a generalization of myhill nerode theorem for fuzzy languages the well known myhill nerode theorem provides a necessary and sufficient condition for a language to be regular in the context of fuzzy languages and automata theory myhill nerode type theorems have been proved for fuzzy languages with finite range this paper introduces a new right equivalence relation on the free monoid of an alphabet based on the notion of factorization of fuzzy languages the index of this relation for a fuzzy language with infinite range can be finite this fact allows us to generalize the myhill nerode theorem for any kind of fuzzy languages in this paper is proved that the following two conditions are mutually equivalent for a given fuzzy language x i there exists a factorization such that the right equivalence relation of x defined via the factorization has a finite index ii the fuzzy language x is recognized by a fuzzy deterministic finite automaton

conditions for minimal fuzzy deterministic finite automata via brzozowski s procedure this paper deals with the application of brzozowski s minimization procedure to fuzzy finite automata with truth values in a complete residuated zero divisor free lattice for a given fuzzy finite automaton mathcal a the procedure computes the automaton d r d r mathcal a where d mathcal a is a fuzzy determinization of mathcal a and r mathcal a is the reverse automaton of mathcal a it is observed that the size of the resulting automaton is strongly dependent on the determinization method used in the procedure consequently this paper studies necessary and sufficient conditions for obtaining minimal fuzzy deterministic finite automata via brzozowski s procedure the study is accomplished for determinization methods based on a generalized notion of accessible fuzzy subset construction the obtained conditions determine that brzozowski s procedure returns a minimal fuzzy deterministic finite automaton when the determinization method does not produce proportional fuzzy states this is the reason to obtain minimal fuzzy deterministic finite automata using brzozowski s procedure for fuzzy finite automata with truth values in the product structure when the determinization method is the method based on factorization of fuzzy states

fuzzy recognizers and recognizable sets in this paper the concepts of fuzzy recognizers and recognizable sets are studied and their relationship with various products of fuzzy finite state machines is established the tools necessary for such study are developed the ideas of reversal inverse image accessible part coaccessible part trim part regarding fuzzy recognizers are introduced and their properties are discussed

determinization of weighted finite automata over strong bimonoids we consider weighted finite automata over strong bimonoids where these weight structures can be considered as semirings which might lack distributivity then in general the well known run semantics initial algebra semantics and transition semantics of an automaton are different we prove an algebraic characterization for the initial algebra semantics in terms of stable finitely generated submonoids moreover for a given weighted finite automaton we construct the nerode automaton and myhill automaton both being crisp deterministic which are equivalent to the original automaton with respect to the initial algebra semantics respectively the transition semantics we prove necessary and sufficient conditions under which the nerode automaton and the myhill automaton are finite and we provide efficient algorithms for their construction also for a given weighted finite automaton we show sufficient conditions under which a given weighted finite automaton can be determinized preserving its run semantics

determinization of fuzzy automata via factorization of fuzzy states fuzzy finite state automata over the algebra 0 1 max 0 1 in which the monoid 0 1 1 denotes a continuous triangular norm is not locally finite can accept fuzzy languages of infinite range for a given fuzzy finite state automaton which accepts a fuzzy language of infinite range we define the determinization of the fuzzy automaton via factorization of fuzzy states i e the computation of an equivalent deterministic fuzzy automaton whether it is finite this method of determinization is a generalization of the well known accessible subset construction our main contribution is to determine that the representable cycles property is the necessary and sufficient condition for determinization of a fuzzy finite state automaton via a maximal factorization of fuzzy states this property is more general than the twins property adapted for fuzzy automata which is the sufficient condition for weighted automata over the tropical semiring

a new barrier to the existence of moving kinks in frenkel kontorova lattices an explanation is offered for an observed lower bound on the wave speed of travelling kinks in frenkel kontorova lattices kinks exist at discrete wavespeeds within a parameter regime where there is a resonance with linear waves phonons however they fail to exist even in this codimension one sense whenever there is more than one phonon branch in the dispersion relation inside such bands only quasi kinks with nondecaying oscillatory tails are possible the results are presented for a discrete sine gordon lattice with an onsite potential that has a tunable amount of anharmonicity novel numerical methods are used to trace kinks with topological charge q 1 and 2 in three parameters representing the propagation speed lattice discreteness and anharmonicity although none of the analysis is presented as rigorous mathematics numerical results suggest that the bound on allowable wavespeeds is sharp the results also explain why the vanishing at discrete values of anharmonicity of the peierls nabarro barrier between stationary kinks as discovered by savin et al does not lead to bifurcation of kinks with small wave speed 2003 elsevier b v all rights reserved

canonization of max min fuzzy automata abstract in this paper we propose a canonization method for fuzzy automata i e a determinization method that is able to return a minimal fuzzy deterministic automaton equivalent to the original fuzzy automaton the canonization method is derived from the well known brzozowski s algorithm for ordinary nondeterministic automata for a given fuzzy automaton a we prove that the construction m r n r a returns a minimal fuzzy deterministic automaton equivalent to a in that construction r represents the reversal of a fuzzy automaton n is the determinization of a fuzzy automaton based on fuzzy accessible subset construction and m is the determinization of a fuzzy automaton via factorization of fuzzy states which also includes a simple reduction of a particular case of proportional fuzzy states the method is accomplished for fuzzy automata with membership values over the godel structure also called max min fuzzy automata these fuzzy automata are always determinizable and have been proved useful in practical applications

on fuzzy regular languages abstract we introduce the concept of a max min fuzzy language fl m and a min max fuzzy language fl m recognized by a type of fuzzy automaton m we show that if l1 and l2 are finite valued f regular languages then so are l1 l2 and l1 l2 we give a form of a fuzzified pumping lemma which we use to give a necessary and sufficient condition for fl m to be nonconstant

minimization algorithms for sequential transducers abstract we present general algorithms for minimizing sequential finite state transducers that output strings or numbers the algorithms are shown to be efficient since in the case of acyclic transducers and for output strings they operate in o s e v e v f p max 1 steps where s is the sum of the lengths of all output labels of the resulting transducer e the set of transitions of the given transducer v the set of its states f the set of final states and p max one of the longest of the longest common prefixes of the output paths leaving each state of the transducer the algorithms apply to a larger class of transducers which includes subsequential transducers

asymptotic stability for tracking control of nonlinear uncertain dynamical systems described by differential inclusions in this paper we will study the tracking control problem for nonlinear uncertain dynamical systems two generalized feedback control inputs have been proposed such that the feedback controlled systems satisfy the complete tracking control property with exponential asymptotic stability and the trajectories of the systems are steered to the pre specified observation map with an exponential convergence rate moreover an estimate of the tracking time of the trajectories attaining the observation map has also been given an example inspired from a guided missile problem illustrates the use of our main results

equivalence reduction and minimization of finite fuzzy automata abstract a new algebraic approach to the problem of equivalence reduction and minimization of some kinds of fuzzy automata is given a system of necessary and sufficient conditions for equivalence of weakly initial fuzzy automata is formulated some algorithms considering the equivalence for two fuzzy automata are constructed

origin of multikinks in dispersive nonlinear systems we develop em the first analytical theory of multikinks for strongly em dispersive nonlinear systems considering the examples of the weakly discrete sine gordon model and the generalized frenkel kontorova model with a piecewise parabolic potential we reveal that there are no 2 pi kinks for this model but there exist em discrete sets of 2 pi n kinks for all n 1 we also show their bifurcation structure in driven damped systems

determinization of fuzzy automata with membership values in complete residuated lattices in this paper we introduce a new method for determinization of fuzzy finite automata with membership values in complete residuated lattices in comparison with the previous methods developed by belohlavek r belohlavek determinism and fuzzy automata information sciences 143 2002 205 209 and li and pedrycz y m li w pedrycz fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids fuzzy sets and systems 156 2005 68 92 our method always gives a smaller automaton and in some cases when the previous methods result in infinite automata our method can result in a finite one we also show that determinization of fuzzy automata is closely related to fuzzy right congruences on a free monoid and fuzzy automata associated with them and in particular to the concept of the nerode s fuzzy right congruence of a fuzzy automaton which we introduce and study here

equivalence reduction and minimization of finite automata over semirings abstract a new unified approach to equivalence reduction and minimization of finite automata over semirings based on free semimodules is presented it includes as special cases deterministic nondeterministic stochastic and fuzzy automata a necessary and sufficient condition for the existence of a finite behaviour matrix for a given automaton is proved different algorithmical decidability properties for equivalence reduction and minimization are connected to a noetherian property in the category s mod c of the semimodules over the semiring c

moving kinks and nanopterons in the nonlinear klein gordon lattice we study moving topological solitons kinks and antikinks in the nonlinear klein gordon chain these solitons are shown to exist with both monotonic non oscillating and oscillating asymptotics tails using the pseudo spectral method the anti kink solutions with oscillating background so called nanopterons are found as travelling waves of permanent profile propagating with constant velocity each of these solutions may be considered as a bound state of an anti kink with a background nonlinear periodic wave so that the wave pushes the anti kink over the peierls nabarro barrier the stability of these bound states is confirmed numerically travelling wave solutions of permanent profile are shown to exist depending on the convexity of the on site substrate potential the set of velocities at which the anti kinks with monotonic asymptotics propagate freely is calculated we also find moving non oscillating anti kink profiles with higher topological charges each of which appears to be the bound state of anti kinks with lower topological charge jq jd 1 2000 elsevier science b v all rights reserved

feedback controlled differential inclusions and stabilization of uncertain dynamical systems deterministic design of stabilizing feedback controls for a class of uncertain dynamical systems is considered adopting a problem formulation based on differential inclusions to encompass all possible realizations of uncertainty the authors describe a class of generalized feedback controls with practical analogues in the form of discontinuous feedbacks that guarantees global uniform ultimate boundedness moreover under a matched uncertainty hypothesis this class of controls guarantees global uniform asymptotic stability of the zero state and ultimate attainment of prescribed model behaviour

dynamical transitions between equilibria in a dissipative klein gordon lattice abstract we consider the energy landscape of a dissipative klein gordon lattice with a 4 on site potential our analysis is based on suitable energy arguments combined with a discrete version of the ojasiewicz inequality in order to justify the convergence to a single nontrivial equilibrium for all initial configurations of the lattice then global bifurcation theory is explored to illustrate that in the discrete regime all linear states lead to nonlinear generalizations of equilibrium states direct numerical simulations reveal the rich structure of the equilibrium set consisting of non trivial topological kink shaped interpolations between the adjacent minima of the on site potential and the wealth of dynamical convergence possibilities these dynamical evolution results also provide insight on the potential stability of the equilibrium branches and glimpses of the emerging global bifurcation structure elucidating the role of the interplay between discreteness nonlinearity and dissipation

algebraic aspects of families of fuzzy languages we study operations on fuzzy languages such as union concatenation kleene intersection with regular fuzzy languages and several kinds of iterated fuzzy substitution then we consider families of fuzzy languages closed under a fixed collection of these operations which results in the concept of full abstract family of fuzzy languages or full affl this algebraic structure is the fuzzy counterpart of the notion of full abstract family of languages that has been encountered frequently in investigating families of crisp i e non fuzzy languages some simpler and more complicated algebraic structures such as full substitution closed affl full super affl full hyper affl will be considered as well in the second part of the paper we focus our attention to full affls closed under iterated parallel fuzzy substitution where the iterating process is prescribed by given crisp control languages proceeding inductively over the family of these control languages yields an infinite sequence of full affl structures with increasingly stronger closure properties

stability radius optimization a geometric approach abstract the subject of this paper is a geometric approach to the robustness optimization problem for uncertain finite dimensional linear systems for the structurally perturbed stabilizable pair a b we show that optimally robust feedback stabilization is possible via the class of feedbacks parameterizing a b feedback invariant subspaces of codimension rk b a direct consequence of this geometric approach is a reduction in dimension of the optimization problem

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

translationally invariant discrete kinks from one dimensional maps for most discretizations of the phi4 theory the stationary kink can only be centered either on a lattice site or midway between two adjacent sites we search for exceptional discretizations that allow stationary kinks to be centered anywhere between the sites we show that this translational invariance of the kink implies the existence of an underlying one dimensional map phi n 1 f phi n a simple algorithm based on this observation generates three families of exceptional discretizations

high speed kinks in a generalized discrete phi 4 model we consider a generalized discrete ensuremath phi 4 model and demonstrate that it can support exact moving kink solutions in the form of tanh with an arbitrarily large velocity the constructed exact moving solutions are dependent on the specific value of the propagation velocity we demonstrate that in this class of models given a specific velocity the problem of finding the exact moving solution is integrable namely this problem originally expressed as a three point map can be reduced to a two point map from which the exact moving solutions can be derived iteratively it was also found that these high speed kinks can be stable and robust against perturbations introduced in the initial conditions

exceptional discretizations of the sine gordon equation the method of one dimensional maps was recently introduced as a means of generating exceptional discretizations of the phi 4 theory i e discrete phi 4 models which support kinks centered at a continuous range of positions relative to the lattice in this paper we employ this method to obtain exceptional discretizations of the sine gordon equation i e exceptional frenkel kontorova chains we also use one dimensional maps to construct a discrete sine gordon equation supporting kinks which move with arbitrary velocities without emitting radiation

output feedback control of uncertain coupled systems the stabilization of uncertain dynamical systems on n that admit a decomposition into two coupled subsystems of dimension nc and nr respectively is studied we refer to the nc dimensional sybsystem as the reduced order system and to the nr dimensional subsystem as the residual system the overall n dimensional system is termed the full system the prototype for which is a mechanical model arising in the study of the active control of structures each subsystem is modelled by a differential equation with a linear nominal part and a nonlinear perturbation of a specified class the output available for feedback purposes is an nc valued linear combination of the state components of the full system and is subject to bounded measurement noise with known bound an output feedback strategy is described and sufficient conditions are obtained for the existence of a calculable global uniform compact attractor containing the state origin for controlled systems of this class

standard nearest neighbour discretizations of klein gordon models cannot preserve both energy and linear momentum we consider nonlinear klein gordon wave equations and illustrate that standard discretizations thereof involving nearest neighbours may preserve either standardly defined linear momentum or standardly defined total energy but not both this has a variety of intriguing implications for the non potential discretizations that preserve only the linear momentum such as the self accelerating or self decelerating motion of coherent structures such as discrete kinks in these nonlinear lattices

normal form for travelling kinks in discrete klein gordon lattices abstract we study travelling kinks in the spatial discretizations of the nonlinear klein gordon equation which include the discrete 4 lattice and the discrete sine gordon lattice the differential advance delay equation for travelling kinks is reduced to the normal form a scalar fourth order differential equation near the quadruple zero eigenvalue we show numerically the non existence of monotonic kinks heteroclinic orbits between adjacent equilibrium points in the fourth order equation making generic assumptions on the reduced fourth order equation we prove the persistence of bounded solutions heteroclinic connections between periodic solutions near adjacent equilibrium points in the full differential advance delay equation with the technique of centre manifold reduction existence of multiple kinks in the discrete sine gordon equation is discussed in connection to recent numerical results of aigner et al a a aigner a r champneys v m rothos a new barrier to the existence of moving kinks in frenkel kontorova lattices physica d 186 2003 148 170 and results of our normal form analysis

fuzzy language on free monoid in this paper we shall introduce the notion of fuzzy regular language adjunctive language and dense language and discuss their algebraic character and algebraic properties

on a class of discretizations of hamiltonian nonlinear partial differential equations abstract we present a new class of discretizations of partial differential equations pdes that preserve a momentum like integral of the motion this results in an effective translational invariance for the dynamical problem and the absence of a peierls nabarro barrier that is usually present in discretizations of hamiltonian pdes a general method to construct such discretizations for any nonlinearity is given and the properties of the resulting differential difference equations are analyzed in a number of different cases for nonlinear klein gordon as well for nonlinear schrodinger type systems while for the former nonintegrability of the dynamical problem is evident in the latter case numerical evidence suggests that the behavior is close to the one of integrable systems we also show how static solutions of the equations can be constructed for these discretizations and discuss the similarities and differences of the method with previously reported ones

conditions for the existence of maximal factorizations abstract extending classical algorithms for ordinary weighted or string to string automata to automata with underlying more general algebraic structures is of significant practical and theoretical interest however the generalization of classical algorithms sets certain assumptions on the underlying structure in this respect the maximal factorization turns out to be a sufficient condition for many practical problems e g minimization and canonization recently an axiomatic approach on monoid structures suggested that monoids with most general equalizer mge monoids provide an alternative framework to achieve similar results in this paper we study the fundamental relation between monoids admitting a maximal factorization and mge monoids we describe necessary conditions for the existence of a maximal factorization and provide sufficient conditions for an mge monoid to admit a maximal factorization

bifurcations of travelling wave solutions in the discrete nls equations abstract we study discrete nonlinear schrodinger nls equations which include the cubic nls lattice with on site interactions and the integrable ablowitz ladik lattice standing wave solutions are known to exist in the discrete nls equations outside of the finite spectral band we study travelling wave solutions which have nonlinear resonances with unbounded linear spectrum by using center manifold and normal form reductions we show that a continuous nls equation with the third order derivative term is a canonical normal form for the discrete nls equation near the zero dispersion limit bifurcations of travelling wave solutions near the zero dispersion limit are analyzed in the framework of the third order derivative nls equation we show that there exists a continuous two parameter family of single humped travelling wave solutions in the third order derivative nls equation when it is derived from the integrable ablowitz ladik lattice on the contrary there are no single humped solutions in the third order derivative nls equation when it is derived from the cubic nls equation with on site interactions nevertheless we show that there exists an infinite discrete set of one parameter families of double humped travelling wave solutions in the latter case our results are valid in the neighborhood of the zero dispersion point on the two parameter plane of travelling wave solutions

finite automata theory with membership values in lattices in this paper we study finite automata with membership values in a lattice which are called lattice valued finite automata the extended subset construction of lattice valued finite automata is introduced then the equivalences between lattice valued finite automata lattice valued deterministic finite automata and lattice valued finite automata with e moves are proved a simple characterization of lattice valued languages recognized by lattice valued finite automata is given then it is proved that the kleene theorem holds in the frame of lattice setting a minimization algorithm of lattice valued deterministic finite automata is presented in particular the role of the distributive law for the truth valued domain of finite automata is analyzed the distributive law is not necessary to many constructions of lattice valued finite automata but it indeed provides some convenience in simply processing lattice valued finite automata

fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids we study fuzzy finite automata in which all fuzzy sets are defined by membership functions whose codomain forms a lattice ordered monoid l for these l fuzzy finite automata l ffa for short we provide necessary and sufficient conditions for the extendability of the state transition function it is shown that nondeterministic l ffa nl ffa for short are more powerful than deterministic l ffa dl ffa for short then we give necessary and sufficient conditions for the simulation of an nl ffa by an equivalent dl ffa next we turn to the closure properties of languages defined by l ffas we establish closure under the regular operations and provide conditions for closure under intersection and reversal in particular we show that the family of fuzzy languages accepted by dl ffas is not closed under kleene closure operation and the family of fuzzy languages accepted by nl ffas is not closed under complement operation furthermore we introduce the notions of l fuzzy regular expressions and give the kleene theorem for nl ffas the description of dl ffas by l fuzzy regular expressions is also given finally we investigate the level structures of l ffas our results provide some insight as to what extend properties of l ffas and their languages depend on the algebraic properties of l

viable control for uncertain nonlinear dynamical systems described by differential inclusions in this paper we will study the viable control problem for a class of uncertain nonlinear dynamical systems described by a differential inclusion the goal is to construct a feedback control such that all trajectories of the system are viable in a map moreover for any initial states no viable in the map under the feedback control all solutions of the system are steered to the map with an exponential convergence rate and viable in the map after a finite time t in this case an estimate of the time t of all trajectories attaining the map is given in the nanomedicine system an example inspired from cerebral embolism and cerebral thrombosis problems illustrates the use of our main results

a formal model of computing with words classical automata are formal models of computing with values fuzzy automata are generalizations of classical automata where the knowledge about the system s next state is vague or uncertain it is worth noting that like classical automata fuzzy automata can only process strings of input symbols therefore such fuzzy automata are still abstract devices for computing with values although a certain vagueness or uncertainty are involved in the process of computation we introduce a new kind of fuzzy automata whose inputs are instead strings of fuzzy subsets of the input alphabet these new fuzzy automata may serve as formal models of computing with words we establish an extension principle from computing with values to computing with words this principle indicates that computing with words can be implemented with computing with values with the price of a big amount of extra computations

weighted finite automata over strong bimonoids we investigate weighted finite automata over strings and strong bimonoids such algebraic structures satisfy the same laws as semirings except that no distributivity laws need to hold we define two different behaviors and prove precise characterizations for them if the underlying strong bimonoid satisfies local finiteness conditions moreover we show that in this case the given weighted automata can be determinized

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

determinism and fuzzy automata the concept of a fuzzy automaton as studied in the literature is a direct generalization of the classical concept of a non deterministic automaton in this paper we consider the question of whether there is a deterministic counterpart to the notion of a fuzzy automaton we propose a definition of deterministic fuzzy automata and show that they are equally powerful as fuzzy automata compared to the classical notion of a deterministic automaton the deterministic fuzzy automaton defined in our paper differs essentially only in that the final states form a fuzzy set moreover we show that deterministic fuzzy automata are equally powerful as nested systems of deterministic automata

continuum limit of lattice approximation schemes boundary layer perturbation theory problems are inherently singular however it is known that discretizing the problem by introducing a lattice may convert such problems into regular perturbation problems the singular nature of boundary layer problems is then relegated to and hidden in the continuum limit the subtle limit in which the lattice spacing tends to zero if the lattice is introduced cavalierly then extrapolating to zero lattice spacing gives a sequence of extrapolants that at first approaches the correct limit and then veers off thereby revealing the asymptotic nature of such problems however discretizing the problem following the procedures described here yields lattice approximations that have a smooth and regular continuum limit these ideas are illustrated by three nonlinear ordinary differential equations the cubic equation that describes instantons an oscillator equation having a quadratic nonlinearity and the blasius equation

one parameter localized traveling waves in nonlinear schrodinger lattices we address the existence of traveling single humped localized solutions in the spatially discrete nonlinear schr nls equation a mathematical technique is developed for analysis of persistence of these solutions from a certain limit in which the dispersion relation of linear waves contains a triple zero the technique is based on using the implicit function theorem for solution of an appropriate differential advance delay equation in exponentially weighted spaces the resulting melnikov calculation relies on a number of assumptions on the spectrum of the linearization around the pulse which are checked numerically we apply the technique to the so called salerno model and the translationally invariant discrete nls equation with a cubic nonlinearity we show that the traveling solutions terminate in the salerno model whereas they generally persist in the translationally invariant nls lattice as a one parameter family of solutions these results are found to be in a close correspondence with numerical approximations of traveling solutions with zero radiation tails analysis of persistence also predicts the spectral stability of the one parameter family of traveling solutions under time evolution of the discrete nls equation c 2007 elsevier b v all rights reserved

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

discrete klein gordon models with static kinks free of the peierls nabarro potential for the nonlinear klein gordon type models we describe a general method of discretization in which the static kink can be placed anywhere with respect to the lattice these discrete models are therefore free of the peierls nabarro potential previously reported models of this type are shown to belong to a wider class of models derived by means of the proposed method a relevant physical consequence of our findings is the existence of a wide class of discrete klein gordon models where slow kinks practically do not experience the action of the peierls nabarro potential such kinks are not trapped by the lattice and they can be accelerated by even weak external fields