normal forms in bl algebra and their contribution to universal approximation of functions this paper continues the investigation of approximating properties of generalized normal forms in fuzzy logic the problem is formalized and solved algebraically normal forms are considered in two variants infinite and finite it is proved that infinite normal forms are universal representation formulas whereas finite normal forms are universal approximation formulas for any l valued function where l is a support set of any complete bl algebra the estimation of the quality of approximation is suggested moreover functions which can be precisely represented by the discrete normal forms are considered

approximation to a class of non autonomous systems by dynamic fuzzy inference marginal linearization method in this paper a dynamic fuzzy inference marginal linearization dfiml method is proposed for modeling nonlinear dynamic systems this method can transfer a group of input output data into a time variant fuzzy system with variable coefficients it is shown that solutions of time variant fuzzy systems generalized by dfiml method are universal approximators to solutions of a class of non autonomous systems also the analytical solutions of these time variant fuzzy systems can be obtained finally a simulation example is provided to illustrate the validity and potential of the developed techniques

decomposition property of fuzzy systems and its applications this paper presents the decomposition property of fuzzy systems using a simple constructive decomposition procedure that is by properly dividing the input space into sub input spaces a general fuzzy system is decomposed into several sub fuzzy systems which are the simplest fuzzy systems in the sub input spaces based on the decomposition property of fuzzy systems the analysis of fuzzy systems can be divided into two steps first analyze the properties of the simplest fuzzy systems and then use the decomposition property to extend the results to general fuzzy systems using this idea two applications of the decomposition property are given the first is the application to the representation capability analysis of fuzzy systems the second is the application to the analysis of a class of nonlinear control systems then based on the piecewise affine fuzzy system model the existence condition and the design of a stable control for a class of single input single output siso nonlinear systems are presented

functions of noncommuting self adjoint operators under perturbation and estimates of triple operator integrals abstract we define functions of noncommuting self adjoint operators with the help of double operator integrals we are studying the problem to find conditions on a function f on r 2 for which the map a b f a b is lipschitz in the operator norm and in schatten von neumann norms s p it turns out that for functions f in the besov class b 1 1 r 2 the above map is lipschitz in the s p norm for p 1 2 however it is not lipschitz in the operator norm nor in the s p norm for p 2 the main tool is triple operator integrals to obtain the results we introduce new haagerup like tensor products of l spaces and obtain schatten von neumann norm estimates of triple operator integrals we also obtain similar results for functions of noncommuting unitary operators

functions of almost commuting operators and an extension of the helton howe trace formula abstract let a and b be almost commuting i e the commutator ab ba belongs to trace class self adjoint operators we construct a functional calculus a b for functions in the besov class b 1 1 r 2 this functional calculus is linear the operators a b and a b almost commute for b 1 1 r 2 and a b u a v b whenever s t u s v t we extend the helton howe trace formula for arbitrary functions in b 1 1 r 2 the main tool is triple operator integrals with integrands in haagerup like tensor products of l spaces

stabilization for t s model based uncertain stochastic systems this paper considers the stabilization problem of a class of uncertain ito stochastic fuzzy systems driven by a multidimensional wiener process the uncertainty modeled in the systems is of the linear fractional type which includes the norm bounded uncertainty as a special case the objective is to design a state feedback fuzzy controller such that the closed loop system is robustly asymptotically stable under a stochastic setting by using a stochastic lyapunov approach sufficiency conditions for the stability and stabilization of this class of systems are established based on a novel matrix decomposition technique the derived stability conditions are then employed to design controllers which stabilize the uncertain ito stochastic fuzzy systems two simulation examples are given to illustrate the effectiveness of the approaches proposed

hierarchical ts fuzzy system and its universal approximation an efficient tool to deal with the rule explosion problem is the hierarchical system by which a fuzzy system can be decomposed into a number of hierarchically connected low dimensional systems in this paper a generalized hierarehical tagaki sugeno ts system is built it is shown that the input output i o relationship of this generalized hierarehical system can be represented as one of a standard ts fuzzy system and the system approximation capability is analyzed by taking piecewise linear functions as a bridge by constructive method it is proven that the hierarchical fuzzy systems hfs s can be universal approximators for the given approximation accuracy an estimation formula about the number of the rules needed in the hfs is established finally some simulation examples confirm that the hfs s with smaller size rule base can approximate the given functions with high accuracy the results obtained here provide us with the theoretical basis for various applications of hfs s

analyses for l p μ norm approximation capability of generalized mamdani fuzzy systems abstract the connected piecewise linear functions are introduced to study the l p norm approximation capability of the generalized mamdani fuzzy system it is shown that such general fuzzy systems are the universal approximators to the p integrable functions with l p norm for the given degree of accuracy an upper bound of the fuzzy rule numbers needed in the fuzzy rule base of the system is estimated finally a simulation example is analyzed to demonstrate our conclusions

multidimensional operator multipliers we introduce multidimensional schur multipliers and characterise them generalising well known results by grothendieck and peller we define a multidimensional version of the two dimensional operator multipliers studied recently by kissin and shulman the multidimensional operator multipliers are defined as elements of the minimal tensor product of several c algebras satisfying certain boundedness conditions in the case of commutative c algebras the multidimensional operator multipliers reduce to continuous multidimensional schur multipliers we show that the multipliers with respect to some given representations of the corresponding c algebras do not change if the representations are replaced by approximately equivalent ones we establish a non commutative and multidimensional version of the characterisations by grothendieck and peller which shows that universal operator multipliers can be obtained as certain weak limits of elements of the algebraic tensor product of the corresponding c algebras

random neural networks with negative and positive signals and product form solution we introduce a new class of random neural networks in which signals are either negative or positive a positive signal arriving at a neuron increases its total signal count or potential by one a negative signal reduces it by one if the potential is positive and has no effect if it is zero when its potential is positive a neuron fires sending positive or negative signals at random intervals to neurons or to the outside positive signals represent excitatory signals and negative signals represent inhibition we show that this model with exponential signal emission intervals poisson external signal arrivals and markovian signal movements between neurons has a product form leading to simple analytical expressions for the system state

fuzzy control of model car this paper discusses fuzzy control of a model car fuzzy control rules are derived by modelling an expert s driving actions experiments are performed using a model car with a sensing unit and a micro computer

fuzzy control of unknown multiple input multiple output plants abstract a self learning fuzzy logic system is given for control of unknown multiple input multiple output mimo plants some nomenclature and several data types are introduced to develop a general formulation for controller activity a random optimization algorithm is given to train the controller a plant model is not required for training instead training is guided by observations of plant responses to inputs

estimates of operator moduli of continuity abstract in aleksandrov and peller 2010 2 we obtained general estimates of the operator moduli of continuity of functions on the real line in this paper we improve the estimates obtained in aleksandrov and peller 2010 2 for certain special classes of functions in particular we improve estimates of kato 1973 18 and show that s t c s t log 2 log s t s t for all bounded operators s and t on hilbert space here s def s s 1 2 moreover we show that this inequality is sharp we prove in this paper that if f is a nondecreasing continuous function on r that vanishes on 0 and is concave on 0 then its operator modulus of continuity f admits the estimate f const e f t d t t 2 log t 0 we also study the problem of sharpness of estimates obtained in aleksandrov and peller 2010 2 3 we construct a c function f on r such that f l 1 f lip 1 and f const log 2 0 1 in the last section of the paper we obtain sharp estimates of f a f b in the case when the spectrum of a has n points moreover we obtain a more general result in terms of the e entropy of the spectrum that also improves the estimate of the operator moduli of continuity of lipschitz functions on finite intervals which was obtained in aleksandrov and peller 2010 2

operator holder zygmund functions abstract it is well known that a lipschitz function on the real line does not have to be operator lipschitz we show that the situation changes dramatically if we pass to holder classes namely we prove that if f belongs to the holder class r with 0 1 then f a f b const a b for arbitrary self adjoint operators a and b we prove a similar result for functions f in the zygmund class 1 r for arbitrary self adjoint operators a and k we have f a k 2 f a f a k const k we also obtain analogs of this result for all holder zygmund classes r 0 then we find a sharp estimate for f a f b for functions f of class def f f const for an arbitrary modulus of continuity in particular we study moduli of continuity for which f a f b const a b for self adjoint a and b and for an arbitrary function f in we obtain similar estimates for commutators f a q q f a and quasicommutators f a q q f b finally we estimate the norms of finite differences j 0 m 1 m j m j f a j k for f in the class m that is defined in terms of finite differences and a modulus continuity of order m we also obtain similar results for unitary operators and for contractions

compensatory neurofuzzy systems with fast learning algorithms in this paper a new adaptive fuzzy reasoning method using compensatory fuzzy operators is proposed to make a fuzzy logic system more adaptive and more effective such a compensatory fuzzy logic system is proved to be a universal approximator the compensatory neural fuzzy networks built by both control oriented fuzzy neurons and decision oriented fuzzy neurons cannot only adaptively adjust fuzzy membership functions but also dynamically optimize the adaptive fuzzy reasoning by using a compensatory learning algorithm the simulation results of a cart pole balancing system and nonlinear system modeling have shown that 1 the compensatory neurofuzzy system can effectively learn commonly used fuzzy if then rules from either well defined initial data or ill defined data 2 the convergence speed of the compensatory learning algorithm is faster than that of the conventional backpropagation algorithm and 3 the efficiency of the compensatory learning algorithm can be improved by choosing an appropriate compensatory degree

approximation of stochastic processes by t s fuzzy systems fuzzy systems can provide us with universal approximation models of deterministic input output relationships but in the stochastic environment few achievements related to the subject have so far achieved in the paper a novel stochastic takagi sugeno t s fuzzy system is introduced to represent approximately existing randomness in many real world systems by recapitulating the general architecture of the stochastic t s fuzzy rule based system we analyze systematically approximating capability of the stochastic system to a class of stochastic processes by the canonical representation of the stochastic processes the stochastic fuzzy system is capable of with arbitrary accuracy providing the approximation to the stochastic processes in mean square sense finally an efficient algorithm for the stochastic t s fuzzy system is developed a simulation example demonstrates how a stochastic t s fuzzy system can be constructed to realize the given stochastic process approximately

functions of operators under perturbations of class sp this is a continuation of our paper 2 we prove that for functions f in the holder class r and 1 p the operator f a f b belongs to sp whenever a and b are self adjoint operators with a b sp we also obtain sharp estimates for the schatten von neumann norms f a f b sp in terms of a b sp and establish similar results for other operator ideals we also estimate schatten von neumann norms of higher order differences j 0m 1 m j mj f a jk we prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions then we find necessary conditions on f for f a f b to belong to sq under the assumption that a b sp we also obtain schatten von neumann estimates for quasicommutators f a r rf b and introduce a spectral shift function and find a trace formula for operators of the form f a k 2f a f a k

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

structure identification of fuzzy model abstract the problems of structure identification of a fuzzy model are formulated a criterion for the verification of a structure is discussed using the criterion an algorithm for identifying a structure is suggested further a successive identification algorithm of the parameters is suggested the proposed methods are applied to an example

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

functional calculus for a pair of almost commuting selfadjoint operators abstract let a b be a pair of almost commuting selfadjoint operators on hilbert space the aim of this paper is to construct a functional calculus defined on a class of functions two variables and satisfying natural properties linearity and multiplicativity modulo the trace class the problem is to make this class as big as possible it is proved that it is impossible to construct such a functional calculus on the class of all continuously differentiable functions on the other hand we find a class c of functions in two real variables and construct a functional calculus on this class which extends functional calculi constructed earlier it is proved that for this functional calculus the trace formula of helton howe holds

on the carey helton howe pincus trace formula abstract in this article we give a new proof of the carey helton howe pincus trace formula using kato s theory of relatively smooth operators and krein s trace formula

second order sliding fuzzy interval type 2 control for an uncertain system with real application a new second order sliding mode type 2 fuzzy controller for nonlinear uncertain perturbed systems is developed in this paper to overcome the constraint on the knowledge of the system model we have used local models that are related to some operating points to synthesize a type 2 nominal fuzzy global model the control is based on the super twisting algorithm which is among second order sliding mode controls smcs moreover two adaptive fuzzy type 2 systems have been introduced to generate the two super twisting signals to avoid both the chattering and the constraint on the knowledge of disturbances and uncertainties upper bounds these adaptive fuzzy type 2 systems are adjusted online by two adaptation laws that are deduced from the stability analysis in the lyapunov sense it has only one input i e the sliding surface and one output i e the optimal values of the gains control which are hard to compute with the original algorithm many results of the one link manipulator are obtained first by the simulation in order to compare the performances of the proposed method with that given by levant and then in a real time application in order to confirm the efficiency of the proposed approach the experimentation and simulation are done for the tracking control problem

a neuro fuzzy system modeling with self constructing rule generationand hybrid svd based learning we propose an approach for neuro fuzzy system modeling a neuro fuzzy system for a given set of input output data is obtained in two steps first the data set is partitioned automatically into a set of clusters based on input similarity and output similarity tests membership functions associated with each cluster are defined according to statistical means and variances of the data points included in the cluster then a fuzzy if then rule is extracted from each cluster to form a fuzzy rule base second a fuzzy neural network is constructed accordingly and parameters are refined to increase the precision of the fuzzy rule base to decrease the size of the search space and to speed up the convergence we develop a hybrid learning algorithm which combines a recursive singular value decomposition based least squares estimator and the gradient descent method the proposed approach has advantages of determining the number of rules automatically and matching membership functions closely with the real distribution of the training data points besides it learns faster consumes less memory and produces lower approximation errors than other methods

removable singularities in yang mills fields we show that a field satisfying the yang mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite we obtain the result that every yang mills field overr4 with bounded functional l2 norm may be obtained from a field ons4 r4 hodge or coulomb gauges are constructed for general small fields in arbitrary dimensions including 4

kernel shapes of fuzzy sets in fuzzy systems for function approximation the shapes of if part fuzzy sets affect the approximating capability of fuzzy systems in this paper the fuzzy systems with the kernel shaped if part fuzzy sets are built directly from the training data it is proved that these fuzzy systems are universal approximators and their uniform approximation rates can be estimated in the single input single output siso case on the basis of these rates the relationships between the approximating capability and the shapes of if part fuzzy sets are developed for the fuzzy systems furthermore the sinc functions that serve as input membership functions are proved to have the almost best approximation property in a particular class of membership functions the theoretical results are confirmed from the simulation data in addition the estimations of the uniform approximation rates are extended to the multi input single output miso case

sugeno controllers with a bounded number of rules are nowhere dense abstract in literature various results can be found claiming that fuzzy controllers are universal approximators in terms of topology this means that fuzzy controllers as subsets of adequate function spaces are dense in this paper the topological structure of fuzzy controllers composed of a bounded number of rules is investigated it turns out that these sets are nowhere dense a topological notion indicating that the sets are almost discrete this means that it is just the number of rules and e g not the great variety of parameters of fuzzy controllers why fuzzy controllers are universal approximators

general siso takagi sugeno fuzzy systems with linear rule consequent are universal approximators takagi sugeno ts fuzzy systems have been employed as fuzzy controllers and fuzzy models in successfully solving difficult control and modeling problems in practice virtually all the ts fuzzy systems use linear rule consequent at present there exist no results qualitative or quantitative to answer the fundamentally important question that is especially critical to ts fuzzy systems as fuzzy controllers and models are ts fuzzy systems with linear rule consequent universal approximators if the answer is yes then how can they be constructed to achieve prespecified approximation accuracy and what are the sufficient renditions on systems configuration in this paper we provide answers to these questions for a general class of single input single output siso fuzzy systems that use any type of continuous input fuzzy sets ts fuzzy rules with linear consequent and a generalized defuzzifier containing the widely used centroid defuzzifier as a special case we first constructively prove that this general class of siso ts fuzzy systems can uniformly approximate any polynomial arbitrarily well and then prove by utilizing the weierstrass approximation theorem that the general ts fuzzy systems can uniformly approximate any continuous function with arbitrarily high precision furthermore we have derived a formula as part of sufficient conditions for the fuzzy approximation that can compute the minimal upper bound on the number of input fuzzy sets and rules needed for any given continuous function and prespecified approximation error bound an illustrative numerical example is provided

multiple operator integrals and higher operator derivatives abstract in this paper we consider the problem of the existence of higher derivatives of the function t a tk where is a function on the real line a is a self adjoint operator and k is a bounded self adjoint operator we improve earlier results by sten kin in order to do this we give a new approach to multiple operator integrals this approach improves the earlier approach given by sten kin we also consider a similar problem for unitary operators

operator lipschitz functions in schatten von neumann classes this paper resolves a number of problems in the perturbation theory of linear operators linked with the 45 year old conjecure of m g kre n in particular we prove that every lipschitz function is operator lipschitz in the schatten von neumann ideals s 1 0 such that left f a f b right alpha leqslant c alpha left f right text lip 1 left a b right alpha where f is a lipschitz function with left f right text lip 1 mathop sup limits lambda ne mu lambda mu in mathbb r left frac f left lambda right f left mu right lambda mu right infty left cdot right alpha is the norm is s and a and b are self adjoint linear operators such that a b in s alpha

normal forms for fuzzy logic functions and their approximation ability abstract aiming at developing theoretical backgrounds of fuzzy logic we consider an mv algebra of fuzzy logic functions two kinds of formulas regarded as disjunctive and conjunctive normal forms are introduced as a formal translation of if then rules their ability for an approximate representation of uniformly continuous functions are investigated the problem of transformation of any formula of propositional fuzzy logic into either of the normal forms is discussed

fuzzy modeling and control of multilayer incinerator abstract a method to design a fuzzy controller based on a fuzzy model is suggested the identification method of a fuzzy model is also discussed the suggested method is applied for the modelling and control of a multilayer incinerator the performance of the designed fuzzy controller is tested by computer simulation using the identified process model the results are satisfactory

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

classification of singular sobolev connections by their holonomy for a connection on a principalsu 2 bundle over a base space with a codimension two singular set a limit holonomy condition is stated in dimension four finite action implies that the condition is satisfied and an a priori estimate holds which classifies the singularity in terms of holonomy if there is no holonomy then a codimension two removable singularity theorem is obtained

functions of perturbed n tuples of commuting self adjoint operators let a1 an and b1 bn be n tuples of commuting self adjoint operators on hilbert space for functions f on rn satisfying certain conditions we obtain sharp estimates of the operator norms or norms in operator ideals of f a1 an f b1 bn in terms of the corresponding norms of aj bj 1 j n we obtain analogs of earlier results on estimates for functions of perturbed self adjoint and normal operators it turns out that for n 3 the methods that were used for self adjoint and normal operators do not work we propose a new method that works for arbitrary n we also get sharp estimates for quasicommutators f a1 an r rf b1 bn in terms of norms of ajr rbj 1 j n for a bounded linear operator r

fuzzy control as a universal control tool abstract it is known that fuzzy control is a universal control tool because an arbitrary control strategy in particular a control strategy that is in some sense optimal can be obtained in principle by applying a fuzzy control methodology to some set of rules this result has already been proved e g 2 7 20 21 for the case when a plant is described by finitely many parameters and a special type of fuzzy control methodology is used in this paper we prove it for arbitrary plants including plants that are distributed systems i e plants whose state requires infinitely many parameters to describe and arbitrary fuzzy control methodologies we also prove that there exists a universal fuzzy controller that generates an appropriate control from an input description of a plant and the desired objective mathematically we prove that fuzzy systems can approximate arbitrary continuous functionals thus generalizing a known result about continuous functions

approximation theory of fuzzy systems spl minus siso case in this paper the approximation properties of mimo fuzzy systems generated by the product inference are discussed we first give an analysis of fuzzy basic functions fbf s and present several properties of fbf s based on these properties of fbf s we obtain several basic approximation properties of fuzzy systems 1 basic approximation property which reveals the basic approximation mechanism of fuzzy systems 2 uniform approximation bounds which give the uniform approximation bounds between the desired control or decision functions and fuzzy systems 3 uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions and 4 universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect the similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds

method for higher order polynomial sugeno fuzzy inference systems a method for higher order polynomial sugeno fuzzy inference systems formation is presented compared to other existing higher order sugeno implementations it uses fewer parameters and compared to zero and 1st order sugeno fuzzy systems it has overall improved model performance while best models are not always obtained via a higher order representation in our proposed method it is possible to choose the polynomial order which best fits the desired data its input is a previously established model found by a clustering algorithm subtractive algorithm in this case afterward parameters of all higher order polynomials are adjusted using recursive least square algorithm for experimental validation multiple benchmark datasets are tested using hold out and k fold validation as well as data forecasting various performance measures are used although akaike information criterion is used as a primary measure to demonstrate that our proposed higher order polynomials have overall better model performance over zero and 1st order polynomials

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

approximation of uncertain functional relationships in this paper we consider methods for the linguistic modelling of vague functional relationships where the linguistic model is supposed to be given as a set of rules two approaches for deriving a fuzzy valued function from such models are proposed it is argued that both approaches can be interpreted as a generalization of ordinary methods for the interpolation and approximation of functions mathematical properties of fuzzy functions associated with a set of rules are investigated particularly for the second approach a universal approximation property is shown

functions of normal operators under perturbations abstract in peller 1980 27 peller 1985 28 aleksandrov and peller 2009 2 aleksandrov and peller 2010 3 and aleksandrov and peller 2010 4 sharp estimates for f a f b were obtained for self adjoint operators a and b and for various classes of functions f on the real line r in this paper we extend those results to the case of functions of normal operators we show that if a function f belongs to the holder class r 2 0 1 of functions of two variables and n 1 and n 2 are normal operators then f n 1 f n 2 const f n 1 n 2 we obtain a more general result for functions in the space r 2 f f 1 f 2 const 1 2 for an arbitrary modulus of continuity we prove that if f belongs to the besov class b 1 1 r 2 then it is operator lipschitz i e f n 1 f n 2 const f b 1 1 n 1 n 2 we also study properties of f n 1 f n 2 in the case when f r 2 and n 1 n 2 belongs to the schatten von neumann class s p

stability and the fuzzy controller abstract this paper corrects a part of the main result theorem 1 in buckley fuzzy sets and systems 77 1996 167 173