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

multiple solutions of nonlinear schrodinger equations via flow invariance and morse theory we prove the existence of multiple bound states of the nonlinear schrodinger equation u v x u f u here the linear potential v is continuous and bounded from below and the nonlinearity f is of asymptotically linear type we show that under certain assumptions on the spectrum of the schrodinger operator v and the asymptotic behaviour of f u u the above equation has at least four non trivial solutions two of them sign changing

decompositions and range for additive fuzzy measures the aim of the present paper is to prove the existence of hahn and lebesgue decompositions for additive fuzzy measures in the sense of 3 it is shown that the radon nikodym theorem can be carried over from classical measures to additive fuzzy measures with their corresponding integrals these results are applied to obtain a generalization of liapounoff s convexity theorem 12 whose utility in the theory of fuzzy games is argued in the last section

spectral resolutions in dedekind σ complete l groups abstract using recent results on a generalized form of the loomis sikorski theorem a dvurecenskij loomis sikorski theorem for complete mv algebras and l groups j austral math soc ser a 68 2000 261 277 d mundici tensor product and the loomis sikorski theorem for mv algebras adv appl math 22 1999 227 248 it is shown that a unital dedekind complete l group is a compatible rickart comgroup in the sense of d j foulis d j foulis spectral resolutions in a rickart comgroup rep math phys 54 2004 229 250 in particular elements in unital dedekind complete l groups and consequently elements in mv algebras admit uniquely defined spectral resolutions similar to spectral resolutions of self adjoint operators a functional calculus and spectra of elements are considered in relation with the loomis sikorski representation by functions

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

construction of fuzzy σ algebras using triangular norms abstract generalizing the definitions given by the author fuzzy sets and systems4 1980 83 93 we introduce and study t fuzzy algebras t being any triangular norm the main result is that for a large class of triangular norms each t fuzzy algebra is generated i e consists of all functions x 0 1 being measurable with respect to some algebra on x

fuzzy probability measures abstract in 4 hohle has defined fuzzy measures on g fuzzy sets 2 where g stands for a regular boolean algebra consequently since the unit interval is not complemented fuzzy sets in the sense of zadeh 8 do not fit in this framework in a straightforward manner it is the purpose of this paper to continue the work started in 5 which deals with 0 1 fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space 5 we give necessary and sufficient conditions for such a measure to be a classical integral as in 9 in the case the space is generated a counterexample in the general case is also presented finally it is shown that a fuzzy probability measure is always an integral if the space is generated if we replace the operations and by the t norm to and its dual s0 see 6

characterization of fuzzy measures constructed by means of triangular norms abstract following the ideas presented by the author e p klement j math anal appl 85 1982 543 565 finite t fuzzy measures are introduced t being a measurable triangular norm we show that a t fuzzy measure is always a fuzzy measure as considered earlier e p klement j math anal appl 25 1980 330 339 then we study the relation to the integral with respect to some classical measure finally for some special triangular norms t we give precise characterizations of the corresponding classes of t fuzzy measures

superlinear systems of second order ode s abstract we discuss the existence of positive solutions of the system u f t u v u v in 0 1 v g t u v u v in 0 1 u 0 u 1 v 0 v 1 0 where the nonlinearities f and g satisfy a superlinearity condition at both 0 and our main result is the proof of a priori bounds for the eventual solutions as an application we consider the dirichlet problem in an annulus for systems of semilinear elliptic equations with nonlinearities depending on the gradient as well as a second application we consider fourth order elastic beam equations with dependence also on the derivatives u u u

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

correspondence between fuzzy measures and classical measures abstract in this paper we study the question whether given a fuzzy measure as defined in 3 and 4 there exists a classical measure such that the fuzzy measure of a measurable fuzzy set equals the classical measure of the area below the membership function of the results are that in the case of finite additivity there is a one to one correspondence between classical measures and fuzzy measures whereas in the case of countable additivity this result only holds for generated fuzzy algebras finally some connections of that problem with the existence of an extension of a fuzzy measure defined on an arbitrary fuzzy algebra to the generated fuzzy algebra are discussed

high dimensional entanglement in states with positive partial transposition genuine high dimensional entanglement i e the property of having a high schmidt number constitutes an instrumental resource in quantum communication overcoming limitations of low dimensional systems states with a positive partial transpose ppt are generally considered weakly entangled as they can never be distilled into pure entangled states this naturally raises the question of whether high schmidt numbers are possible for ppt states volume estimates suggest that optimal i e linear scaling in the local dimension should be possible albeit without providing insight into the possible slope we provide the first explicit construction of a family of ppt states that achieves linear scaling in the local dimension and we prove that random ppt states typically share this feature our construction also allows us to prove a recent conjecture of chen et al on the existence of ppt states whose schmidt number increases by an arbitrarily large amount upon partial transposition finally we link the schmidt number to entangled sub block matrices of a quantum state we use this connection to prove that quantum states that are either i invariant under partial transposition on the smallest of their two subsystems or ii absolutely ppt cannot have a maximal schmidt number overall our findings shed new light on some fundamental problems in entanglement theory

optimal entanglement criterion for mixed quantum states we develop a strong and computationally simple entanglement criterion the criterion is based on an elementary positive map phi which operates on state spaces with even dimension n or 4 it is shown that phi detects many entangled states with a positive partial transposition ppt and that it leads to a class of optimal entanglement witnesses this implies that there are no other witnesses which can detect more entangled ppt states the map phi yields a systematic method for the explicit construction of high dimensional manifolds of bound entangled states

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

existence of changing sign solutions for some semilinear problems with jumping nonlinearities at zero we study the existence of changing sign solutions of an elliptic semilinear boundary value problem which arises as a limiting equation of the two species lotka volterra competing equations system using variational methods and a result of d aujourd hui we find conditions which are both sufficient and necessary for this existence problem

the entropy of fuzzy dynamical systems general scheme and generators in this paper some fuzzy modifications of the kolmogorov sinaj entropy are studied we introduce the general scheme which includes the mentioned entropies finally the kolmogorov sinaj theorem on generators is proved for special cases

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

a bahri lions theorem revisited abstract in 1988 a bahri and p l lions a bahri p l lions morse index of some min max critical points i application to multiplicity results comm pure appl math 41 1988 1027 1037 studied the following elliptic problem u u p 2 u f x u u h 0 1 where is a bounded smooth domain of r n 2 p 2 n 2 n 2 and f x u is not assumed to be odd in u they proved the existence of infinitely many solutions under an appropriate growth restriction on f in the present paper we improve this result by showing that under the same growth assumption on f the problem admits in fact infinitely many sign changing solutions in addition we derive an estimate on the number of their nodal domains we also deal with the corresponding fourth order equation 2 u u p 2 u f x u with both dirichlet and navier boundary conditions as well as with strongly coupled elliptic systems

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

t evaluators and s evaluators the scalar evaluation of the aggregation of elements from a complete lattice via the operation meet join is compared with the aggregation of scalar evaluations of these elements via a t norm t conorm based on this comparison the notion of a t evaluator s evaluator is introduced and its basic properties are studied the relationship between t s evaluators fuzzy measures and generalized filters is investigated an application of t evaluators in fuzzy preference modelling is presented

on superquadratic elliptic systems in this article we study the existence of solutions for the elliptic system updelta u frac partial h partial v left u v x right text in upomega updelta v frac partial h partial v left u v x right text in upomega u 0 v 0 text on partial upomega where upomega is a bounded open subset of mathbb r n with smooth boundary partial upomega and the function h mathbb r 2 times overline upomega to mathbb r is of class c 1

principles of inclusion and exclusion for interval valued fuzzy sets and if sets abstract the principle of inclusion and exclusion is one of basic tools of classical set theory many previous papers have presented its generalizations to fuzzy sets or if sets atanassov s intuitionistic fuzzy sets with godel or product operations in our previous work we clarified these questions for fuzzy sets with intersections and unions based on continuous triangular norms and conorms here we extend these results to interval valued fuzzy sets and if sets

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

principles of inclusion and exclusion for fuzzy sets abstract many previous papers have presented generalizations of the principle of inclusion and exclusion to fuzzy sets or if sets atanassov s intuitionistic fuzzy sets with godel or product operations we clarify for which fuzzy intersections and unions based on continuous triangular norms and conorms the principle of inclusion and exclusion holds

additive fuzzy measures and integrals iii abstract a point of view concerning fuzzy measures is explained to this end a new concept of disjointness for fuzzy is introduced and studied also a concept of an additive class of fuzzy sets is defined to be a class of fuzzy sets closed under some additive operations the fuzzy measures are defined to be sum preserving real functions over such additive classes some basic properties of the fuzzy measures are derived in contrast with other homonymous concepts studied in literature our fuzzy measures lead to an additive fuzzy integral see the part ii of the paper

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

sharp and unsharp observables on σ mv algebras a comparison with the hilbert space approach we prove that any observable with a standard value space on a probability mv algebra m can be extended from a mapping defined on measurable subsets with values in m to a mapping defined on bounded real valued measurable functions with values in the set of sharp observables associated with m our method of proof is quite different from the method of the proof of analogous result for pov observables on a hilbert space we show that for pov observables with commuting range both approaches give the same results

on the existence of many solutions for a class of superlinear elliptic systems abstract we seek multiple solutions to systems of semilinear elliptic equations of the type u x h x u x x u x 0 x where r n n 1 is a bounded regular domain and u u1 u2 r 2 for a class of potentials where the variables are strongly coupled as in the model h x u1 u2 u1 1 u2 2 with i 1 we provide the existence of nine nontrivial solutions characterized by sign properties of each component

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

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

unextendible product bases uncompletable product bases and bound entanglement we report new results and generalizations of our work on unextendible product bases upb uncompletable product bases and bound entanglement we present a new construction for bound entangled states based on product bases which are only completable in a locally extended hilbert space we introduce a very useful representation of a product basis an orthogonality graph using this representation we give a complete characterization of unextendible product bases for two qutrits we present several generalizations of upbs to arbitrary high dimensions and multipartite systems we present a sufficient condition for sets of orthogonal product states to be distinguishable by separable superoperators we prove that bound entangled states cannot help increase the distillable entanglement of a state beyond its regularized entanglement of formation assisted by bound entanglement

characterization of measures based on strict triangular norms as a natural generalization of a measure space butnariu and klement 4 introduced t tribes of fuzzy sets with t measures they gave a complete characterization of t measures for a frank triangular norm t here we characterize t measures with respect to non frank strict triangular norms we show the specific roles of frank triangular norms and a newly introduced family nearly frank triangular norms

non atomic fuzzy measures and games in this paper liapounoff s theorem for fuzzy measures is proved and it is applied to extend a theorem of aumann and shapley 2 concerning the existence of the value for non atomic games precisely it is proved that the range of any non atomic n vector fuzzy measure in the sense of 6 is convex and compact and it is shown that this result allows to extend theorem b from 2 p 23 from games with unambiguous coalitions to games with fuzzy coalitions called fuzzy games defined on s algebras of fuzzy sets which are not necessarily s algebras of ideal sets

a characterization of the ordering of continuous t norms abstract given two t norms t 1 and t 2 it is quite often difficult or even impossible to check directly whether t 1 t 2 in schweizer and sklar 1983 a necessary and sufficient condition is given for continuous archimedean t norms to be comparable we simplify this result proving a sufficient condition which involves only one argument of a one place function rather than two arguments this result is applied to show the monotonicity of some well known classes of t norms next we generalize the result of schweizer and sklar to the case of all continuous t norms and present also a sufficient condition which is again easier to check

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

markov kernels and tribes abstract we define an ordering on the set of bounded markov kernels associated with a tribe of fuzzy sets we show that under this order the set of bounded markov kernels is a dedekind complete lattice in addition we define a sum of bounded markov kernels such that the set of bounded markov kernels is a lattice ordered semigroup if we concentrate only to sharp bounded markov kernels then this set is even a dedekind complete l group with strong unit we show that our methods work also for bounded markov kernels associated with t s tribes of fuzzy sets where t s is any frank t norm and s 0

on hudetz entropy localization abstract in rahimi and riazi 2014 22 a type of entropy of continuous maps on compact metric spaces is defined via integrating a local entropy map this average entropy is represented in terms of the entropy in the sense of dumitrescu but the equality of these two types of entropy is left as an open problem in 22 in this paper we modify the problem stated in 22 by replacing the dumitrescu entropy by the hudetz entropy since in our case the two previous types of entropy take an infinite value and the problem stated in 22 reduces to a trivial equality then we conclude the paper by proving the equality of the modified average entropy and the hudetz entropy the key tool is a version of jaccob s theorem for the hudetz entropy

evaluating convex roof entanglement measures we show a powerful method to compute entanglement measures based on convex roof constructions in particular our method is applicable to measures that for pure states can be written as low order polynomials of operator expectation values we show how to compute the linear entropy of entanglement the linear entanglement of assistance and a bound on the dimension of the entanglement for bipartite systems we discuss how to obtain the convex roof of the three tangle for three qubit states we also show how to calculate the linear entropy of entanglement and the quantum fisher information based on partial information or device independent information we demonstrate the usefulness of our method by concrete examples

quantum states with a positive partial transpose are useful for metrology we show that multipartite quantum states that have a positive partial transpose with respect to all bipartitions of the particles can outperform separable states in linear interferometers we introduce a powerful iterative method to find such states we present some examples for multipartite states and examine the scaling of the precision with the particle number some bipartite examples are also shown that possess an entanglement very robust to noise we also discuss the relation of metrological usefulness to bell inequality violation we find that quantum states that do not violate any bell inequality can outperform separable states metrologically we present such states with a positive partial transpose as well as with a nonpositive partial transpose

a note on a generalized frank functional equation abstract we study a generalized frank functional equation in the broader framework of associative aggregation functions and show that up to the two projections we obtain exactly the same set of solutions as in the original paper by m j frank aequationes mathematicae 19 1979

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

fuzzy measurability and integrability developpement d une nouvelle theorie relative au probleme de la mesure des objets flous continuite des fonctions sigma additives sur une classe sigma additive d ensembles flous extension de la notion d integrale floue methode de calcul des integrales floues problemes ouverts

bell inequalities for graph states we investigate the nonlocal properties of graph states to this aim we derive a family of bell inequalities which require three measurement settings for each party and are maximally violated by graph states in turn for each graph state there is an inequality maximally violated only by that state we show that for certain types of graph states the violation of these inequalities increases exponentially with the number of qubits we also discuss connections to other entanglement properties such as the positivity of the partial transpose or the geometric measure of entanglement

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