generalized drazin invertibility of combinations of idempotents abstract the paper serves as a correction to j math anal appl 359 2009 731 738 dealing with the drazin invertibility of combinations of idempotents p q in a banach algebra as the arguments employed to do this work equally well for both the usual and generalized drazin inverse the latter is included in the discussion which covers the equivalence of the drazin invertibility of 1 pq p q p q and in general of p q with 0 as well as the equivalence of the drazin invertibility of the commutator pq qp and anticommutator pq qp of p q

an alternative proof for the recursive formulae for computing the moore penrose m inverse of a matrix in this paper we provide an alternative proof for the recursive formulae for the determination of the generalized moore penrose m inverse of a matrix through the direct verification of the four properties of moore penrose m inverse

the point residual and continuous spectrum of an upper triangular operator matrix abstract in this paper for given operators a b h and b b k we completely describe the set of all c b k h such that m c is injective r m c is not dense in h k respectively and describe residual and continuous spectrum of the upper triangular operator matrix m c

matrices a such that aa a a are nonsingular abstract in this paper we study the class of square matrices a such that aa a a is nonsingular where a stands for the moore penrose inverse of a among several characterizations we prove that for a matrix a of order n the difference aa a a is nonsingular if and only if r a r a c n 1 where r denotes the range space also we study matrices a such that r a r a

some results on the drazin invertibility and idempotents in this paper several equivalent conditions for the drazin invertibility of the sum difference and the product of idempotents in banach algebra are established

automorphisms of b h with respect to minus partial order abstract let h be an infinite dimensional hilbert space and b h the algebra of all bounded linear operators on h we discuss possible extensions of the concept of the minus partial order from matrix to operator algebras in particular we show that mitra s unified theory of matrix partial orders based on generalized inverses can be modified in such a way that we get a unified approach to partial orders on b h or even more general algebras and rings then we choose the most natural among possible definitions of minus partial order on b h and describe the structure of corresponding automorphisms

geometry structures of the generalized inverses of block two by two matrices abstract in this paper we study the invertibility and the moore penrose invertibility of 2 2 block matrix with schur complement having closed range the necessary and sufficient conditions for the existence as well as the expressions for the inverses and the moore penrose inverses are obtained

on parallel sum of operators abstract the basic properties of the parallel sum were developed for nonnegative matrices in finite dimensional spaces however without suitable restrictions very little of the preceding theories will hold for bounded linear operators a and b acting in hilbert space in this paper generalization to non selfadjoint operators is considered and various properties of parallel sum a b a a b b are given meanwhile the common upper and lower bounds of positive operators by using the parallel sum are given and some relationships between the parallel sum of projection operators are obtained

state estimation for fuzzy cellular neural networks with time delay in the leakage term discrete and unbounded distributed delays this paper deals with the problem of state estimation for fuzzy cellular neural networks fcnns with time delay in the leakage term discrete and unbounded distributed delays in this paper leakage delay in the leakage term is used to unstable the neuron states it is challenging to develop a delay dependent condition to estimate the unstable neuron states through available output measurements such that the error state system is globally asymptotically stable by constructing the lyapunov krasovskii functional which contains a triple integral term an improved delay dependent stability criterion is derived in terms of linear matrix inequalities lmis however by using the free weighting matrices method a simple and efficient criterion is derived in terms of lmis for estimation the restriction such as the time varying delay which was required to be differentiable or even its time derivative which was assumed to be smaller than one are removed instead the time varying delay is only assumed to be bounded finally numerical examples and its simulations are given to demonstrate the effectiveness of the derived results

a domain attraction criterion for interval fuzzy neural networks in this paper we study qualitative properties of equilibrium points in a class of interval fuzzy neural networks and obtain an estimate on the domain of robust attraction of locally exponentially stable equilibrium points both the conditions and the estimate are formulated in terms of the parameter intervals so they are easily verifiable the result can be used for the evaluation of fault tolerance capability for interval fuzzy neural networks

on disjoint range matrices abstract for a square complex matrix f and for f being its conjugate transpose the class of matrices satisfying r f r f 0 where r denotes range column space of a matrix argument is investigated besides identifying a number of its properties several functions of f such as f f f f ff f f and f f are considered particular attention is paid to the moore penrose inverses of those functions and projectors attributed to them it is shown that some results scattered in the literature whose complexity practically prevents them from being used to deal with real problems can be replaced with much simpler expressions when the ranges of f and f are disjoint furthermore as a by product of the derived formulae one obtains a variety of relevant facts concerning for instance rank and range

on the product of orthogonal projectors abstract in this paper we introduce a canonical form to represent the product of two orthogonal projectors this form is useful for deriving many known results which are scattered in the literature we summarize many of those results and present new ones

characterizations of the commutators and the anticommutator involving idempotents in this paper we characterize the idempotency the invertibility or generalized invertibility the positivity and the range relations of the commutator c pq qp and the anticommutator d pq qp involving idempotents particular attention is paid to the idempotent operators p and q and some further characteristics when p and q are positive are derived as well

periodic bidirectional associative memory neural networks with distributed delays some sufficient conditions are obtained for the existence and global exponential stability of a periodic solution to the general bidirectional associative memory bam neural networks with distributed delays by using the continuation theorem of mawhin s coincidence degree theory and the lyapunov functional method and the young s inequality technique these results are helpful for designing a globally exponentially stable and periodic oscillatory bam neural network and the conditions can be easily verified and be applied in practice an example is also given to illustrate our results

sharp norm estimations for moore penrose inverses of stable perturbations of hilbert c module operators it is well known that norm upper bound vert bar t dag vert leq frac vert t dag vert 1 vert t dag vert cdot vert delta t vert plays a fundamental role in stable resp rank preserving perturbation analysis for moore penrose inverses of hilbert space operators resp matrices in this paper in the general setting of hilbert c module operators we provide a new approach to the study of norm upper bounds of bar t dag and obtain a sharp estimation for vert bar t dag vert this obtained estimation is applied to the study of the linear least squares problem and of perturbation analysis for the schur complement of a positive semidefinite operator matrix some new norm upper bounds for schur complements of positive semidefinite operator matrices are derived

star partial order on b h abstract let h be an infinite dimensional complex hilbert space and let b h be the algebra of all bounded linear operators on h in the paper the equivalent definition of the star partial order on b h using selfadjoint idempotent operators is introduced also some properties of the generalized concept of order relations on b h defined with the help of idempotent operators are investigated

constructing projections on sums and intersections abstract in this note we list several formulas for computing orthogonal projections onto the linear sum and intersection of two subspaces of c n in terms of the projections on the individual subspaces

characterizations of normal hyponormal and ep operators abstract in this paper normal and hyponormal operators with closed ranges as well as ep operators are characterized in arbitrary hilbert spaces all characterizations involve generalized inverses thus recent results of s cheng and y tian s cheng y tian two sets of new characterizations for normal and ep matrices linear algebra appl 375 2003 181 195 are extended to infinite dimensional settings

existence and stability of almost periodic solution for bam neural networks with delays by using the banach fixed point theorem and constructing suitable lyapunov function some sufficient conditions are obtained ensuring existence uniqueness and global stability of almost periodic solution of the bam neural networks with variable coefficients and delays these results are helpful to design global exponential stable bam networks and almost periodic oscillatory bam networks

a characteristic point based fuzzy inference system aimed to minimize the number of fuzzy rules this paper presents a characteristic point based fuzzy inference system cpfis for fuzzy modeling from training data the aim of the cpfis is not only satisfactory precision performance but also to employ as few purely linguistic fuzzy rules as possible by using a minimization based systematic training method characteristic points cps are defined as the few data points among the original training data which when they are directly mapped to fuzzy rules and thus form the entire rule base allow the underlying system to be effectively modeled three minimization based algorithms in a sequence are proposed to train the cpfis a gradient projection method a gauss jordan elimination based column elimination and back propagation the cps are determined by iterative computations of the first two minimization algorithms after which the resulting fuzzy sets are further fine tuned by the third algorithm experiments conducted on three benchmark problems showed that the cpfis used one of the smallest number of fuzzy rules among the reported results for other methods the gaussian membership functions in both the input and output fuzzy sets and the small number of fuzzy rules make the rule interpretation of the cpfis much easier than that of other methods thus enhancing human computer cooperation in knowledge discovery

a gentle guide to the basics of two projections theory abstract this paper is a survey of the basics of the theory of two projections it contains in particular the theorem by halmos on two orthogonal projections and roch silbermann gohberg and krupnik s theorem on two idempotents in banach algebras these two theorems which deliver the desired results usually very quickly and comfortably are missing or wrongly cited in many recent publications on the topic the paper is intended as a gentle guide to the field the basic theorems are precisely stated some of them are accompanied by full proofs others not but precise references are given in each case and many examples illustrate how to work with the theorems

asymptotic normality of series estimators for nonparametric and semiparametric regression models this paper establishes the asymptotic normality of series estimators for nonparametric regression models gallant s fourier flexible form estimators trigonometric series estimators and polynomial series estimators are prime examples of the estimators covered by the results the results apply to a wide variety of estimates in the regression model under consideration including derivatives and integrals of the regression function the errors in the model may be homoskedastic or heteroskedastic the paper also considers series estimators for additive interactive regression semiparametric regression and semiparametric index regression models and shows them to be consistent and asymptotically normal copyright 1991 by the econometric society

range inclusion and operator equations abstract letaanddbe positive operators on a complex hilbert spaceh in this work we show that the operator equationldx x a a l 0 has a unique solutionclsatisfying certain conditions if there exists a positive operatorbsuch thatab d b a 0 this implies several previously known results in special cases on this class of operator equations also along with some definitions a counterexample to a previous conjecture will be given

approximation capabilities of hierarchical fuzzy systems derived from practical application in location analysis and pricing and based on the approach of hierarchical structure analysis of continuous functions this paper investigates the approximation capabilities of hierarchical fuzzy systems by first introducing the concept of the natural hierarchical structure it is proved that continuous functions with natural hierarchical structure can be naturally and effectively approximated by hierarchical fuzzy systems to overcome the curse of dimensionality in both the number of rules and parameters then based on kolmogorov s theorem it is shown that any continuous function can be represented as a superposition of functions with the natural hierarchical structure and can then be approximated by hierarchical fuzzy systems to achieve the universal approximation property further the conditions under which the hierarchical fuzzy approximation is superior to the standard fuzzy approximation in overcoming the curse of dimensionality are analyzed

sufficient conditions to impose derivative constraints on miso takagi sugeno fuzzy logic systems a fuzzy logic system fls can be completely described through its behavioral function if the analytical expression of this function is known then the system synthesis consists in the realization of a structure implementing it in the most convenient way so the problem of the fls design can be considered as a function approximation problem many methods have been proposed in literature to minimize a certain error parameter nevertheless sometimes the specifications of the problem need some further analytical properties of the approximating function this paper analyzes multiple input single output miso flss with polynomial membership functions inspecting the case of two input variables in particular it will be shown what characteristics such systems should have in order to respect the constraints imposed on the derivatives of the implemented function such as their continuity up to a desired order and even their value in the grid points of the input space furthermore we prove that these systems can uniformly approximate under certain conditions any multivariate function with a desired approximation accuracy both on the target function and its partial derivatives proper illustrative examples will show the behavior of these flss comparing their performance with other methods proposed in literature

generalized inverses of partitioned matrices in banachiewicz schur form abstract the problem of developing conditions under which generalized inverses of a partitioned matrix can be expressed in the so called banachiewicz schur form is reconsidered theorem of marsaglia and styan sankhy ser a 36 1974 437 concerning the class of all generalized inverses the class of reflexive generalized inverses and the moore penrose inverse is strengthened and new results are established for the classes of outer inverses least squares generalized inverses and minimum norm generalized inverses

fuzzy systems as universal approximators an additive fuzzy system can uniformly approximate any real continuous function on a compact domain to any degree of accuracy an additive fuzzy system approximates the function by covering its graph with fuzzy patches in the input output state space and averaging patches that overlap the fuzzy system computes a conditional expectation e y x if we view the fuzzy sets as random sets each fuzzy rule defines a fuzzy patch and connects commonsense knowledge with state space geometry neural or statistical clustering systems can approximate the unknown fuzzy patches from training data these adaptive fuzzy systems approximate a function at two levels at the local level the neural system approximates and tunes the fuzzy rules at the global level the rules or patches approximate the function

bilateral shorted operators and parallel sums in this paper we study shorted operators relative to two different subspaces for bounded operators on infinite dimensional hilbert spaces we define two notions of complementability in the sense of ando for operators and study the properties of the shorted operators when they can be defined we use these facts in order to define and study the notions of parallel sum and substraction in this hilbertian context

nonsingularity of linear combinationsof idempotent matrices abstract gros and trenkler siam j matrix anal appl 21 1999 390 pointed out that if a difference of idempotent matrices p 1 and p 2 is nonsingular then so is their sum and koliha et al linear algebra appl in press expressed explicitly a condition which combined with the nonsingularity of p 1 p 2 ensures the nonsingularity of p 1 p 2 in the present note these results are strengthened by showing that the nonsingularity of p 1 p 2 is in fact equivalent to the nonsingularity of any linear combination c 1 p 1 c 2 p 2 wherein c1 c2 0 some other nonsingularity type relationships referring to linear combinations of p 1 and p 2 are also established

monotonic smooth takagi sugeno fuzzy systems with fuzzy sets with compact support in this paper sufficient conditions on monotonicity of takagi sugeno t s fuzzy systems with linear consequent parts with smooth membership functions with compact support i e equal to zero outside a compact set are derived the presented conditions considerably reduce conservatism of those known for t s fuzzy systems utilizing membership functions with the support that is not compact besides two illustrative examples will demonstrate that the monotonicity requirement imposed on fuzzy systems may significantly improve their extrapolation capability

hierarchical fuzzy systems for function approximation on discrete input spaces with application this paper investigates the capabilities of hierarchical fuzzy systems to approximate functions on discrete input spaces first it is shown that any function on a discrete space has an arbitrary separable hierarchical structure and can be naturally approximated by hierarchical fuzzy systems as a by product of this result a discrete version of kolmogorov s theorem is obtained second it is proven that any function on a discrete space can be approximated to any degree of accuracy by hierarchical fuzzy systems with any desired separable hierarchical structure that is functions on discrete spaces can be approximated more simply and flexibly than those on continuous spaces third a hierarchical fuzzy system identification method is proposed in which human knowledge and numerical data are combined for system construction and identification finally the proposed method is applied to the market condition performance modeling problem in site selection decision support and shows the better performance in both accuracy and interpretability than the regression and neural network approaches in additions the reason and mechanism why hierarchical fuzzy systems outperform regression and neural networks in this type of application are analyzed

range inclusion and the uup property abstract let a and b be positive operators on a complex hilbert space h the parallel sum a b of a and b is the strong operator limit as e 0 of the net a e a b 2e 1 b e if this net converges in norm then following green and morley we say that the pair a b is of class uup it is known that if range a range a b then a b is a uup pair in this work we establish that range a range a b if and only range b range a b we also show that if ab ba b2 0 then range a range a b for all real with 1 and that if ab ba 0 then range a range a b for all 0 this implies several previously known results e g that if ab ba then a b is a uup pair finally we establish a number of other results which relate the class uup to range inclusion

on nonsingularity of block two by two matrices national natural science foundation 11271308 91118001 natural science foundation of fujian province for distinguished young scholars 2010j06002 program for new century excellent talents in universities pr china hundred talent project of chinese academy of sciences national basic research program 2011cb309703 national natural science foundation for creative research groups pr china 11021101

stability of asymmetric hopfield networks in this paper we discuss dynamical behaviors of recurrently asymmetrically connected neural networks in detail we propose an effective approach to study global and local stability of the networks many of well known existing results are unified in our framework which gives much better test conditions for global and local stability sufficient conditions for the uniqueness of the equilibrium point and its stability conditions are given too

the stone weierstrass theorem and its application to neural networks the stone weierstrass theorem and its terminology are reviewed and neural network architectures based on this theorem are presented specifically exponential functions polynomials partial fractions and boolean functions are used to create networks capable of approximating arbitrary bounded measurable functions a modified logistic network satisfying the theorem is proposed as an alternative to commonly used networks based on logistic squashing functions

on disjoint range operators in a hilbert space for a bounded linear operator m in a hilbert space h various relations among the ranges r m r m r m m and the null spaces n m n m are considered from the point of view of their relations to the known classes of operators such as ep co ep weak ep gp dr or sr particular attention is paid to the range projectors of the operators m m and some further characteristics of these projectors are derived as well

fuzzy basis functions comparisons with other basis functions fuzzy basis functions fbf s which have the capability of combining both numerical data and linguistic information are compared with other basis functions because a fbf network is different from other networks in that it is the only one that can combine numerical and linguistic information comparisons are made when only numerical data is available in particular a fbf network is compared with a radial basis function rbf network from the viewpoint of function approximation their architectural interrelationships are discussed additionally a rbf network which is implemented using a regularization technique is compared with a fbf network from the viewpoint of overcoming ill posed problems a fbf network is also compared with specht s probabilistic neural network and his general regression neural network grnn from an architectural point of view a fbf network is also compared with a gaussian sum approximation in which gaussian functions play a central role finally we summarize the architectural relationships between all the networks discussed in this paper

how to express a parallel sum of k matrices abstract we present an alternative expression for the parallel sum of k hermitian nonnegative definite matrices by using the moore penrose inverse of a block matrix

more on generalized inverses of partitioned matrices with banachiewicz schur forms abstract necessary and sufficient conditions are derived for a 2 by 2 partitioned matrix to have 1 1 2 1 3 1 4 inverses and the moore penrose inverse with banachiewicz schur forms as applications the banachiewicz schur forms of 1 1 2 1 3 1 4 inverses and the moore penrose inverse of a 2 by 2 partitioned hermitian matrix are also given

generalized browder s and weyl s theorems for banach space operators abstract we find necessary and sufficient conditions for a banach space operator t to satisfy the generalized browder s theorem we also prove that the spectral mapping theorem holds for the drazin spectrum and for analytic functions on an open neighborhood of t as applications we show that if t is algebraically m hyponormal or if t is algebraically paranormal then the generalized weyl s theorem holds for f t where f h t the space of functions analytic on an open neighborhood of t we also show that if t is reduced by each of its eigenspaces then the generalized browder s theorem holds for f t for each f h t

the moore penrose inverses of m n block matrices and their applications abstract this article gives the expressions for the moore penrose inverses of m n block matrices when they satisfy rank additivity conditions and presents some of their special cases and applications

inverses of 2 2 block matrices abstract in this paper the authors give explicit inverse formulae for 2 2 block matrices with three different partitions then these results are applied to obtain inverses of block triangular matrices and various structured matrices such as hamiltonian per hermitian and centro hermitian matrices

reverse order law for the moore penrose inverse abstract in this paper we present new results related to the reverse order law for the moore penrose inverse of operators on hilbert spaces some finite dimensional results are extended to infinite dimensional settings

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

eigenvalues of functions of orthogonal projectors by representing two orthogonal projectors in a finite dimensional vector space as partitioned matrices several characterizations concerning eigenvalues of various functions of the pair are obtained these results substantially extend the ones already available in the literature additionally some related results dealing with the functions of a pair of orthogonal projectors are provided with the emphasis laying on the problem of invertibility

robust stability for interval hopfield neural networks with time delay the conventional hopfield neural network with time delay is intervalized to consider the bounded effect of deviation of network parameters and perturbations yielding a novel interval dynamic hopfield neural network idhnn model a sufficient condition related to the existence of unique equilibrium point and its robust stability is derived

moore penrose inverses and commuting elements of c algebras abstract let a be an element of a c algebra a satisfying a a a a where a is the moore penrose inverse of a and let b a we study several spectral properties concerning the commutativity of a and b we deduce some new and old results concerning ep matrices

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

constructive granular systems with universal approximation and fast knowledge discovery conventional gradient descent learning algorithms for soft computing systems have the learning speed bottleneck problem and the local minima problem to effectively solve the two problems the n variable constructive granular system with high speed granular constructive learning is proposed based on granular computing and soft computing and proved to be a universal approximator the fast granular constructive learning algorithm can highly speed up granular knowledge discovery by directly calculating all parameters of the n variable constructive granular system using training data and then construct the n variable constructive granular system with any required accuracy using a small number of granular rules predictive granular knowledge discovery simulation results indicate that the direct calculation based granular constructive algorithm is better than the conventional gradient descent learning algorithm in terms of learning speed learning error and prediction error

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

reverse order laws for the generalized inverses of multiple matrix products by using the ranks of matrices this article gives necessary and sufficient conditions for the reverse product a n a 2a 1of the moore penrose inverses of a1 a2 an to a 1 inverse inner inverse 1 2 inverse reflexive inner inverse 1 3 inverse 1 4 inverse 1 2 3 inverse 1 2 4 inverse or moore penrose inverse of matrix product a a1a2 an

a note on smooth approximation capabilities of fuzzy systems modeling and prediction in some systems requires the simultaneous approximation of mappings and their derivatives to a certain finite order in this paper universal approximation capabilities of fuzzy systems are extended to this situation by showing the denseness of some general classes of fuzzy models in appropriate function spaces where distance between functions is defined in terms of their derivatives requirements are generally very mild and are often limited to imposing sufficient differentiability conditions on the classes of fuzzy systems to be used the cases covered by this paper include additive fuzzy systems with gaussian membership functions and general additive fuzzy models with realistic membership functions some potential application fields are considered including examples from economics and time series prediction

explicit expressions of the generalized inverses and condensed cramer rules in this paper we obtain an explicit representation of the 2 inverse a 2 t s of a matrix a c m n with the prescribed range t and null space s as special cases new expressions for the moore penrose inverse a and drazin inverse a d are derived through explicit expressions we re derive the condensed cramer rules of werner for minimal norm least squares solution of linear equations ax b and propose two new condensed cramer rules for the unique solution of a class of singular system ax b x r a k b r a k k ind a finally condensed determinantal expressions for a a d a a a a a ndaa d are also presented

the difference and sum of projectors abstract the aim of this paper is to present new results on the invertibility of the sum of projectors and new relations between the nonsingularity of the difference and the sum of projectors we also give simple proofs without referring to rank theory of some results on the nonsingularity of the difference of projectors obtained by gros and trenkler siam j matrix anal appl 21 1999 390

approximation capability of fuzzy systems using translations and dilations of one fixed function as membership functions this paper reports on a related study on approximation theory of fuzzy systems first some basic principles are presented to construct membership functions then an approach is proposed to form membership functions by using translations and dilations of one fixed function called a basis function which is very similar to that in wavelets analysis the properties of this type of membership function reflect the advantages of the given approach finally it is proved that fuzzy systems based on such membership functions are universal approximators under certain mild conditions on the basis function this conclusion expands the family of fuzzy systems which can be universal approximators

an elasticity can be estimated consistently without a priori knowledge of functional form we consider an open question in applied price theory without a priori knowledge of a firm s cost function or a consumer s indirect utility function is it possible to estimate price and substitution elasticities consistently by observing a demand system as the work of white 30 guilkey lovell and sickles 19 and others has shown ordinary flexible functional forms such as the translog cannot achieve this objective we find that if one is prepared to assume that elasticities of substitution cannot oscillate wildly over the region of interest then consistent estimation is possible using the fourier flexible form provided the number of fitted parameters increases as the number of observations increases this result obtains with any of the commonly used statistical methods as for example multivariate least squares maximum likelihood and three stage least squares it obtains if the number of fitted parameters is chosen adaptively by observing the data or chosen deterministically according to some fixed rule we approach the problem along the classical lines of estimability considerations as used in the study of less than full rank linear statistical models and thereby discover that the problem has a fascinating structure which we explore in detail

simultaneous solution of fuzzy models an application to economic equilibrium analysis abstract the fuzzy modeling technology which uses a rule based description of the relationship between variables is discussed we then turn to the problem of the simultaneous solution of relationships involving fuzzy models we illustrate this problem by considering the equilibrium problem in economics in this case we have two relationships one between price and supply and the other between price and demand and we desire to find the price for which the supply equals the demand it is shown that if the fuzzy models have antecedent fuzzy sets which are of the trapezoidal triangular type then a solution technique can be found

unbounded or bounded idempotent operators in hilbert space abstract a densely defined closed linear operator f in a hilbert space is said to be idempotent if ran f dom f and f f f we show that such an idempotent operator is written as f p p q 1 2 p q 1 2 where p and q are the orthoprojections to ran f and ker f respectively when f is bounded this becomes f p p q 1 further we show that for any 0 the operator p q is invertible and f p p q 1 in addition to the known results we present several descriptions of the norm f in terms of p q p q 1 or p q 1

the moore penrose inverses of products and differences of projections in a c algebra abstract for two given projections p and q in a c algebra we investigate how to express moore penrose inverses of products pq and differences p q and pq qp moreover it is shown that pq qp is moore penrose invertible if and only if pq and p q are moore penrose invertible in addition some related topics are considered