migrative uninorms and nullnorms over t norms and t conorms in this paper the notions of migrative uninorms and nullnorms over a fixed t norm t and over a fixed t conorm s are introduced and studied all cases when the uninorm u lies in any one of the most usual classes of uninorms are analyzed characterizing all solutions of the migrativity equation for all possible combinations of u and t and for all possible combinations of u and s a similar study is done for nullnorms

coarse stability and bifurcation analysis using time steppers a reaction diffusion example evolutionary pattern forming partial differential equations pdes are often derived as limiting descriptions of microscopic kinetic theory based models of molecular processes e g reaction and diffusion the pde dynamic behavior can be probed through direct simulation time integration or more systematically through stability bifurcation calculations time stepper based approaches like the recursive projection method shroff g m keller h b 1993 siam j numer anal 30 1099 1120 provide an attractive framework for the latter we demonstrate an adaptation of this approach that allows for a direct effective coarse bifurcation analysis of microscopic kinetic based models this is illustrated through a comparative study of the fitzhugh nagumo pde and of a corresponding lattice boltzmann model

migrativity property for uninorms the migrative uninorms over a fixed uninorm with different neutral elements are presented all cases when both uninorms lay in any one of the most usual classes of uninorms are analyzed characterizing all solutions of the migrativity equation for some possible combinations the solutions obtained generalize the results where both uninorms have the same neutral elements

a generalization of the migrativity property of aggregation functions this paper brings a generalization of the migrativity property of aggregation functions suggested in earlier work of some of the present authors by imposing the a migrativity property of durante and sarkoci for all values of a instead of a single one replacing the algebraic product by an arbitrary aggregation function b naturally leads to the properties of a b migrativity and b migrativity this generalization establishes a link between migrativity and a particular case of aczel s general associativity equation already considered by cutello and montero as a recursive formula for aggregation following a basic investigation emphasis is put on aggregation functions that can be represented in terms of an additive generator more specifically strict t norms strict t conorms and representable uninorms

on properties of uninorms with underlying t norm and t conorm given as ordinal sums uninorms as binary operations on the unit interval have been widely applied in the fuzzy set theory this paper presents some properties of uninorm like operations for which the underlying operations are given by ordinal sums if the underlying operations of a uninorm are given by ordinal sums then the cartesian product of the union of two arbitrary intervals one in 0 e and the other in e 1 where e is the neutral element of the uninorm is a set closed under the uninorm when transposing such a cartesian product to the unit square one obtains a uninorm like operation as a result we have described the uninorm like operations for which the underlying operations are basic pseudo t norms and pseudo t conorms and one of them is idempotent

characterization of idempotent n uninorms abstract the structure of idempotent n uninorms is studied we show that each idempotent 2 uninorm can be expressed as an ordinal sum of an idempotent uninorm possibly also of a countable number of idempotent semigroups with operations min and max and a 2 uninorm from class 1 possibly restricted to open or half open unit square similar results are shown also for idempotent n uninorms further it is shown that idempotent n uninorms are in one to one correspondence with special lower semi lattices defined on the unit interval the z ordinal sum construction for partially ordered semigroups is also defined

on the structure of 2 uninorms abstract in this paper we describe the structure of 2 uninorms defined on the unit interval we divide the class of 2 uninorms into five mutually exclusive subclasses based on their boundary behaviour as determined by the absorbing element and the conjunctive disjunctive nature of the underlying uninorms which every 2 uninorm is known to have for each of these subclasses we fully characterize the structure of its members albeit for two subclasses only under an additional continuity assumption a common practice in the study of associative operations

convex combinations of uninorms and triangular subnorms abstract the problem whether a non trivial convex combination of two continuous triangular norms can be a t norm was extensively studied in the last decades bringing the negative answer in all examined special cases on the other hand in the field of non continuous t norms including left continuous t norms several examples were found including mainly a convex combination of an migrative t norm and the drastic product t norm or a convex combination of ordinal sums of product related t subnorms recently convex combinations of uninorms were discussed in 18 since any uninorm with the neutral element e 0 1 restricted to 0 e 2 is isomorphic to a t norm this investigation extends the original problem of a convex combination of t norms given by alsina frank and schweizer in this work we will study convex combinations of uninorms further covering uninorms with continuous underlying functions uninorms with different neutral elements and representable uninorms we will also show that convex combinations of triangular subnorms play a major role in this investigation

deterministic continuation of stochastic metastable equilibria via lyapunov equations and ellipsoids numerical continuation methods for deterministic dynamical systems have been one of the most successful tools in applied dynamical systems theory continuation techniques have been employed in all branches of the natural sciences as well as in engineering to analyze ordinary partial and delay differential equations here we show that the deterministic continuation algorithm for equilibrium points can be extended to track information about metastable equilibrium points of stochastic differential equations we stress that we do not develop a new technical tool but that we combine results and methods from probability theory dynamical systems numerical analysis optimization and control theory into an algorithm that augments classical equilibrium continuation methods in particular we use ellipsoids defining regions of high concentration of sample paths it is shown that these ellipsoids and the distances between them can be efficiently calculated using iterative methods that take advantage of the numer

the modularity condition for uninorms revisited abstract functional equations involving aggregation functions play an important role in fuzzy sets and fuzzy logic theory the modularity equation is a kind of restricted general associative equations in the literature the already known results concerning the modularity equation for two uninorms are based on the assumption that both uninorms lie in one of the most studied classes of uninorms in this study we explore the modularity equation involving uninorms in a most general setting specifically we investigate the modularity between two uninorms in the cases when one uninorm belongs to any of the most studied classes of uninorms but the other one is any uninorm with no further assumptions further many new solutions appear from this new point of view that were not included in previous approaches

some properties of weak uninorms a weak uninorm is not only a generalization of a uninorm but also of a nullnorm in this paper we will show that if a weak uninorm is archimedean then it is an archimedean nullnorm and if a weak uninorm is e archimedean then it is an archimedean uninorm moreover the de morgan triples of the weak uninorms will be characterized too

experimental bifurcation analysis of an impact oscillator determining stability abstract we propose and investigate three different methods for assessing stability of dynamical equilibrium states during experimental bifurcation analysis using a control based continuation method the idea is to modify or turn off the control at an equilibrium state and study the resulting behavior as a proof of concept the three methods are successfully implemented and tested for a harmonically forced impact oscillator with a hardening spring nonlinearity and controlled by electromagnetic actuators we show that under certain conditions it is possible to quantify the instability in terms of finite time lyapunov exponents as a special case we study an isolated branch in the bifurcation diagram brought into existence by a 1 3 subharmonic resonance on this isola it is only possible to determine stability using one of the three methods which is due to the fact that only this method guarantees that the equilibrium state can be restored after measuring stability

migrativity equation for uninorms with continuous underlying operators abstract functional equations involving aggregation functions play an important role in fuzzy sets and fuzzy logic theorem as a kind of restricted general associative equation the migrativity equation has been proven to be useful in a wide range of fields like decision making aggregation of information image processing and so on in the literature the already existing results concerning the migrativity equation for two uninorms are based on the assumption that one of the uninorms belongs to one of the most studied classes of uninorms or that they have the same neutral element in this study we will explore the migrativity equation involving uninorms in a most general setting to be specific we investigate the migrativity between two uninorms with only continuous underlying operators then many new solutions of the equation are fully characterized from this new point of view

characterizing set valued functions of uninorms with continuous underlying t norm and t conorm abstract the uninorms with continuous underlying functions were characterized by their set of discontinuity points in the previous work of the author using the characterizing set valued function in this paper properties of this characterizing set valued function are studied it is shown that the type of the monotonicity of such a set valued function is always changed in idempotent points of the corresponding uninorm several additional properties of the characterizing set valued function of a uninorm with continuous underlying functions are shown and an example is included results shown in this paper are used for a complete characterization of uninorms with continuous underlying functions via the ordinal sum construction which is shown in the consecutive paper

numerical continuation methods a perspective in this historical perspective the principal numerical approaches to continuation methods are outlined in the framework of the mathematical sources that contributed to their development notably homotopy and degree theory simplicial complexes and mappings submanifolds defined by submersions and singularity and foldpoint theory

distributivity and conditional distributivity of a uninorm with continuous underlying operators over a continuous t conorm this paper deals with the distributivity and conditional distributivity of uninorms with continuous underlying operators over continuous triangular conorms in particular the involved triangular conorm is either maximum operator or an ordinal sum with only one summand in which the corresponding triangular conorm is strict from the obtained results it is deduced that distributivity and conditional distributivity are equivalent moreover we obtain the full characterization of some cases of this class of uninorms of which either the underlying triangular norm or triangular conorm is strict

associativity of triangular norms characterized by the geometry of their level sets associativity of triangular norms is an algebraic property which unlike for example their commutativity is usually understood as hardly visually interpretable this problem has been studied intensively in the last decade and as a result geometric symmetries of triangular norms with involutive level sets have been revealed the presented paper intends to introduce a different approach which gives more general results the inspiration is taken from web geometry a branch of differential geometry and its concept of reidemeister closure condition which is known to provide a geometric characterization of associativity of loops the paper shows that this concept can be adopted successfully for triangular norms so that it characterizes their associativity in a similar way moreover the offered adaptation preserves the beneficial transparency and simplicity of the reidemeister closure condition this way a visual characterization of the associativity based on the geometry of the level sets is provided for general continuous and continuous archimedean triangular norms

basic concepts for a theory of evaluation the aggregative operator abstract starting with an explication of the aggregative concept and deducing a general structure which satisfies a number of minimal requirements properties of clustering the main features of a new mathematical theory called theory of evaluation are developed the theory sheds new light on such well known concepts as membership conjunction and disjunction and seems to be a very promising tool to handle representation problems as they grow from the fields of theory of fuzzy set and its many applications of human decision making and of multicriteria analysis

on the distributivity property for uninorms abstract distributivity between two operations is a property that was already posed many years ago and that is especially interesting in the framework of logical connectives for this reason the distributivity property has been extensively studied for several families of operations like triangular norms and conorms some kinds of uninorms and nullnorms also called t operators and even for some generalizations of them in this paper we investigate the distributivity equation involving two uninorms lying in any one of the most studied classes of uninorms leading to many new solutions

convex combinations of nilpotent triangular norms abstract in this paper we deal with the open problem of convex combinations of continuous triangular norms stated by alsina frank and schweizer c alsina m j frank b schweizer problems on associative functions aequationes math 66 2003 128 140 problems 5 and 6 they pose a question whether a non trivial convex combination of triangular norms can ever be a triangular norm the main result of this paper gives a negative answer to the question for any pair of continuous archimedean triangular norms with different supports with the help of this result we show that a non trivial convex combination of nilpotent t norms is never a t norm the main result also gives an alternative proof to the result presented by ouyang and fang y ouyang j fang some observations about the convex combination of continuous triangular norms nonlinear anal 68 11 2008 3382 3387 theorem 3 1 in proof of the main theorem we utilize the reidmeister condition known from the web geometry

some properties of idempotent uninorms on a special class of bounded lattices abstract binary operations have many applications in fuzzy set theory one of them is uninorms in this paper we study some properties of idempotent uninorms on bounded lattices it is shown that idempotent uninorms on an arbitrary bounded lattice need not always be internal with the extended definition of the term internal we give a sufficient condition for the bounded lattice to any idempotent uninorm be internal in addition that it is given many properties of idempotent uninorms in this case

a single point characterization of representable uninorms we study uninorms of which both the underlying t norm and underlying t conorm are strict such uninorms are the only candidates for being representable by an additive generator we prove that the representability of such a uninorm depends solely on its value at a single arbitrary point in the remaining open part of the unit square more explicitly such a uninorm turns out to be representable if and only if this single value is located strictly between the minimum and the maximum of the corresponding arguments if this single value coincides with one of these bounds then the value of the uninorm at any point in the remaining open part is determined by the same bound

cauchy like functional equations for uninorms continuous in 0 1 2 abstract commutativity is an important property in two step information merging procedure it is shown that the result obtained from the procedure should not depend on the order in which signal steps are performed in the case of a bisymmetric aggregation operator with the neutral element saminger et al have provided a full characterization of commutative n ary operator by means of unary distributive functions further characterizations of these unary distributive functions can be viewed as resolving a kind of the cauchy like equations f x y f x f y where f 0 1 0 1 is a monotone function is a bisymmetric aggregation operator with the neutral element in this paper we are still devoted to investigating and fully characterizing the cauchy like equation f u x y u f x f y where f 0 1 0 1 is an unknown function but not necessarily monotone u is a uninorm continuous in 0 1 2 these results show the key technology is how to find a transformation from this equation into several known cases moreover this equation has completely different and non monotone solutions in comparison with the obtained results

experimental bifurcation analysis continuation for noise contaminated zero problems abstract noise contaminated zero problems involve functions that cannot be evaluated directly but only indirectly via observations in addition such observations are affected by a non deterministic observation error noise we investigate the application of numerical bifurcation analysis for studying the solution set of such noise contaminated zero problems which is highly relevant in the context of equation free analysis coarse grained analysis and bifurcation analysis in experiments and develop specialized algorithms to address challenges that arise due to the presence of noise as a working example we demonstrate and test our algorithms on a mechanical nonlinear oscillator experiment using control based continuation which we used as a main application and test case for development of the coco compatible matlab toolbox continex that implements our algorithms

open problems from the 12th international conference on fuzzy set theory and its applications eighteen open problems posed during fsta 2014 liptovsk jan slovakia are presented these problems concern fuzzy logics fuzzy partitions copulas triangular norms and related aggregation functions some open problems concerning effect and mv algebras are also included

on migrativity property for uninorms recently u 0 migrative uninorms u were introduced and discussed where u and u 0 are two uninorms with neutral element e in this paper we will further study u 2 migrative uninorms u 1 where u 1 and u 2 are two uninorms with different neutral elements e 1 and e 2 respectively all cases when the uninorm u 1 and u 2 are in different families out of the most usual classes of uninorms are analyzed characterizing all solutions of the migrativity equation for some possible combinations of u 1 and u 2

ordinal sums of representable uninorms abstract we investigate properties of an ordinal sum of uninorms in the case that the summands are proper representable uninorms we show sufficient and necessary conditions for a uninorm to be an ordinal sum of representable uninorms an example is also included

characterizations of uni nullnorms with continuous archimedean underlying t norms and t conorms abstract a uni nullnorm is a special case of 2 uninorms obtained by letting a uninorm and a nullnorm share the same underlying t conorm in this paper we give complete descriptions of uni nullnorms with continuous archimedean underlying t norms and t conorms

branch switching in bifurcation problems for ordinary differential equations the problem of switching branches in boundary value problems of ordinary differential equations is considered three non local methods for calculating emanating solutions near a nontrivial bifurcation point are proposed these methods calculate one solution on an emanating branch without a priori exact knowledge of the bifurcation point other solutions on the branch can be obtained by global continuation the methods are convenient as they consist in solving boundary value problems by standard software the construction of an initial approximation of the emanating solution is outlined a characteristic feature of the proposed methods is that they can be easily automated the user can avoid nearly all preparatory work the methods are tested on several examples arising in different application areas

rotation invariant t norms the rotation invariance property revisited focusing on conjunctive left continuous increasing 0 1 2 0 1 functions t we redefine the rotation invariance property in terms of contour lines under the assumption of the existence of a neutral element email protected 0 1 this rotation invariance property requires some partial commutativity and associativity the functions that are rotation invariant w r t all of their contour lines are characterized

van melle s combining function in mycin is a representable uninorm an alternative proof abstract recently tsadiras and margaritis have shown that the function implemented in the expert system mycin for combining certainty factors is a uninorm in this paper we present a compact alternative to the given direct and lengthy proof based on the construction of representable uninorms by means of an additive generator consequently we show that van melle s combining function is a representable uninorm in fact it is a rescaled version of the representable uninorm on the unit interval constructed from the algebraic product and the probabilistic sum

a characterization of discrete uninorms having smooth underlying operators in this paper discrete uninorms u such that their underlying t norm t and t conorm s are smooth are characterized the different cases combining when t is the minimum or the ukasiewicz t norm and s is the maximum or the ukasiewicz t conorm are given separately and the number of discrete uninorms with these underlying operators is given the same is done for the general case when t and or s are smooth but ordinal sums

deciding the nature of the coarse equation through microscopic simulations the baby bathwater scheme recent developments in multiscale computation allow the solution of coarse equations for the expected macroscopic behavior of microscopically evolving particles without ever obtaining these coarse equations in closed form the closure is obtained on demand through appropriately initialized bursts of microscopic simulation the effective coupling of microscopic simulators with macroscopic behavior requires certain decisions about the nature of the unavailable coarse equation such decisions include a the highest spatial derivative active in the coarse equation b whether the equation satisfies certain conservation laws or c whether the coarse dynamics is hamiltonian or dissipative these decisions affect the number and type of boundary conditions as well as the algorithms employed in the absence of an explicit formula for the temporal derivative we propose implement and validate a simple scheme for deciding these and other similar questions about the coarse equation using only the microscopic simulator simulations under periodic boundary conditions are carried out for appropriately chosen families of random initial conditions evaluating the sample variance of certain statistics over the simulation ensemble allows us to infer the highest order of spatial derivatives active in the coarse equation in the same spirit we show how to determine whether a certain coarse conservation law exists or not and we discuss plausibility tests for the existence of a coarse hamiltonian or integrability we believe that such schemes constitute an important part of the equation free approach to multiscale computation

conditional distributivity equation for uninorms with continuous underlying operators our investigations are motivated by distributive logical connectives and their generalizations used in fuzzy set theory and focused also by klement in the linz seminar 2000 closing session this paper is mainly devoted to solving the functional equations of conditional distributivity of u 1 over u 2 where both u 1 and u 2 are uninorms with continuous underlying operators then the almost complete characterization of such a pair u 1 u 2 of uninorms is given except for only a small fraction of the unit square next we give the complete characterization of the previous pair under some appropriate restriction that is the second operator u 2 in mathbf u max cup mathbf u min finally we also construct a counterexample to illustrate that the obtained results are necessary but not sufficient for general cases with unlimited condition

ground states of dispersion managed nonlinear schrodinger equation an exact pulse for the parametrically forced nonlinear schrodinger equation nls is isolated the equation governs wave envelope propagation in dispersion managed fiber lines with positive residual dispersion the pulse is obtained as a ground state of an averaged variational principle associated with the equation governing pulse dynamics the solutions of the averaged and original equations are shown to stay close for a sufficiently long time a properly adjusted pulse will therefore exhibit nearly periodic behavior in the time interval of validity of the averaging procedure furthermore we show that periodic variation of dispersion can stabilize spatial solitons in a kerr medium and one dimensional solitons in the nls with quintic nonlinearity the results are confirmed by numerical simulations

on the conditional distributivity of nullnorms over uninorms conditional distributivity of nullnorm over uninorm with continuous operators is discussed the structure of uninorms with idempotent operators is discussed the problem of conditional distributivity for different two aggregation operators plays an important role in many different fields such as utility theory and integration theory this paper deals with the conditional distributivity of a continuous nullnorm with respect to a uninorm with the continuous underlying t norm and t conorm

averaging of dispersion managed solitons existence and stability we consider existence and stability of dispersion managed solitons in the two ap proximations of the periodic nonlinear schrodinger nls equation i a dynamicalsystem for a gaussian pulse and ii an average integral nls equation we apply normal form transformations for finite dimensionaland infinite dimensionalhamil tonian systems with periodic coefficients first order corrections to the leading order averaged hamiltonian are derived explicitly for both approxi mations bifurcations of soliton solutions and their stability are studied by analysis of critical points of the first order averaged hamiltonians the validity of the averaging procedure is verified and the presence of ground states corresponding to dispersion managed solitons in the averaged hamiltonian is established

choquet like integrals abstract integrals based on a pseudo addition and a pseudo multiplication with properties similar to those of the choquet and the sugeno integral respectively are introduced the structure of convenient pseudo arithmetical operations leading to a choquet like integral is studied

equation free analysis of agent based models and systematic parameter determination abstract agent based models abm s are increasingly used in social science economics mathematics biology and computer science to describe time dependent systems in circumstances where a description in terms of equations is difficult yet few tools are currently available for the systematic analysis of abm behaviour numerical continuation and bifurcation analysis is a well established tool for the study of deterministic systems recently equation free ef methods have been developed to extend numerical continuation techniques to systems where the dynamics are described at a microscopic scale and continuation of a macroscopic property of the system is considered to date the practical use of ef methods has been limited by 1 the over head of application specific implementation 2 the laborious configuration of problem specific parameters and 3 large ensemble sizes potentially leading to computationally restrictive run times in this paper we address these issues with our tool for the ef continuation of stochastic systems which includes algorithms to systematically configuration problem specific parameters and enhance robustness to noise our tool is generic and can be applied to any black box simulator and determines the essential ef parameters prior to ef analysis robustness is significantly improved using our convergence constraint with a corrector repeat c3r method this algorithm automatically detects outliers based on the dynamics of the underlying system enabling both an order of magnitude reduction in ensemble size and continuation of systems at much higher levels of noise than classical approaches we demonstrate our method with application to several abm models revealing parameter dependence bifurcation and stability analysis of these complex systems giving a deep understanding of the dynamical behaviour of the models in a way that is not otherwise easily obtainable in each case we demonstrate our systematic parameter determination stage for configuring the system specific ef parameters

vector soliton collision dynamics in nonlinear optical fibers we consider the interactions of two identical orthogonally polarized vector solitons in a nonlinear optical fiber with two polarization directions described by a coupled pair of nonlinear schrodinger equations we study a low dimensional model system of hamiltonian ordinary differential equations odes derived by ueda and kath and also studied by tan and yang we derive a further simplified model which has similar dynamics but is more amenable to analysis sufficiently fast solitons move by each other without much interaction but below a critical velocity the solitons may be captured in certain bands of initial velocities the solitons are initially captured but separate after passing each other twice a phenomenon known as the two bounce or two pass resonance we derive an analytic formula for the critical velocity using matched asymptotic expansions for separatrix crossing we determine the location of these resonance windows numerical simulations of the ode models show they compare quite well with the asymptotic theory

uninorms on bounded lattices with the underlying t norms and t conorms abstract uninorms are a generalization of the concepts of triangular norms t norms and triangular conorms t conorms that admit a neutral element as any element in a bounded lattice in this study we discuss the characterization of uninorms on bounded lattices we introduce some new constructions for uninorms on bounded lattices which are derived from both t norms and t conorms in particular two idempotent uninorms on bounded lattices are obtained we also provide some corresponding examples to help understand the structure of these new classes of uninorms in addition we provide an answer to the open problem proposed by cayli 2019 14

dispersion managed solitons via an averaged variational principle constrained minimization is used as a computational strategy to approximate and study dispersion managed solitons through their characterization as minima of an averaged variational principle a basis of hermite gaussian functions is used and the constrained minimization procedure is carried out to find such pulses this method produces more accurate pulse shapes than the usual gaussian approximations propagating with less noise and providing a far better representation of the tail of the pulse the success of this procedure provides confirmation that the dispersion managed soliton is truly a minimum of the constrained variational principle when the residual dispersion is negative the dispersion managed soliton can no longer be a minimum a fact that is confirmed by our numerical simulations even in this case however approximate pulse shapes can be obtained using this finite dimensional approximation additionally we find critical points other than the ground state the most interesting is an excited bound state corresponding to a symmetric bisoliton 2003 elsevier b v all rights reserved

on the structure of special classes of uninorms in this paper the concept of the reidemeister closure condition is adopted in order to characterize associativity of uninorms with a special attention paid to the class of representable uninorms thus conditions replacing the associativity requirement for such uninorms are given further the attention is focused on the uninorms where the underlying t norm and t conorm are idempotent or induce an involutive negator based on the aforementioned results the structure of these classes of uninorms is fully described

new construction of t norms and t conorms on bounded lattices abstract this article studies the construction of ordinal sums of triangular norms t norms and triangular conorms t conorms on bounded lattices first we present a new class of t norms and t conorms on an arbitrary bounded lattice some examples are provided to illustrate that our new construction method is different from some existing methods for the construction of ordinal sums of t norms and t conorms on an arbitrary bounded lattice then we illustrate that our new construction method can be generalized by induction to a modified ordinal sum construction for t norms and t conorms on an arbitrary bounded lattice finally for any bounded sublattice s of a bounded lattice l we provide a sufficient condition that does not need s and l to have the same bottom and top elements such that an ordinal sum function is a t norm on l whereas the function is determined by an arbitrary selection of bounded sublattices as carriers for arbitrary summand t norms in particular we give a necessary and sufficient condition that needs s to have the same bottom element as l such that an ordinal sum function is a t norm on l

numerical continuation of bifurcations of limit cycles in matlab rm cl matcont and matcont are matlab continuation packages for the interactive numerical study of a range of parameterized nonlinear dynamical systems in particular odes matcont is an interactive graphical package and rm cl matcont is a commandline version both packages allow us to compute curves of equilibria limit points hopf points limit cycles flip fold and torus bifurcation points of limit cycles we discuss computational details of the continuation of limit cycles and flip fold and torus bifurcations of limit cycles in matcont and rm cl matcont using orthogonal collocation instead of the more commonly used fully extended systems we use minimally extended systems we further describe the use of the matlab sparse matrix routines and the initialization and adaptation of the bordering vectors that are essential in minimally extended systems finally we compare the use of the minimally and the fully extended systems in the matlab environment

the migrativity equation for uninorms revisited abstract the migrativity equation with interesting applications in decision making and image processing has been extensively discussed involving different kinds of aggregation functions from t norms and t conorms to uninorms nullnorms and some generalizations of them in recent papers the already known results concerning the migrativity of two uninorms are based on the assumption that both uninorms belong to one of the most studied classes of uninorms in this paper we will explore the migrativity equation involving uninorms in a most general setting specifically we will study the migrativity between two uninorms in the cases when the second uninorm lies in any of the most studied classes of uninorms but the first one is any uninorm with no further assumptions we will show along the paper that many new solutions appear from this new point of view that were not included in the previous approaches

a note on distributivity and conditional distributivity of a uninorm with continuous underlying operators over a continuous t conorms abstract in this paper the study of the distributivity equation involving uninorms and t conorms given in li and liu 2016 3 is revised some errors in the proof of theorems in the mentioned reference are pointed out and their right versions are offered furthermore the class of uninorms with continuous underlying operators which are conditionally distributive over a continuous t conorm is characterized

modularity conditions between overlap grouping function and uni nullnorm or null uninorm abstract overlap and grouping functions as two kinds of special continuous commutative aggregation functions have wide applications in image processing and classification modularity conditions between the class of overlap grouping functions satisfying o 1 e e g 0 e e and special classes of uninorms and those of another special class of overlap grouping functions over nullnorms have been studied in 49 in this paper we extend these results to any overlap grouping function moreover we study the modularity conditions between overlap grouping functions and uni nullnorms null uninorms which are the generalization of uninorms nullnorms

some results on the convex combination of uninorms abstract it is well known that under some conditions a convex combination of the drastic product t d and the strict triangular norm is still a triangular norm uninorms as a special kind of generalization of both triangular norms and triangular conorms have been widely used in fuzzy set theory in this paper the convex combination of uninorms is studied some properties of the class of uninorms with the underlying triangular norm t d are presented firstly then the convex combination of this class of uninorms is discussed and it is shown that only in some trivial cases the convex combination is still a uninorm

uninorms continuous on 0 e 2 e 1 2 uninorms continuous on 0 e 2 e 1 2 are discussed archimedean uninorms continuous and cancellative on 0 e 2 e 1 2 are characterized and related non archimedean uninorms are also discussed the problem when is the border continuous projection of a t norm associative i e a t subnorm is also answered

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

the distributivity equation for uninorms revisited abstract the distributivity equation has been widely studied involving different classes of aggregation functions from t norms and t conorms to uninorms nullnorms and generalizations of them it is important in the framework of logical connectives because of its applications in fuzzy logic and approximate reasoning as well as in image processing since uninorms have been used in these topics the study of the distributivity between two uninorms becomes specially interesting in a recent paper by the same authors the already known solutions were compiled and completed when the first uninorm is in any of the most studied classes of uninorms and the second uninorm is anyone in this paper we want to achieve this study by focusing on the reverse direction that is for the cases when the second uninorm lies in any of the most studied classes of uninorms and the first one is any uninorm we show along the paper that this new point of view leads to many new solutions

on properties of uninorms locally internal on the boundary abstract recently a class of uninorms locally internal on the boundary was proposed by mas et al during the study of the migrativity property for uninorms in this paper some properties of this class of uninorms are presented and the characterizations are obtained for different cases

an extension of the migrative property for uninorms in this paper the notion of a migrative uninorms over a fixed uninorm u 0 with the same neutral element is introduced all uninorms u that are a migrative over a uninorm u 0 are studied all cases when u and u 0 have continuous underlying operations and belong to any one of the most usual classes of uninorms are analyzed characterizing all solutions of the migrativity equation for all possible combinations

properties of uninorms with the underlying operations given as ordinal sums abstract uninorms as mixed aggregation operations on the unit interval have been widely used as logical connectives in fuzzy set theory this paper presents some properties of uninorms for which the underlying operations are given as ordinal sums specially we present the relations between two arbitrary summands and the value on cartesian product of their underlying intervals one in 0 e and the other in e 1 where e is neutral element of the uninorm whose results bring us a step closer to the structure of uninorms that have continuous underlying operations as a byproduct we characterize several special classes of uninorms that are different from the well studied classes of uninorms

migrativity property for uninorms and semi t operators in this paper the notions of migrative uninorms over semi t operators and migrative semi t operators over uninorms are introduced and investigated all solutions of the migrativity equations for all possible combinations of semi t operators and uninorms are characterized

characterization of uninorms with continuous underlying t norm and t conorm by their set of discontinuity points uninorms with continuous underlying t norm and t conorm are discussed and properties of the set of discontinuity points of such a uninorm are shown this set is proved to be a subset of the graph of a special symmetric u surjective nonincreasing set valued function which gives us a necessary condition for a uninorm to have continuous underlying functions a sufficient condition for a uninorm to have continuous underlying operations is also given several examples are included

a note on an extension of the migrative property for uninorms this paper shows by an example that proposition 15 iv c in mas et al 2013 about migrative property for uninorms in u c o s m i n is false and corrects these propositions

an equivalence relation based on the u partial order abstract in this paper an equivalence relation on the class of uninorms induced by the u partial order is discussed defining the set of all incomparable elements w r t the u partial order this set is deeply investigated and some relations with the sets of all incomparable elements w r t the orders induced by the corresponding underlying t norm and t conorm are presented also the set of all incomparable elements with a fixed element w r t the u partial order is defined and studied in detail

associative monotonic operations in fuzzy set theory abstract the properties of binary operations in a real interval are considered and used in the discussion of generalized operations on fuzzy sets on fuzzy numbers and on fuzzy probabilistic sets

a note on uninorms with continuous underlying operators abstract our investigations are motivated by the fact uninorms as mixed aggregation operations on the unit interval have been widely used in fuzzy set theory this paper is mainly to characterize the inner structure of uninorms with continuous underlying operators given by ordinal sums and then presents some of their properties to be specific for a uninorm we describe the relations of its two arbitrary summands and characterize the structure on cartesian products of its underlying intervals corresponding to those two summands mentioned previously one in 0 e and the other in e 1 where e is the neutral element of the uninorm these results bring us a step closer to get the full structure of a uninorm with continuous underlying operators

on a new class of t norms and t conorms on bounded lattices abstract the ordinal sum of triangular norms on unit interval has been used to construct other triangular norms but in general ordinal sum construction of triangular norms and triangular conorms may not work on an arbitrary bounded lattice in this study we focus on ordinal sums of triangular norms and triangular conorms on an arbitrary bounded lattice moreover we show a complete generalization of our ordinal sums of triangular norms and triangular conorms on an arbitrary bounded lattice and some illustrative examples are added for clarity

on the relationships between the orders induced by uninorms and nullnorms abstract in this paper some relationships between the orders induced by uninorms and nullnorms are discussed some relations between the sets admitting some incomparability w r t those orders are investigated also some relationships between the algebraic structures obtained from those orders are determined

on locally internal monotonic operations this paper deals with monotonic binary operations f 0 1 2 0 1 with the property called locally internal property that the value at any point x y is always one of its arguments x y after stating a theorem that characterizes this kind of operations some special cases are studied in detail by considering additional properties of the operation commutativity existence of a neutral element and associativity in case of locally internal associative monotonic operations with neutral element a characterization theorem gives an improvement of a well known theorem of czogala and drewniak on idempotent associative and increasing operations with neutral element as well as an improvement of a characterization theorem for left and right continuous idempotent uninorms

on distributivity equations for uninorms over semi t operators recently dryga generalized nullnorms and t operators and introduced semi t operators by eliminating commutativity from the axioms of t operators distributivity equations were investigated in families of certain operations e g triangular norms conorms uninorms and nullnorms or called t operators in this paper we give out the solutions of distributivity equations for uninorms over semi t operators previous results about distributivity equations for uninorms over nullnorms can be obtained as corollaries

on the structure of uninorms on bounded lattices abstract uninorms are a generalization of triangular norms and triangular conorms with a neutral element in the interior of a bounded lattice in this study we present two methods to construct uninorms from a given triangular norm and triangular conorm on bounded lattices under some constraints as a by product of these methods we obtain new types of idempotent uninorms with a neutral element on bounded lattices in addition some illustrative examples for the constructions of uninorms on bounded lattices are provided

a characterization of a class of uninorms with continuous underlying operators in this paper all uninorms locally internal in the region a e given by the complement in 0 1 2 of 0 e 2 e 1 2 where e is the neutral element of the uninorm having continuous underlying operators are studied and characterized by distinguishing some cases when the underlying t norm and t conorm are not given by ordinal sums it is proved that uninorms locally internal in a e are in fact all possible uninorms with these underlying operators except when both the t norm and the t conorm are strict in which case there is also the class of representable uninorms leading to a finite number of possibilities when at least one of the continuous underlying operators is given by an ordinal sum again there are other possible uninorms than those that are locally internal in a e but all uninorms with this property are also characterized in this case infinitely many possibilities can appear depending on the set of idempotent elements of the uninorm

on a new class of uninorms on bounded lattices we study and propose some new construction methods to obtain uninorms on bounded lattices considering an arbitrary bounded lattice l we show the existence of idempotent uninorms on l for any element e l 0 1 playing the role of a neutral element by our construction method we obtain the smallest idempotent uninorm and the greatest idempotent uninorm with the neutral element e l 0 1 we see that the obtained uninorms are conjunctive and disjunctive uninorms respectively on the other hand if l is not a chain we also provide an example of an idempotent uninorm which is neither conjunctive nor disjunctive

equivalence relations induced by the u partial order abstract in this paper an equivalence relation on the class of uninorms with the same neutral element induced by a u partial order is provided and discussed the equivalence classes linked to some special uninorms are characterized furthermore the set of incomparable elements with respect to the u partial order is deeply investigated finally another equivalence relation on the class of uninorms with the same neutral element is introduced and studied

aggregation operators and fuzzy systems modeling abstract a general class of aggregation operators called mica having the properties of monotonicity symmetry and an identity element are introduced we stress the significance of the choice of identity in characterizing the operator we show that the t norm and t conorm are special cases of these operators other classees of these operators are introduced notably an additive class which is very much in the spirit of the kind of aggregation used in neural networks it is shown how a general description of the fuzzy system modeling technique can be obtained using these operators

multi polar t conorms and uninorms abstract transformation of uninorms defined on the unit interval to the bipolar scale yields bipolar t conorms and transformation of nullnorms on the unit interval yields bipolar t norms we use this relation in the investigation of the structure of bipolar and multi polar t norms and t conorms we use the same transformation of the unit interval into the bipolar scale in order to obtain bipolar aggregation operators from ordinal sum t norms and t conorms the extensions of these special bipolar aggregation operators generalize both multi polar t norms and t conorms and therefore we call them multi polar uninorms several examples of multi polar t conorms and uninorms are also presented

distributivity and conditional distributivity of a uninorm and a continuous t conorm the open problem recalled by klement in the linz2000 closing session related to distributivity and conditional distributivity of a uninorm and a continuous t conorm is solved for the most usual known classes of uninorms from the obtained results it is deduced that distributivity and conditional distributivity are equivalent for these cases it is remarkable that solutions appear involving not only strict t conorms but also ordinal sums of the maximum with a strict t conorm conversely the distributivity of a t conorm over a uninorm is also studied leading only to already known solutions moreover the dual case of distributivity and conditional distributivity involving uninorms and continuous t norms is also solved proving again the equivalence of both kinds of distributivities

some notes on the partial orders induced by a uninorm and a nullnorm in a bounded lattice abstract in this paper an equivalence on the class of nullnorms on a bounded lattice is discussed some relationships between the u and v partial orders are investigated a necessary and sufficient condition making a bounded lattice also a lattice w r t the v partial order is given moreover the relation between the algebraic structures related to the u and v partial orders is studied

experimental bifurcation analysis of an impact oscillator tuning a non invasive control scheme abstract we investigate a non invasive locally stabilizing control scheme necessary for an experimental bifurcation analysis our test rig comprises a harmonically forced impact oscillator with hardening spring nonlinearity controlled by electromagnetic actuators and serves as a prototype for electromagnetic bearings and other machinery with build in actuators we propose a sequence of experiments that allows one to choose optimal control gains filter parameters and settings for a continuation method without a priori study of a model depending on the algorithm for estimating the jacobian required by newton s method we find two almost disjoint sets of suitable control parameters control based continuation succeeds reliably in producing the full bifurcation diagram including both stable and unstable equilibrium states for an appropriately tuned controller

ordering based on uninorms in this paper an order induced by uninorms on bounded lattices is given and discussed by defining such an order the t partial order is extended to a more general form the relationship between the order induced by the underlying t norm and t conorm and the order induced by a uninorm is studied finally the order is extended to the order induced by n uninorms on chains

on a certain class of aggregative operators abstract here our starting point is a study of connection with dombi aggregative operators uninorms strict t norms and t conorms we present a new representation theorem of strong negations that explicitly contains the neutral value then the relationships for aggregative operators and strong negations are verified as well as those for the t norm and t conorm using the pan operator concept we introduce the multiplicative pliant concept and give the necessary and sufficient conditions for it we study a certain class of weighted aggregative operators representable uninorms which build a self demorgan class with infinitely many negations we provide the necessary and sufficient conditions for these operators

remarks on uninorm aggregation operators it is proved that continuous uninorm aggregation operators are only t norms or t conorms some general constructions about uninorm aggregation operators are given