t2 fss 2 proves that set theoretic operations for t2 fss can be computed using very simple alpha plane computations that are the set theoretic operations for interval t2 it2 fss 3 reviews how the centroid of a t2 fs can be computed using alpha plane computations that are also very simple because they can be performed using existing karnik mendel algorithms that are applied to each alpha plane 4 shows how many theoretically based geometrical properties can be obtained about the centroid even before the centroid is computed 5 provides examples that show that the mean value defuzzified value of the centroid can often be approximated by using the centroids of only 0 and 1 alpha planes of a t2 fs 6 examines a triangle quasi t2 fuzzy logic system q t2 fls whose secondary membership functions are triangles and for which all calculations use existing t1 or it2 fs mathematics and hence they may be a good next step in the hierarchy of flss from t1 to it2 to t2 and 7 compares t1 it2 and triangle q t2 flss to forecast noise corrupted measurements of a chaotic mackey glass time series

design of interval type 2 fuzzy sliding mode controller in this paper an interval type 2 fuzzy sliding mode controller it2fsmc is proposed for linear and nonlinear systems the proposed it2fsmc is a combination of the interval type 2 fuzzy logic control it2flc and the sliding mode control smc which inherits the benefits of these two methods the objective of the controller is to allow the system to move to the sliding surface and remain in on it so as to ensure the asymptotic stability of the closed loop system the lyapunov stability method is adopted to verify the stability of the interval type 2 fuzzy sliding mode controller system the design procedure of the it2fsmc is explored in detail a typical second order linear interval system with 50 parameter variations an inverted pendulum with variation of pole characteristics and a duffing forced oscillation with uncertainty and disturbance are adopted to illustrate the validity of the proposed method the simulation results show that the it2fsmc achieves the best tracking performance in comparison with the type 1 fuzzy logic controller t1flc the it2flc and the type 1 fuzzy sliding mode controller t1fsmc

an interval valued fuzzy controller for complex dynamical systems with application to a 3 psp parallel robot in this paper we present a novel interval valued fuzzy model based controller for handling the effects of uncertainty in controlling a complex dynamical system theoretically model based controllers may be the ideal control mechanisms however they are highly sensitive to model uncertainties and lack robustness these controllers are also computationally intensive rendering them unusable for many real world applications in this work we incorporate an interval fuzzy logic paradigm into a computed torque controller for a 3 psp parallel robot this paradigm aims to handle the uncertainties in the robot model the proposed approach benefits from algebraic operations on type i fuzzy numbers to enhance its capability in dealing with uncertainty the simulations prove the superiority of the proposed controller in the presence of uncertainty furthermore comparisons with a competing type i reduced controller as well as a pd controller show this superiority to be more pronounced especially when noise level is remarkably high moreover the designed controller satisfies the computational complexity constraints for real time implementation

real hypersurfaces orders of contact and applications a fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the cauchy riemann equations suppose that d is an open domain in cn and that its boundary m is a smooth real submanifold of cn how does the geometry of m influence the function theory on d one approach to this question is the a neumann problem 6 there one studies the inhomogeneous cauchy riemann equations au a where a is a 0 q form with distribution coefficients on the closed domain d u m satisfying the necessary compatibility condition a a 0 the a neumann problem constructs a particular solution u which is orthogonal to the null space of the complex laplacian kohn 10a has solved this problem and has shown that good local regularity properties for u in terms of a follow from so called subelliptic estimates it is natural to seek geometric conditions on m that are necessary and sufficient for these estimates in this paper we will be concerned with a certain geometric condition on m that we feel is the right one for subellipticity although the motivation for the questions discussed here comes from partial differential equations the techniques come from algebraic geometry to see why we state theorems of kohn greiner and catlin concerning subellipticity kohn 1oa has discovered a sufficient condition for a subelliptic estimate near a fixed point p in m in case d is pseudoconvex and is contained in c2 this condition is equivalent to the finiteness of order of tangency of every complex analytic one dimensional manifold with m at p greiner 13 established the necessity of this condition in

centroid of a type 2 fuzzy set abstract in this paper we introduce the centroid and generalized centroid of a type 2 fuzzy set both of which are essential for implementing a type 2 fuzzy logic system and explain how to compute them for practical use we show how to compute the centroid of interval and gaussian type 2 fuzzy sets an exact computation procedure is provided for an interval type 2 set whereas an approximation is provided for both interval and gaussian type 2 sets examples are given that compare the exact computational results with the approximate results

maximal abelian subalgebras of c algebras associated with complex dynamical systems and self similar maps abstract we consider an analogy among markov shifts complex dynamical systems and self similar maps their dynamics are given by 0 1 matrices a rational functions r and self similar maps on a compact metric space k respectively if the 0 1 matrix a is irreducible and not a permutation then the cuntz krieger algebra o a is simple and purely infinite similarly if the rational function r is restricted to the julia set j r and the self similar map satisfies the open set condition respectively then the associated c algebras o r j r and o k are simple and purely infinite let a be the associated infinite path space for the 0 1 matrix a then c a is known to be a maximal abelian subalgebra of o a in this paper we shall show that c j r is a maximal abelian subalgebra of o r j r and c k is a maximal abelian subalgebra of o k

general type 2 fuzzy logic systems made simple a tutorial the purpose of this tutorial paper is to make general type 2 fuzzy logic systems gt2 flss more accessible to fuzzy logic researchers and practitioners and to expedite their research designs and use to accomplish this the paper 1 explains four different mathematical representations for general type 2 fuzzy sets gt2 fss 2 demonstrates that for the optimal design of a gt2 fls one should use the vertical slice representation of its gt2 fss because it is the only one of the four mathematical representations that is parsimonious 3 shows how to obtain set theoretic and other operations for gt2 fss using type 1 t1 fs mathematics cuts play a central role 4 reviews mamdani and tsk interval type 2 it2 flss so that their mathematical operations can be easily used in a gt2 fls 5 provides all of the formulas that describe both mamdani and tsk gt2 flss 6 explains why center of sets type reduction should be favored for a gt2 fls over centroid type reduction 7 provides three simplified gt2 flss two are for mamdani gt2 flss and one is for a tsk gt2 fls all of which bypass type reduction and are generalizations from their it2 fls counterparts to gt2 flss 8 explains why gradient based optimization should not be used to optimally design a gt2 fls 9 explains how derivative free optimization algorithms can be used to optimally design a gt2 fls and 10 provides a three step approach for optimally designing flss in a progressive manner from t1 to it2 to gt2 each of which uses a quantum particle swarm optimization algorithm by virtue of which the performance for the it2 fls cannot be worse than that of the t1 fls and the performance for the gt2 fls cannot be worse than that of the it2 fls

holder estimates for homotopy operators on strictly pseudoconvex domains with c 2 boundary we derive a new homotopy formula for a strictly pseudoconvex domain of c 2 boundary in mathbf c n by using a method of lieb and range and obtain estimates in lipschitz spaces for the homotopy operators for r 1 and q 0 we obtain a lambda r 1 2 solution u to overline partial u f for a overline partial closed 0 q form f of class lambda r in the domain we apply the estimates to obtain boundary regularities of mathcal d solutions for a domain in mathbf c n times mathbf r m

on the existence of conformal measures a general notion of conformal measure is introduced and some basic properties are studied sufficient conditions for the existence of these measures are obtained using a general construction principle the geometric properties of conformal measures relate equilibrium states and hausdorff measures this is shown for invariant subsets of s under expanding maps

similarity measures for general type 2 fuzzy sets based on the α plane representation abstract similarity measures are very important concepts in fuzzy sets fss theory there are many different definitions of similarity measures for both type 1 t1 fss and interval type 2 it2 fss in this paper one similarity measure for it2 fss which is extended from the jaccard similarity measure for t1 fss is reviewed then based on the plane representation for a general type 2 gt2 fs this similarity measure is generalized to such t2 fss some examples that demonstrate how to compute the similarity measures for different t2 fss are given

finite digraphs and kms states abstract the paper contains a description of the kms states of a generalized gauge action on the c algebra of a finite graph

kms states on pimsner algebras associated with c dynamical systems abstract we examine the theory of the kms states on pimsner algebras arising from multivariable unital c dynamical systems as an application we show that pimsner algebras of piecewise conjugate classical systems attain the same kms states even though it is still an open problem whether or not they are isomorphic

new results about the centroid of an interval type 2 fuzzy set including the centroid of a fuzzy granule the centroid of an interval type 2 fuzzy set it2 fs provides a measure of the uncertainty of such a fs its calculation is very widely used in interval type 2 fuzzy logic systems in this paper we present properties about the centroid of an it2 fs we also illustrate many of the general results for a t2 fuzzy granule fg in order to develop some understanding about the uncertainty of the fg in terms of its vertical and horizontal dimensions at present the t2 fg is the only it2 fs for which it is possible to obtain closed form formulas for the centroid and those formulas are in this paper

long term learning for type 2 neural fuzzy systems abstract the development of a new long term learning framework for interval valued neural fuzzy systems is presented for the first time in this article the need for such a framework is twofold to address continuous batch learning of data sets and to take advantage the extra degree of freedom that type 2 fuzzy logic systems offer for better model predictive ability the presented long term learning framework uses principles of granular computing grc to capture information knowledge from raw data in the form of interval valued sets in order to build a computational mechanism that has the ability to adapt to new information in an additive and long term learning fashion the latter is to accommodate new input output mappings and new classes of data without significantly disturbing existing input output mappings therefore maintaining existing performance while creating and integrating new knowledge rules this is achieved via an iterative algorithmic process which involves a two step operation iterative rule base growth capturing new knowledge and iterative rule base pruning removing redundant knowledge for type 2 rules the two step operation helps create a growing but sustainable model structure the performance of the proposed system is demonstrated using a number of well known non linear benchmark functions as well as a highly nonlinear multivariate real industrial case study simulation results show that the performance of the original model structure is maintained and it is comparable to the updated model s performance following the incremental learning routine the study is concluded by evaluating the performance of the proposed framework in frequent and consecutive model updates where the balance between model accuracy and complexity is further assessed

inverse controller design for fuzzy interval systems this paper aims at designing and analyzing an inverse controller for stable inversible minimum phase fuzzy interval linear and or multilinear systems the controller is designed from the fuzzy interval ranges of the system parameters using an spl alpha cut methodology indeed for a given spl alpha cut of the fuzzy system parameters representing an uncertainty level the control objective can be viewed as maintaining the system output within a tolerance envelope around the exact trajectory specified by the degree of preference spl alpha on the fuzzy trajectory the stability is ensured in the way that the controller restricts the system output divergence within the tolerance envelope the validity of the proposed method is illustrated by simulation examples

hausdorff dimension of julia sets of feigenbaum polynomials with high criticality we consider unimodal polynomials with feigenbaum topological type and critical points whose orders tend to infinity it is shown that the hyperbolic dimensions of their julia set go to 2 furthermore that the hausdorff dimensions of the basins of attraction of their feigenbaum attractors also tend to 2 the proof is based on constructing a limiting dynamics with a flat critical point

a type 2 fuzzy approach to linguistic summarization of data this paper introduces an application of type 2 fuzzy sets in data linguistic summarization the original approach by yager 1982 based on representing natural language statements via type 1 i e the zadeh fuzzy sets is generalized with type 2 fuzzy sets applied as models of linguistically expressed quantities and or properties of objects type 2 sets extend the known summarization procedures by handling fuzzy values stored in databases and allow to represent a linguistic term via a few different membership functions e g provided by different experts which makes the method more general and human consistent furthermore quality measures for type 2 summaries are discussed in order to evaluate the informativeness of the messages generated finally two prototype applications are presented and the success of the new method is discussed

the collapsing method of defuzzification for discretised interval type 2 fuzzy sets this paper proposes a new approach for defuzzification of interval type 2 fuzzy sets the collapsing method converts an interval type 2 fuzzy set into a type 1 representative embedded set res whose defuzzified values closely approximates that of the type 2 set as a type 1 set the res can then be defuzzified straightforwardly the novel representative embedded set approximation resa to which the method is inextricably linked is expounded stated and proved within this paper it is presented in two forms simple resa this approximation deals with the most simple interval fou in which a vertical slice is discretised into 2 points interval resa this approximation concerns the case in which a vertical slice is discretised into 2 or more points the collapsing method simple resa version was tested for accuracy and speed with excellent results on both criteria the collapsing method proved more accurate than the karnik mendel iterative procedure kmip for an asymmetric test set for both a symmetric and an asymmetric test set the collapsing method outperformed the kmip in relation to speed