the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

algorithms for solving fuzzy relational equations in a probabilistic setting abstract we discuss methods suitable for finding approximate solutions to fuzzy relational equations defined in finite universe of discourse and taking the form of either a system of equations x k r y k k 1 2 k with r being unknown fuzzy relation or a single equation x r y with x being looked for the key idea of the proposed algorithms is to incorporate some statistics concerning the solution of a single equation and utilize it in the final process of approximation of the unknown fuzzy relation set numerical considerations support the established algorithms

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

the fu yau equation with negative slope parameter the fu yau equation is an equation introduced by fu and yau as a generalization to arbitrary dimensions of an ansatz for the strominger system as in the strominger system it depends on a slope parameter alpha prime the equation was solved in dimension 2 by fu and yau in two successive papers for alpha prime 0 and for alpha prime 0 in the present paper we solve the fu yau equation in arbitrary dimension for alpha prime 0 to our knowledge these are the first non trivial solutions of the fu yau equation in any dimension strictly greater than 2

ensemble and trajectory statistics in a nonlinear partial differential equation we have examined the influence of parametric noise on the solution behavioru t x of a nonlinear initial value problem arising in cell kinetics in terms of ensemble statistics the eventual limiting solution mean mathop xi limits u and variance mathop sigma u 2 limits are well characterized functions of the noise statistics and mathop xi limits u and mathop sigma u 2 limits depend on when noise is continuously present along the trajectory mathop xi limits u and mathop sigma u 2 limits are independent of the noise statistics and however in their evolution toward mathop xi limits u and mathop sigma u 2 limits both u t x and u2 t x depend on the noise and

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

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

solution of fuzzy equations with extended operations abstract assuming that is any operation defined on a product set x y and taking values on a set z it can be extended to fuzzy sets by means of zadeh s extension principle given a fuzzy subset c of z it is here shown how to solve the equation a b c or a b c when a fuzzy subset a of x or a fuzzy subset b of y is given the methodology we provide includes as a special case the resolution of fuzzy arithmetical operations i e when stands for or extended to fuzzy numbers fuzzy subsets of the real line the paper is illustrated with several examples in fuzzy arithmetic

relational equations in totally ordered lattices and their complete resolution abstract the set of all solutions of a composite fuzzy relation equation of sanchez inform and control 30 1976 defined on finite spaces is studied by determining and characterizing all the lower solutions of such an equation

fuzzy sets in pattern recognition methodology and methods abstract an objective of the paper is to discuss a state of the art of methodology and algorithms of fuzzy sets in the field of pattern recognition in real world recognition and classification problems we are faced with fuzziness that is connected with diverse facets of cognitive activity of the human being an origin of sources of fuzziness is related to labels expressed in feature space as well as to labels of classes taken into account in classification procedures an evident difference between a way of information processing by means of probability and fuzzy set theory and a way of interpretation of results is explained in detail in sequel methods of pattern recognition are studied in two main streams namely supervised and unsupervised learning different approaches to designing of classification schemes as e g relation calculus decision making approach etc are put into account a method of feature selection with the aid of fuzzy measure and fuzzy integral is introduced in clustering techniques methods relying on minimization of objective function are analyzed in detail also a problem of cluster validity expressed in terms of clustering indices is addressed numerical examples are also provided

an alternative approach to the resolution of fuzzy relation equations abstract in this paper we propose a solution procedure by means of a quasi characteristic matrix the matrix is so defined that not only the solution set can be determined but also the type of solution set such as the non empty set full set unique minimal solution and unique solution can be identified before the solution set is determined therefore it provides an insight into the structure of fuzzy relation equations

the fu yau equation on compact astheno kahler manifolds abstract in this paper we study the fu yau equation on compact hermitian manifolds and prove the existence of solutions of equation on astheno kahler manifolds we also prove the uniqueness of solutions of fu yau equation when the slope parameter is negative

hypersurfaces of prescribed mean curvature in lorentzian manifolds we give a new existence proof for closed hypersurfaces of prescribed mean curvature in lorentzian manifolds

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

applications of fuzzy relational equations for methods of reasoning in presence of fuzzy data the paper deals with implementation of a general scheme for fuzzy logic inference with the aid of fuzzy relational equations it is shown that antecedents and consequents are related in terms of a fuzzy relation various implementation methods of generalized modus ponens and modus tollens are put into extensive study it is pointed out that fuzzy relational equations form an effective and plausible tool for implementations of reasoning methods with fuzzy premises two examples of fuzzy controllers being a direct application of schemes of approximate reasoning are also discussed

solution algorithms for fuzzy relational equations with max product composition the conditions for the existence of an inverse solution to the max min composition of fuzzy relational equations have been well documented since the original work by sanchez 30 31 these same existence theorems have been extended to the t norm composition of relational equations in which the max product composition is a member 5 13 26 several studies 8 15 24 33 34 38 have shown that the max min operator may not always be the most desirable fuzzy relational composition and in fact the max product operator was superior in these instances this paper reviews the algorithms necessary to determine the complete solution of the inverse for fuzzy relational equations with max product composition

diagnosis problem solving using fuzzy relations this paper deals with the restoration and the identification of the causes diagnoses through the observed effects symptoms on the basis of fuzzy relations and zadeh s compositional rule of inference we propose an approach for building fuzzy systems of diagnosis which enables solving fuzzy relational equations together with design and tuning of fuzzy relations on the basis of expert and experimental information the essence of tuning consists of the selection such membership functions for fuzzy causes and effects and also fuzzy relations which minimize the difference between model and experimental results of diagnosis the genetic algorithm is used for solving the optimization problem the proposed approach is illustrated by the computer experiment and the real example of diagnosis

an algorithm for solving fuzzy relation equations with max t composition operator this paper studies the problem of solving a max t composite finite fuzzy relation equation where t is a special class of pseudo t norms if the equation is solvable then its set of feasible solutions is determined by the greatest solution and a finite number of minimal solutions some necessary conditions are presented for the minimal solutions in terms of the maximum solution and zero value under these conditions some minimal solutions of the system can be obtained easily some procedures are also proposed in order to simplify the original system the simplified system is then decomposed if possible into several subsystems with smaller dimensions which are very easy to solve furthermore a method is presented to solve each subsystem by combining the method and those procedures an efficient algorithm is proposed to obtain the set of feasible solutions of the original system two examples are also given to illustrate the algorithm

on the set of solutions of composite fuzzy relation equations in this paper we introduce some algorithms which have minimization properties about the fuzziness of solutions in the maxmin fuzzy relation equations

global c2 estimates for convex solutions of curvature equations we establish a c2 a priori estimate for convex hypersurfaces whose principal curvatures 1 n satisfy k x f x x the weingarten curvature equation we also obtain such an estimate for admissible 2 convex hypersurfaces in the case k 2 our estimates resolve a longstanding problem in geometric fully nonlinear elliptic equations 2015 wiley periodicals inc

on the fuzziness of solutions of fuzzy relation equations on finite spaces let be the set of solutions of a fuzzy relation equation defined on finite spaces in this paper we use some algorithms already introduced in our foregoing work in order to determine some relations of which have a minimal value of fuzziness measure by taking into account the fuzzy entropy concept we build a new lattice structure and some fuzzy topological results are also presented

on estimates for the fu yau generalization of a strominger system we study an equation proposed by fu and yau as a natural n dimensional generalization of a strominger system that they solved in dimension 2 it is a complex hessian equation with right hand side depending on gradients building on the methods of fu and yau we obtain c0 c2 and c2 a priori estimates we also identify difficulties in extending the fu yau arguments for non degeneracy from dimension 2 to higher dimensions

an interior estimate for convex solutions and a rigidity theorem abstract we establish an interior c 2 estimate for k 1 convex solutions to dirichlet problems of k hessian equations we also use such estimate to obtain a rigidity theorem for k 1 convex entire solutions of k hessian equations in euclidean space

a new look at solving a system of fuzzy relational equations abstract solving a system of fuzzy relational equations exactly has been studied for many years a usual understanding on the solvable conditions is that the input fuzzy sets must be normal and pairwise disjoint such understanding is reexamined in this paper we show that the usual pairwise disjoint condition is too conservative and a system of max t fuzzy relational equations can be solved exactly when the input fuzzy sets are semi overlapped a condition commonly found in most rule based system applications in addition the boundary condition of solvability with respect to the compactness of the input fuzzy sets is derived if it cannot be satisfied we show that the system of equations could still be solved almost exactly by specifying the t norm as drastic product the results have been applied to study the capacity of fuzzy relations

a class of hessian quotient equations in euclidean space abstract in this paper we consider a class of hessian quotient equations in euclidean space under some sufficient conditions we obtain an existence result using standard degree theory based on a priori estimates for solutions to hessian quotient equations

a variational theory of the hessian equation by studying a negative gradient flow of certain hessian functionals we establish the existence of critical points of the functionals and consequently the existence of ground states to a class of nonhomogenous hessian equations to achieve this we derive uniform first and second order a priori estimates for the elliptic and parabolic hessian equations our results generalize well known results for semilinear elliptic equations and the monge ampere equation c 2001 john wiley sons inc

an approach to the analysis of fuzzy systems this paper deals with a formal description of ill defined processes fuzzy systems by the use of fuzzy relational equations it is pointed out that fuzzy relational equations form a generalized version of the difference equations widely considered in control theory some equivalence between these two kinds of description is presented basic problems of fuzzy systems e g identification prediction sensitivity and stability are shown and numerical algorithms are given indices of each method are introduced especially the degree of fuzziness the sensitivity index which makes it possible to express the quality of each of them

on the existence of solutions of systems of fuzzy equations abstract a necessary and sufficient condition for the existence of solutions of a fuzzy equation given by e sanchez is extended to systems of fuzzy equations

interval valued fuzzy backward reasoning the importance and efficiency of backward reasoning in nonfuzzy reasoning has been stressed for a long time especially in the case of expert systems and decision support systems the extension of this reasoning method to fuzzy theory however has never been considered in this paper the authors propose a definition of fuzzy backward reasoning based on the generalized modus ponens and show the necessity of considering interval valued fuzzy backward reasoning then the authors propose solving methods for fuzzy backward reasoning in the case of a rule with one or several conditions as well as in the case of several rules

interior c 2 regularity of convex solutions to prescribing scalar curvature equations we establish interior c2 estimates for convex solutions of the scalar curvature equation and the 2 hessian equation we also prove interior curvature estimates for isometrically immersed hypersurfaces mn g rn 1 with positive scalar curvature these estimates are consequences of interior estimates for these equations obtained under a weakened condition

on measures of fuzziness of solutions of fuzzy relation equations with generalized connectives by using a general class of fuzzy connectives of yager fuzzy sets and systems 4 1980 235 242 pedrycz fuzzy relational equations with generalized convectives and their applications fuzzy sets and systems 10 1983 185 201 has shown that the classical fuzzy relation equations of sanchez in fuzzy automata and decision processes m m gupta g n saridis and b r gaines eds pp 221 234 north holland amsterdam 1977 can be considered as a particular case of a more extensive class of fuzzy equations for such types of equations in this paper the solutions having the greatest energy measure and the smallest possible entropy measure of fuzziness are characterized

concepts of probabilistic sets abstract in the field of pattern recognition or decision making theory the important subjects are as follows 1 ambiguity of property of objects 2 variety of character of objects 3 subjectivity of observers 4 evolution of knowledge of observers i e learning considering these points the concept of probabilistic set is proposed it is based on both probability theory and fuzzy concepts a probabilistic set on a total space is defined by a point wise measurable function from a parameter space which is a probability space to a characteristic space which is a measurable space it is shown that the family of all probabilistic sets constitutes a complete pseudo boolean algebra moment analysis is possible by using a probability measure of the parameter space other useful concepts are also mentioned such as probabilistic mappings and expected cardinal numbers

approximate solutions of fuzzy relational equations in the previous chs we have discussed various problems concerning solutions of fuzzy relation equations obviously an underlying assumption is that there exists a nonempty set of solutions a situation may occurr and indeed it is quite common in which no solution exists nevertheless even in this case one might be interested to obtain an approximate solution and know to which extent it can be viewed as a solution this stream of investigations is particularly interesting for applicational purposes contrary to the topics already discussed in the previous chs this is a field of research which has not been developed enough so far it concerns studies on solvability properties of fuzzy relation equations in this ch we shall try to answer by using several techniques the following basic question how difficult is it to attain a situation in which the system of equations has solutions and then how to measure this property

analysis and synthesis of fuzzy systems by the use of probabilistic sets the paper deals with an application of probabilistic sets in system theory especially in identification problems in systems described by means of max min fuzzy relational equations the identification procedures discussed are based on some ideas of iterative clustering techniques isodata and fuzzy c means which lead to a concrete method of determination of probabilistic sets a vagueness function associated with the fuzzy relation of the system forms a validity indicator of the identification algorithm numerical examples containing fuzzy and nonfuzzy data form an illustration of the methods provided

heterogeneous fuzzy logic networks fundamentals and development studies the recent trend in the development of neurofuzzy systems has profoundly emphasized the importance of synergy between the fundamentals of fuzzy sets and neural networks the resulting frameworks of the neurofuzzy systems took advantage of an array of learning mechanisms primarily originating within the theory of neurocomputing and the use of fuzzy models predominantly rule based systems being well established in the realm of fuzzy sets ideally one can anticipate that neurofuzzy systems should fully exploit the linkages between these two technologies while strongly preserving their evident identities plasticity or learning abilities to be shared by the transparency and full interpretability of the resulting neurofuzzy constructs interestingly this synergy still becomes a target yet to be satisfied this study is an attempt to address the fundamental interpretability challenge of neurofuzzy systems our underlying conjecture is that the transparency of any neurofuzzy system links directly with the logic fabric of the system so the logic fundamentals of the underlying architecture become of primordial relevance having this in mind the development of neurofuzzy models hinges on a collection of logic driven processing units named here fuzzy logic neurons these are conceptually simple logic oriented elements that come with a well defined semantics and plasticity owing to their diversity such neurons form essential building blocks of the networks the study revisits the existing categories of logic neurons provides with their taxonomy helps understand their functional features and sheds light on their behavior when being treated as computational components of any neurofuzzy architecture the two main categories of aggregative and reference neurons are deeply rooted in the fundamental operations encountered in the technology of fuzzy sets including logic operations linguistic modifiers and logic reference operations the developed heterogeneous networks come with a well defined semantics and high interpretability which directly translates into the rule based representation of the networks as the network takes advantage of various logic neurons this imposes an immediate requirement of structural optimization which in this study is addressed by utilizing various mechanisms of genetic optimization genetic algorithms we discuss the development of the networks elaborate on the interpretation aspects and include a number of illustrative numeric examples

fuzzy modelling fundamentals construction and evaluation abstract the paper is devoted to fundamental aspects of fuzzy modelling namely modelling articulated with the use of methodology of fuzzy sets open issues are addressed which summarize advantages of fuzzy models and point out some methodological deficiencies blurring existing approaches the problem of relevancy of fuzzy models and its direct impact on a suitable interpretation of derived results is studied in detail finally we analyze some particular fuzzy models developed in terms of relational equations

neuron inspired learning rules for fuzzy relational structures abstract fuzzy relational equations are a suitable framework for fuzzy modelling however their constructive resolution methods suffer from known drawbacks in order to determine approximate solutions both conventional and adaptive gradient based learning methods are proposed namely for extended versions of fuzzy relational structures simulation results show that for both learning methods good approximate solutions are found however the adaptive version shows considerably quicker convergence rates the problem of process fuzzy identification using the proposed framework is then outlined as an illustrative application

fuzzy system identification via probabilistic sets abstract the paper deals with the applications of probabilistic sets in system theory especially in system identification and the design of fuzzy logic controllers fuzzy systems described by means of fuzzy relational equations and fuzzy systems are discussed the identification procedure is based on some ideas of clustering techniques and probabilistic sets numerical examples using fuzzy and nonfuzzy data are used to illustrate the theory

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

solvability of fuzzy relational equations and manipulation of fuzzy data abstract the problem of characterization of fuzzy relational equations with respect to their solvability property is studied some aspects of manipulation of fuzzy data with the aid of fuzzy relational equations are considered two stages of manipulation process are indicated combining pieces of evidence and inferring their mutual correspondence both of them are formulated and solved by the use of fuzzy relational equations the role of a solvability index is extensively studied numerical considerations give an illustration of the approach we propose and deal with some real data used for fuzzy controllers

curvature estimates for immersed hypersurfaces in riemannian manifolds we establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in riemannian manifold n n 1 bar g through regularity study of a degenerate fully nonlinear curvature equation in general riemannian manifold the estimate has a direct consequence for the weyl isometric embedding problem of mathbb s 2 g in 3 dimensional warped product space n 3 bar g we also discuss isometric embedding problem in spaces with horizon in general relativity like the anti de sitter schwarzschild manifolds and the reissner nordstrom manifolds

strong ergodic properties of a first order partial differential equation on etudie l equation differentielle d ordre 1 u t c x u x f x u pour t x 0 0 1 avec la condition initiale u 0 x v x pour x 0 1 on demontre l existence d une mesure invariante exacte ayant certaines proprietes supplementaires

fuzzy relation equations and inequalities abstract this paper generalizes a method of solution of fuzzy relation equations introduced by sanchez 1976 and developed by pedrycz 1982 we find the greatest or the least solution of a composite l fuzzy relation equation with arbitrary monotone composition

fuzzy set framework for development of a perception perspective abstract the question of perception attracts a lot of investigation concerning fundamental issues of knowledge representation and particularly application oriented techniques we clearly indicate the context dependent character of information processed in cognitive processes as opposed to statistical analysis in which no context is taken into account the problem being addressed here is translated into an optimization scheme under which algorithms of data analysis are worked out a general class of techniques dedicated to linguistic analysis of records of numerical data is introduced its essence relies on determining the level of data and fuzzy labels

products of fuzzy subgroups fuzzy subgroups are different from ordinary subgroups in that one cannot tell with certainty which group elements belong and which do not of course this requires an appropriate modification of the closure property which takes the form of an inequality in this paper a study of products of fuzzy subgroups is begun some results are expected the product of fuzzy subgroups is a fuzzy subgroup products with the same but permuted factors are isomorphic and there is a natural isomorphism from any factor into the product functions called t norms are used to construct products of fuzzy subgroups different t norms yielding different products since a fuzzy subgroup must satisfy an inequality generalizing closure and the strength of that inequality is important some results relate the t norm used in forming the product to the strength of the ensuing inequality when can complicated fuzzy subgroups be expressed as products of simple ones this question is addressed for fuzzy subgroups of finite abelian groups

the complex hessian equations with gradient terms on hermitian manifolds abstract in this paper we consider the complex hessian equation involving additional gradient terms on both left and right hand sides we derive the second order estimates for plurisubharmonic solutions to this equation on compact hermitian manifolds such an estimate may be helpful to study the existence and regularity for the solution of the equation

a rule based method to calculate exactly the widest solution sets of a max min fuzzy relational inequality abstract in this paper we propose a rule based method to calculate exactly all the widest solution sets of a fuzzy relational inequality viewed as an extension of fuzzy relational equations fuzzy relational inequalities can be applied to many fields of fuzzy logic yet they provide with more flexibility as they enable to get rid of the sensitivity of fuzzy relational equations to numerical values up to now however fuzzy relational inequalities have received few attention from researchers and few methods have been proposed to solve these inequalities the proposed analytical methods furthermore do not calculate exactly all the widest solutions sets of a fuzzy relational inequality and often lead to redundant solution sets the rule based method we propose in this paper determines easily exactly all the widest solution sets of a fuzzy relational inequality