asymmetric subsethood product fuzzy neural inference system asupfunis this work presents an asymmetric subsethood product fuzzy neural inference system asupfunis that directly extends the supfunis model by permitting signal and weight fuzzy sets to be modeled by asymmetric gaussian membership functions the asymmetric subsethood product network admits both numeric as well as linguistic inputs input nodes which act as tunable feature fuzzifiers fuzzify numeric inputs with asymmetric gaussian fuzzy sets and linguistic inputs are presented as is the antecedent and consequent labels of standard fuzzy if then rules are represented as asymmetric gaussian fuzzy connection weights of the network the model uses mutual subsethood based activation spread and a product aggregation operator that works in conjunction with volume defuzzification in a gradient descent learning framework despite the increase in the number of free parameters the proposed model performs better than supfunis on various benchmarking problems both in terms of the performance accuracy and architectural economy and compares excellently with other various existing models with a performance better than most of them

reply to comment on invariants of differential equations defined by vector fields we undertook the determination of fundamental sets of invariants for differential equations defined by vector fields in a recent paper using the equivariant moving frame method a more exhaustive list for these invariants was found in a subsequent comment by francis valiquette we reply to that paper and point out some essential inconsistencies in the comment

approximation analysis of feedforward regular fuzzy neural network with two hidden layers the closure fuzzy mapping is employed to analyze approximation capability of feedforward regular fuzzy neural networks fnns with two hidden layers the bridge to do that is the fuzzy valued bernstein polynomial by the fuzzy polynomial approximation theorem this paper establishes some equivalent conditions for the fuzzy functions that can be approximated by the regular fnns to any degree of accuracy thus universal approximation of regular fnns based on zadeh s extension principle and fuzzy arithmetic can be studied systematically finally a simulation example is used to demonstrate the constructive procedure of approximating fnns

fuzzy continuous input output controllers are universal approximators the aim of this paper is twofold i to analyse why the fuzzy input output controllers proposed by buckley and hayashi fuzzy sets and systems 58 1993 273 are hardly continuous and based on this analysis a special necessary and sufficient condition for the controllers being continuous is obtained ii to amend buckley hayashi s controllers so that to make them unconditionally continuous

fuzzy nonlinear regression with fuzzified radial basis function network a fuzzified radial basis function network frbfn is a kind of fuzzy neural network that is obtained by direct fuzzification of the well known neural model rbfn a frbfn contains fuzzy weights and can handle fuzzy in fuzzy out data this paper shows that a frbfn can also be interpreted as a kind of fuzzy expert system hence it owns the advantages of simple structure and clear physical meaning some metrics for fuzzy numbers have been extended to the metrics for n dimensional fuzzy vectors which are applicable to computations in frbfns the corresponding metric spaces for n dimensional fuzzy vectors are proved to be complete further frbfns are proved to be able to act as universal function approximators for any continuous fuzzy function defined on a compact set this paper applies the proposed frbfn to nonparametric fuzzy nonlinear regression problems for multidimensional lr type fuzzy data fuzzy nonlinear regression with frbfns can be formulated as a nonlinear mathematical programming problem two training algorithms are proposed to quickly solve the two types of problems under different criteria and constraint conditions namely the two stage and bp back propagation training algorithms simulation studies are carried out to verify the feasibility and demonstrate the advantages of the proposed approaches

numerical solution of a system of fuzzy polynomials by fuzzy neural network in this paper a new approach for solving systems of fuzzy polynomials based on fuzzy neural network fnn is presented this method can also lead to improve numerical methods in this work an architecture of fuzzy neural networks is also proposed to find a real root of a system of fuzzy polynomials if exists by introducing a learning algorithm finally we illustrate our approach by numerical examples

characterization of compact subsets of fuzzy sets a characterization of compact subsets is presented for the metric space of normal fuzzy convex fuzzy sets on the base space rn the metric for which is the supremum over the hausdorff distances between corresponding level sets it is shown that a closed subset is compact if and only if it is uniformly support bounded and the corresponding set of support functions is equileftcontinuous in the membership grade variable

fuzzy input output controllers are universal approximators abstract in this paper we show that a fuzzy input output controller can approximate to any degree of accuracy a continuous fuzzy system this generalizes the known result that fuzzy controllers crisp input output can approximate continuous functions

spectral geometry of symplectic spinors symplectic spinors form an infinite rank vector bundle dirac operators on this bundle were constructed recently by habermann k the dirac operator on symplectic spinors ann global anal geom 13 155 168 1995 here we study the spectral geometry aspects of these operators in particular we define the associated distance function and compute the heat trace asymptotics

iterative inversion of fuzzified neural networks the inversion of a neural network is a process of computing inputs that produce a given target when fed into the neural network the inversion algorithm of crisp neural networks is based on the gradient descent search in which a candidate inverse is iteratively refined to decrease the error between its output and the target in this paper we derive an inversion algorithm of fuzzified neural networks from that of crisp neural networks first we present a framework of learning algorithms of fuzzified neural networks and introduce the idea of adjusting schemes for fuzzy variables next we derive the inversion algorithm of fuzzified neural networks by applying the adjusting scheme for fuzzy variables to total inputs in the input layer finally we make three experiments on the parity three problem examine the effect of the size of training sets on the inversion and investigate how the fuzziness of inputs and targets of training sets affects the inversion

a hybrid genetic fuzzy neural network algorithm designed for classification problems involving several groups we propose a multigroup classification algorithm based on a hybrid genetic fuzzy neural net gfnn framework recent results on evolutionary computation and fuzzy neural network methodology are combined to effectively adapt the membership functions of the fuzzifier and the defuzzifier to the data set separate membership functions are defined for each dimension in the fuzzifier and for each fuzzy output group in the defuzzifier the signal inherent in the fuzzifier is aggregated by a suitable t norm and transmitted to the defuzzifier the defuzzifier aggregates the response i e the predicted group membership by a suitable conorm if misclassifications occur during training the membership functions of both the fuzzifier and the defuzzifier are adapted by a systematic robust procedure the algorithm is successfully tested with real economic data in total the gfnn performs as good as the best of the competing methods in our test the results suggest economically meaningful interpretations

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

evaluative linguistic expressions vs fuzzy categories in this paper we discuss the distinction between categories characterized by verbal labels taken from a fuzzy rating scale and special class of linguistic expressions called evaluative the latter form a general class of expressions that includes gradable and evaluative adjectives and their hedges first we will provide a brief linguistic analysis of them then we outline basic principles for construction of the mathematical model of semantics of evaluative expressions in section 3 we will analyze the concepts of rating scale with verbal labels fuzzy rating scale their semantics and demonstrate that the latter cannot be identified with the semantics of evaluative expressions finally we suggest a method for construction of fuzzy rating scales on the basis of evaluative linguistic expressions the main goal of this paper is to demonstrate the power of fuzzy set theory and to show that it is capable of providing tools for advanced modeling of linguistic semantics

stability analysis of neural net controllers using fuzzy neural networks neural networks can only be trained with a crisp and finite data set therefore the approximation quality of a trained network is hard to verify so a common way in proving stability of a trained neural net controller is to demonstrate the existence of a lyapunov function in this article we propose a new method how stability of a neural net controller used as a closed loop feedback controller can be proven instead of finding a lyapunov function conditions for a fuzzy training set are developed if a fuzzy neural net is trained using this training set special properties of fuzzy neural nets can be used for estimating the generalization error after defuzzification of the fuzzy net finite stability of the process can be concluded

fuzzified neural network based on fuzzy number operations the fuzzified neural network based on fuzzy number operations is presented as a powerful modelling tool here we systematically introduce ideas and concepts of a novel neural network based on fuzzy number operations first we suggest how to compute the results of addition subtraction multiplication and division for two fuzzy numbers second we propose a learning algorithm and present some ideas about the choice of fuzzy weights and fuzzy biases and a numerical scheme for the calculation of outputs of the fuzzified neural network finally we show some results of computer simulations

a neural network approach to the fuzzy transform the paper deals with the f transform technique which was introduced as a method for an approximate representation of continuous functions the same task is solved by many methods from different areas neural networks also belong to techniques which have been proved to be powerful for approximation objectives they provide us with many advantages especially incremental way of learning parameters the paper inherits neural approaches to the f transform method and presents experiments justifying the proposed approach incremental way of determination of certain parameters of the f transform method which were up to now given by batch formula enriches possible areas of application of the method by fast on line processes moreover the neural approach is demonstrated to be an appropriate one for finding such fuzzy partition of a domain which respects better a given set of measured samples which are to be approximated by a continuous function with no predetermined shape

analysis of approximation of continuous fuzzy functions by multivariate fuzzy polynomials for the given fuzzy valued function j rn f0 r the fuzzy valued bernstein polynomial is introduced to show the fact that continuous fuzzy valued functions can be approximately represented as the fuzzy valued polynomials on any compact set u rn some equivalent condition for the continuous fuzzy function f f0 r n f0 r that can be approximated to any degree of accuracy by fuzzy polynomials on arbitrary compact set u f0 r n are presented

approximation capabilities of multilayer fuzzy neural networks on the set of fuzzy valued functions this paper first introduces a piecewise linear interpolation method for fuzzy valued functions based on this we present a concrete approximation procedure to show the capability of four layer regular fuzzy neural networks to perform approximation on the set of all d p continuous fuzzy valued functions this approach can also be used to approximate d continuous fuzzy valued functions an example is given to illustrate the approximation procedure

noncommutative residue and sub dirac operators for foliations in this paper we define lower dimensional volumes associated with sub dirac operators for foliations in some cases we compute these lower dimensional volumes we also prove the kastler kalau walze type theorems for foliations with or without boundary as a corollary we give an explanation of the gravitational action for the robertson walker space a b f m3

vector fields on a disk with mixed boundary conditions vector fields on a disk satisfying two types of boundary conditions are studied these boundary conditions are selected by brst invariance in electrodynamics they also appear in the de rham complex the construction of the harmonic expansion is presented the eigenfunctions of the vector laplace operator are expressed in terms of fields satisfying pure dirichlet or robin boundary conditions for the case of four dimensional disk several first coefficients of the heat kernel expansion are computed an error in the analytical expression by branson and gilkey is corrected

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

universal approximations of continuous fuzzy valued functions by multi layer fuzzy neural networks abstract the fact that four layer feedforward regular fuzzy neural networks with sigmoid function in the first hidden layer are capable of approximately representing continuous fuzzy valued functions on any compact set of r is shown at first bernstein polynomials associated with fuzzy valued functions are employed to approximate continuous fuzzy valued function defined on a compact set secondly by the conclusions related to standard feedforward networks universal approximations of continuous fuzzy valued functions by regular fuzzy neural networks are obtained

heat kernel coefficients for chiral bag boundary conditions we study the asymptotic expansion of the smeared l 2 trace of f e tp 2 where p is an operator of dirac type f is an auxiliary smooth smear ing function which is used to localize the problem and chiral bag boundary conditions are imposed special case calculations functorial methods and the theory of and invariants are used to obtain the boundary part of the heat kernel coefficientsa1 and a2

the fuzzy associative memory of max min fuzzy neural network with threshold abstract in this paper we introduce the max min fuzzy neural network with threshold which generalizes the fuzzy neural network models in 1 4 6 8 10 and show that the storage capacity of the two layer max min fuzzy neural network is identical with one of the max min fuzzy neural network with hidden layers for given fuzzy pattern pairs x 1 y 1 x p y p we obtain an equivalent condition which the family of given fuzzy pattern pairs can be stored in the fuzzy neural network finally we give a example to demonstrate our conclusions

global and local aspects of spectral actions the principal object in noncommutative geometry is the spectral triple consisting of an algebra a hilbert space and a dirac operator field theories are incorporated in this approach by the spectral action principle which sets the field theory action to where f is a real function such that the trace exists and is a cutoff scale in the low energy weak field limit the spectral action reproduces reasonably well the known physics including the standard model however not much is known about the spectral action beyond the low energy approximation in this paper after an extensive introduction to spectral triples and spectral actions we study various expansions of the spectral actions exemplified by the heat kernel we derive the convergence criteria for a commutative spectral triple we compute the heat kernel on the torus up to the second order in gauge connection and consider limiting cases this article is part of a special issue of journal of physics a mathematical and theoretical in honour of stuart dowker s 75th birthday devoted to applications of zeta functions and other spectral functions in mathematics and physics

principal component analysis of fuzzy data using autoassociative neural networks this paper describes an extension of principal component analysis pca allowing the extraction of a limited number of relevant features from high dimensional fuzzy data our approach exploits the ability of linear autoassociative neural networks to perform information compression in just the same way as pca without explicit matrix diagonalization fuzzy input values are propagated through the network using fuzzy arithmetics and the weights are adjusted to minimize a suitable error criterion the inputs being taken as target outputs the concept of correlation coefficient is extended to fuzzy numbers allowing the interpretation of the new features in terms of the original variables experiments with artificial and real sensory evaluation data demonstrate the ability of our method to provide concise representations of complex fuzzy data

an intelligent stock trading decision support system through integration of genetic algorithm based fuzzy neural network and artificial neural network the stock market which has been investigated by various researchers is a rather complicated environment most research only concerned the technical indexes quantitative factors instead of qualitative factors e g political effect however the latter plays a critical role in the stock market environment thus this study develops a genetic algorithm based fuzzy neural network gfnn to formulate the knowledge base of fuzzy inference rules which can measure the qualitative effect on the stock market next the effect is further integrated with the technical indexes through the artificial neural network ann an example based on the taiwan stock market is utilized to assess the proposed intelligent system evaluation results indicate that the neural network considering both the quantitative and qualitative factors excels the neural network considering only the quantitative factors both in the clarity of buying selling points and buying selling performance

spectral action for torsion with and without boundaries we derive a commutative spectral triple and study the spectral action for a rather general geometric setting which includes the skew symmetric torsion and the chiral bag conditions on the boundary the spectral action splits into bulk and boundary parts in the bulk we clarify certain issues of the previous calculations show that many terms in fact cancel out and demonstrate that this cancellation is a result of the chiral symmetry of spectral action on the boundary we calculate several leading terms in the expansion of spectral action in four dimensions for vanishing chiral parameter of the boundary conditions and show that 0 is a critical point of the action in any dimension and at all orders of the expansion

can neural nets be universal approximators for fuzzy functions abstract we first argue that the extension principle is too computationally involved to be an efficient way for a computer to evaluate fuzzy functions we then suggest using cuts and interval arithmetic to compute the valued of fuzzy functions using the method of computing fuzzy functions we then show that neural nets are universal approximators for computable fuzzy functions when we only input non negative or non positive fuzzy numbers

non monotonic set theoretic operations abstract we introduce a new operation called the non monotonic intersection this operation has significant applications in default and other commonsense reasoning systems in addition to being non monotonic this operator is not a pointwise operator furthermore it is neither commutative nor associative but rather supports a concept of priority amongst its arguments we also introduce a related non monotonic union we generalize these ideas with the aid of t norms

spectral triples and manifolds with boundary abstract we investigate manifolds with boundary in noncommutative geometry spectral triples associated to a symmetric differential operator and a local boundary condition are constructed we show that there is no tadpole for classical dirac operators with a chiral boundary condition on spin manifolds

topological properties of fuzzy numbers fuzzy numbers are defined and a metric is assigned to this class of fuzzy sets topological properties of the resulting metric space are studied the notion of q crisp fuzzy numbers is introduced and in particular it is shown that this class of fuzzy numbers is homeomorphic to the hilbert cube

a learning algorithm of fuzzy neural networks with triangular fuzzy weights abstract in this paper first we propose an architecture of fuzzy neural networks with triangular fuzzy weights the proposed fuzzy neural network can handle fuzzy input vectors as well as real input vectors in both cases outputs from the fuzzy neural network are fuzzy vectors the input output relation of each unit of the fuzzy neural network is defined by the extension principle of zadeh next we define a cost function for the level sets i e cuts of fuzzy outputs and fuzzy targets then we derive a learning algorithm from the cost function for adjusting three parameters of each triangular fuzzy weight finally we illustrate our approach by computer simulations on numerical examples

sugeno type controllers are universal controllers abstract we show that given any continuous real valued function on a compact set there is a sugeno type controller that will approximate this function to any degree of accuracy an application of this result to fuzzy control is presented which shows that this type of sugeno controller is a universal controller

universal fuzzy controllers based on generalized t s fuzzy models this paper investigates the universal fuzzy control problem based on generalized t s fuzzy models the universal approximation capability of the generalized t s fuzzy models is shown and an approach to robust controller design for general nonlinear systems based on this kind of generalized t s fuzzy models is developed the results of universal fuzzy controllers for two classes of nonlinear systems are then given and constructive procedures to obtain the universal fuzzy controllers are also provided an example is finally presented to show the effectiveness of our approach

numerical solution of fuzzy polynomials by fuzzy neural network in this paper an architecture of fuzzy neural networks is proposed to find a real root of fuzzy polynomials if exists by introducing a learning algorithm we proposed a learning algorithm from the cost function for adjusting of crisp weights finally we illustrated our approach by numerical examples

fuzzy neural networks a survey abstract in this paper a fuzzy neural network will be a layered feedforward neural net that has fuzzy signals and or fuzzy weights we survey recent results on learning algorithms and applications for fuzzy neural networks

formalized theory of general fuzzy reasoning first a rigorous mathematical theory for calculating the conclusion of general fuzzy reasoning is proposed second a formalized non fuzzy version of the theory of general fuzzy reasoning is founded in classical logics

universal approximation by hierarchical fuzzy system with constraints on the fuzzy rule this paper presents a special hierarchical fuzzy system where the outputs of the previous layer are not used in the if parts but used only in the then parts of the fuzzy rules of the current layer the proposed scheme can be shown to be a universal approximator to any continuous function on a compact set if complete fuzzy sets are used in the if parts of the fuzzy rules with singleton fuzzifier and center average defuzzifier from the simulation of ball and beam control system it is demonstrated that the proposed scheme approximates with good accuracy the model nonlinear controller with fewer fuzzy rules than the centralized fuzzy system and its control performance is comparable to that of the nonlinear controller

fuzzy hyperbolic neural network with time varying delays in this paper the formula of fuzzy basis functions with time varying delays dfbf is firstly given based on the dfbf a novel fuzzy hyperbolic neural network with time varying delays dfhnn and its approximation properties are presented respectively by constructing an appropriate lyapunov krasovskii functional the stability of dfhnn is discussed and some new criteria concerning the global exponential stability of dfhnn are proposed finally three numerical examples are given to demonstrate the effectiveness of the obtained results

the fuzzy neural network approximation lemma abstract it is well known that artificial neural networks are universal approximators but what about fuzzy neural networks only buckley and hayashi 1 presented a theoretical result for these networks they showed that there are fuzzy functions which cannot be approximated by a certain fuzzy neural network in this paper we answer the question for a special type of fuzzy neural networks especially with regard to the fuzzy arithmetic that is used it is simplified in order to minimize the computational expense as well as to simplify the theoretical examinations we prove that the class of fuzzy functions which is identical to the class of all continuous real functions extended by means of the extension principle can be approximated by certain fuzzy neural networks

a neuro fuzzy approach to gear system monitoring the detection of the onset of damage in gear systems is of great importance to industry in this paper a new neuro fuzzy diagnostic system is developed whereby the strengths of three robust signal processing techniques are integrated the adopted techniques are the continuous wavelet transform amplitude and beta kurtosis based on the overall residual signal and the phase modulation by employing the signal average three reference functions are proposed as post processing techniques to enhance the feature characteristics in a way that increases the accuracy of fault detection monitoring indexes are derived to facilitate the automatic diagnoses a constrained gradient reliability algorithm is developed to train the fuzzy membership function parameters and rule weights while the required fuzzy completeness is retained the system output is set to different monitoring levels by using an optimization procedure to facilitate the decision making process the test results demonstrate that the novel neuro fuzzy system because of its adaptability and robustness significantly improves the diagnostic accuracy it outperforms other related classifiers such as those based on fuzzy logic and neuro fuzzy schemes which adopt different types of rule weights and employ different training algorithms

evaluation of fuzzy regression models by fuzzy neural network in this paper a novel hybrid method based on fuzzy neural network for approximate fuzzy coefficients parameters of fuzzy linear and nonlinear regression models with fuzzy output and crisp inputs is presented here a neural network is considered as a part of a large field called neural computing or soft computing moreover in order to find the approximate parameters a simple algorithm from the cost function of the fuzzy neural network is proposed finally we illustrate our approach by some numerical examples

numerical relationships between neural networks continuous functions and fuzzy systems abstract in this survey paper we first discuss computational equivalence between continuous functions regular neural nets fuzzy controllers and discrete fuzzy expert systems we also discuss how to build hybrid neural nets numerically identical to a fuzzy controller or a discrete fuzzy expert system we then discuss computational equivalence between continuous fuzzy functions regular fuzzy neural nets and fuzzy systems we also discuss how to construct hybrid fuzzy neural nets to approximate continuous fuzzy functions

max min fuzzy hopfield neural networks and an efficient learning algorithm abstract we set up a dynamical fuzzy neural network system i e the so called max min fuzzy hopfield network in the paper and prove the lyapunov stability of the equilibrium points attractor of the system also we discuss the uniform stability of the system and show some sufficient conditions with which the given fuzzy pattern is the attractor of the system moreover we obtain a nontrivially attractive basin of the attractor therefore our models have good fault tolerance with an analytic method we design an efficient learning algorithm for connected weights of the networks finally simulation examples demonstrate our conclusions

comment on invariants of differential equations defined by vector fields in ndogmo 2008 j phys a math theor 41 025207 the author claims to have determined a complete list of functionally independent differential invariants up to order 2 for the equivalence group of differential equations defined by vector fields when there are two and three independent variables in this comment we show that this is not the case using the equivariant moving frame method we derive a complete set of functionally independent differential invariants of orders 1 and 2 for an arbitrary number n 2 of independent variables in the particular case n 2 we obtain six functionally independent invariants two of which were not found in ndogmo 2008 j phys a math theor 41 025207 in the case n 3 we get 21 functionally independent invariants six of which are new we also give a complete classification of the differential invariants

analyses of regular fuzzy neural networks for approximation capabilities abstract by introducing a trapezoidal fuzzy function and using the properties of the function we set up a continuously increasing fuzzy function that cannot be arbitrarily closely approximated on a compact set of f 0 r by the regular fuzzy neural network rfnn thus the conclusions in buckley and hayashi fuzzy sets and systems 61 1994 43 51 are improved and the problem if the rfnn is the universal approximator to the class of continuously increasing fuzzy functions is solved finally we obtain the universal approximation to the extended fuzzy functions by rfnn

fuzzy nonlinear regression analysis using a random weight network modeling a fuzzy in fuzzy out system where both inputs and outputs are uncertain is of practical and theoretical importance fuzzy nonlinear regression fnr is one of the approaches used most widely to model such systems in this study we propose the use of a random weight network rwn to develop a fnr model called fnrrwn where both the inputs and outputs are triangular fuzzy numbers unlike existing fnr models based on back propagation bp and radial basis function rbf networks fnrrwn does not require iterative adjustment of the network weights and biases instead the input layer weights and hidden layer biases of fnrrwn are selected randomly the output layer weights for fnrrwn are calculated analytically based on a derived updating rule which aims to minimize the integrated squared error between cut sets that correspond to the predicted fuzzy outputs and target fuzzy outputs respectively in fnrrwn the integrated squared error is solved approximately by riemann integral theory the experimental results show that the proposed fnrrwn method can effectively approximate a fuzzy in fuzzy out system fnrrwn obtains better prediction accuracy in a lower computational time compared with existing fnr models based on bp and rbf networks

fuzzy basis functions universal approximation and orthogonal least squares learning fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions using the stone weierstrass theorem it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy based on the fuzzy basis function representations an orthogonal least squares ols learning algorithm is developed for designing fuzzy systems based on given input output pairs then the ols algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system the fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system and the simulation results show that the control performance is improved by incorporating some common sense fuzzy control rules

universal fuzzy controllers a collection b of fuzzy controllers is a universal fuzzy controller if and only if given any controllable process p it can also be controlled by some fuzzy controller in b this paper identifies a new type of universal fuzzy controller

approximations of continuous functionals by neural networks with application to dynamic systems the paper gives several strong results on neural network representation in an explicit form under very mild conditions a functional defined on a compact set in c a b or l sup p a b spaces of infinite dimensions can be approximated arbitrarily well by a neural network with one hidden layer the results are a significant development beyond earlier work where theorems of approximating continuous functions defined on a finite dimensional real space by neural networks with one hidden layer were given all the results are shown to be applicable to the approximation of the output of dynamic systems at any particular time

theory of the fuzzy controller abstract we first introduce notation in an attempt to standardize the description of a basic fuzzy controller if r 1 r 2 r n are the real number inputs to the fuzzy controller and is its defuzzified output then f r 1 r n for some function f by the theory of the fuzzy controller we mean in this paper the study of the structure of f for elementary fuzzy controllers using this standardized notation we first survey the known results on the theory of the fuzzy controller and then discuss important future research topics in this area

invariants of differential equations defined by vector fields we determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold we also find all invariants and differential invariants for this group up to the second order a result on the characterization of classes of these equations by the invariant functions is also given

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

controllable processes and the fuzzy controller abstract we show that any controllable process can be controlled by the scaled and averaged output of multiple elementary fuzzy controllers operating simultaneously

novel fuzzy inference system fis analysis and design based on lattice theory we introduce novel set and lattice theoretic perspectives and tools for the analysis and design of fuzzy inference systems fiss we present an fis including both fuzzification and defuzzification as a device for implementing a function f rnrarr rm the family of fis functions has cardinality aleph2 2aleph1 where aleph1 is the cardinality of the set r of real numbers hence the fis family is much larger than polynomials neural networks etc furthermore a fis has a capacity for local generalization a formulation in the context of lattice theory allows us to define the set f of fuzzy interval numbers fins which includes both fuzzy numbers and intervals we present a metric dk on f which can introduce tunable nonlinearities fis design based on dk has advantages such as an alleviation of the curse of dimensionality problem and a potential for improved computer memory utilization we present a new fis classifier namely granular self organizing map grsom which we apply to an industrial fertilizer modeling application

universal approximators for fuzzy functions abstract we show how to construct a large class of universal approximators for fuzzy functions which continuously map fuzzy numbers into fuzzy numbers and are the extension principle extensions of continuous real valued functions one important application is that layered feedforward neural nets with real weights and bias terms and fuzzy signals whose output is computed using the extension principle are universal approximators for these functions

maurer cartan equations for lie symmetry pseudogroups of differential equations a new method of constructing structure equations of lie symmetry pseudogroups of differential equations dispensing with explicit solutions of the infinitesimal determining systems of the pseudogroups is presented and illustrated by the examples of the kadomtsev petviashvili and korteweg de vries equations

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

efficient learning algorithms for three layer regular feedforward fuzzy neural networks a key step of using gradient descend methods to develop learning algorithms of a regular feedforward fuzzy neural network fnn is to differentiate max min functions which contain max spl or and min spl and operations the paper aims at several objectives first investigate further the differentiation of spl or spl and functions second employ general fuzzy numbers which include triangular and trapezoidal fuzzy numbers as special cases to define a three layer regular fnn the general fuzzy numbers related can be approximately determined by their corresponding finite level sets so we can approximately represent the input output i o relationship of the regular fnn as functions of the endpoints of all finite level sets third a fuzzy back propagation algorithm is presented and to speed up the convergence of the learning algorithm a fuzzy conjugate gradient algorithm for fuzzy weights and biases is developed furthermore the convergence of the algorithm is analyzed systematically finally some real simulations demonstrate the efficiency of our learning algorithms the regular fnn is applied to the approximate realization of fuzzy inference rules and fuzzy functions defined on given compact sets

on gravity torsion and the spectral action principle abstract we consider compact riemannian spin manifolds without boundary equipped with orthogonal connections we investigate the induced dirac operators and the associated commutative spectral triples in case of dimension four and totally anti symmetric torsion we compute the chamseddine connes spectral action deduce the equations of motions and discuss critical points

neuro fuzzy rule generation survey in soft computing framework the present article is a novel attempt in providing an exhaustive survey of neuro fuzzy rule generation algorithms rule generation from artificial neural networks is gaining in popularity in recent times due to its capability of providing some insight to the user about the symbolic knowledge embedded within the network fuzzy sets are an aid in providing this information in a more human comprehensible or natural form and can handle uncertainties at various levels the neuro fuzzy approach symbiotically combining the merits of connectionist and fuzzy approaches constitutes a key component of soft computing at this stage to date there has been no detailed and integrated categorization of the various neuro fuzzy models used for rule generation we propose to bring these together under a unified soft computing framework moreover we include both rule extraction and rule refinement in the broader perspective of rule generation rules learned and generated for fuzzy reasoning and fuzzy control are also considered from this wider viewpoint models are grouped on the basis of their level of neuro fuzzy synthesis use of other soft computing tools like genetic algorithms and rough sets are emphasized rule generation from fuzzy knowledge based networks which initially encode some crude domain knowledge are found to result in more refined rules finally real life application to medical diagnosis is provided

applying fuzzy adaptive network to fuzzy regression analysis abstract a fuzzy inference system based on the sugeno inference model is first formulated for fuzzy regression analysis this system is then represented by a fuzzy adaptive network this approach combines the power of representation of fuzzy inference system with the ability of learning of the neural network numerical examples are trained and solved to illustrate the approach the results are compared with other approaches

fuzzy least squares abstract several models for simple least squares fitting of fuzzy valued data are developed criteria are given for when fuzzy data sets can be fitted to the models and analogues of the normal equations are derived

a class of hierarchical fuzzy systems with constraints on the fuzzy rules this paper presents a class of hierarchical fuzzy systems where previous layer outputs are used not in if parts but only in then parts of the fuzzy rules of the current layer the proposed scheme is shown to be a universal approximator to any real continuous function on a compact set if complete fuzzy sets are used in the if parts of the fuzzy rules with singleton fuzzifier and center average defuzzifier from the example of ball and beam control system simulation it is demonstrated that the proposed scheme approximates with high accuracy a model nonlinear controller with fewer fuzzy rules than a centralized fuzzy system and its control performance is comparable to that of a nonlinear controller

the relation between inference and interpolation in the framework of fuzzy systems abstract this papers aims at clarifying the meaning of different interpretations of the max min or more generally the max t norm rule in fuzzy systems it turns out that basically two distinct approaches play an important role in fuzzy logic and its applications fuzzy interpolation on the basis of an imprecisely known function and logical inference in the presence of fuzzy information

strong ellipticity and spectral properties of chiral bag boundary conditions we prove strong ellipticity of chiral bag boundary conditions on even dimensional manifolds from a knowledge of the heat kernel in an infinite cylinder some basic properties of the zeta function are analysed on cylindrical product manifolds of arbitrary even dimension

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

a fuzzy neural network algorithm for multigroup classification we propose a multigroup classification algorithm based on a hybrid fuzzy neural net framework a key feature of the approach is the adaptation of membership functions to new data in this way learning is reflected in the shape of the membership functions by defining separate membership functions for each fuzzy output class we allow dynamic adjustment of the functions during training the algorithm is successfully tested with real economic data the results suggest economically meaningful interpretations

fuzzy i o controller abstract we describe a fuzzy controller which accepts and produces fuzzy data linguistic variables as well as crisp data real numbers for the control of man machine systems

approximation of fuzzy functions by regular fuzzy neural networks in this paper we investigate the ability of regular fuzzy neural networks to provide approximations to fuzzy functions since the operation of regular fuzzy neural networks is based on zadeh s extension principle we first present a level characterization of the zadeh s extensions of level continuous fuzzy valued functions and consider the continuity of these extensions on the basis of this we give characterizations of fuzzy functions which can be approximated by a class of four layer regular fuzzy neural networks according to supremum metric and level convergence

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space

can fuzzy neural nets approximate continuous fuzzy functions abstract we first argue that regular fuzzy neural nets based on standard fuzzy arithmetic and the extension principle cannot be universal approximators this negative result is surprising since regular neural nets are universal approximators we then argue that hybrid fuzzy neural nets not necessarily based on standard fuzzy arithmetic can be universal approximators