on approximate reasoning with graded rules this contribution presents a comprehensive view on problems of approximate reasoning with imprecise knowledge in the form of a collection of fuzzy if then rules formalized by approximating formulas of a special type two alternatives that follow from the dual character of approximating formulas are developed in parallel the link to the theory of fuzzy control systems is also explained

a note on the infinitary action of triangular norms and conorms in this note the action of triangular norms and conorms on subsets of the unit interval with arbitrary cardinality is defined and the properties of these generalized operations are also considered

triangular norms position paper ii general constructions and parameterized families this second part out of three of a series of position papers on triangular norms for part i see triangular norms position paper i basic analytical and algebraic properties fuzzy sets and systems in press deals with general construction methods based on additive and multiplicative generators and on ordinal sums also included are some constructions leading to non continuous t norms and a presentation of some distinguished families of t norms

generalized lagrangian dynamics of physical and non physical systems abstract in this paper we show how to study the evolution of a complex system given imprecise knowledge about the state of the system and the dynamics laws it will be shown that dynamics of these systems is equivalent to lagrangian or hamiltonian mechanics in a n 1 dimensional space where n is a system s dimensionality in some cases however the corresponding lagrangian is more general than the usual one and could depend on the action in this case lagrange s equations gain a non zero right side proportional to the derivative of the lagrangian with respect to the action examples of such systems are unstable systems systems with dissipation and systems which can remember their history moreover in certain situations the lagrangian could be a set valued function the corresponding equations of motion then become differential inclusions instead of differential equations we will also show that the principal of least action is a consequence of the causality principle and the local topology of the state space and not an independent axiom of classical mechanics we emphasize that our adaptation of lagrangian mechanics does not use or depend on specific properties of the physical system being modeled therefore this lagrangian approach may be equally applied to non physical systems an example of such an application is presented as well

on triangular norm based propositional fuzzy logics abstract fuzzy logics based on triangular norms and their corresponding conorms are investigated an affirmative answer to the question whether in such logics a specific level of satisfiability of a set of formulas can be characterized by the same level of satisfiability of its finite subsets is given tautologies contradictions and contigencies with respect to such fuzzy logics are studied in particular for the important cases of min max and ukasiewicz logics finally fundamental t norm based fuzzy logics are shown to provide a gradual transition between min max and ukasiewicz logics

geometrical methods in wajsberg hoops abstract the aim of this paper is to show that certain geometric methods originally developed for investigating ukasiewicz many valued logics may be profitably applied to the study of classes of partially ordered residuated monoids we substantiate this claim by characterizing all free wajsberg hoops in terms of mcnaughton functions over the n cube and by giving complete axiomatizations for all varieties of wajsberg hoops

polynomial first integrals of quadratic vector fields abstract we classify all quadratic polynomial differential systems having a polynomial first integral and provide explicit normal forms for such systems and for their first integrals

invariant algebraic curves and conditions for a centre conditions for the existence of a centre in two dimensional systems are considered along the lines of darboux we show how these methods can be used in the search for maximal numbers of bifurcating limit cycles we also extend the method to include more degenerate cases such as are encountered in less generic systems these lead to new classes of integrals in particular the kukles system is considered and new centre conditions for this system are obtained

t norm based logics with an independent involutive negation in this paper we investigate the addition of arbitrary independent involutive negations to t norm based logics we deal with several extensions of mtl and establish general completeness results indeed we will show that given any t norm based logic satisfying some basic properties its extension by means of an involutive negation preserves algebraic and finite strong standard completeness we will deal with both propositional and predicate logics

fuzzy logics with an additional involutive negation this paper surveys the present state of knowledge on propositional fuzzy logics extending sbl with an additional involutive negation the involutive negation is added as a new propositional connective in order to improve the expressive power of the standard mathematical fuzzy logics based on continuous triangular norms

compactness of fuzzy logics compactness is an important property of classical logic it states that simultaneous satisfiability of an infinite set of formulas is equivalent to the satisfiability of all its finite subsets in fuzzy logics we have different degrees of satisfiability hence the questions of compactness become more complicated here we give an overview of the recent results on compactness and we extend them to various fuzzy logics

erratum to phase portraits of quadratic polynomial vector fields having a rational first integral of degree 3 nonlinear anal 70 2009 3549 3560 erratum erratum to phase portraits of quadratic polynomial vector fields having a rational first integral of degree 3 nonlinear anal 70 2009 3549 3560 jaume llibre a regilene d s oliveira b a departament de matematiques universitat autonoma de barcelona 08193 bellaterra barcelona spain b departamento de matematica instituto de ciencias matematicas e computacao universidade de sao paulo c p 668 13 560 970 sao carlos sp brazil

down and up operators associated to fuzzy relations and t norms a definition of fuzzy semi ideals by means of the use of a t norm t on a complete lattice l and of a l fuzzy relation r on a set x there are stated definitions of lf semi ideal and of its dual lf semi filter of the set x it is shown that both the poset of the lf semi ideals and one of the lf semi filters determined by the respective restrictions of the usual order in the class of l fuzzy sets are complete lattices another characterization of these concepts as down and up l fuzzy sets is made via properties of the l fuzzy relation r at the end we give some constructive methods with which the set of lf semi ideals and the set of down l fuzzy sets can be determined examples and particular cases of the theory are presented

quadratic hamiltonian vector fields abstract let h be a cubic polynomial in two variables over r then h defines a quadratic hamiltonian vector field h y h x the purpose of this paper is to prove that there are exactly 28 non equivalent topologic phase portraits of quadratic hamiltonian vector fields

residuated fuzzy logics with additional connectives and their validation sets in this paper we generalize the classical notions of tautology and contradiction to the case of fuzzy logic this leads to the notion of a validation set the validation set of a formula in a fuzzy logic is the set of all truth values which this formula may achieve we summarize and extend the recent results on characterizations of validation sets in various fuzzy logics

on the redundancy of fuzzy partitions abstract we introduce a concept of redundancy for families of fuzzy subsets based on a left continuous t norm and its corresponding operator and study its basic properties including a characterization of families with redundancy zero in some important special cases examples illustrate that in the case of t norms with zero divisors several definitions for fuzzy partitions from the literature are insensitive to the phenomenon of redundancy finally a geometric interpretation of the redundancy based on the ukasiewicz t norm and an upper bound for its value are given

mathematical fuzzy logic as a tool for the treatment of vague information the paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language by a semantics and by a calculus for the derivation of formulas besides this logical approach also a more algebraic approach is discussed and the paper ends with some hints toward applications which are based upon these theoretical considerations

liouvillian first integrals of differential equations liouvillian functions are functions that are built up from rational functions using exponentiation integration and algebraic functions we show that if a system of differential equations has a generic solution that satisfies a liouvillian relation that is there is a liouvillian function of several variables vanishing on the curve defined by this solution then the system has a liouvillian first integral that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set we can refine this result in special cases to show that the first integral must be of a very special form

planar quadratic differential systems with invariant straight lines of total multiplicity four in this article we prove a classification theorem of real planar quadratic vector fields which possess four invariant straight lines including the line at infinity and including their multiplicities this classification which is taken modulo the action of the group of real affine transformations and time rescaling is given in terms of algebraic invariants and comitants and also geometrically using cycles on the complex projective plane and on the line at infinity the algebraic invariants and comitants allow us to verify for any given real quadratic system whether or not it has invariant lines of total multiplicity four and to specify its configuration of lines endowed with their corresponding real singularities of this system the calculations can be implemented on computer this classification is instrumental in studying the problem of darboux integrability of such systems work which is in progress

graded many valued resolution with aggregation abstract the aim of this paper is to give a sound resolution deduction rule in many valued logic with truth value range 0 1 and with arbitrary connectives and graded information we introduce and study the resolution truth function which will evaluate the truth value of the result of resolution conditions under which this function yields an aggregation operator will be given

on the integrability of two dimensional flows this paper deals with the notion of integrability of flows or vector fields on two dimensional manifolds we consider the following two key points about first integrals 1 they must be defined on the whole domain of definition of the flow or vector field or defined on the complement of some special orbits of the system 2 how are they computed we prove that every local flow on a two dimensional manifold m always has a continuous first integral on each component of m where is the set of all separatrices of we consider the inverse integrating factor and we show that it is better to work with it instead of working directly with a first integral or an integrating factor for studying the integrability of a given two dimensional flow or vector field finally we prove the existence and uniqueness of an analytic inverse integrating factor in a neighborhood of a strong focus of a non resonant hyperbolic node and of a siegel hyperbolic saddle

geometry of quadratic differential systems in the neighborhood of infinity in this article we give a complete global classification of the class qsess of planar essentially quadratic differential systems i e defined by relatively prime polynomials and whose points at infinity are not all singular according to their topological behavior in the vicinity of infinity this class depends on 12 parameters but due to the action of the affine group and re scaling of time the family actually depends on five parameters our classification theorem theorem 7 1 gives us a complete dictionary connecting very simple integer valued invariants which encode the geometry of the systems in the vicinity of infinity with algebraic invariants and comitants which are a powerful tool for computer algebra computations helpful in the route to obtain the full topological classification of the class qs of all quadratic differential systems

fuzzy dynamics as an alternative to statistical mechanics abstract we propose a description of continuous evolution of systems with uncertainty based on min or product t norms of fuzzy logic we show that the dynamics has a hamiltonian form in the extended state space composed from physical and information components if the min t norm is used this description is useful when available information on a system behavior is mainly ordinal and not numerical it was shown also that if the product t norm is used the dynamics on a continuous universe is similar to the stochastic processes dynamics but a probability distribution is replaced by a possibility density we show that such replacement leads to a principal extension of the stochastic process theory and correspondence differential equation of evolution for the possibility density cover most basic evolution equations in physics

quadratic systems with a polynomial first integral a complete classification in the coefficient space r12 abstract in this paper we are going to apply the invariant theory to give invariant conditions on the coefficients of any non degenerate quadratic system in order to determine if it has or not a polynomial first integral without using any normal form we obtain that the existence of polynomial first integral is directly related with the fact that all the roots of a convenient cubic polynomial are rational and negative the coefficients of this cubic polynomial are invariants related with some geometric properties of the system

quadratic differential systems with complex conjugate invariant lines meeting at a finite point abstract in this article we study the class qs2cil of quadratic differential systems with two complex conjugate invariant lines meeting at a finite point from the literature we know that quadratic systems with invariant lines of total multiplicity at least four or with the line at infinity filled up with singularities are integrable via the method of darboux and hence they have no limit cycles these could only occur if we have only the two complex lines and the line at infinity all simple we first find all integrable systems in qs2cil due to the presence of invariant lines we next indicate a gap in the 1986 proof of suo and chen that systems in qs2cil have at most one limit cycle and we give a complete proof of this result finally we give the topological classification of qs2cil yielding 22 phase portraits three of which with a limit cycle

a triangular norm hierarchy abstract triangular norms and their hierarchy are investigated a localisation of classical t s norms and various other norms is obtained the class of t norms proposed by zimmermann fuzzy set theory and its applications kluwer dordrecht 1991 is placed in the above hierarchy and a proof that it is weaker see butnariu fuzzy sets and systems 69 1995 241 255 then t 1 concludes the paper

four dimensional conformal field theory models with rational correlation functions recently established rationality of correlation functions in a globally conformal invariant quantum field theory satisfying wightman axioms is used to construct a family of soluble models in four dimensional minkowski spacetime we consider in detail a model of a neutral scalar field of dimension two it depends on a positive real parameter c an analogue of the virasoro central charge and admits for all finite c an infinite number of conserved symmetric tensor currents the operator product algebra of is shown to coincide with a simpler one generated by a bilocal scalar field v x1 x2 of dimension 1 1 the modes of v together with the unit operator span an infinite dimensional lie algebra v whose vacuum i e zero energy lowest weight representations only depend on the central charge c wightman positivity i e unitarity of the representations of v is proven to be equivalent to c

triangular norm based measures and their markov kernel representation abstract we approach the problem whether left continuous triangular norm based valuations called t measures or t probability measures defined on triangular normbased tribes of the unit cube can be disintegrated by markov kernels we prove that each t measure based on a fundamental triangular norm these triangular norms t together with their corresponding triangular conorms s satisfy the functional equation t x y s x y x y can be uniquely represented as a sum of a disintegrable t measure and a hard core which is either identically zero or which is monotonically irreducible i e cannot be disintegrated

on the order of triangular norms comments on a triangular norm hierarchy by e cretu in cretu fuzzy sets and systems 120 2001 371 the members of the families of frank dubois prade yager and hamacher t norms respectively are compared in a pointwise way with the minimum the product and the lukasiewicz t norm all these results are well known and trivial moreover these families of t norms cannot only be compared with the three basic t norms above but all these families are monotone with respect to their index a fact which is also well known and straightforward to prove with the exception of the family of frank t norms whose monotonicity has been first proven in butnariu and klement triangular norm based measures and games with fuzzy coalitions kluwer dordrecht 1993

evolution of systems under fuzzy dynamic laws abstract the global behavior of a system with information on fuzzy described short term evolution laws is proposed and investigated in the paper for deterministic or stochastic models such problems are solved by use of differential equations describing the short term laws but if the short term laws of evolution are known only in a fuzzy form and the information on the initial state of the system is also fuzzy or if we do not have a reasonable statistical model describing the systems environment the fuzzy approach is more appropriate we will show that the introduction of the fuzzy state fuzzy velocity and use of the fuzzy logic rules for obtaining the equation of evolution of the system leads to a consistent description of the fuzzy dynamics our approach is based on the fuzzy master equation an analog of the equation describing the markov process in stochastic dynamics for continuous evolution we obtain differential equations describing the flows of the fuzzy state two cases of weak and strong fuzziness are considered a comparison with other approaches to fuzzy dynamics is performed

harmonic bilocal fields generated by globally conformal invariant scalar fields the twist two contribution in the operator product expansion of phi 1 rm x 1 phi 2 rm x 2 for a pair of globally conformal invariant scalar fields of equal scaling dimension d in four space time dimensions is a field v1 x1 x2 which is harmonic in both variables it is demonstrated that the huygens bilocality of v1 can be equivalently characterized by a single pole property concerning the pole structure of the rational correlation functions involving the product phi 1 rm x 1 phi 2 rm x 2 this property is established for the dimension d 2 of phi 1 phi 2 as an application we prove that any system of gci scalar fields of conformal dimension 2 in four space time dimensions can be presented as a possibly infinite superposition of products of free massless fields