lagrangian hamiltonian unified formalism for field theory the rusk skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems it incorporates all the characteristics of lagrangian and hamiltonian descriptions of these systems including dynamical equations and solutions constraints legendre map evolution operators equivalence etc in this work we extend this unified framework to first order classical field theories and show how this description comprises the main features of the lagrangian and hamiltonian formalisms both for the regular and singular cases this formulation is a first step toward further applications in optimal control theory for partial differential equations

multivector fields and connections setting lagrangian equations in field theories the integrability of multivector fields in a differentiable manifold is studied then given a jet bundle j1e e m it is shown that integrable multivector fields in e are equivalent to integrable connections in the bundle e m that is integrable jet fields in j1e this result is applied to the particular case of multivector fields in the manifold j1e and connections in the bundle j1e m that is jet fields in the repeated jet bundle j1j1e in order to characterize integrable multivector fields and connections whose integral manifolds are canonical lifting of sections these results allow us to set the lagrangian evolution equations for first order classical field theories in three equivalent geometrical ways in a form similar to that in which the lagrangian dynamical equations of nonautonomous mechanical systems are usually given then using multivector fields we discuss several aspects of these evolution equations both for the regular and singular cases namely the existence and nonuniqueness o

rewriting systems over similarity and generalized pseudometric spaces and their properties we present a study of confluence and related properties of fuzzy relations defined over similarity spaces the ordinary confluence is an essential property of relations connected to the idea of rewriting and substituting which appears in abstract rewriting systems this paper is a continuation of our previous paper where we have introduced analogous notions related to substitutability in graded setting using residuated lattices as structures of truth degrees leaving the ordinary notions a particular case when the underlying structure is the two valued boolean algebra in this paper we further extend our previous results by developing the notions of confluence and related properties respecting a given similarity relation we also present definitions of confluence and related properties of relations on a generalized pseudometric spaces using the well known link between generalized pseudometrics and similarities we describe the connection of the notions defined on generalized pseudometric spaces to the corresponding notions on similarity spaces the introduced notions are further illustrated by example of rewriting based on graded if then rules

the universal r matrix for u_q rm sl 3 and beyond ther matrices for the quantised lie algebrasa n are constructed through the quantum double procedure given by drinfel d 6 the case ofu q sl 3 is thoroughly analysed initially to demonstrate the more subtle points of the calculation the ease of the calculation fora n is very dependent on a choice of generators for the borel subalgebrau q b and its dual and a certain ordering imposed on these generators which is related to the length of a certain word in the weyl group

fuzzy equivalence relations and their equivalence classes in this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice we give certain characterizations of fuzzy semi partitions and fuzzy partitions over a complete residuated lattice as well as over a linearly ordered complete heyting algebra in the latter case for a fuzzy equivalence relation over a linearly ordered complete heyting algebra we construct an algorithm for calculation of a minimal family of its equivalence classes which generates it most of the presented results are new but some of them are generalizations of known results given in a way which simplifies and clarifies them

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

hamilton jacobi theory in multisymplectic classical field theories the geometric framework for the hamilton jacobi theory developed in the studies of carinena et al int j geom methods mod phys 3 7 1417 1458 2006 carinena et al int j geom methods mod phys 13 2 1650017 2015 and de leon et al variations geometry and physics nova science publishers new york 2009 is extended for multisymplectic first order classical field theories the hamilton jacobi problem is stated for the lagrangian and the hamiltonian formalisms of these theories as a particular case of a more general problem and the classical hamilton jacobi equation for field theories is recovered from this geometrical setting particular and complete solutions to these problems are defined and characterized in several equivalent ways in both formalisms and the equivalence between them is proved the use of distributions in jet bundles that represent the solutions to the field equations is the fundamental tool in this formulation some examples are analyzed and in particular the h

fuzzy implication operators for difference operations for fuzzy sets and cardinality based measures of comparison in this paper we determine by means of fuzzy implication operators two classes of difference operations for fuzzy sets and two classes of symmetric difference operations for fuzzy sets which preserve properties of the classical difference operation for crisp sets and the classical symmetric difference operation for crisp sets respectively the obtained operations allow us to construct as in b de baets h de meyer transitivity preserving fuzzification schemes for cardinality based similarity measures european journal of operational research 160 2005 726 740 cardinality based similarity measures which are reflexive symmetric and transitive fuzzy relations and to propose two classes of distances metrics which are fuzzy versions of the well known distance of cardinality of the symmetric difference of crisp sets

building a class of fuzzy equivalence relations in this paper we propose a practical method given a strict triangular norm with a convex additive generator for deriving a fuzzy equivalence relation whose reflexivity condition generalizes ruspini s definition of fuzzy partitions the properties of the relations their comparison their transitivity the construction of fuzzy equivalence relations on cartesian products are presented a large part of the paper is devoted to applications with fuzzy partitions defined on the real line several examples including the fairy tale problem from de cock and kerre on un suitable fuzzy relations to model approximate equality fuzzy sets and systems 133 2 2003 137 153 the comparison of colored objects and comfort situations are proposed

a note on approximate equality versus the poincare paradox the purpose of this note is to comment on a paper by m de cock and e e kerre that was recently accepted for publication in fuzzy sets and systems the paper in contrast to the view of several other researchers states that fuzzy equivalence relations are not always able to resolve the poincare paradox this brief paper discusses this controversial contribution from the present author s subjective point of view

multivector field formulation of hamiltonian field theories equations and symmetries we state the intrinsic form of the hamiltonian equations of first order classical field theories in three equivalent geometrical ways using multivector fields jet fields and connections thus these equations are given in a form similar to that in which the hamiltonian equations of mechanics are usually given then using multivector fields we study several aspects of these equations such as the existence and non uniqueness of solutions and the integrability problem in particular these problems are analysed for the case of hamiltonian systems defined in a submanifold of the multimomentum bundle furthermore the existence of first integrals of these hamiltonian equations is considered and the relation between cartan noether symmetries and general symmetries of the system is discussed noether s theorem is also stated in this context both the classical version and its generalization to include higher order cartan noether symmetries finally the equivalence between the lagrangian and hamiltonian formalisms is also discussed

transitivity bounds in additive fuzzy preference structures transitivity plays a crucial role in preference modeling and related fields in this paper we discuss this property in the general context of additive fuzzy preference structures of particular interest is the decomposition of a large preference relation r in its symmetric part i indifference relation and its asymmetric part p strict preference relation by means of a so called indifference generator i given the type of transitivity of a large preference relation r w r t a conjunctor and a generator we establish basic lower bounds and general upper bounds on the transitivity of p and i these bounds are due to the careful design of generic counterexamples moreover we identify the situations in which these bounds are effectively reached thereby establishing connections with interesting properties such as dominance bisymmetry the 1 lipschitz property and rotation invariance

i fuzzy equivalence relations and i fuzzy partitions a t fuzzy equivalence relation is a fuzzy binary relation on a set x which is reflexive symmetric and t transitive for a t norm t recently mesiar et al r mesiar b reusch h thiele fuzzy equivalence relations and fuzzy partitions j multi valued logic soft comput 12 2006 167 181 have generalised the t norm t to any general conjunctor c and investigated the minimal assumptions required on such operations called duality fitting conjunctors such that the fuzzification of the equivalence relation admits any value from the unit interval and also the one one correspondence between the fuzzy equivalence relations and fuzzy partitions is preserved in this work we conduct a similar study by employing a related form of c transitivity viz i transitivity where i is an implicator we show that although every i fuzzy equivalence relation can be shown to be a c fuzzy equivalence relation there exist c fuzzy equivalence relations that are not i fuzzy equivalence relations and hence these concepts are not equivalent most importantly we show that the class of duality fitting implicators i is much richer than the residuals of the duality fitting conjunctors in the study of mesiar et al we also show that the i fuzzy partitions have a constant wise structure

on the construction of fuzzy betweenness relations from metrics abstract we consider the problem of constructing a fuzzy betweenness relation from a metric more precisely given a continuous archimedean triangular norm we present two construction methods for a fuzzy betweenness relation from a metric by making use of the pseudo inverse of either a continuous additive generator or a continuous multiplicative generator of the triangular norm in case the metric is bounded and given a 1 lipschitz continuous triangular norm we present a third construction method for a fuzzy betweenness relation from a metric by making use of the residual implication of the triangular norm since the ukasiewicz and product triangular norms are both continuous archimedean and 1 lipschitz continuous all three construction methods may be used interestingly the construction method based on the residual implication is proved to coincide with that based on a continuous additive generator for the ukasiewicz triangular norm and with that based on a continuous multiplicative generator for the product triangular norm we end by noting that all three construction methods result in a fuzzy prebetweenness relation when considering a pseudometric instead of a metric

approximate solutions of fuzzy relational equations and a characterization of t norms that define metrics for fuzzy sets abstract if one considers the membership degrees of fuzzy sets as truth values of a many valued logic there is a canonical way to define a fuzzified equality for fuzzy sets this generalized equality relation can be used to define a degree to which some value of the unknown variable is a solution of the fuzzy equation as well as a degree of solvability of the whole equation furthermore there is an intimate relationship between such a degree to which some fuzzy set is a solution and the quality to which it is an approximate solution the background is that the negation of the fuzzified equality measures a kind of distinctness of fuzzy sets formally the definition of this fuzzified equality contains as parameter some t norm which acts as truth functions of a conjunction connective for some such t norms this negation of the fuzzified equality really has the properties of a metric in the mathematical sense of the word the paper gives a necessary and sufficient condition which characterizes all those t norms which in that way yield a metric for fuzzy sets

products of elements in vague semigroups and their implementations in vague arithmetic vague arithmetic different from the present literature of fuzzy arithmetic has been proposed in demirci internat j uncertainty fuzziness and knowledge based systems 10 1 2002 25 internat j general systems 32 2 2003 157 177 to model vaguely defined arithmetic operations resulting from the indistinguishability of real numbers the main motivating problem of this paper is to introduce the notion of vague product sum of a finite number of real numbers in vague arithmetic and to point out their fundamental properties from a more abstract mathematical point of view the vague product sum of a finite number of real numbers in vague arithmetic and their properties can be considered as the vague product of a finite number of elements in vague semigroups and their relevant properties for this reason a large part of this paper is devoted to the vague product of a finite number of elements in vague semigroups and their elementary properties as a direct implementation of the present results it is shown that the vague product sum of a finite number of real numbers in vague arithmetic can be easily evaluated in terms of the underlying many valued equivalence relations furthermore various non trivial examples for the vague product sum of a finite number of real numbers in vague arithmetic are designed and a simple technique for the construction of such non trivial examples is stated

symmetric difference operators based on thresholds abstract here we study two formulas that define the symmetric difference operators of fuzzy sets in fuzzy logic we call this an exclusive or operator an interesting open problem was to find a certain class of operator systems where the two well known definitions of the symmetric difference operator are equivalent and it is associative here we state a functional equation which characterizes the equivalence of these two forms we present a general class of operators which fulfills the associativity requirement and we show that the two definitions are equivalent we generalize the symmetric difference operator to the n variable case and based on the latter the weighted form can be obtained we will give concrete forms of these operators and we will also show that they have different types of transitivity properties

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

integrable three particle systems hidden symmetries and deformations of the calogero moser system the properties of some three particle systems are studied using as an approach the theory of generalized or hidden symmetries it is proven that the existence of a family that possesses in addition to the energy function two nonlinear constants of motion in involution this family of integrable systems is obtained by considering deformations of the lagrangian of the calogero moser system finally the generalization of the results to the general case of n particles is discussed

metrics and t equalities abstract the relationship between metrics and t equalities is investigated the latter are a special case of t equivalences a natural generalization of the classical concept of an equivalence relation it is shown that in the construction of metrics from t equalities triangular norms with an additive generator play a key role conversely in the construction of t equalities from metrics this role is played by triangular norms with a continuous additive generator or equivalently by continuous archimedean triangular norms these results are then applied to the biresidual operator e t of a triangular norm t it is shown that e t is a t equality on 0 1 if and only if t is left continuous furthermore it is shown that to any left continuous triangular norm t there correspond two particular t equalities on f x the class of fuzzy sets in a given universe x one of these t equalities is obtained from the biresidual operator e t t by means of a natural extension procedure these t equalities then give rise to interesting metrics on f x

t equalities as fuzzy non archimedean metrics and δ fuzzy equivalences of type s abstract the starting point of our approach is the remark that a fuzzy t equality can be regarded as a fuzzy non archimedean metric accordingly we propose a generalization of the notions of t equivalence and t equality by replacing the numbers e x y with generalized distribution functions thus we introduce the concepts of fuzzy t equivalence of type s and fuzzy t equality of type s where t is a menger norm and s is an operation on the nonnegative extended reals by using specific methods from probabilistic metric spaces theory concrete formulas for generated metrics slightly extending some similar results of de baets and mesiar are also given

a realization of quantum algebras some applications some applications of a new realization of quantum algebras are given finite dimensional modules are constructed in a new basis with explicit matrix elements for q a root of 1 the structure of weyl modules for is analysed in detail as a corollary a partial result on tensor product structure is obtained generalizations to other quantum algebras are briefly discussed realizations of in terms of difference operators and some unusual representations are also presented

geometry of multisymplectic hamiltonian first order field theories in the jet bundle description of field theories multisymplectic models in particular there are several choices for the multimomentum bundle where the covariant hamiltonian formalism takes place as a consequence several proposals for this formalism can be stated and on each one of them the differentiable structures needed for setting the formalism are obtained in different ways in this work we make an accurate study of some of these hamiltonian formalisms showing their equivalence in particular the geometrical structures canonical or not needed for the hamiltonian formalism are introduced and compared and the derivation of hamiltonian field equations from the corresponding variational principle is shown in detail furthermore the hamiltonian formalism of systems described by lagrangians is performed both for the hyper regular and almost regular cases finally the role of connections in the construction of hamiltonian field theories is clarified

the generalized associative law in vague groups and its applications ii as a continuation of part i vague integral powers of elements in vague groups and their representation properties are introduced in this paper thereafter some rudimentary algebraic properties of vague integral powers of elements obtained from the generalized vague associative law formulated in part i are established the present paper particularly provides the abstract foundations of integral powers and multiples of real numbers in vague arithmetic for this reason special attention is also paid to the calculation of integral powers and multiples of real numbers in vague arithmetic and some practical applications related to the discrete structure of measurement instruments are also given

constrained variational calculus for higher order classical field theories we develop an intrinsic geometrical setting for higher order constrained field theories as a main tool we use an appropriate generalization of the classical skinner rusk formalism some examples of applications are studied in particular to the geometrical description of optimal control theory for partial differential equations

k cosymplectic manifolds and lagrangian field theories a geometrical description of classical field theories of first order is given the underlying k cosymplectic structure permits to derive the corresponding field equations

the dominance relation in some families of continuous archimedean t norms and copulas the dominance relation in several families of continuous archimedean t norms and copulas is investigated on the one hand the contribution provides a comprehensive overview on recent conditions and properties of dominance as well as known results for particular cases of families on the other hand it contains new results clarifying the dominance relationship in five additional families of continuous archimedean t norms and copulas

reduction of polysymplectic manifolds the aim of this paper is to generalize the classical marsden weinstein reduction procedure for symplectic manifolds to polysymplectic manifolds in order to obtain quotient manifolds which inherit the polysymplectic structure this generalization allows us to reduce polysymplectic hamiltonian systems with symmetries such as those appearing in certain kinds of classical field theories as an application of this technique an analogue to the kirillov kostant souriau theorem for polysymplectic manifolds is obtained and some other mathematical examples are also analyzed our procedure corrects some mistakes and inaccuracies in previous papers gunther 1987 j differ geom 25 23 53 munteanu et al 2004 j math phys 45 1730 51 on this subject

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

similarity based fuzzy limits in this paper the concept of similarity based fuzzy limits in metric spaces is introduced and it is shown that it is an application of limit maps of topologies on 0 1 valued sets introduced by hohle and sostak furthermore it is observed that the similarity based setting introduced is a more general approach to fuzzy limits than the one proposed by burgin

weight conservation and quantum group construction of the braid group representation it is proved that the braid group representation constructed in terms of the quantum group can be covered by that obtained from the proposal of weight conservation by the extended kauffman diagram technique the non standard braid group representations associated with su 2 are obtained by adding the terms of weight conservation to the standard universal r matrix

some results on the degree of symmetry of fuzzy relations abstract in this paper we investigate the degree of the symmetry of fuzzy relations on a set x based on the fuzzy e equality derived from uninorms with unit element e the degree of the symmetry of fuzzy relations are defined by two different approaches the type i degree of symmetry and the type ii degree of symmetry the main work of this paper include first we discuss some basic properties of those two degrees especially for the relationship between them in particular for continuous t norms we give a necessary and sufficient condition such that the type i degree of symmetry and the type ii degree of symmetry are equal and then we obtain that the type ii degree of symmetry is more appropriate to character the degree of the symmetry of fuzzy relations than the type i degree of symmetry with respect to whether they preserve the fuzzy e equalities or not in the meantime for a special case with continuous t norms we provide a necessary and sufficient condition such that the type i degree of symmetry preserves the fuzzy equalities finally for conjunctive left continuous idempotent uninorms with neutral element e 0 1 we find out a symmetric fuzzy relation s which is close enough to the given fuzzy relation r with respect to fuzzy e equality such that the degree of fuzzy e equality of s and r is the type ii degree of symmetry of r in particular we obtain analogous results for continuous t norms and one class of left continuous t norms

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

an integrable three particle system a study of the existence of some integrable systems with nonlinear constants of motion is presented using the approach of the theory of generalized dynamical or hidden symmetries two lagrangians are considered both obtained by modifying the toda lagrangian first a two particle system is studied and then the results are generalized to a three particle system it is shown that in both cases the lagrangians possess nonlinear constants of motion in involution and thus they are integrable

on approximations of zadeh s extension principle in this paper we provide several results concerning the approximation of zadeh s extension of a given function we provide several simple observations due to which we provide a general procedure allowing to approximate zadeh s extension of any continuous map among other things we demonstrated that the technique of f transform can also be used to approximate this extension further we studied how the choice of the metric on the space of fuzzy sets resp fuzzy numbers can affect the quality of the approximation and we clarified that in comparison to recent habits the choice of the levelwise metric need not be the most appropriate one and that the approximation process could be simplified our results are general in the sense that we mostly require nothing but the continuity of the original map

a non standard quantum double as the l state boson algebra and its r matrices for the quantum yang baxter equation in this paper we construct a new quantum double by endowing the l state boson algebra with a non trivial hopf algebra structure which is not a q deformation of the lie algebra or superalgebra the universal r matrix for the yang baxter equation associated with this new quantum group structure is obtained explicitly by building the representations of this quantum double we obtain some r matrices which can result in new representations of the braid group

maxima and minima in fuzzified linear orderings the paper studies the notions of fuzzy maxima minima suprema and infima in the setting of fuzzified linear orderings the main result shows that in fuzzified linear orders satisfying certain reasonable assumptions the fuzzy maxima and minima of all non empty fuzzy sets are non empty

l approximation spaces and l fuzzy quasi uniform spaces we introduce the notion of generalized l upper and generalized l lower approximation spaces under a family of information systems we define l upper and l lower quasi approximation operators in fuzzy rough sets and l fuzzy quasi uniform spaces as a view point of the entourage approach in a strictly two sided commutative bi quantale we investigate the relations between l quasi approximation spaces and l fuzzy quasi uniform spaces

a distance metric for a space of linguistic summaries producing linguistic summaries of large databases or temporal sequences of measurements is an endeavor that is receiving increasing attention these summaries can be used in a continuous monitoring situation like eldercare where it is important to ascertain if the current summaries represent an abnormal condition it is therefore necessary to compute the distance between summaries as a basis for such a determination in this paper we propose a dissimilarity measure between summaries based on fuzzy protoforms and prove that this measure is a metric we take into account not only the linguistic meaning of the summaries but also two quality evaluations namely the truth values and the degrees of focus we present examples of how the distance metric behaves and show that it corresponds with intuition

design of natural classification kernels using prior knowledge a new class of kernels has been designed to enhance the usability of prior knowledge prior knowledge is shown to improve the generalization ability of kernel algorithms for binary classification problems the prior knowledge is expressed in natural language via fuzzy rules first the concepts of a fuzzy rule base and prior confidence region are proposed to formulate the prior knowledge then the new kernels which are referred to as natural classification kernels ncks are represented by fuzzy equivalence relations based on the formulation of prior knowledge an nck is interpreted as a measure of similarities between samples it is proven that ncks have two desired properties 1 transitivity with respect to the triangular norms and 2 the ability to provide higher similarities to spatially closer samples from the same class using transitivity a large number of ncks may be directly obtained by means of triangular norms additionally the theoretical results show that the second property makes it possible for the support vector machine svm and convex hull separation algorithm to generalize from training samples to test samples in the prior confidence region finally some synthetic datasets and a benchmark dataset are employed to validate the proposed approach

hamiltonian systems on k cosymplectic manifolds the hamiltonian framework on symplectic and cosymplectic manifolds is extended in order to consider classical field theories to do this the notion of k cosymplectic manifold is introduced and a suitable hamiltonian formalism is developed so that the field equations for scalar and vector hamiltonian functions are derived

on a kind of noether symmetries and conservation laws in k cosymplectic field theory this paper is devoted to studying symmetries of certain kinds of k cosymplectic hamiltonian systems in first order classical field theories thus we introduce a particular class of symmetries and study the problem of associating conservation laws to them by means of a suitable generalization of noether s theorem

the polysymplectic hamiltonian formalism in field theory and calculus of variations i the local case on presente un formalisme hamiltonien geometrique invariant pour des problemes variationnels a integrale multiple et des theories de champ le formalisme est base sur la notion de forme polysymplectique qui est une generalisation a valeur vectorielle des formes symplectiques on etudie des equations hamiltoniennes des transformations canoniques des systemes de lagrange des symetries des applications en pulsion de theorie du champ et une classification des systemes de theorie du champ g homogenes sur une generalisation des orbites coadjointes

on the compactness of admissible transformations of fuzzy partitions in terms of t equivalence relations this paper analyzes constraints in terms of fuzzy logical concepts for transforming families of fuzzy sets in order to guarantee linguistically interpretable solutions of fine tuned or optimized fuzzy systems and further to make the whole process of fine tuning a well posed problem for this purpose transformations for fuzzy sets are introduced which are constructed by means of the compositional rule of inference it is investigated under which conditions on these transformations various characteristics of fuzzy partitions and fuzzy sets like redundancy convexity and topological aspects remain invariant it is argued that further restrictions like the compactness of the set of admissible transformations are required in order to make the process of fine tuning well posed as main result it is demonstrated that compactness of this function space endowed with the uniform norm can be characterized in terms of purely fuzzy logical concepts that is by means of fuzzy equivalence relations

fuzzy preorder and fuzzy topology this paper presents by means of alexandrov topology for fuzzy preordered sets and specialization order for fuzzy topological spaces a systematic investigation of the interrelationship between fuzzy preordered sets topological spaces and fuzzy topological spaces

the gunther s formalism in classical field theory momentum map and reduction the polysymplectic formalism in local field theory developed by gunther j diff geom 25 23 1987 is revised a new approach and new results on momentum maps and reduction are given

k symplectic structures the basic properties of k symplectic structures are introduced and developed in analogy with the well known symplectic differential geometry examples of such structures related to k symplectic lie algebras are given

relations in fuzzy class theory this paper studies fuzzy relations in the graded framework of fuzzy class theory fct this includes i rephrasing existing work on graded properties of binary fuzzy relations in the framework of fuzzy class theory and ii generalizing existing crisp results on fuzzy relations to the graded framework our particular aim is to demonstrate that fuzzy class theory is a powerful and easy to use instrument for handling fuzzified properties of fuzzy relations this paper does not rephrase the whole theory of fuzzy relations instead it provides an illustrative introduction showing some representative results with a strong emphasis on fuzzy preorders and fuzzy equivalence relations

structure of regular representation of quantum universal enveloping algebra slq 2 with qp 1 the authors construct the regular representation quantum lie algebra of slq 2 and its quotient representations and then analyse the reducible structures of these representations when qp 1 for an integer p they show how the indecomposable representations of slq 2 are obtained by a purely algebraic method

an introduction to vague complemented ordered sets abstract the purpose of this paper is to introduce a theory of fuzzily defined complement operations on nonempty sets equipped with fuzzily defined ordering relations many valued equivalence relation based fuzzy ordering relations also called vague ordering relations provide a powerful and a comprehensive mathematical modelling of fuzzily defined partial ordering relations for this reason starting with a nonempty set x equipped with a many valued equivalence relation and a vague ordering relation a fuzzily defined complement operation called a vague complement operation on x will be formulated by means of the underling many valued equivalence relation and vague ordering relation because of the fact that the practical implementations of vague complement operations basically depend on their representation properties a considerable part of this paper is devoted to the representations of vague complement operations in addition to this the present paper provides various nontrivial examples for vague complements and introduces a many valued logical interpretation of quantum logic as a real application of vague complements

dominance in the family of sugeno weber t norms the dominance relationship between two members of the family of sugeno weber t norms is proven by using a quantifier elimination algorithm further it is shown that dominance is a transitive and therefore also an order relation on this family of t norms

superintegrable n 2 systems quadratic constants of motion and potentials of drach the properties of superintegrable systems in two degrees of freedom possessing three independent globally defined constants of motion are studied using as an approach the existence of hidden symmetries and the generalized noether s theorem the potentials are obtained as solution of a system of two partial differential equations first the case of standard lagrangians is studied and then the method is applied to the case of lagrangians with a pseudo euclidean kinetic term finally the results are related with other approaches and with a family of potentials admitting a second integral of motion cubic in the velocities obtained by drach

a generalization of the mulholland inequality for continuous archimedean t norms it is well known that dominance between strict t norms is closely related to the mulholland inequality which can be seen as a generalization of the minkowski inequality however strict t norms constitute only one part of the class of continuous archimedean t norms the basic elements from which all continuous t norms are composed in this paper dominance between continuous archimedean t norms is shown to be related to a generalization of the mulholland inequality we provide sufficient and necessary conditions for its fulfillment

a kernel based framework for learning graded relations from data driven by a large number of potential applications in areas such as bioinformatics information retrieval and social network analysis the problem setting of inferring relations between pairs of data objects has recently been investigated intensively in the machine learning community to this end current approaches typically consider datasets containing crisp relations so that standard classification methods can be adopted however relations between objects like similarities and preferences are often expressed in a graded manner in real world applications a general kernel based framework for learning relations from data is introduced here it extends existing approaches because both crisp and graded relations are considered and it unifies existing approaches because different types of graded relations can be modeled including symmetric and reciprocal relations this framework establishes important links between recent developments in fuzzy set theory and machine learning its usefulness is demonstrated through various experiments on synthetic and real world data the results indicate that incorporating domain knowledge about relations improves the predictive performance

extended hamiltonian systems in multisymplectic field theories we consider hamiltonian systems in first order multisymplectic field theories we review the properties of hamiltonian systems in the so called restricted multimomentum bundle including the variational principle which leads to the hamiltonian field equations in an analogous way to how these systems are defined in the so called extended symplectic formulation of nonautonomous mechanics we introduce hamiltonian systems in the extended multimomentum bundle the geometric properties of these systems are studied the hamiltonian equations are analyzed using integrable multivector fields the corresponding variational principle is also stated and the relation between the extended and the restricted hamiltonian systems is established all these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almost regular field theories

higher order noether symmetries and constants of the motion the authors identify a class of symmetries for lagrangian systems called higher order noether symmetries which yield a corresponding first integral without further integrations first a version of noether s theorem is recalled in which noether symmetries are considered to be symmetries of the two form d theta theta being the cartan form nth order noether symmetries are then defined by the requirement lynd theta 0 the picture is further generalised by exploring various ways in which the computation of repeated lie derivatives of d theta can lead to the identification of first integrals a number of illustrative examples are discussed

a theory of vague lattices based on many valued equivalence relations i general representation results the present work introduces a new and general theory of ordering relations and lattices based on many valued equivalence relations under the name vague ordering relations and vague lattices respectively representations and constructions of vague ordering relations and vague lattices are the main subjects of this paper and various desirable results in this direction are established furthermore algebraic characterizations of vague lattices are studied

on the t transitivity of kernels abstract in this paper we present a view of kernels from a fuzzy set theoretical perspective indeed it turns out that kernels which are positive definite functions have to fulfill a consistency property given by the so called t transitivity of a fuzzy t equivalence relation with respect to the triangular norm t as a result we introduce a triangular norm t cos which is characterized as being the greatest one for which all kernels mapping to the unit interval and having constant 1 in its diagonal are t cos equivalences finally some applied relevant aspects are discussed as a starting point for further research

representations of quantum matrix algebra mq 2 and its q boson realization in this paper it is proved that the nontrivial finite dimensional irreducible representations of the quantum matrix algebra mq 2 or the coordinate ring of the quantum group glq 2 exist only when q is a root of unity and thereby construct such representations for mq 2 some finite dimensional indecomposable representations of mq 2 are also explicitly presented finally the q boson realization of mq 2 is discussed and it is shown that it works in this case as well as in the quantum algebra case

q analogues of verma representations of quantum algebras for q a root of unity the case of uq sl 3 verma representation theory for classical lie algebra is extended to study the representation of quantum universal enveloping algebra quantum algebra for the nongeneric case that q is a root of unity on certain subspaces and quotient spaces of the verma space finite and infinite dimensional irreducible or indecomposable representations of slq 3 uq sl 3 are obtained in explicit matrix forms

measures of specificity over continuous spaces under similarity relations we introduce the concept of specificity and indicate its importance as a measure of uncertainty for information represented using fuzzy sets or possibility distributions we provide formal measures of specificity for variables whose domain is an interval of the real line similarity relations are discussed and formalized the class of width based similarity relations is introduced we extend the measure of specificity to allow for the effect of an underlying similarity relationship the manifestation of this extension for a number of different similarity relations is investigated particularly notable results are obtained for width based similarity relationships more generally we note the connection between the size of a granule of information and its uncertainty motivated by this we investigate the effect of a similarity relationship on the perception of distance in the underlying space and use this to develop an extension of the specificity measure