enumerating fuzzy switching functions and free kleene algebras abstract we investigate the free kleene algebra on n free generators fk n this algebra is essentially the same as the set of n variable fuzzy switching functions the cardinality of fk n has been previously determined only for n 1 2 and 3 we present a general method for enumeration problems of this kind we show fk 4 has 160 297 985 276 elements the final section contains new upper and lower bounds for the cardinality of fk n we also consider the number of regular functions and show that there are 11 197 129 615 and 430 904 428 717 such functions in 1 2 3 and 4 variables

prolongation structures of complex quasi polynomial evolution equations the authors use wahlquist estabrook prolongation theory 1976 to investigate second order complex equations of generalised nls dnls type they isolate a number of new integrable cases

cech de rham theory for leaf spaces of foliations we present a new cech de rham model for the cohomology of the classifying space of a foliated manifold this model enables us to lift the construction of known characteristic classes in the cohomology of the manifold to the cohomology of the classifying space by standard geometric methods and without making any reference to necessarily non hausdorff groupoids and their classifying spaces we also show how the cech de rham model can be used to prove some other known formulas as well as a version of poincare duality for foliations

painleve analysis of the non linear schrodinger family of equations the authors apply the painleve test to the generalised derivative non linear schrodinger equation iut uxx iauu ux ibu2ux cu3u 2 where u denotes the complex conjugate of u and a b and c are real constants to determine under what conditions the equation might be completely integrable it is shown that apart from a trivial multiplicative factor this equation possesses the painleve property for partial differential equations as formulated by weiss tabor and carnevale 1983 only if c 1 4b 2b a when then this relation holds this is equivalent under a gauge transformation to the derivative non linear schrodinger equation dnls of kaup and newell which is known to be completely integrable or else to a linear equation

factorisation of backlund transformations the factorisation of the l operator in a lax pair l p has played an important role in the theory of integrable systems for example the miura transformation can be immediately obtained from the factorisation of the schrodinger operator delta x2 u k2 delta x a delta x b requires that u k2 ax a2 and a b for zero curvature representations this procedure is not available however the author attempts to produce an analogous theory of factorisations for zero curvature representations

the prolongation structures of quasi polynomial flows we look closely at the process of finding a wahlquist estabrook prolongation structure for a given system of nonlinear evolution equation s there are two main steps in this calculation the first to reduce the problem to the investigation of a finitely generated free lie algebra with constraints the second to find a finite dimensional linear representation of these generators we discuss some of the difficulties that arise in this calculation for quasi polynomial flows defined later we give an algorithm for the first step we do not totally solve the problems of the second step but do give an algebraic framework and a number of techniques that are quite generally applicable we illustrate these methods with many examples several of which are new

lax pairs of time dependent gross pitaevskii equation we calculate the lax pairs of homogeneous and inhomogeneous one dimensional time dependent gross pitaevskii equations with time dependent scattering length the inhomogeneity corresponds to linear and quadratic potentials our approach introduces a systematic method of searching for the lax pair corresponding to a given differential equation we derive known lax pairs for the gross pitaevskii equation with homogeneous and quadratic potentials and time dependent scattering length we also derive new lax pairs corresponding to a gross pitaevskii equation with a linear potential using the resulting lax pairs the darboux transformation can be performed and exact solutions of the gross pitaevskii equation can be obtained for experimentally relevant cases such as solitonic solutions

flat bundles and residues for foliations in this note we compute the residues for a vector field preserving a foliation we assume the vector field disappears only to first order along its singular set and that its singular set consists of closed separated leaves under these conditions the residue is completely determined by the characteristic classes of the flat bundle v o which is the normal bundle v of the foliation restricted to the singular set and the action of the vector field on v o for a foliation of even codimension q whose normal bundle has euler class zero we interpret the pontrjagin ring of the normal bundle in dimension 2q as an obstruction to the existence of such a vector field we show that if the euler class of the normal bundle of a foliation is zero and it admits such a vector field then many of its secondary characteristic classes must be also be zero this is one of the few results relating the geometry of a foliation to its characteristic classes we will make extensive use of the material in 6 it will be very helpful to the reader to be familiar with this paper

regular ternary logic functions ternary logic functions suitable for treating ambiguity a special group of ternary functions called regular ternary logic functions are defined these functions are useful in switching theory programming languages algorithm theory and many other fields if we are concerned with the indefinite state in such fields this correspondence describes the fundamental properties and representations of the regular ternary logic functions

integrability of kersten krasil shchik coupled kdv mkdv equations singularity analysis and lax pair the integrability of a coupled kdv mkdv system is tested by means of singularity analysis the true lax pair associated with this system is obtained by the use of prolongation technique

integrability of a coupled kdv system painleve property lax pair and backlund transformation the integrability of a coupled kdv system is studied by prolongation technique and singularity analysis as a result backlund transformation and linear spectral problem associated with this system are obtained some special solutions of the system are also proposed

multiple valued logic its status and its future multiple valued logic in which the number of discrete logic levels is not confined to two has been the subject of much research over many years the practical objective of this work has been to increase the information content of the digital signals in a system to a higher value than that provided by binary operation in this tutorial survey paper we will review the historical developments in this field both in circuit realizations and in methods of handling multiple valued design data and consider the present state of the art and future expectations

prolongation structures and exact solutions of k m n equations in this paper rosenau and hyman s phys rev lett 70 564 1993 k m n equations are studied by prolongation technique it is proved that k m n equations are lax integrable only for certain special parameters m n the nontrivial prolongation structures and lax pairs for the integrable cases are given finally as an example the one and two soliton solutions for the k 12 12 equation are derived by means of inverse scattering method

miura type transformations and homogeneous spaces we relate miura type transformations mts over an evolution system to its zero curvature representations with values in lie algebras we prove that certain homogeneous spaces of produce mts and show how to distinguish these spaces for a scalar translation invariant evolution equation this allows us to classify all mts in terms of homogeneous spaces of the wahlquist estabrook algebra of the equation for other evolution systems this allows us to construct some mts as an example we study mts over the kdv equation a fifth order equation of harry dym type and the coupled kdv mkdv system of kersten and krasilshchik

integrability of the coupled kdv equations derived from two layer fluids prolongation structures and miura transformations abstract the lax integrability of the coupled kdv equations derived from two layer fluids s y lou b tong h c hu x y tang coupled kdv equations derived from two layer fluids j phys a math gen 39 2006 513 527 is investigated by means of prolongation technique as a result the lax pairs of some painleve integrable coupled kdv equations and several new coupled kdv equations are obtained finally the miura transformations and some coupled modified kdv equations associated with the lax integrable coupled kdv equations are derived by an easy way

discrete interval truth values logic and its application in this paper we focus on functions defined on a special subset of the power set of 0 1 r 1 the elements in the subset will be called discrete interval truth values and operations on the truth values the operations discussed in this paper will be called regular because they are one of the extensions of the regularity which was first introduced by kleene in his ternary logic mukaidono investigated some properties of ternary functions which can be represented by the regular operations he called such ternary functions regular ternary logic functions regular ternary logic functions are useful for representing and analyzing ambiguities such as transient states and or initial states in binary logic circuits that boolean functions cannot cope with furthermore they are also applied to studies of fail safe systems for binary logic circuits in this paper we will discuss an extension of regular ternary logic functions to functions on the discrete interval truth values first we will suggest an extension of the regularity in the sense of kleene into operations on the discrete interval truth values we will then present some mathematical properties of functions on the discrete interval truth values consisting of regular operations and one application of these functions

a representation of exponential form on fuzzy logic abstract this paper proposes a representation of exponential form ef in fuzzy logic by ef any fuzzy value a in fuzzy valued logic or fuzzy linguistic valued logic can be represented as b c where b is called the truth base and c the confidence exponent the basic idea of this form and its properties are discussed by using a different truth base the exponential form can be applied to represent positive and negative logic in fuzzy valued logic as well as fuzzy linguistic valued logic some applications of ef for approximate reasoning and fuzzy neural networks are also introduced in this paper

a comparative analysis of painleve lax pair and similarity transformation methods in obtaining the integrability conditions of nonlinear schrodinger equations we derive the integrability conditions of nonautonomous nonlinear schrodinger equations using the lax pair and similarity transformation methods we present a comparative analysis of these integrability conditions with those of the painleve method we show that while the painleve integrability conditions restrict the dispersion nonlinearity and dissipation gain coefficients to be space independent and the external potential to be only a quadratic function of position the lax pair and the similarity transformation methods allow for space dependent coefficients and an external potential that is not restricted to the quadratic form the integrability conditions of the painleve method are retrieved as a special case of our general integrability conditions we also derive the integrability conditions of nonautonomous nonlinear schrodinger equations for two and three spacial dimensions

scalar second order evolution equations possessing an irreducible sl2 valued zero curvature representation we find all scalar second order evolution equations that possess an sl2 valued zero curvature representation irreducible to a proper subalgebra of sl2 none of these zero curvature representations depends on a parameter that could serve as the spectral parameter

phase engineering modulational instability and solitons of gross pitaevskii type equations in 1 1 dimensions motivated by recent proposals of collisionally inhomogeneous bose einstein condensates becs which have a spatially modulated scattering length we introduce a phase imprint into the macroscopic order parameter governing the dynamics of becs with spatiotemporal varying scattering length described by a cubic gross pitaevskii gp equation and then suitably engineer the imprinted phase to generate the modified gp equation also called the cubic derivative nonlinear schrodinger nls equation this equation describes the dynamics of condensates with two body attractive and repulsive interactions in a time varying quadratic external potential we then carry out a theoretical analysis which invokes a lens type transformation that converts the cubic derivative nls equation into a modified nls equation with only explicit temporal dependence our analysis suggests a particular interest in a specific time varying potential with the strength of the magnetic trap similar to 1 t t 2 for a time varying quadratic external potential of this kind an explicit expression for the growth rate of a purely growing modulational instability is presented and analyzed we point out the effect of the imprint parameter and the parameter t on the instability growth rate as well as on the solitary waves of the becs doi 10 1103 physreve 87 022914

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

derivative nonlinear schrodinger equations and hermitian symmetric spaces it is shown that to each hermitian symmetric space there corresponds an integrable system of generalised derivative nonlinear schrodinger equations nls the nonlinear terms are related to the curvature tensor of the associated symmetric space the hamiltonian form of the equations is presented the results are an extension of those presented in an earlier paper 1983 on generalised nls equations associated with symmetric and reductive homogeneous spaces

nonautonomous integrable nonlinear schrodinger equations with generalized external potentials inhomogeneous nonlinear schrodinger equations are constructed systematically using the covariance with respect to the darboux transformation the darboux invariants and the spectral parameter are functions of time and space which result in equations describing nonautonomous solitons moving in generalized external potentials all the known inhomogeneous nonlinear schrodinger equations as well as new ones of a more generalized form can be derived from the constructed equation one and two soliton solutions are explicitly constructed using the darboux transformation

korteweg de vries burgers equation and the painleve property it is shown that the korteweg de vries burgers equation possesses the painleve property conditionally using an algorithmic approach a travelling wave solution is reproduced

some fundamental properties of multiple valued kleenean functions and determination of their logic formulas multiple valued kleenean functions that are models of a kleene algebra and are logic functions expressed by logic formulas composed of variables constants and logic operations and or and not are discussed the set of kleenean functions is a model with the largest number of logic functions among existing models of a kleene algebra such as fuzzy logic functions regular ternary logic functions and b ternary logic functions mainly it is shown that any p valued kleenean function is derived from a monotonic ternary input functions and any p valued unate function is derived from a unate binary input function the mapping relations between them and the method to determine the logic formula of the kleenean function and unate function from that of the monotonic ternary input function and unate binary input function respectively are classified 7 or less valued kleenean functions and unate functions of 3 or fewer variables are enumerated it is known that the number of p valued kleenean functions increases stepwise and that of unate functions increases smoothly as p becomes larger

the dynamics of open foliated manifolds and a vanishing theorem for the godbillon vey class let m f be a c3 foliated n manifold with leaves of dimension n 1 let gv f e h3 m ir denote the characeristic class introduced by godbillon and vey in g v considerable effort has been made to discover the geometric significance of this invariant recall thurston s helical wobble th which can be interpreted via the formulas in r w in addition to celebrated examples of foliations for which gv f does not vanish g v th br the list of foliations for which it does has been growing steadily by work of herman her and wallet wi gv sr 0 for foliated circle bundles and interval bundles over t2 this result has interesting consequences that are far from being obvious thus morita and tsuboi have deduced the vanishing of gv x for closed foliated manifolds without holonomy m t and mizutani morita and tsuboi have extended this to foliations almost without holonomy m m t in all of the above vanishing theorems the leaves of the foliations have polynomial growth other classes of foliations with all leaves polynomial and gv f 0 were found by nishimori n sergiescu ser and tsuchiya ts in a preprint c c2 that became obsolete within a few months of its distribution the present authors proved the vanishing theorem for all closed foliated manifolds in which every leaf has polynomial growth indeed we proved that if every leaf is nonresilient l and if no leaf lies at infinite level then gv st 0

similarity transformations for nonlinear schrodinger equations with time dependent coefficients abstract in this paper we obtain a similarity transformation connecting nonlinear schrodinger equations with time varying coefficients with the autonomous cubic nonlinear schrodinger equation as applications we construct exact breathing solutions to multidimensional non autonomous nonlinear schrodinger equations and discuss how to construct time dependent coefficients leading to solutions which collapse weakly in three dimensional scenarios our results are applicable to the study of the dynamics of bose einstein condensates in the mean field limit and dispersion managed optical systems

the prospects for multivalued logic a technology and applications view advances in multiple valued logic mvl have been inspired in large part by advances in integrated circuit technology multiple valued logic has matured to the point where four valued logic is now part of commercially available vlsi ic s besides reduction in chip area mvl offers other benefits such as the potential for circuit test this paper describes the historical and technical background of mvl and areas of present and future application it is intended as well to serve as a tutorial for the nonspecialist

meaningful special classes of ternary logic functions regular ternary logic functions and ternary majority functions special ternary logic functions namely regular functions suitable for treating ambiguity and majority functions based on an extended majority principle are considered the one to one correspondence between n ary monotone regular ternary logic functions and n 1 ary monotone boolean functions is investigated as well as the one to one correspondence between n ary monotone ternary majority functions and n 1 ary monotone binary threshold functions the ternary majority functions are enumerated

algebraic structures in fuzzy logic abstract common features in various fuzzy logical systems are investigated it turns out that adjoint couples and residuated lattices are very often present though not always explicitly expressed some minor new results are also proved

global invariants for measured foliations new exotic invariants for measured foliations are constructed which we call the u classes of a pair f u the dependence of the s classes on the geometry of the foliation gis examined and the dynamics of a foliation is shown to determine the u classes in many cases we use the u classes to study the classifying space bfsl of foliations with a transverse invariant volume form and we show the homotopy groups of bfsl are uncountably generated starting in degrees q 3 new invariants for groups of volume preserving diffeomorphisms also arise from the t classes these invariants are nontrivial and related to the geometric aspects of the group action relations between the tl classes and the secondary classes of a foliation are exhibited 0 introduction in this paper we construct new exotic characteristic classes for foliations which have a transverse invariant measure these invariants which we call the t classes are similar to the secondary classes and take values in the real cohomology h m of the foliated manifold m theorem 1 6 the values of the m classes are strongly related to the geometry of the foliation proposition 3 4 the u classes have applications to the study of foliations with a transverse invariant volume form theorems 5 1 and 5 2 to the computations of the homology groups of the discrete group of volume preserving diffeomorphisms of rw theorem 7 4 and to the problem of relating the secondary classes of a foliation with its global geometry 15 201 the a classes arose from an attempt to obtain general invariants of a foliation which are intermediate between the leaf and secondary classes they are a special case of a family of universal invariants associated to a c2 foliation the weil operators in codimension one the introduction of the weil operator is a decisive ingredient in duminy s elegant proof of the sullivan conjecture about the godbillon vey class 9 the properties of the weil operators in higher codimensions are given in 15 where they are used to study the residuable secondary classes we briefly describe the construction of the weil operators and their relations with the l classes and other foliation invariants in 2 for a foliation with all leaves compact we note that the weil operators are identically zero proposition 2 6 we denote by f a c2 foliation of codimension q on a paracompact manifold m and i denotes a transverse invariant measure for sf recall that ti assigns to each received by the editors june 28 1982 and in revised form november 9 1982 1980 mathematics subject classification primary 57r30 57r20 58f1 1 secondary 58h10 28d15

integrable nonautonomous nonlinear schrodinger equations are equivalent to the standard autonomous equation a class of nonautonomous nonlinear schrodinger equations claiming to be novel integrable systems with rich properties continues to appear in the literature all such equations are shown to be not new but equivalent to the standard autonomous equation which trivially explains their integrability features

p functions ternary logic functions capable of correcting input failures and suitable for treating ambiguities the authors consider a new class of special ternary logic functions the p functions capable of correcting input failures based on the regular ternary logic there are three major results first the p functions are a special subset of the regular functions with no information loss second a ternary logic function defined by the all prime implicant disjoint of any boolean connective is a p function third a p function may be identified from the irredundant disjunctive form the relationship between p functions and other meaningful ternary logic functions is discussed

relationship between p valued majority functions and p valued threshold functions in a previous paper trans iece japan vol j63 d p 493 500 1980 vol j64 d p 172 3 1981 and vol e 67 p 47 8 1984 the authors defined a new class of multiple valued logic functions called multiple valued majority functions the authors clarifies the distinction of multiple valued majority functions from multiple valued threshold functions through the difference between a number function and an inner product of an input vector and a weight vector

interval valued fuzzy sets representation on multiple antecedent fuzzy s implications and reasoning abstract interval valued fuzzy sets are used to represent multiple antecedent s implications based on normal forms where linguistic connectives antecedents and consequents are assumed to be fuzzy these intervals are represented equivalently with their lower and upper bounds general forms are given for the bounds of multiple antecedent fuzzy s implications bounds for multiple antecedent reasoning are also investigated based on the bounds of multiple antecedent s implications it is suggested that modification functions utilized in approximation analogical reasoning aar be determined in such a way that consequents deduced by aar fall within the bounds of the reasoning an example is given to show how the bounds might be used to choose a modification scheme in aar

comparison of fuzzy reasoning methods l a zadeh e h mamdani and m mizumoto et al have proposed methods for fuzzy reasoning in which the antecedent involves a fuzzy conditional proposition if x is a then y is b with a and b being fuzzy concepts mizumoto et al have investigated the properties of their methods in the case of generalized modus ponens this paper deals with the properties of their methods in the case of generalized modus tollens and investigates the other new fuzzy reasoning methods obtained by introducing the implication rules of many valued logic systems finally the properties of syllogism and contrapositive are investigated under each fuzzy reasoning method

the extended estabrook wahlquist method abstract variable coefficient korteweg de vries vckdv modified korteweg de vries vcmkdv and nonlinear schrodinger nls equations have a long history dating from their derivation in various applications a technique based on extended lax pairs has been devised recently to derive time and space dependent coefficient generalizations of various such lax integrable nlpde hierarchies which are thus more general than almost all cases considered earlier via methods such as the painleve test bell polynomials and similarity methods however this technique although operationally effective has the significant disadvantage that for any integrable system with spatiotemporally varying coefficients one must guess a generalization of the structure of the known lax pair for the corresponding system with constant coefficients motivated by the somewhat arbitrary nature of the above procedure we embark in this paper on an attempt to systematize the derivation of lax integrable systems with variable coefficients we consider the estabrook wahlquist ew prolongation technique a relatively self consistent procedure requiring little prior information however this immediately requires that the technique be significantly generalized in several ways including solving matrix partial differential equations instead of algebraic ones as the structure of the lax pair is systematically computed and also in solving the constraint equations to deduce the explicit forms for various coefficient matrices the new and extended ew technique which results is illustrated by algorithmically deriving generalized lax integrable versions of the nls generalized fifth order kdv mkdv and derivative nonlinear schrodinger dnls equations we also show how this method correctly excludes the existence of a nontrivial lax pair for a nonintegrable nlpde such as the variable coefficient cubic quintic nls

prolongation structures and lie algebra real forms a technique useful as an aid for finding finite dimensional representations of a wahlquist estabrook prolongation algebra is introduced the technique is illustrated with an examination of equations of the form 2za x1 x2 fa zb

traveling waves to a burgers korteweg de vries type equation with higher order nonlinearities abstract in this paper first we survey some recent advances in the study of traveling wave solutions to the burgers korteweg de vries equation and some comments are given then we study a burgers korteweg de vries type equation with higher order nonlinearities a qualitative analysis to a two dimensional autonomous system which is equivalent to the burgers kdv type equation is presented and indicates that under certain conditions the burgers korteweg de vries type equation has neither nontrivial bell profile solitary waves nor periodic waves finally a solitary wave solution is obtained by means of the first integral method which is based on the ring theory of commutative algebra