quantum algebras as quantizations of dual poisson lie groups a systematic computational approach for the explicit construction of any quantum hopf algebra uz g z starting from the lie bialgebra g that gives the first order deformation of the coproduct map z is presented the procedure is based on the well known quantum duality principle namely the fact that any quantum algebra can be viewed as the quantization of the unique poisson lie structure g g on the dual group g which is obtained by exponentiating the lie algebra g defined by the dual map from this perspective the coproduct for uz g is just the pull back of the group law for g and the poisson analogues of the quantum commutation rules for uz g are given by the unique poisson lie structure g on g whose linearization is the poisson analogue of the initial lie algebra g this approach is shown to be a very useful technical tool in order to solve the lie bialgebra quantization problem explicitly since once a lie bialgebra g is given the full dual poisson lie group g can be obtained either by applying standard poisson lie group techniques or by implementing the algorithm presented here with the aid of symbolic manipulation programs as a consequence the quantization of g will give rise to the full uz g quantum algebra provided that ordering problems are appropriately fixed through the choice of certain local coordinates on g whose coproduct fulfils a precise quantum symmetry property the applicability of this approach is explicitly demonstrated by reviewing the construction of several instances of quantum deformations of physically relevant lie algebras such as the 2 1 anti de sitter algebra so 2 2 and the poincare algebra in 3 1 dimensions

approximations to the zoeppritz equations and their use in avo analysis to efficiently invert seismic amplitudes for elastic parameters pseudoquartic approximations to the zoeppritz equations are derived to calculate p p wave reflection and transmission coefficients as a function of the ray parameter p these explicit expressions have a compact form in which the coefficients of the p2 and p4 terms are given in terms of the vertical slownesses the amplitude coefficients are also represented as a quadratic function of the elastic contrasts at an interface and are compared to the linear approximation used in conventional amplitude variation with offset avo analysis which can invert for only two elastic parameters numerical analysis with the second order approximation shows that the condition number of the frechet matrix for three elastic parameters is improved significantly from using a linear approximation therefore those quadratic approximations can be used directly with amplitude information to estimate not only two but three parameters p wave velocity contrast s wa

a simplification of the zoeppritz equations the compressional wave reflection coefficient r given by the zoeppritz equations is simplified to the following r r0 a0r0 1 2 sin2 1 2 vpvp tan2 sin2 the first term gives the amplitude at normal incidence 0 the second term characterizes r at intermediate angles and the third term describes the approach to critical angle the coefficient of the second term is that combination of elastic properties which can be determined by analyzing the offset dependence of event amplitude in conventional multichannel reflection data if the event amplitude is normalized to its value for normal incidence then the quantity determined is a a0 1 1 2 r0 a0 specifies the normal gradual decrease of amplitude with offset its value is constrained well enough that the main information conveyed is r0 where is the contrast in poisson s ratio at the reflecting interface and r0 is the amplitude at normal incidence this simplified formula for r accounts for all of the relations between r

an r matrix approach to the quantization of the euclidean group e 2 the r matrices for two different deformations of the euclidean group e 2 calculated in a two dimensional representation are used to determine the deformed hopf algebra of the representative functions the duality of the latter with the initial quantum algebras is explicitly proved and the relationship between the two quantum groups is discussed and clarified

on integration of fuzzy mappings abstract in this paper we introduce the concept of an integral of a fuzzy mapping this generalises the integral introduced by matloka 10 and also extends riemann stieltjes integral over a closed interval to fuzzy mappings

a survey of viability theory some theorems of viability theory which are relevant to nonlinear control problems with state constraints and state dependent control constraints are motivated and surveyed they all deal with viable solutions to nonlinear control problems i e solutions satisfying at each instant given state constraints of a general and diverse nature some classical results on controlled invariance of smooth nonlinear systems are adopted to the nonsmooth case including inequality constraints bearing on the state and state dependent constraints on the controls for instance existence of a viability kernel of a closed set corresponding to the largest controlled invariant manifold is provided under general conditions even when the zero dynamics algorithm does not converge the concepts of slow and heavy viable solutions are introduced providing concrete ways of regulating viable solutions by closed loop feedbacks and closed loop dynamical feedbacks viability theorems also allow the extension of lyapunov s second meth

fixed points for fuzzy mappings in this paper the problem of the existence and computation of fixed points for fuzzy mappings is approached a fuzzy mapping r over a set x is defined to be a function attaching to each x in x a fuzzy subset r of x an element x of x is called fixed point of r iff its membership degree to r is at least equal to the membership degree to r of any y x i e r r y y x two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described the convergence theorem of our algorithm is proved under the restrictive assumption that for any x in x the membership function of r has a complementary function examples of fuzzy mappings having this property are given but the problem of proving general criteria for a function to have a complementary remain open

fuzzy linear regression and its applications to forecasting in uncertain environment abstract linear regression has been used for many years in developing mathematical models for application in marketing management and sales forecasting in this paper two different sales forecasting techniques are discussed the first technique involves non fuzzy abstract methods of linear regression and econometrics a study of the international market sales of cameras done in 1968 by john scott armstrong utilized these non fuzzy forecasting techniques the second sales forecasting technique uses fuzzy linear regression introduced by h tanaka s uejima and k asai in 1980 in this paper a study of the computer and peripheral equipment sales in the united states is discussed using fuzzy linear regression moreover fuzzy linear regression is applied to forecasting in an uncertain environment finally some possible improvements and suggestions for further study are mentioned

classification of three dimensional lie bialgebras the paper contains a classification of the three dimensional real and complex lie bialgebras with a description of the corresponding drinfel d doubles

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

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

fuzzy differential systems and malliavin calculus abstract we consider fuzzy differential systems or differential equations with fuzzy inputs and fuzzy initial conditions using a fuzzy sample path approach we derive partial differential equations for the resulting membership functions we also demonstrate sufficient continuity conditions under which the solutions exist and are fuzzy convex the membership functions are shown to be identical to those for a time varying deterministic differential equation with a fuzzy initial condition under sufficient stability conditions we also consider the membership functions in steady state

eigen fuzzy number sets in this paper we extend certain results of sanchez resolution of eigen fuzzy sets equations fuzzy sets and systems 1 1978 69 75 to the context of fuzzy numbers a slight modification of the dubois prade definition of fuzzy numbers is used

numerical methods for calculating the fuzzy integral abstract a general procedure for calculating the fuzzy integral fi is developed for monotonic and single maxima functions in the case of monotonic functions an iterative algorithm based on the standard iteration method sim is shown to converge to the fi independently of the initial value an alternative newton type procedure can be applied for both monotonic and single maxima functions provided that an appropriate initial approximation is used

the eq 2 group via direct quantization of the lie poisson structure and its lie algebra the quantum group dual to the deformed eq 2 algebra obtained by celeghini et al j math phys 31 2548 1990 is constructed

toward a general theory of fuzzy variables abstract a general framework for a theory is presented that encompasses both statistical uncertainty which falls within the province of probability theory and nonstatistical uncertainty which relates to the concept of a fuzzy set and possibility theory l a zadeh j fuzzy sets 1 1978 3 28 the concept of a fuzzy integral is used to define the expected value of a random variable properties of the fuzzy expectation are stated and a mean value theorem for the fuzzy integral is proved comparisons between the fuzzy and the lebesgue integral are presented after a new concept of dependence is formulated various convergence concepts are defined and their relationships are studied by using a chebyshev like inequality for the fuzzy integral the possibility of using this theory in bayesian estimation with fuzzy prior information is explored

measurablility concepts for fuzzy mappings abstract in a series of recent works alternative measurability concepts for fuzzy mappings were introduced and studied in this paper we show that some of these concepts are equivalent or strongly inter related

the three dimensional euclidean quantum group e 3 q and its r matrix a contraction procedure starting from so 4 q is used to determine the quantum analog e 3 q of the three dimensional euclidean group and the structure of its representations a detailed analysis of the contraction of the r matrix is then performed and its explicit expression has been found the classical limit of r is shown to produce an integrable dynamical system by means of the r matrix the pseudogroup of the noncommutative representative functions is considered it will finally be shown that a further contraction made on e 3 q produces the two dimensional galilei quantum group and this in turn can be used to give a new realization of e 3 q and e 2 1 q

topological properties of the spaces of regular fuzzy sets abstract by identifying the space f 0 x of regular fuzzy sets with a subspace t 0 x of some compact set valued function space we define a uniform topology on f 0 x and discuss several topological properties such as first countability separability connectedness etc when x is a reflexive banach space f 0 x is imbedded into a locally convex topological vector space and a definition of integration and differentiation for f 0 x valued functions is given

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

on the validity of the peano theorem for fuzzy differential equations abstract kaleva 6 attempted to derive the validity of the classical peano theorem over a locally compact subset of en d based on this attempt kloeden 8 concluded that the peano theorem is also valid over en dp this paper challenges both attempts by counterexamples

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

solutions of algebraic equations involving generalized fuzzy numbers abstract this paper deals with algebraic equations involving generalized fuzzy numbers with continuous membership functions the generalized fuzzy number defined in this paper is a general name for fuzzy numbers fuzzy intervals crisp numbers and interval numbers three important properties of the generalized fuzzy numbers with continuous membership functions are introduced as the basis for deriving the solvability criterion the sufficient and necessary condition for solving a x c is derived the sufficient and necessary condition for solving a x c and a x c when a and c are nonzero generalized fuzzy numbers is also derived in this paper instead of the inverse operation method which fails for fuzzy sets inverse operations on the boundary points of level set intervals for continuous membership functions are used when the derived criteria of solvability are satisfied a unique solution rather than a solution set is obtained

towards fuzzy differential calculus part 1 integration of fuzzy mappings abstract this paper deals with fuzzy set valued mappings of the real line and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line these mappings can also be viewed as fuzzy relations using zadeh s extension principle the integral of such fuzzy mappings over a crisp interval is defined provided a special analytical representation of the fuzzy mapping the practical computation of such an integral is shown to be easy practically speaking it yields the fuzzy surface of a fuzzily bounded area

comparison of topological and uniform structures for fuzzy numbers and the fixed point problem abstract the goal of this paper is to compare certain topological and uniform structures on sets of fuzzy numbers since fuzzy sets can be viewed as a kind of subsets they can be endowed with uniform structures in ways similar to those used for classic subsets the hausdorff construction is one such approach the structures we study are the myope structure the uniform convergence of the cuts in the sense of hausdorff and a structure induced by a metric defined by goetschel and voxman we show how these structures are related to fixed point problems for some classes of fuzzy numbers

quantum 1 1 extended galilei algebras from lie bialgebras to quantum r matrices and integrable systems the lie bialgebras of the 1 1 extended galilei algebra are obtained and classified into four multiparametric families their quantum deformations are obtained together with the corresponding deformed casimir operators for the coboundary cases quantum universal r matrices are also given applications of the quantum extended galilei algebras to classical integrable systems are explicitly developed

three dimensional quantum groups from contractions of su 2 q contractions of lie algebras and of their representations are generalized to define new quantum groups an explicit and complete exposition is made for the one dimensional heisenberg h 1 q and the two dimensional euclidean quantum group e 2 q obtained by contracting su 2 q

quantumsu 2 ande 2 groups contraction procedure in 3 it was shown in the framework of deformed enveloping algebras that quantumsu 2 ande 2 groups are related by the contraction procedure we consider the same problem on thec level as a result we find a number of formulae coupling the comultiplications in quantumsu 2 ande 2 in particular we show that the comultiplications in both groups are implemented by partial isometries an unexpected feature of quantume 2 is discovered and the corresponding strange behavior of quantumsu 2 is described

on the observability of fuzzy dynamical control systems ii in this paper we continue to study the fuzzy logic system from the aspect of fuzzy differential equations and discuss the concept of observability given the rule base for the input and the output we can determine rule base for initial value by a formula the significance of this result is that given a deterministic system and fuzzy input and output rule base we can determine the rule base for initial value without solving the equation

fixed point theorems for fuzzy numbers abstract in this paper we study the problem of the existence of fixed points for fuzzy maps we first define uniform structures on sets of fuzzy subsets and describe the closed convergence and myope convergence structures which have properties close to the corresponding traditional ones we then give two ways in order to obtain fixed point results we finally give applications of these results particularly to fuzzy markov processes

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

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

classical deformations poisson lie contractions and quantization of dual lie bialgebras a poisson hopf algebra of smooth functions is simultaneously constructed on the two dimensional euclidean poincare and heisenberg groups by using a classical r matrix which is invariant under contraction the quantization for this algebra of functions is developed and its dual hopf algebra is also computed contractions on these quantum groups are studied it is shown that within this setting classical deformations are transformed into quantum ones by hopf algebra duality and the quantum heisenberg algebra is derived by means of a dual poisson lie quantization that deforms the standard moyal weyl product

classification of low dimensional lie super bialgebras a thorough analysis of lie super bialgebra structures on lie super algebras osp 1 2 and super e 2 is presented a combined technique of computer algebraic computations and subsequent identification of equivalent structures is applied in all the cases poisson lie brackets on supergroups are found the possibility of quantizing them in order to obtain quantum groups is discussed it turns out to be straightforward for all but one structure for the super e 2 group

classification of real three dimensional poisson lie groups all real three dimensional poisson lie pl groups are explicitly constructed and fully classified under group automorphisms by making use of their one to one correspondence with the complete classification of real three dimensional lie bialgebras given in gomez 2000 j math phys 41 4939 many of these 3d pl groups are non coboundary structures whose poisson brackets are given here for the first time casimir functions for all three dimensional pl groups are given and some features of several pl structures are commented

lie bi algebra structures for centrally extended two dimensional galilei algebra and their lie poisson counterparts all bi algebra structures for centrally extended galilei algebra are classified the corresponding lie poisson structures on centrally extended galilei group are found

quantum e 2 groups and lie bialgebra structures lie bialgebra structures on e 2 are classified for two lie bialgebra structures which are not coboundaries i e those which are not determined by a classical r matrix we solve the cocycle condition find the lie poisson brackets and obtain quantum group relations there is one to one correspondence between lie bialgebra structures on e 2 and possible quantum deformations of u e 2 and e 2

on fuzzy number lattice r abstract we give a systematic development of fuzzy number lattice theory many of our results generalize to number lattice over the lattice algebra our first main result is that the real number chain r is a homomorphic image of the fuzzy number lattice r that is r r and the congruence class r is a component of r thus we can be specialized to describe the structure of the fuzzy number lattice r next we study the structure and representation of the congruence class

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

a note on inverses in arithmetic with fuzzy numbers abstract algorithms for systematical search of inverse in arithmetic operations with fuzzy numbers are discussed the fuzzy number derived is considered either in a discrete representation of its support or described by the parameters of the corresponding membership function formula both algorithms are easy to be programmed for desk calculators

numerical solutions of fuzzy differential equations abstract numerical algorithms for solving fuzzy ordinary differential equations fode are considered a scheme based on the classical euler method is discussed in detail and this is followed by a complete error analysis the algorithm is illustrated by solving several linear and nonlinear fuzzy cauchy problems

fuzzy eigenvalues and input output analysis abstract we first generalize the perron frobenius results for nonnegative matrices to nonnegative fuzzy matrices which says that these fuzzy matrices have a dominant nonnegative fuzzy eigenvalue this result is then used to extend leontief s closed input output analysis to fuzzy economies and obtain a fuzzy model of the world s economy

fuzzy random variables expectation and martingales abstract banach space valued fuzzy random variables on a complete finite measure space f are studied their fuzzy valued integral is defined and its properties are examined some connections between fuzzy integral and conditional expectation are given the definition and some properties of fuzzy martingales are investigated

some notes on the characterization of compact sets in e n d p in this paper our purpose is to modify the characterization of compact sets in en dp 1 p shown in p diamond and p kloeden metric spaces of fuzzy sets fuzzy sets and systems 35 1990 241 249

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

an inversion method for seafloor elastic parameters abstractamplitude variation with offset avo inversion is a common approach to estimate elastic parameters in conventional avo inversion various approximations of the zoeppritz equation have been used however the weak elastic contrast assumption of those approximations leads to big errors when dealing with the seafloor situation in which the contrast of the elastic parameters across the interface is very strong to extend the application of avo to seafloor elastic parameters estimation as a linear problem we developed a high contrast approximation of the p wave reflection coefficient to the zoeppritz equation via a series truncation procedure model tests determined that the approximation we derived was more applicable than conventional approximations when considering the seafloor case any approximation limits avo inversion to a small incidence angle range a two step inversion method using the joint application of approximation and exact equation was evaluated to overcome the angle limit then we

random fuzzy number integrals in banach spaces abstract in this paper a new kind of random fuzzy number integral is introduced by the pettis aumann integral of the set valued functions in banach spaces the existence of the random fuzzy integral is also shown levi s convergence theorem of the integral is given a special mention

eigenvalues and eigenvectors of linearized elastic inversion small contrasts in the parameters allow the linearization of elastic modeling and inversion although this assumption simplifies parameter estimation it also impoverishes information a linearized model besides being an approximation requires shrinking the data space to precritical angles this work investigates the two basic questions regarding the type of information that can be retrieved and how to retrieve it the method is the singular value decomposition of the transfer function between data and parameters this tool allows a simple interpretation of the information contained in reflections and it outlines a general estimation procedure a medium characterized by a uniform background is considered reflections are linearized according to the born approximation p p reflections are analyzed the simplicity of the model allows a theoretical study of the effects of velocity errors in the overburden the analysis is performed for two different sets of elastic parameters in both cases the conclusion

towards fuzzy differential calculus part 2 integration on fuzzy intervals abstract the problem of integrating mappings over fuzzy domains has already received some attention in the literature however there is not a unique approach but several definitions have been proposed each carrying its own meaning this paper tries to emphasize the relationships and the differences between the various points of view

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

hopf superalgebra contractions and r matrix for fermions different contractions are applied to the quantum superalgebra ospq 1 mod 2 in the first of them the graded analogue s hq 1 of the one dimensional heisenberg quantum algebra is obtained and its r matrix explicitly calculated in the second contraction a z2 graded version of the q oscillator is proposed and finally the supersymmetric two dimensional euclidean quantum algebra s eq 2 is found

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

remarks on peano like theorems for fuzzy differential equations the validity of the classical peano theorem is noted for differential equations on the metric space en dp of normal fuzzy convex sets in rn where dp is the p th mean of the hausdorff distances between corresponding level sets this contrasts with differential equations on the space en d where d is the max metric which is not locally compact so more than just continuity is required to ensure the existence of solutions some additional properties differing from that given by kaleva are mentioned

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

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

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

amplitude variation with angle inversion using the exact zoeppritz equations theory and methodology to overcome the weaknesses of conventional prestack amplitude variation with angle inversion based on various linear or quasi linear approximations we have conducted a nonlinear inversion method using the exact zoeppritz matrix ezai however the inversion using the exact zoeppritz matrix was highly nonlinear and often unstable if not properly treated to tackle these issues we have used an iteratively regularizing levenberg marquardt scheme irlm which regularizes the inversion problem within an algorithm that minimizes the misfit between the observed and the modeled data at the same time by incorporating the tikhonov regularization method as a result the new ezai method solved using the irlm scheme is feasible for seismic data sets with large incidence angles even up to or beyond the critical angle as well as strong parameter contrasts single and multilayered synthetic examples were used to test these features these tests also showed that ezai is robust on noisy gathers for parameter extraction and has weak dependence on the initial model for the influence of inaccurate amplitudes dominant frequencies and phase angles we found that ezai is less sensitive to the variation in amplitude and phase shifts than to the dominant frequencies specifically the inversion results of ezai for p and s wave velocities and density were reliable if the inaccurate range for the amplitude was within 20 or the angle of the phase shift was no more than 20 the superiority of ezai makes it a very promising method for the estimation of subsurface elastic parameters