boundary behavior of solutions to some singular elliptic boundary value problems abstract let be a bounded domain with smooth boundary in r n n 1 for the more general weight b some nonlinearities f and singularities g by two kinds of nonlinear transformations a new perturbation method which was introduced by garcia melian in j garcia melian boundary behavior of large solutions to elliptic equations with singular weights nonlinear anal 67 2007 818 826 and comparison principles we show that the boundary behavior of solutions to a boundary blow up elliptic problem w b x f w w 0 x w and a singular dirichlet problem u b x g u u 0 x u 0 has the same form under the nonlinear transformations which can be determined in terms of the inverses of the transformations

hedge algebras an algebraic approach to structure of sets of linguistic truth values abstract it is shown that any sets of linguistic values of linguistic variables can be axiomatized which leads to a notion of hedge algebras some intuitive properties of linguistic hedges which are a basis of the axiomatization of the hedge algebras are discussed informally the characteristics of the hedge algebras depend on such a discussion and they seem to reflect the natural structure of sets of linguistic values of linguistic variables some fundamental properties of hedge algebras are examined they constitute a foundation of further development of our theory of hedge algebras and of some kind of fuzzy logic

which logic is the real fuzzy logic this paper is a contribution to the discussion of the problem whether there is a fuzzy logic that can be considered as the real fuzzy logic we give reasons for taking imtl bl email protected and ev l fuzzy logic with evaluated syntax as those fuzzy logics that should be indeed taken as the real fuzzy logics

a second order estimate for blow up solutions of elliptic equations abstract we investigate second term asymptotic behavior of boundary blow up solutions to the problems u b x f u x subject to the singular boundary condition u x in a bounded smooth domain r n b x is a non negative weight function the nonlinearly f is regularly varying at infinity with index 1 that is lim u f u f u for every 0 and the mapping f u u is increasing on 0 the main results show how the mean curvature of the boundary appears in the asymptotic expansion of the solution u x our analysis relies on suitable upper and lower solutions and the karamata regular variation theory

boundary blow up in nonlinear elliptic equations of bieberbach rademacher type the research of the first author was carried out at victoria university melbourne with the support of the australian government through detya the second author has been supported by grant 2 cex06 11 18 2006

a new version of 2 tuple fuzzy linguistic representation model for computing with words in this paper we provide a new proportional 2 tuple fuzzy linguistic representation model for computing with words cw which is based on the concept of symbolic proportion this concept motivates us to represent the linguistic information by means of 2 tuples which are composed by two proportional linguistic terms for clarity and generality we first study proportional 2 tuples under ordinal contexts then under linguistic contexts and based on canonical characteristic values ccvs of linguistic labels we define many aggregation operators to handle proportional 2 tuple linguistic information in a computational stage for cw without any loss of information our approach for this proportional 2 tuple fuzzy linguistic representation model deals with linguistic labels which do not have to be symmetrically distributed around a medium label and without the traditional requirement of having equal distance between them moreover this new model not only provides a space to allow a continuous interpolation of a sequence of ordered linguistic labels but also provides an opportunity to describe the initial linguistic information by members of a continuous linguistic scale domain which does not necessarily require the ordered linguistic terms of a linguistic variable being equidistant meanwhile under the assumption of equally informative which is defined by a condition based on the concept of ccv we show that our model reduces to herrera and mart spl inodot spl acute nez s translational 2 tuple fuzzy linguistic representation model

fuzzy logic of lukasiewicz logic a clarification pavelka 10 had shown in 1979 that the only natural way of formalizing fuzzy logic for truth value in the unit interval 0 1 is by using lukasiewicz s implication operator a b min 1 1 a b or some isomorphic form of it a considerable number of other papers around the same time had attempted to formulate alternative definitions for a b by giving intuitive justifications for them there continues to be some confusion however even today about the right notion of fuzzy logic much of this has its origin in the use of improper and or and the not operations and a misunderstanding of some of the key differences between proofs or inferencing in fuzzy logic and those in lukasiewicz s logic we point out the need for defining the strong conjunction operator in connection with fuzzy modus ponens rule and why we do not need the fuzzy syllogism rule we also point out the shortcomings of many of the alternative definitions of a b which indicate further support for pavelka s result we hope that these discussions help to clarify the misconceptions about fuzzy logic

boundary blow up elliptic problems with nonlinear gradient terms by karamata regular variation theory and perturbation method we show the exact asymptotical behaviour of solutions near the boundary to nonlinear elliptic problems u u q b x g u u 0 in u where is a bounded domain with smooth boundary in rn q 0 g c1 0 g 0 0 g is regularly varying at infinity with index with 0 and b is nonnegative nontrivial in which may be vanishing on the boundary

are fuzzy sets a reasonable tool for modeling vague phenomena in the paper the indeterminacy phenomenon is discussed that is a phenomenon having two facets uncertainty and vagueness we argue that fuzzy sets are a reasonable mathematical tool for modeling of the latter the necessary sound foundations of their theory can now be more easily established because of significant progress reached in the formal theory of fuzzy logic further direction in the development of fuzzy set theory is also discussed

linguistic truth valued lattice valued propositional logic system l p x based on linguistic truth valued lattice implication algebra in the semantics of natural language quantification may have received more attention than any other subject and syllogistic reasoning is one of the main topics in many valued logic studies on inference particularly lattice valued logic a kind of important non classical logic can be applied to describe and treat incomparability by the incomparable elements in its truth valued set in this paper we first focus on some properties of linguistic truth valued lattice implication algebra secondly we introduce some concepts of linguistic truth valued lattice valued propositional logic system p x whose truth valued domain is a linguistic truth valued lattice implication algebra then we investigate the semantic problem of p x finally we further probe into the syntax of linguistic truth valued lattice valued propositional logic system p x and prove the soundness theorem deduction theorem and consistency theorem

optimal uniqueness theorems and exact blow up rates of large solutions in this paper we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non linearity and combine these results with the localization method introduced in lopez gomez the boundary blow up rate of large solutions j differential equations 195 2003 25 45 to prove that any large solution l of u a x up p 1 a 0 must satisfy limx x0l x fx0 dist x i0 pp 1p 1p 1p 1p 1foreachx0 where fx0 t t 0sfx01p 1 p 1p 1ds i0 limt 0fx0 t fx0 t fx0 t 2 and fx0 is any smooth extension of the boundary normal section of a at x0 i e fx0 t a x0 tnx0 t 0 t 0 subsequently nx0 stands for the outward unit normal at x0 therefore the theory can be extended to cover the general case when fx0 l1p 1

symmetry of large solutions of nonlinear elliptic equations in a ball let g be a locally lipschitz continuous real valued function which satisfies the keller osserman condition and is convex at infinity then any large solution of u g u 0 in a ball is radially symmetric

blow up rates of large solutions for elliptic equations in this paper we mainly study the boundary behavior of solutions to boundary blow up elliptic problems for more general nonlinearities f which may be rapidly varying at infinity u b x f u x u where is a bounded domain with smooth boundary in rn and b c which is positive in and may be vanishing on the boundary and rapidly varying near the boundary further when f s sp f1 s for s sufficiently large where p 1 and f1 is normalized regularly varying at infinity with index p1 0 p we show the influence of the geometry of on the boundary behavior for solutions to the problem we also give the existence and uniqueness of solutions

a necessary and sufficient condition for existence of large solutions to semilinear elliptic equations we consider the semilinear equation u p x f u on a domain rn n 3 where f is a nonnegative nondecreasing continuous function which vanishes at the origin and p is a nonnegative continuous function with the property that any zero of p is contained in a bounded domain in such that p is positive on its boundary for bounded we show that a nonnegative solution u satisfying u x as x exists if and only if the function s s0 f t dt satisfies 1 s 1 2 ds for unbounded including rn we show that a similar result holds where u x as x within and u x as x if p x decays to zero rapidly as x

a topological completion of refined hedge algebras and a model of fuzziness of linguistic terms and hedges the main aim of the investigation of hedge algebras is to discover an algebraic structure in the category of universal algebras of terms domains of linguistic variables or to establish an algebraic approach to the semantic structure of terms domains the meaning of linguistic terms in general is vague however vagueness is a concept which is difficult to be defined formally and reasonably motivated by introducing a mathematical foundation of qualitative and quantitative fuzziness of linguistic terms this paper is aimed to introduce and investigate a completion of refined hedge algebras and obtain a class of complete ones they will be defined by adding limit elements to refined hedge algebras in an axiomatization way it is shown that a formal model of fuzziness of linguistic terms and hedges can be defined reasonably in this semantic structure and there is a closed relation between fuzziness of terms and semantic based topology of these algebras

an axiomatizable logical foundation for lattice ordered qualitative linguistic approach for reasoning with words it is commonsense that people do express think reason and make decision directly using linguistic terms in natural language rather than using numerical quantification how to establish the formalized approach imitating the common way of human being s to manipulate directly linguistic terms without numerical approximation to provide the rational decision is still an open research area investigations on the algebraic structure of linguistic term set for varied decision making applications especially in social science still lack a formalism for development of strict linguistic valued logic system as a theoretical foundation and its approximate reasoning scheme in practice to attain this goal we characterize and construct a typical structure of commonly used linguistic term sets in natural language by a lattice ordered algebra structure lattice implication algebra lia where lukasiewicz implication algebra as a special case of lia plays a substantial role by using lukasiewicz logic s axiomatizability in terms of pavelka type fuzzy logic we propose a new axiomatizable lattice ordered qualitative linguistic truth valued logic system based on lia to place an important foundation for further establishing formal linguistic truth valued logic based approximate reasoning and decision making with applications this proposed logic system has a distinct feature of handling comparable or incomparable linguistic terms directly without numerical quantification will be especially beneficial for perception based decision making processes it attempts to enhance the quantitative theory of decision science with qualitative algebraic and logic oriented approaches to achieve reasoning with words

a dynamic consensus scheme based on a nonreciprocal fuzzy preference relation modeling in this paper we propose a dynamical consensus scheme which makes use of an index of comparability and of an index of concordance to regulate the information flow in the discussion among the experts in addition to the traditional use of a concordance index to identify the discordant expert who can be invited to review his opinion or to explain his judgment to the other experts here an index of comparability is utilized to identify the experts that have faced difficulties in expressing their preferences with this information at hand a moderator can invite the experts that are confident on their opinions to support the experts that are hesitating in the construction of the fuzzy preference relations

an algebraic approach to linguistic hedges in zadeh s fuzzy logic the paper addresses the mathematical modelling of domains of linguistic variables i e term sets of linguistic variables in order to obtain a suitable algebraic structure for the set of truth values of zadeh s fuzzy logic we shall give a unified algebraic approach to the natural structure of domains of linguistic variables which was proposed by ho and wechler fuzzy sets and systems 35 1990 281 and then by ho and nam proc ncst vietnam 9 1 1997 15 logic algebra and computer science vol 46 banach center publications pwn warsaw 1999 p 63 in this approach every linguistic domain can be considered as an algebraic structure called hedge algebra because properties of its unary operations reflect semantic characteristics of linguistic hedges many fundamental properties of refined hedge algebras rh algebras are examined especially it is shown that every rh algebra of a linguistic variable with a chain of the primary terms is a distributive lattice rh algebras with exactly two distinct primary terms one being an antonym of the other will also be investigated and they will be called symmetrical rh algebras it is shown that a class of finite symmetrical rh algebras has a rich enough algebraic structure

the second expansion of large solutions for semilinear elliptic equations abstract in this paper we analyze the influence of the domain geometry in the second expansion of solutions to the boundary blow up elliptic problem u b x f u u 0 x u where is a bounded domain with c 4 smooth boundary in r n b c which is positive in and may be vanishing on the boundary and f is normalized regularly varying at infinity with index p p 1

extended hedge algebras and their application to fuzzy logic this paper continues our investigation on hedge albebras 6 we extend hedge algebras by two additional operations corresponding to infimum and supremum of the so called concept category of an element x i e the set which is generated from x by means of the hedge operations it is shown that every extended hedge algebra with a lattice of the primary generators is a lattice in the symmetrical extended hedge algebras we are able to define negation and implication called concept negation and concept implication furthermore it is proved that there exists an isomorphism from a subaigebra of a symmetrical extended hedge algebra of a linguistic truth variable into the closed unit interval 0 1 under which the concept negation and the concept implication correspond to the negation and a kind of implication in multiple valued logic based on the unit interval 0 1

blow up boundary solutions of semilinear elliptic problems we consider the following semilinear elliptic equation u p x f u in u 0 u 0 in 1 where rn n 3 is a smooth domain bounded or possibly unbounded with compact possibly empty boundary we assume throughout this paper that p is a non negative function such that p c0 if is bounded and p c loc otherwise the nonlinearity f is assumed to ful ll f 1 f c1 0 f 0 f 0 0 and f 0 on 0 and the keller osserman condition see 7 13

a fuzzy linguistic methodology to deal with unbalanced linguistic term sets many real problems dealing with qualitative aspects use linguistic approaches to assess such aspects in most of these problems a uniform and symmetrical distribution of the linguistic term sets for linguistic modeling is assumed however there exist problems whose assessments need to be represented by means of unbalanced linguistic term sets i e using term sets that are not uniformly and symmetrically distributed the use of linguistic variables implies processes of computing with words cw different computational approaches can be found in the literature to accomplish those processes the 2 tuple fuzzy linguistic representation introduces a computational model that allows the possibility of dealing with linguistic terms in a precise way whenever the linguistic term set is uniformly and symmetrically distributed in this paper we present a fuzzy linguistic methodology in order to deal with unbalanced linguistic term sets to do so we first develop a representation model for unbalanced linguistic information that uses the concept of linguistic hierarchy as representation basis and afterwards an unbalanced linguistic computational model that uses the 2 tuple fuzzy linguistic computational model to accomplish processes of cw with unbalanced term sets in a precise way and without loss of information

expanding the asymptotic explosive boundary behavior of large solutions to a semilinear elliptic equation the main goal of this paper is to study the asymptotic expansion near the boundary of the large solutions of the equation delta u lambda u m f in omega where lambda 0 m 1 f is an element of c omega f 0 and omega is an open bounded set of r n n 1 with boundary smooth enough roughly speaking we show that the number of explosive terms in the asymptotic boundary expansion of the solution is finite but it goes to infinity as in goes to 1 we prove that the expansion consists in two eventual geometrical and non geometrical parts separated by a term independent on the geometry of partial derivative omega but dependent on the diffusion for low explosive sources the non geometrical part does not exist all coefficients depend on the diffusion and the geometry of the domain by means of well known properties of the distance function dist x partial derivative omega for high explosive sources the preliminary coefficients relative to the non geometrical part are independent on omega and the diffusion finally the geometrical part does not exist for very high explosive sources consists in two eventual geometrical and non geometrical parts separated by a term independent on the geometry of partial omega but dependent on the diffusion for low explosive sources the non geometrical part does not exist all coefficients depend on the diffusion and the geometry of the domain by means of well known properties of the distance function rm dist x partial omega dist x for high explosive sources the preliminary coefficients relative to the non geometrical part are independent on omega and the diffusion finally the geometrical part does not exist for very high explosive sources

algebraic structures in fuzzy logic abstract common features in various fuzzy logical systems are investigated it turns out that adjoint couples and residuated lattices are very often present though not always explicitly expressed some minor new results are also proved

computing the numerical scale of the linguistic term set for the 2 tuple fuzzy linguistic representation model when using linguistic approaches to solve decision problems we need the techniques for computing with words cw together with the 2 tuple fuzzy linguistic representation models i e the herrera and martinez model and the wang and hao model some computational techniques for cw are also developed in this paper we define the concept of numerical scale and extend the 2 tuple fuzzy linguistic representation models under the numerical scale we find that the key of computational techniques based on linguistic 2 tuples is to set suitable numerical scale with the purpose of making transformations between linguistic 2 tuples and numerical values by defining the concept of the transitive calibration matrix and its consistent index this paper develops an optimization model to compute the numerical scale of the linguistic term set the desired properties of the optimization model are also presented furthermore we discuss how to construct the transitive calibration matrix for decision problems using linguistic preference relations and analyze the linkage between the consistent index of the transitive calibration matrix and one of the linguistic preference relations the results in this paper are pretty helpful to complete the fuzzy 2 tuple representation models for cw

boundary behaviour for solutions of boundary blow up problems in a borderline case we investigate boundary blow up solutions of the equation u f u in a bounded domain rn under the condition that f t has a relatively slow growth as t goes to infinity we show how the mean curvature of the boundary appears in the asymptotic expansion of the solution u x in terms of the distance of x from

boundary behavior for large solutions to elliptic equations with singular weights abstract in this paper we analyze the boundary behavior of large positive solutions to some semilinear elliptic equations which include a singular weight the most important point is that the growth of the solutions can be determined in terms of the solution to a one dimensional first order equation we also consider the questions of existence and uniqueness of positive solutions

boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms abstract in this paper we study the boundary behavior of solutions to boundary blow up elliptic problems u u q b x f u x u where is a bounded domain with smooth boundary in r n q 0 b c which is positive in and may be vanishing on the boundary and rapidly varying near the boundary and f is rapidly varying or normalized regularly varying at infinity

a view of averaging aggregation operators aggregation operators have been used and studied for a long time more than ten different types of means including the arithmetic geometric and harmonic ones were already studied 2000 years ago nevertheless this topic has gained relevance in recent years this is partly due to the increasing need of methods and operators for fusing information within computer programs in this paper we will present a personal view of the field focusing on the application issues and on averaging aggregation operators we will point out some research topics and open lines for future research

fuzziness measure on complete hedge algebras and quantifying semantics of terms in linear hedge algebras in the paper we shall examine the fuzziness measure fm of terms or of complete and linear hedge algebras of a linguistic variable the notion of semantically quantifying mappings sqms previously examined by the first author will be redefined more generally and a closed relation between the fm of linguistic terms and a family of sqms with the parameters to be the fm of primary terms and linguistic hedges will be established a semantics based topology of hedge algebras and a closed and interesting relation between this topology the fm and the above family of sqms will be discovered and examined an applicability of the fm and sqms will be shown by an examination of some application examples

uniqueness of positive solutions for a boundary blow up problem abstract in this paper we prove the uniqueness of the positive solution for the boundary blow up problem u f u in u on where is a c 2 bounded domain in r n under the hypotheses that f t is nondecreasing in t 0 and f t t p is increasing for large t and some p 1 we also consider the uniqueness of a related problem when the equation includes a nonnegative weight a x

eigenvalue problems of linear hamiltonian systems arising from h filtering abstract from the viewpoint of differential eigenvalue problem of hamiltonian system a linear finite time h filtering problem can be addressed by computing eigenvalue and solving riccati differential equation of an associated linear hamiltonian system this paper shows how a problem of determining the minimum induced norm of the h filter is formulated as a hamiltonian differential eigenvalue problem the h filters concerned here include central filter perturbed filter and decentralised filter the methods presented in the paper are based on characteristics of eigensolution of the corresponding hamiltonian system and riccati differential equation with eigensolution of the hamiltonian system arising from the central h filtering problem variational methods are proposed to compute eigenvalues of perturbed hamiltonian systems and large scale hamiltonian systems derived from perturbed h filters and decentralised h filters respectively then eigenvalues can be obtained by calculating stationary values of corresponding extended rayleigh s quotient with dual argument functions which is essentially different from the well known rayleigh s quotient of lagrange systems with only one independent argument function numerical examples are also presented to illustrate the variational approaches presented in this paper