on relating two approaches to fractional calculus abstract in this paper two methods are discussed for defining fractional powers of the nth order ordinary differential expression l x a 1 dx a 2 d x a n dx a n 1 d d dx where each ai i 1 2 n 1 is a complex number and x is real and positive an important role is played by the number m i 1 n 1 a i n when m 0 spectral theory is used to obtain a suitable formula for l c while the case m 0 considered in a previous paper by a c mcbride proc london math soc 3 45 1982 519 546 is dealt with by expressing l in terms of erdelyi kober operators we give accounts of each approach and go on to consider how the results for m 0 can be obtained as limiting cases of the results for m 0

radial basis function based adaptive fuzzy systems and their applications to system identification and prediction abstract in this paper we describe a neuro fuzzy system with adaptive capability to extract fuzzy if then rules from input and output sample data through learning the proposed system called radial basis function rbf based adaptive fuzzy system afs employs the gaussian functions to represent the membership functions of the premise part of fuzzy rules three architectural deviations of the rbf based afs are also presented according to different consequence types such as constant first order linear function and fuzzy variable these provide versatility of the network to handle arbitrary fuzzy inference schemes we present examples of system identification and time series prediction to illustrate how to solve these problems and to demonstrate its validity and effectiveness using the rbf based afs

path integral quantization of the dyonium the dyonium is solved exactly by path integration the green s function for the dyonium is separated into the monopole harmonics and the radial path integral and the radial green s function is found in closed form the exact energy spectrum is also obtained dirac s charge quantization condition is seen to be essential for performing path integration

distributional fractional powers and riesz fractional integrals abstract a dual space fractional power theory is developed which follows naturally from the frechet space fractional power theory considered by s e schiavone proc roy soc edinburgh sect a 112 1989 237 247 properties of such dual space fractional powers are derived in addition an application to the riesz fractional integral associated with the three space dimensional wave equation facilitates the solution of a generalised cauchy problem conditions are established for the existence of a weak solution to this problem

a distributional theory of fractional calculus in this paper a theory of fractional calculus is developed for certain spaces d p of generalised functions the theory is based on the construction of fractionalpowers of certain simple differential and integral operators with the parameter suitably restricted these fractional powers are shown to coincide with the riemann liouville and weyl operators of fractional integration and differentiation standard properties associated with fractional integrals and derivatives follow immediately from results obtained previously by the author on fractional powers of operators see 6 7 some spectral properties are also obtained

fuzzy set based models of neurons and knowledge based networks we will introduce and study different fuzzy set oriented computational models of neurons the generic topologies of the neurons emerging there are significantly influenced by basic logic operators and or not encountered in the theory of fuzzy sets the logical flavor of the proposed constructs is expressed in terms of operators used in their formalization and a way of their superposition in the neurons the two broad categories of neurons embrace basic aggregation neurons named and and or neurons and referential processing units such as matching dominance inclusion neurons the specific features of the neurons are flexibly modeled with the aid of triangular norms the inhibitory and excitatory characteristics are captured by embodying direct and complemented negated input signals we will propose various topologies of neural networks put together with the use of these neurons and demonstrate straightforward relationships coming off between the problem specificity and the resulting architecture of the network this limpid way of mapping the domain knowledge onto the structure of the network contributes significantly toward enhancements in learning processes in the network and substantially facilitates its interpretation

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

path integral treatment of the hydrogen atom in a curved space of constant curvature ii hyperbolic space for pt i see ibid vol 20 p 6271 the path integral treatment of the hydrogen atom in a hyperbolic space is discussed the authors show by mapping the radial path integral into the su 1 1 group manifold that the system has a dynamical su 1 1 symmetry the energy spectrum and normalised energy eigenfunctions are calculated in the flat space limit the standard hydrogen spectrum and corresponding normalised wavefunctions are regained

fuzzy regression analysis using neural networks abstract in this paper we propose simple but powerful methods for fuzzy regression analysis using neural networks since neural networks have high capability as an approximator of nonlinear mappings the proposed methods can be applied to more complex systems than the existing lp based methods first we propose learning algorithms of neural networks for determining a nonlinear interval model from the given input output patterns a nonlinear interval model whose outputs approximately include all the given patterns can be determined by two neural networks next we show two methods for deriving nonlinear fuzzy models from the interval model determined by the proposed algorithms nonlinear fuzzy models whose h level sets approximately include all the given patterns can be derived last we show an application of the proposed methods to a real problem

a fractional power approach to fractional calculus abstract a theory of fractional powers of operators on an arbitrary frechet space is discussed as special cases multivariable fractional integrals and derivatives defined on certain spaces of test functions and generalised functions are obtained in particular properties of the two dimensional riesz fractional integral are determined and used to solve a distributional initial value problem involving the wave operator

a study on relationship between generalization abilities and fuzziness of base classifiers in ensemble learning we investigate essential relationships between generalization capabilities and fuzziness of fuzzy classifiers viz the classifiers whose outputs are vectors of membership grades of a pattern to the individual classes the study makes a claim and offers sound evidence behind the observation that higher fuzziness of a fuzzy classifier may imply better generalization aspects of the classifier especially for classification data exhibiting complex boundaries this observation is not intuitive with a commonly accepted position in traditional pattern recognition the relationship that obeys the conditional maximum entropy principle is experimentally confirmed furthermore the relationship can be explained by the fact that samples located close to classification boundaries are more difficult to be correctly classified than the samples positioned far from the boundaries this relationship is expected to provide some guidelines as to the improvement of generalization aspects of fuzzy classifiers

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

rspop rough set based pseudo outer product fuzzy rule identification algorithm system modeling with neuro fuzzy systems involves two contradictory requirements interpretability verses accuracy the pseudo outer product pop rule identification algorithm used in the family of pseudo outer product based fuzzy neural networks popfnn suffered from an exponential increase in the number of identified fuzzy rules and computational complexity arising from high dimensional data this decreases the interpretability of the popfnn in linguistic fuzzy modeling this article proposes a novel rough set based pseudo outer product rspop algorithm that integrates the sound concept of knowledge reduction from rough set theory with the pop algorithm the proposed algorithm not only performs feature selection through the reduction of attributes but also extends the reduction to rules without redundant attributes as many possible reducts exist in a given rule set an objective measure is developed for popfnn to correctly identify the reducts that improve the inferred consequence experimental results are presented using published data sets and real world application involving highway traffic flow prediction to evaluate the effectiveness of using the proposed algorithm to identify fuzzy rules in the popfnn using compositional rule of inference and singleton fuzzifier popfnn cri s architecture results showed that the proposed rough set based pseudo outer product algorithm reduces computational complexity improves the interpretability of neuro fuzzy systems by identifying significantly fewer fuzzy rules and improves the accuracy of the popfnn

fuzzy regression by fuzzy number neural networks in this paper we describe a method for nonlinear fuzzy regression using neural network models in earlier work strong assumptions were made on the form of the fuzzy number parameters symmetric triangular asymmetric triangular quadratic trapezoidal and so on our goal here is to substantially generalize both linear and nonlinear fuzzy regression using models with general fuzzy number inputs weights biases and outputs this is accomplished through a special training technique for fuzzy number neural networks the technique is demonstrated with data from an industrial quality control problem

on fuzzy associative memory with multiple rule storage capacity kosko s fuzzy associative memory fam is the very first neural network model for implementing fuzzy systems despite its success in various applications the model suffers from very low storage capacity i e one rule per fam matrix a lot of hardware and computations are usually required to implement the model and hence it is limited to applications with small fuzzy rule base in this paper the inherent property for storing multiple rules in a fam matrix is identified a theorem for perfect recalls of all the stored rules is established and based upon which the hardware and computation requirements of the fam model can be reduced significantly furthermore we have shown that when the fam model is generalized to the one with max bounded product composition single matrix implementation is possible if the rule base is a set of semi overlapped fuzzy rules rule modification schemes are also developed and the inference performance of the established high capacity models is reported through a numerical example

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

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

incorporating fuzzy membership functions into the perceptron algorithm the perceptron algorithm one of the class of gradient descent techniques has been widely used in pattern recognition to determine linear decision boundaries while this algorithm is guaranteed to converge to a separating hyperplane if the data are linearly separable it exhibits erratic behavior if the data are not linearly separable fuzzy set theory is introduced into the perceptron algorithm to produce a fuzzy algorithm which ameliorates the convergence problem in the nonseparable case it is shown that the fuzzy perceptron like its crisp counterpart converges in the separable case a method of generating membership functions is developed and experimental results comparing the crisp to the fuzzy perceptron are presented

is bias dispensable for fuzzy neural networks it is well known that a conventional feedforward neural network always has bias terms which is necessary for it to solve classification or approximation problems but for fuzzy neural networks fnns it seems not quite clear yet whether the bias is dispensable or not some authors introduce the bias while the others do not this note tries to partly answer this question for two simple building blocks of fnns fuzzy perceptron and max min fnn it is shown that the bias is basically dispensable for fuzzy perceptrons max min fnns plus a sign function which are usually used for classification on the other hand the bias is dispensable for max min fnns which are used for both approximation and classification if and only if a special condition is valid but this special condition is generally not valid or not easy to justify in practice so the bias is generally indispensable for max min fnns

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

t sum of bell shaped fuzzy intervals the usual arithmetic operations on real numbers can be extended to arithmetical operations on fuzzy intervals by means of zadeh s extension principle based on a t norm t a t norm is called consistent with respect to a class of fuzzy intervals for some arithmetic operation if this arithmetic operation is closed for this class it is important to know which t norms are consistent with a particular type of fuzzy intervals recently dombi and gyorbiro proved that addition is closed if the dombi t norm is used with two bell shaped fuzzy intervals a result proved by mesiar on a strict t norm based shape preserving additions of lr fuzzy intervals with unbounded support is recalled as applications we define a broader class of bell shaped fuzzy intervals then we study t norms which are consistent with these particular types of fuzzy intervals dombi and gyorbiro s results are special cases of the results described in this paper

cascaded fuzzy neural network model based on syllogistic fuzzy reasoning in recent years there has been an increasing interest in the fusion of neural networks and fuzzy logic most of the existing fuzzy neural network fnn models have been proposed to implement different types of single stage fuzzy reasoning mechanisms single stage fuzzy reasoning however is only the most basic among a human being s various types of reasoning mechanisms syllogistic fuzzy reasoning where the consequence of a rule in one reasoning stage is passed to the next stage as a fact is essential to effectively build up a large scale system with high level intelligence in view of the fact that the fusion of syllogistic fuzzy logic and neural networks has not been sufficiently studied a new fnn model based on syllogistic fuzzy reasoning termed cascaded fnn cfnn is proposed in this paper from the stipulated input output data pairs the model can generate an appropriate syllogistic fuzzy rule set through structure genetic learning and parameter back propagation learning procedures proposed in this paper in addition we particularly discuss and analyze the performance of the proposed model in terms of approximation ability and robustness as compared with single stage fnn models the effectiveness of the proposed cfnn model is demonstrated through simulating two benchmark problems in fuzzy control and nonlinear function approximation domain

interpretability issues in data based learning of fuzzy systems this paper presents a method for an automatic and complete design of fuzzy systems from data the main objective is to build fuzzy systems with a user controllable trade off between accuracy and interpretability whereas criteria for accuracy mostly follow straightforwardly from the application definition of interpretability and its criteria are subject to controversial discussion for this reason a set of interpretability criteria is given which guide the design process consequently interpretability is maintained by structural choices regarding the type of membership functions rules and inference mechanism on the one hand and by including interpretability criteria in the rule rule base evaluation on the other hand an application in instrumented gait analysis to characterize a certain group of patients in comparison to healthy subjects illustrates the proposed algorithm

residuation in fuzzy algebra and some applications abstract the system 0 1 max min has a number of well known applications many of which involve the inversion of linear relations we show the advantage of residuation theory for this process and apply it to the analysis of the problem of chebychev approximation

a general framework for fuzzy morphological associative memories fuzzy associative memories fams can be used as a powerful tool for implementing fuzzy rule based systems the insight that fams are closely related to mathematical morphology mm has recently led to the development of new fuzzy morphological associative memories fmams in particular implicative fuzzy associative memories ifams as the name fmam indicates these models belong to the class of fuzzy morphological neural networks fmnns thus each node of an fmam performs an elementary operation of fuzzy mm clarifying several misconceptions about fmams that have recently appeared in the literature we provide a general framework for fmams within the class of fmnn we show that many well known fam models fit within this framework and can therefore be classified as fmams moreover we employ certain concepts of duality that are defined in the general theory of mm in order to derive a large class of strategies for learning and recall in fmams

solving systems of linear fuzzy equations abstract in this paper we wish to construct solutions for x to the fuzzy matrix equation ax b when the elements in a and b are triangular fuzzy numbers a is square and always non singular we argue that the previous method of solving for x based on the extension principle and regular fuzzy arithmetic should be abandoned since it too often fails to produce a solution we present six new solutions of which we show that five are identical we then adopt the common value x of these five new solutions as our solution to ax b we show that x is a fuzzy vector which is the generalization to rn of real fuzzy numbers we also show how these results pertain to and extend previous research on this problem within the area of interval analysis

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

an architecture of neural networks with interval weights and its application to fuzzy regression analysis abstract in this paper we first propose an architecture of neural networks that have interval weights and interval biases a neural network with the proposed architecture maps an input vector of real numbers to an output interval the target output is also given as an interval next we define a cost function using the interval output from the neural network and the corresponding target output a learning algorithm is derived from the cost function in a similar manner as the bp back propagation algorithm we also show two variations of the learning algorithm for the proposed architecture which lead to the inclusion relation between the interval output and the interval target last we apply the learning algorithms to fuzzy regression analysis where target outputs are given as fuzzy numbers

fuzzy nonlinear regression analysis of dose response relationships abstract a general methodology for fuzzy regression is developed and illustrated by analyzing a dose response relationship fuzzy regression is used as an alternative to statistical regression analysis because the relationship between variables is imprecise data are uncertain and sample sizes are insufficient three goodness of fit criteria are used namely the maximum the average vagueness and the prediction vagueness criterion in the case of a nonlinear relationship the calculation of fuzzy regression parameters leads to a nonlinear programming problem this technique is illustrated by the relationship between the dose of a n nitroso compound and the probability of tumor development in rats this relationship is nonlinear imprecise and based on only six data points in the present case three common dose response models are fitted using nonlinear fuzzy regression techniques with the six data points regression parameters have been obtained by minimizing a nonlinear objective function under a set of specific nonlinear constraints the methodology is relatively simple and offers a viable alternative to statistical regression techniques

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

path integration over compact and noncompact rotation groups applications of group theoretical methods in the path integral formalism of nonrelativistic quantum theory are considered analysis of the symmetry of the lagrangian leads to the expansion of the short time propagator in matrix elements of unitary irreducible representations of the symmetry group identification of the coordinates with the group parameters transforms the path integral to integrals over the group manifold the integration is performed using the orthogonality of the representations compact and noncompact rotation groups are considered where the corresponding path integral is embedded in euclidean and pseudo euclidean spaces respectively the unit sphere and unit hyperboloid may either be viewed as the group manifold itself or at least as a group quotient in the first case fourier analysis leads to an expansion in group characters in the second case an expansion in zonal spherical functions is obtained as examples the groups so n su 2 so n 1 1 and su 1 1 are explicitly discussed the path integral on so n m and so n m in bispherical coordinates is also treated

implicative fuzzy associative memories associative neural memories are models of biological phenomena that allow for the storage of pattern associations and the retrieval of the desired output pattern upon presentation of a possibly noisy or incomplete version of an input pattern in this paper we introduce implicative fuzzy associative memories ifams a class of associative neural memories based on fuzzy set theory an ifam consists of a network of completely interconnected pedrycz logic neurons with threshold whose connection weights are determined by the minimum of implications of presynaptic and postsynaptic activations we present a series of results for autoassociative models including one pass convergence unlimited storage capacity and tolerance with respect to eroded patterns finally we present some results on fixed points and discuss the relationship between implicative fuzzy associative memories and morphological associative memories

path integrals with a periodic constraint the aharonov bohm effect the aharonov bohm effect is formulated in terms of a constrained path integral the path integral is explicitly evaluated in the covering space of the physical background to express the propagator as a sum of partial propagators corresponding to homotopically different paths the interference terms are also calculated for an infinitely thin solenoid which are found to contain the usual flux dependent shift as the dominant observable effect and an additional topological shift unnoticeable in the two slit interference experiment

path integral on s2 the rosen morse oscillator guided by the group chain so 3 contains implies o 2 the authors construct an angular path integral for the symmetric rosen morse oscillator on s2 by explicit path integration they obtain the normalised energy eigenfunction as well as the exact energy spectrum

algebraic approach to the scattering matrix a purely algebraic procedure is presented for calculating recursion relations for s matrices belonging to problems associated with the group su 1 1 the procedure involves supplementing a recently introduced group theoretic approach to scattering by an algebraic framework that characterizes asymptotic behavior our formulation makes use of the euclidean group which contains the symmetry transformations for a free particle the coulomb and poeschl teller potentials are discussed to illustrate our methods

identification of fuzzy relational equations by fuzzy neural networks abstract although the identification of a fuzzy system by a feedforward neural network is obviously possible whenever the system was continuous however when we use fuzzy max min neural network some problems on the learning of the network arise in this paper we present a method to identify a fuzzy relation by a fuzzy max min neural network we adapt the learning algorithm backpropagation to learning of max min neural network by using one which we will denote as smooth derivative finally we present some examples and a comparison with a similar method

on multistage fuzzy neural network modeling in the past couple of years there has been increasing interest in the fusion of neural networks and fuzzy logic most of the existing fuzzy neural network fnn models have been proposed to implement different types of single stage fuzzy reasoning mechanisms and inevitably they suffer from the dimensionality problem when dealing with complex real world problems to address the problem fnn modeling based on multistage fuzzy reasoning msfr is pursued here and two hierarchical network models namely incremental type and aggregated type are introduced the new models called multistage fnn msfnn model a hierarchical fuzzy rule set that allows the consequence of a rule passed to another as a fact through the intermediate variables from the stipulated input output data pairs they can generate an appropriate fuzzy rule set through structure and parameter learning procedures proposed in this paper in addition we have particularly addressed the input selection problem of these two types of multistage network models and proposed two efficient methods for them the effectiveness of the proposed msfnn models in handling high dimensional problems is demonstrated through various numerical simulations

fractional klein gordon equations and related stochastic processes this paper presents finite velocity random motions driven by fractional klein gordon equations of order alpha in 0 1 a key tool in the analysis is played by the mcbride s theory which converts fractional hyper bessel operators into erdelyi kober integral operators special attention is payed to the fractional telegraph process whose space dependent distribution solves a non homogeneous fractional klein gordon equation the distribution of the fractional telegraph process for alpha 1 coincides with that of the classical telegraph process and its driving equation converts into the homogeneous klein gordon equation fractional planar random motions at finite velocity are also investigated the corresponding distributions obtained as well as the explicit form of the governing equations fractionality is reflected into the underlying random motion because in each time interval a binomial number of deviations b n alpha with uniformly distributed orientation are considered the parameter n of b n alpha is itself a random variable with fractional poisson distribution so that fractionality acts as a subsampling of the changes of direction finally the behaviour of each coordinate of the planar motion is examined and the corresponding densities obtained extensions to n dimensional fractional random flights are envisaged as well as the fractional counterpart of the euler poisson darboux equation to which our theory applies

novel neural algorithms based on fuzzy d rules for solving fuzzy relation equations part iii abstract in this paper we first design a fuzzy neuron which possesses some generality this fuzzy neuron is founded by replacing the operators of the traditional neuron with a pair of abstract fuzzy operators as which we call fuzzy neuron operators for example it may be or etc it is an extended fuzzy neuron and a network composed o such neurons is an extended fuzzy neural network then we discuss the relationship between the fuzzy neuron operators and t norm and t conorm and point out fuzzy neuron operators are based on t norm but much wider than t norm in this paper we will emphatically discuss a two layered network and its training algorithm which will have to satisfy a set of various operators this work is very related to solving fuzzy relation equations so it can be used to resolve fuzzy relation equations furthermore the new fuzzy neural algorithm is found to be stronger than other existing methods to some degree some simulation results will be reported in detail

genetic algorithm for the design of a class of fuzzy controllers an alternative approach a simple easy to implement alternative method for designing fuzzy logic controllers flcs with symmetrically distributed fuzzy sets in a universe of discourse is introduced the design parameters include the parameters of the membership functions of the inputs and outputs and the rule base the method is based on a network implementation of the flc with real and binary weights with constraints due to the presence of the binary weights the backpropagation technique cannot be used the learning problem is cast as a mixed integer constrained dynamic optimization problem and solved using the genetic algorithm ga the crossover and mutation are slightly disrupted in order to cope with the constraints on the binary weights training of the controller is carried out in a closed loop simulation with the controller in the loop

fuzzy regression using asymmetric fuzzy coefficients and fuzzified neural networks in this paper first we explain several versions of fuzzy regression methods based on linear fuzzy models with symmetric triangular fuzzy coefficients next we point out some limitations of such fuzzy regression methods then we extend the symmetric triangular fuzzy coefficients to asymmetric triangular and trapezoidal fuzzy numbers we show that the limitations of the fuzzy regression methods with the symmetric triangular fuzzy coefficients are remedied by such extension several formulations of linear programming problems are proposed for determining asymmetric fuzzy coefficients from numerical data finally we show how fuzzified neural networks can be utilized as nonlinear fuzzy models in fuzzy regression in the fuzzified neural networks asymmetric fuzzy numbers are used as connection weights the fuzzy connection weights of the fuzzified neural networks correspond to the fuzzy coefficients of the linear fuzzy models nonlinear fuzzy regression based on the fuzzified neural networks is illustrated by computer simulations where type i and type ii membership functions are determined from numerical data

path integral treatment of the hydrogen atom in a curved space of constant curvature the path integral treatment of the hydrogen atom in a spherical space is discussed the dynamical group su 1 1 of the system is used for path integration by mapping the radial path integral onto the su 1 1 manifold the energy spectrum and the normalised wavefunctions are obtained in the flat space limit the standard hydrogen spectrum and the corresponding normalised energy eigenfunctions are recovered the scattering states are also found in the limit

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

hybrid identification in fuzzy neural networks this paper introduces an identification method for nonlinear models in the form of fuzzy neural networks fnn in this model we use two forms of the fuzzy inference methods a simplified and linear fuzzy inference and exploit a standard error back propagation learning algorithm the fnn modeling and identification environment realizes parameter identification through a synergistic usage of clustering techniques genetic optimization and a complex search method we use a hard c means hcm clustering algorithm to determine initial apexes of the membership functions of the information granules used in this fuzzy model the parameters such as apexes of membership functions learning rates and momentum coefficients are then adjusted using hybrid algorithm the proposed hybrid identification algorithm is carried out by combining both genetic optimization genetic algorithm ga and the improved complex method an aggregate objective function performance index with a weighting factor is introduced to achieve a sound balance between approximation and generalization of the model according to the selection and adjustment of the weighting factor of this objective function we reveal how to design a model with sound approximation and generalization abilities the proposed model is experimented with using several time series data gas furnace sewage treatment process and nox emission process data of gas turbine power plant

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

neuro fuzzy systems for function approximation abstract we present a neuro fuzzy architecture for function approximation based on supervised learning the learning algorithm is able to determine the structure and the parameters of a fuzzy system the approach is an extension to our already published nefcon and nefclass models which are used for control or classification purposes the proposed extended model which we call nefprox is more general and can be used for any application based on function approximation

solving fuzzy equations we start in the next section 10 2 with looking at possible solutions to the simple fuzzy linear equation overline a cdot overline x overline b overline c we discuss three different types of solution which we have studied before in solving fuzzy equations then we present a fourth type of solution based on our fuzzy monte carlo method in section 10 2 2 this new solution is based on random fuzzy numbers in section 10 3 we look at only classical solutions to the fuzzy quadratic equation and apply our fuzzy monte carlo method to obtain new solutions then in section 10 4 we consider the fuzzy matrix equation overline a cdot overline x overline b and a number of solution types for overline x and then another solution based on fuzzy monte carlo techniques the last section contains a brief summary and our conclusions

path integrals for potential problems with δ function perturbation the author presents several examples of potential problems with a function perturbation by means of path integrals the idea is to sum a perturbation series expansion resulting in an energy dependent green function g e the energy levels en of the perturbed problem are determined by the equation one dimensional case ig v a a en gamma where g v is the green function of the unperturbed problem gamma is the strength of the potential and a its location in r in d dimensional radial problems with a spherically shaped delta function located at r a this equation changes into ig v l a a en ad 1 where l denotes the angular momentum number

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

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