fractional brownian motion and fractional gaussian noise in subsurface hydrology a review presentation of fundamental properties and extensions recent studies have shown that fractional brownian motion fbm and fractional gaussian noise fgn are useful in characterizing subsurface heterogeneities in addition to geophysical time series although these studies have led to a fairly good understanding of some aspects of fbm fgn a comprehensive introduction to these stochastic fractal functions is still lacking in the subsurface hydrology literature in this paper efforts have been made to define fbm fgn and present a development of their mathematical properties in a direct yet rigorous manner use of the spectral representation theorem allows one to derive spectral representations for fbm fgn even though these functions do not have classical fourier transforms the discrete and truncated forms of these representations have served as a basis for synthetic generation of fbm fgn the discrete spectral representations are developed and various implications discussed in particular it is shown that a discrete form of the fbm spectral representation is equivalent to the well known weierstrass mandelbrot random fractal function although the full implications are beyond the scope of the present paper it is observed that discrete spectral representations of fbm constitute stationary processes even though fbm is nonstationary a new and general spectral density function is introduced for construction of complicated anisotropic 3 d fractals including those characterized by vertical fgn and horizontal fbm such fractals are useful for modeling anisotropic subsurface heterogeneities but cannot be generated with existing schemes finally some basic properties of fractional levy motion and concepts of universal multifractals which can be considered as generalizations of fbm fgn are reviewed briefly

a fuzzy dynamic model based state estimator systems containing uncertainty are traditionally analyzed with probabilistic methods however for non linear non gaussian systems solutions can sometimes be very difficult to obtain the focus of this work is to determine if in such cases fuzzy dynamic system models may provide an alternative approach that more easily leads us to a good solution in this paper we present a fuzzy estimator whose system model is a fuzzy dynamic system we show that for the linear gaussian case the fuzzy estimator produces the same result as the kalman filter more importantly we show that the fuzzy estimator can succeed for some non gaussian nonlinear systems finally we illustrate the application of the fuzzy estimator on a non linear non gaussian time varying rocket launch problem where we show that it performs better than the extended kalman filter from a broad perspective this paper essentially shows how to build on zadeh s seminal ideas in fuzzy sets logic and systems and use kalman s seminal ideas on optimal estimators to construct a novel fuzzy estimator for non linear estimation problems while this seems to reconcile some of the fundamental ideas of zadeh and kalman it is unfortunate that the fuzzy estimator can be very computationally complex to implement for practical applications

fixed points of functions with max weighted quasi arithmetic mean operator abstract let 0 1 and f be a continuous strictly monotone real valued function the weighted quasi arithmetic mean of two numbers a b is defined by a b f 1 f a 1 f b let a a i j be an n n real matrix and x x 1 x 2 x n t r n we construct a function a 1 a 2 a n a r n r n by j a x max 1 l n x l a l j for all j 1 2 n in this paper we show that a has a unique fixed point x a moreover it can be shown that for each x r n the sequence x a k generated by the following iterative scheme x a 0 x and x a k a x a k 1 for all k 1 converges to the unique fixed point x a besides some properties of the fixed point are derived as an application our results imply that the max weighted quasi arithmetic mean powers of any matrix are always convergent the continuity of the function m n n r r n defined by a x a is proposed as well

fuzzy relational equations and the inverse problem abstract the inverse problem concerned with fuzzy relations is investigated the conditions for the existence of a solution are shown and an analytical solution is given a method for the improvement of the solution is proposed

on model selection criteria in multimodel analysis 1 hydrologic systems are open and complex rendering them prone to multiple conceptualizations and mathematical descriptions there has been a growing tendency to postulate several alternative hydrologic models for a site and use model selection criteria to 1 rank these models 2 eliminate some of them and or 3 weigh and average predictions and statistics generated by multiple models this has led to some debate among hydrogeologists about the merits and demerits of common model selection also known as model discrimination or information criteria such as aic aicc bic and kic and some lack of clarity about the proper interpretation and mathematical representation of each criterion we examine the model selection literature to find that 1 all published rigorous derivations of aic and aicc require that the true model having generated the observational data be in the set of candidate models 2 though bic and kic were originally derived by assuming that such a model is in the set bic has been rederived by cavanaugh and neath 1999 without the need for such an assumption and 3 kic reduces to bic as the number of observations becomes large relative to the number of adjustable model parameters implying that it likewise does not require the existence of a true model in the set of alternatives we explain why kic is the only criterion accounting validly for the likelihood of prior parameter estimates elucidate the unique role that the fisher information matrix plays in kic and demonstrate through an example that it imbues kic with desirable model selection properties not shared by aic aicc or bic our example appears to provide the first comprehensive test of how aic aicc bic and kic weigh and rank alternative models in light of the models predictive performance under cross validation with real hydrologic data

numerical and applicational aspects of fuzzy relational equations we discuss numerical methods leading to a resolution of fuzzy relational equations which create a formal description of ill defined systems an applicability of the numerical approach is shown in solving some problems in fuzzy systems such as identification and control an idea of fuzzy discretization fuzzy chopping as a meaningful tool for handling various problems is also investigated

fuzzy relational equations with generalized connectives and their applications the idea of fuzzy relational equation with generalized connectives is introduced and algorithms of resolution of this class equations are presented in details it is clearly shown that some types of well known fuzzy relational equations sup min inf max can be treated as special cases of a wise class of equations under discussion applicational aspects are carefully investigated presenting the advantages of the proposed approach

on the power sequence of a fuzzy matrix with convex combination of max product and max min operations in the literature the powers of a fuzzy matrix with max min max product max archimedean t norm max t norm max arithmetic mean compositions have been studied it turns out that the limiting behavior of the powers of a fuzzy matrix depends on the composition involved in this paper we consider the powers of a fuzzy matrix with convex combination of max product and max min operations we show that the powers of a fuzzy matrix a with such operations are asymptotical p period if and only if the powers of an associated boolean matrix a are p periodic moreover necessary and sufficient conditions for such a fuzzy matrix to be nilpotent are proposed

indices and periods of incline matrices abstract inclines are the additively idempotent semirings in which products are less than or equal to factors thus inclines generalize boolean algebra fuzzy algebra and distributive lattice and the boolean matrices the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices i e the matrices over inclines this paper studies the power sequence of incline matrices in detail a necessary and sufficient condition for the incline matrix to have index is given and the indices of some incline matrices with indices are estimated it is proved that the period of n n incline matrix with index is a divisor of n and that the set of periods of n n incline matrices with indices is not bounded from above in the sense of a power of n for all n an equivalent condition and some sufficient conditions for the incline matrix to converge in finite steps are established the stability of the orbits of an incline matrix is considered and a theorem in fuzzy sets and systems 81 1996 227 is pointed out being false the results in the present paper include some previous results in the literatures which were obtained for the boolean matrices the fuzzy matrices and the lattice matrices among their special cases

some theoretical aspects of fuzzy relation equations describing fuzzy systems abstract this paper contains some theoretical results on fuzzy relation equations and applications to analysis of fuzzy systems as well as identification and control problems if ai 1 i k are fuzzy inputs b is a fuzzy output and r is a fuzzy relation describing the system the above problems are illustrated by means of a max min multidimensional equation of type b a 1 a k r furthermore we investigate some properties of fuzzy systems described by fuzzy relation equations focusing our attention on energy and entropy measures of fuzziness of solutions of the class of equations under discussion

on the sequence of consecutive powers of a fuzzy matrix with max archimedean t norms in the literature the limiting behavior of powers of a fuzzy matrix has been studied with max product composition pang and guu proposed a simple and effective characterization for the limiting behavior with the notion of asymptotic period in this paper we shall extend pang and guu s work to the realm with the max archimedean t norm composition to which the max product belongs moreover some sufficient conditions for the powers to converge in finitely many steps will be established

calculation and application of a minimum fuzzy relational matrix abstract this paper compares the maximum and minimum inverse formulations for fuzzy identification of a relational matrix given a series of non normal input output data baboshin and naryshkin 1990 showed for the case of normal input data that identification of the relational matrix using an estimate of the minimum inverse resulted in a smaller hamming distance between actual and predicted output values than the mamdani identification method this paper extends their work to include unrestricted or non normal input data and confirms that the estimate of the minimum inverse for this case is again better than mamdani s method it is then proven that the relative rank of identification algorithms for minimization of the hamming distance between the actual and predicted output values is the maximum inverse calculation of sanchez 1976 the minimum inverse formulation initial formulation by sanchez 1977 and extended to a series of input output data by sessa 1984 the estimate of the minimum inverse and then the mamdani method however this ranking applies only in the fuzzy domain and not to the discrete or defuzzified domain relevant to many practical applications

convergence of max arithmetic mean powers of a fuzzy matrix fuzzy matrices provide convenient representations for fuzzy relations on finite universes in the literature the powers of a fuzzy matrix with max min max product max archimedean t norm compositions have been studied it turns out that the limiting behavior of the powers of a fuzzy matrix depends on the composition involved in this paper the max arithmetic mean composition is considered for the fuzzy relations we show that the max arithmetic mean powers of a fuzzy matrix always are convergent

on generating conductivity fields with known fractal dimension and nonstationary increments 1 fractional brownian motion fbm is a stochastic process that has stationary increments with long range correlations and known fractal dimension we study a multiple dimensional extension of fbm with nonstationary increments that allows for trends in the statistical structure while maintaining the gaussian nature and fractal dimension of fbm two methods for simulating this extension are employed and described in detail one approach combines cholesky decomposition with a generalization of random midpoint displacement the other makes repeated use of the cholesky decomposition the resulting fields can be employed in various geophysical settings e g as log conductivity fields in hydrology and topographic elevation in geomorphology

ubiquity of and geostatistics for nonstationary increment random fields 1 nonstationary random fields such as fractional brownian motion and fractional levy motion have been studied extensively in the hydrology literature on the other hand random fields that have nonstationary increments have seen little study a mathematical argument is presented that demonstrates processes with stationary increments are the exception and processes with nonstationary increments are far more abundant the abundance of nonstationary increment processes has important implications e g in kriging where a translation invariant variogram implicitly assumes stationarity of the increments an approach to kriging for processes with nonstationary increments is presented and accompanied by some numerical results

an on line identification algorithm for fuzzy systems abstract based on the pedrycz s reference fuzzy sets the paper presents an on line identification algorithm for fuzzy systems which can on line update the fuzzy model the method proposed is useful for further analysis and development of the fuzzy model an example is given to illustrate the on line identification of the fuzzy model

fuzzy prediction based on regression models abstract we use a linguistic variable to represent imprecise information to be inserted in regression models used for prediction we show how one can obtain probabilistic statements about the forecasted variables

weak hopf algebras i integral theory and c structure we give an introduction to the theory of weak hopf algebras proposed as a coassociative alternative of weak quasi hopf algebras we follow an axiomatic approach keeping as close as possible to the classical theory of hopf algebras the emphasis is put on the new structure related to the presence of canonical subalgebras al and ar in any weak hopf algebra a that play the role of non commutative numbers in many respects a theory of integrals is developed in which we show how the algebraic properties of a such as the frobenius property or semisimplicity or innerness of the square of the antipode are related to the existence of non degenerate normalized or haar integrals in case of c weak hopf algebras we prove the existence of a unique haar measure h a and of a canonical grouplike element g a implementing the square of the antipode and factorizing into left and right elements g glg 1r gl al gr ar further discussion of the c case will be presented in part ii

nonparametric estimation of nonstationary spatial covariance structure abstract estimation of the covariance structure of spatial processes is a fundamental prerequisite for problems of spatial interpolation and the design of monitoring networks we introduce a nonparametric approach to global estimation of the spatial covariance structure of a random function z x t observed repeatedly at times ti i 1 t at a finite number of sampling stations xi i 1 2 n in the plane our analyses assume temporal stationarity but do not assume spatial stationarity or isotropy we analyze the spatial dispersions var z xi t z xj t as a natural metric for the spatial covariance structure and model these as a general smooth function of the geographic coordinates of station pairs xi xj the model is constructed in two steps first using nonmetric multidimensional scaling mds we compute a two dimensional representation of the sampling stations for which a monotone function of interpoint distances ij approximates the spatial dispersions mds transforms the problem

new fuzzy learning model with recursive estimation for dynamic systems abstract a systematic design technique for a discrete time self learning fuzzy model based on fuzzy relational equations is presented with some novel features firstly the model utilizes a probabilistic fuzzy relation obtained from referential fuzzy sets on the input and output spaces secondly the conventional input output formulation of the fuzzy relation is extended by introducing internal fuzzy states in order to model the internal dynamics of the given dynamic system thirdly the fuzzy relation is used to estimate the output as a response to a given stimulus by means of the compositional rule of inference extended to include recursive estimation of the internal fuzzy state as a result the new model is shown to be a fuzzy automaton first and second order versions of the fuzzy model are validated by means of a well known set of industrial data

standard eigenvectors of incline matrices abstract inclines are the additively idempotent semirings in which products are less than or equal to either factor thus inclines generalize boolean algebra fuzzy algebra and distributive lattice this paper studies the standard eigenvectors of incline matrices i e matrices over inclines in detail as the main results for an incline matrix with index and for a lattice matrix the structures of all the standard eigenvectors are given and the greatest standard eigenvector is determined also it is proved that for an incline matrix with index the set of all the standard eigenvectors of it coincides with the set of all the standard eigenvectors of its transitive closure the power convergence of an incline matrix with index is characterized in terms of the sets of standard eigenvectors the results in the present paper include some previous results for the boolean matrices the fuzzy matrices and the lattice matrices among their special cases

on identification in fuzzy systems and its applications in control problems abstract the problem of identification in fuzzy systems described by the use of fuzzy equation is considered the identification method and its performance index is also presented the formal procedure of the identification algorithm is illustrated by means of a numerical example the possibility of using the proposed algorithm for the solution of a control problem is given as well

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

on the max contraction powers of a fuzzy matrix abstract in the literature the powers of a fuzzy matrix with max arithmetic mean convex combination of max min and max arithmetic mean convex combination of max product and max min compositions have been studied it turns out that the limiting behavior of the powers of a fuzzy matrix depends on the composition involved in this paper we consider the max contraction powers of a fuzzy matrix which is an extension of the max arithmetic mean convex combination of max min and max arithmetic mean convex combination of max product and max arithmetic mean operation we show that the powers of such a fuzzy matrix are always convergent and the limit matrix has the feature that all elements of each column are identical

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

adaptive fuzzy relational predictive control abstract the performance of two fuzzy relational controllers namely the self learning predictive fuzzy controller slpfc bourke and fisher proc 5th int conf on fuzzy systems new orleans 1996 pp 464 1470 and the fuzzy relational long range predictive controller frlrpc postlethwaite iee proc d 138 3 1991 199 206 are evaluated experimentally on a laboratory scale non linear interacting tank process both fuzzy controllers gave good closed loop control performance but the slpfc gave slightly better results it was also found that the choice of the on line fuzzy relational identification scheme has a large impact on control performance

stochastic interpolation of aquifer properties using fractional levy motion this paper describes a new technique for conditional simulation of aquifer properties in unobserved regions the technique utilizes a recently introduced model for heterogeneity based on the levy stable family of probability distributions to achieve a higher degree of realism than is possible with gaussian based techniques specifically permeability and porosity variations are modeled using fractional levy motion a new algorithm for producing fractional levy motion conditioned to match known data is introduced two dimensional simulations using actual permeability measurements as input demonstrate that this algorithm produces random fields with the specified properties simulations using a subset of a detailed permeability map of an outcrop as input illustrate how the new method can be used to estimate important flow and transport properties and to quantify uncertainties in these estimates

reconstruction of weak quasi hopf algebras abstract all rational semisimple braided tensor categories are representation categories of weak quasi hopf algebras to prove this result we construct for any given category of this kind a weak quasi tensor functor to the category of finite dimensional vector spaces this allows us to reconstruct a weak quasi hopf algebra with the given category as its representation category

an identification algorithm in fuzzy relational systems abstract the paper presents an effective identification method in fuzzy relational systems we propose an algorithm for constructing models on the basis of fuzzy and nonfuzzy data with the aid of fuzzy discretization and clustering techniques the usefulness of the method provided is demonstrated by means of two numerical examples also a possible way of generating a linguistic decision making algorithm is discussed

design of fuzzy logic controllers based on generalized t operators abstract since zadeh first proposed the basic principle of fuzzy logic controllers in 1968 the min and max operators have been popular in the design of fuzzy logic controllers in this paper the general concept of t operators is introduced into the conventional design methods for fuzzy logic controllers so that a general and flexible methodology for the design of these fuzzy logic controllers is available then by computer simulations studies are made so as to determine the relations between the various t operators and the performance of a fuzzy logic controller it is concluded that the performance of the fuzzy logic controller for a given class of plants very much depends upon the choice of the t operators

fuzzy adaptive control of a first order process abstract fuzzy adaptive control of a first order process with a varying gain and time constant is demonstrated the fuzzy adaptive controller is based on a fuzzy model based controller that uses an explicit fuzzy model of the process adaptation is achieved by the addition of on line identification of the fuzzy process model starting with a crude initial model the adaptive controller matches the model to the process to self tune the controller the model is further modified when the operating point is changed to adapt the controller to the new process conditions two different fuzzy identification algorithms are both shown to provide a successful fuzzy adaptive controller the robustness of the algorithms in the presence of process noise is investigated and improved by the addition of weighting factors

fuzzy model identification for classification of gait events in paraplegics fuzzy system identification was applied to a biomedical system for classification purposes gait achieved through functional electrical stimulation fes of paraplegics was divided using sensor measurements of kinematic variables as inputs to five discrete events two identification algorithms were used to estimate the system model both max min and max product composition were used membership functions were either trapezoidal or triangular and all membership functions in a particular universe of discourse had the same shape and size the universe of discourse was varied by altering the overlap between membership functions the classification performance was assessed quantitatively by measuring the percentage of time steps in which the correct event was found and qualitatively by observing types of errors the identification algorithm affected system performance no difference in classification was found between max min and max product composition the performance was dependent on membership function overlap a comparison of the classification found using the fuzzy rule base versus that found using a traditional look up table demonstrated that the fuzzy approach was superior it is speculated that the use of fuzzy logic decreased errors stemming from sensor noise and or small variations in the input signals the performance of this approach was compared to that of a feedforward neural network and the fuzzy system is found superior

on the max nilpotent t norm powers of a fuzzy matrix fuzzy matrices have been proposed to represent fuzzy relations in finite universes various studies have evaluated the powers of a fuzzy matrix with max min max product max archimedean t norm max t norm max arithmetic mean compositions indicating that the limiting behavior of the powers of a fuzzy matrix depends on its composition in this paper max nilpotent composition is considered for the fuzzy relations we demonstrate that the max nilpotent powers of a fuzzy matrix either converge or oscillate in a finite period moreover the max nilpotent t norm powers of a fuzzy matrix a are p periodic if and only if the powers of an associated boolean matrix a c are p periodic finally necessary and sufficient conditions are proposed for nilpotent fuzzy matrices that exhibit max nilpotent composition

fractional brownian motion run with a nonlinear clock we construct a family of stochastic processes with nonstationary correlated increments which allow a priori independent selections of both fractal dimension and mean square displacement the family is essentially fractional brownian motion fbm run with a nonlinear clock fbm nlc the fractal dimension of fbm nlc is shown to be the same as that of the underlying fbm process we also compute the p variation and discuss the problems in using this to differentiate between diffusive processes the fbm nlc process illustrates that the range of anomalous diffusive processes has not been adequately explored

new turaev braided group categories and weak co quasi turaev group coalgebras in order to construct a class of new braided crossed g categories with nontrivial associativity and unit constraints we study the g graded monoidal category over a family of algebras h g and introduce the notion of a weak co quasi turaev g co algebra then we prove that the category of co representations of co quasitriangular weak co quasi turaev co algebras is exactly a braided crossed g category in fact this co quasitriangular structure provides a solution to a generalized quantum yang baxter type equation

fuzzy relational predictive identification abstract fuzzy relational models typically generate poorer long range predictions than conventional discrete input output models in this paper a new on line fuzzy relational identification algorithm frpi is developed which minimizes the prediction error over a multi point user defined prediction horizon the new algorithm gave better long range predictions than the single point fuzzy relational identification algorithm of shaw and kruger 1992 and the recursive algorithm of yi and chung 1993

fuzzy relational inequalities and equations fuzzy quasi orders closures and openings of fuzzy sets the paper deals with systems of fuzzy relation equations known in the literature as eigen fuzzy set equations as well as with systems of related fuzzy relation inequalities our main results are algorithms for computing the greatest and the smallest solutions to the considered systems of fuzzy relation inequalities and algorithms for computing the greatest solutions to the considered systems of fuzzy relation equations the systems are studied in the framework of complete residuated lattices as the structure of membership values by means of fuzzy quasi orders and closures and openings on a collection of fuzzy sets

convergence eigen fuzzy sets and stability analysis of relational matrices abstract if stable control applications are formulated with a max product composition then instability in the relational matrix under any conditions is undesirable this paper reviews the stability analysis of relational matrices combined with the max min composition 19 and then presents an analysis of the stability of relational matrices combined with the max product composition this analysis includes results defining the convergence properties of the relational matrix determination of the eigen fuzzy sets of the stable matrices and algorithmic solutions for several important cases a method is also presented to stabilize unstable relational matrices for some control applications