short communication some remarks on the category fuz h of m eytan the aim of this paper is to disprove a result asserted by m eytan in his paper fuzzy sets a topos logical point of view in this journal vol 5 pp 47 67 namely that the category fuz h is a topos for any complete heyting algebra h it is proved that this result can be true only when h is a boolean algebra

partial ordering in l underdeterminate sets abstract the purpose of this paper is to improve results on fuzzy partial orderings obtained by zadeh in 9 we overcome the difficulties connected with the axioms of antisymmetry and linearity moreover if the underlying lattice l is a complete residuated lattice we establish a szpilrajn theorem i e any l fuzzy partial ordering has a linear extension in opposition to zadeh s our point of view is that an axiom of antisymmetry without a reference to a concept of equality is meaningless therefore we first introduce the category lus cf 2 which can be considered as a mathematical model of fuzzy equality and subsequently we specify the axioms of l fuzzy partial orderings with respect to the frame given by lus the axioms we use clearly display the usefulness of having a zadeh like complementation and as a consequence the usefulness of a positivistic and nonintuitionistic frame of study an example concerning l rn which we give clearly shows that the lus version of the szpilrajn theorem cannot be reduced to a fuzzification of an already existing theorem but provides us with additional information

wronskian solutions of the constrained kadomtsev petviashvili hierarchy the integrable kadomtsev petviashvili kp hierarchy is compatible with generalized k constraints of the type lk i qi 1xri a large class of solutions among them solitons can be represented by wronskian determinants of functions satisfying a set of linear equations in this paper we shall obtain additional conditions for these functions imposed by the constraints

on the constrained b type kadomtsev petviashvili hierarchy hirota bilinear equations and virasoro symmetry we derive the bilinear equations of the constrained bkp hierarchy from the calculus of pseudodifferential operators the full hierarchy equations can be expressed in hirota s bilinear form characterized by the functions and besides we also give a modification of the original orlov schulman additional symmetry to preserve the constrained form of the lax operator for this hierarchy the vector fields associated with the modified additional symmetry turn out to satisfy a truncated centerless virasoro algebra

constrained kp hierarchy and bi hamiltonian structures the kakomtsev petviashvili kp hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions constraining the kp flows in terms of squared eigenfunctions one obtains 1 1 dimensional integrable equations with scattering problems given by pseudo differential lax operators the bi hamiltonian nature of these systems is shown by a systematic construction of two general poisson brackets on the algebra of associated lax operators gauge transformations provide miura links to modified equations these systems are constrained flows of the modified kp hierarchy for which again a general description of their bi hamiltonian nature is given the gauge transformations are shown to be poisson maps relating the bi hamiltonian structures of the constrained kp hierarchy and the modified kp hierarchy the simplest realization of this scheme yields the akns hierarchy and its miura link to the kaup broer hierarchy

short communication the categories set h and fuz h this paper follows up d ponasse s paper some remarks on the category fuz h of m eytan in this journal 4 in particular it gives a new proof that fuz h cannot have a subobject classifier if h is a complete chain with at least three elements

a survey of fuzzy set and topos theory abstract this paper is a comparison and contrast of approaches to many valued mathematics offered by fuzzy set theory and topos theory it gives a survey of the categorical foundations of fuzzy set theory and related topoi topoi are not a basis for fuzzy set theory butthey do suggest appropriate directions to go and questions to ask for a synthesis which does provide a foundation one possible structure which has a lopos like internal logic and the rich variety of loggical connectives used in fuzzy sets is included

solitons from dressing in an algebraic approach to the constrained kp heirachy the algebraic matrix hierarchy approach based on affine lie sl n algebras leads to a variety of 1 1 soliton equations by varying the rank of the underlying sl n algebra as well as its gradation in the affine setting one encompasses the set of the soliton equations of the constrained kp hierarchy the soliton solutions are then obtained as elements of the orbits of the dressing transformations constructed in terms of representations of the vertex operators of the affine sl n algebras realized in the unconventional gradations such soliton solutions exhibit non trivial dependence on the kdv odd time flows and kp odd and even time flows which distinguishes them from the conventional structure of the darboux backlund wronskian solutions of the constrained kp hierarchy

on a categorical analysis of zadeh generalized subsets of sets ii how can fuzzy sets be viewed as generalized subsets of actual sets an answer to this question is given via a categorical analysis of the synchronic approach of fuzzy set theory starting from an abstract definition of a category with fuzzy subsets we establish necessary and sufficient conditions on a posst to be that of truth values for such a category these conditions justify zadeh s original choice of the unit segment as a set of truth values some defects concerning universal constructions in a non trivial category with fuzzy subsets are mentioned and a natural best toposophical approximation of such a category is proposed

reconstruction possibilities for fuzzy sets towards a new level of understanding abstract using the methodological approach of rational reconstruction some possible explications of the notion of fuzzy set are critically discussed with respect to their intuitive meaning plausibility and consistency together with a historical survey we try to provide a better understanding of the development of various concepts within traditional fuzzy set theory based on the approaches of 39 and 18 we show that zadeh s original intuition of a fuzzy set and its formalization as generalized membership function can be reinterpreted in a plausible and consistent way and thus are supported not only on a formal level by mathematical results but can also be justified in principle on a semantical level hence both the operational as well as the traditional valuational viewpoint have their own rights in their corresponding areas of modeling reality and it is shown how the latter can be given a sound foundation based upon the former

fuzzy sets a topos logical point of view abstract benabou deduction categories are defined with a set of additional assumptions that define categories with formal finite limits resp formal regular categories formal logoi formal topoi they are shown to be generalized structures in which higher order many sorted languages can be realized the corresponding gentzen type higher order calculus of sequents is explicited and the soundness theorem is formulated a construction is given which associates to each deduction category with formal properties a real category with the corresponding real properties in a universal way the corresponding sounddess and completeness properties are formulated for the real categories thus obtained fuzzy sets as generalized by goguen are introduced considered as the objects of a category fuz h which turns out to be the real category associated to a very simple formal topos and thus to be itself a topos furthermore this is proved to be a grothendieck topos which is a strictly full epireflective subcategory of higgs category of h valued sets topoi are proposed as generalized fuzzy sets and deductio0 categories as generalized 2 fuzzy sets some related topics such as arbib manes fuzzy theories probability many valued and fuzzy logics intensional logic are very briefly touched upon

fuzzy sets do not form a topos abstract the relationship between fuzzy sets and h valued sets is briefly examined and a claim of eytan s that the former comprise the objects of a topos is corrected

gauge transformations for the constrained ckp and bkp hierarchies a special chain of the gauge transformations of the two component constrained kp is introduced and then the transformed components and the function of this hierarchy are presented based on this by the different reductions the gauge transformations of the constrained bkp and the constrained ckp hierarchies are obtained furthermore the transformed component functions and functions of them are expressed by the determinants with the help of the determinant representation of the gauge transformation operators

1 1 dimensional hamiltonian systems as symmetry constraints of the kadomtsev petviashvili equation the authors consider linear problems associated with the kadomtsev petviashvili equation they prove that the linear problems are 1 1 dimensional hamiltonian systems under the symmetry constraints moreover they find that the hamiltonian flows of the linear problems are commutative

fuzzy sets and toposes towards higher order fuzzy logic abstract a series of paper which have appeared in fuzzy sets and systems over the last three years are reviewed in an expository manner the claims and counter claims made in them are dealt with some of the problems of representing knowledge in the first order predicate calculus are discussed briefly and a case is made for exposing the relationship between topoi and fuzzy mathematics with a view to making a true fuzzy logic both consistent complete and respectable the paper is an attempt to present the controversy to the non specialist and especially to practitioners in expert systems and artificial intelligence

solutions for the vector k constrained kp hierarchy a class of 1 1 dimensional integrable systems which are constrained from the kadomtsev petviashvili kp hierarchy is studied the bilinear equations of these integrable systems are shown first and then by using the free fermion operators their rational and soliton solutions which are closely related to the functions of the kp hierarchy are obtained the result implies that a large family of solutions of the 2 1 dimensional kp hierarchy can be obtained by solving these 1 1 dimensional integrable systems in particular the n soliton solutions of the kp hierarchy can be deduced from the soliton solutions of these 1 1 dimensional integrable systems

gauge transformations of constrained kp flows new integrable hierarchies integrable systems in 1 1 dimensions arise from the kp hierarchy as symmetry reductions involving square eigenfunctions exploiting the residual gauge freedom in these constraints new integrable systems are derived they include generalizations of the hierarchy of the kundu eckhaus equation and higher order extensions of the yajima oikawa and melnikov hierarchies constrained modified kp flows yield further integrable equations such as the hierarchies of the derivative nls equation the gerdjikov ivanov equation and the chen lee liu equation

the applications of the gauge transformation for the bkp hierarchy abstract in this paper we investigated four applications of the gauge transformation for the bkp hierarchy firstly it is found that the orbit of the gauge transformation for the constrained bkp hierarchy defines a special 2 1 dimensional toda lattice equation structure then the tau function of the bkp hierarchy generated by the gauge transformation is shown to be the pfaffian and the higher fay like identities for the bkp hierarchy is also obtained through the gauge transformation at last the compatibility between the additional symmetry and the gauge transformation of the bkp hierarchy is proven

categorical properties of f set theory abstract in the first part of the paper we show that the interpretation of a fuzzy set as a bundle of sets over a base space makes the category of fuzzy sets into a topos f set v i e to paraphrase pitts 5 fuzzy sets do form a topos the second part exhibit close connections to operations proposed for fuzzy set theory from an intuitive and operational viewpoint by toth 7 these two results together seem to indicate that there are two essentially different ways to understand the concept of a fuzzy set either as an evaluation over a universe or as a collection of evaluated objects the latter one being a set theoretical interpretation and henceforth leading to a topos

a new bidirectional generalization of 2 1 dimensional matrix k constrained kadomtsev petviashvili hierarchy we introduce a new bidirectional generalization of 2 1 dimensional k constrained kadomtsev petviashvili kp hierarchy 2 1 bdk ckph this new hierarchy generalizes 2 1 dimensional k ckp hierarchy ta b and a b matrix hierarchies 2 1 bdk ckph contains a new matrix 1 1 k constrained kp hierarchy some members of 2 1 bdk ckph are also listed in particular it contains matrix generalizations of davey stewartson ds systems 2 1 dimensional modified korteweg de vries equation and the nizhnik equation 2 1 bdk ckph also includes new matrix 2 1 dimensional generalizations of the yajima oikawa and melnikov systems binary darboux transformation dressing method is also proposed for construction of exact solutions for equations from 2 1 bdk ckph as an example the exact form of multi soliton solutions for vector generalization of the ds system is given

multicomponent integrable reductions in the kadomtsev petviashvilli hierarchy new types of reductions of the kadomtsev petviashvili kp hierarchy are considered on the basis of sato s approach within this approach the kp hierarchy is represented by infinite sets of equations for potentials u2 u3 of pseudodifferential operators and their eigenfunctions and adjoint eigenfunctions the kp hierarchy was studied under constraints of the following type ni 1 i i x s x where s x are symmetries for the kp equation and i i i i are eigenfunctions with eigenvalue i it is shown that for the first three cases 2 3 4 these constraints give rise to hierarchies of 1 1 dimensional commuting flows for the variables u2 1 n 1 n bi hamiltonian structures for the new hierarchies are presented

preordered sets valued in a gl monoid let m l be a gl monoid an m valued preordered set is an l subset endowed with a reflexive and m transitive l relation it is essentially a category enriched in a quantaloid generated by m this paper presents a study of m valued preordered sets with emphasis on symmetrization and the cauchy completion the main result states that symmetrization and the cauchy completion of m valued preordered sets commute up to a natural isomorphism

solving the kp hierarchy by gauge transformations we show that it is convenient to use gauge transformations sometimes called explicit backlund transformations to generate new solutions for the kp hierarchy two particular kinds of gauge transformation operators constructed out of the initial wave functions are of fundamental importance in this approach through such gauge transformations a very simple formula for the tau function is obtained encompassing and unifying all kinds of existing solutions the corresponding free fermion representation and baker functions for the new function can also be constructed

two choices of the gauge transformation for the akns hierarchy through the constrained kp hierarchy on the basis of the equivalence between the akns hierarchy and the ckp hierarchy with the constraint k 1 we point out that there exist two choices to keep the form of the lax operator when we perform the gauge transformation for the akns hierarchy which results in two classes of functions to trigger the gauge transformation for the second choice two theorems for two types of gauge transformation are established several new and more general forms of tau functions for the akns hierarchy are obtained by means of gauge transformations of both types the union of the two choices leads to new forms of functions we generate the akns hierarchy from the free lax operator l 0 via a chain of gauge transformations

solving the constrained kp hierarchy by gauge transformations we solve the constrained kadomtsev petviashvili ckp hierarchy by using the gauge transformation technique we show that there are two kinds of gauge transformations which preserve the form of the lax operator of the ckp hierarchy one of them is differential type and the other is integral type through two such gauge transformations we obtain not only the wronskian type functions for the ckp hierarchy but also the binary type functions which have not been obtained before

robust normalization and guaranteed cost control for a class of uncertain descriptor systems this paper investigates the problem of robust normalization and guaranteed cost control for descriptor systems with uncertainties the parameter uncertainties are assumed to be time varying norm bounded appearing not only in both the state and input matrices but also in the derivative matrix the quadratic performance index is nonstandard which includes the additional derivative state term attention is focused on the proportional plus derivative state feedback controller design based on the linear matrix inequality method which guarantees the closed loop system to be normal and quadratically stable with a prescribed upper bound of the cost function

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

darboux theorems and wronskian formulas for integrable systems i constrained kp flows abstract generalizations of the classical darboux theorem are established for pseudo differential scattering operators of the form l i 0 n u i i i 1 m i 1 i i iteration of the darboux transformations leads to a gauge transformed operator with coefficients given by wronskian formulas involving a set of eigenfunctions of l nonlinear integrable partial differential equations are associated with the scattering operator l which arise as a symmetry reduction of the multicomponent kp hierarchy with a suitable linear time evolution for the eigenfunctions the darboux transformation is used to obtain solutions of the integrable equations in terms of wronskian determinants

binary darboux backlund transformations for the manin radul super kdv hierarchy we construct the supersymmetric extensions of the darboux backlund transformations dbts for the manin radul super korteweg de vries kdv hierarchy using the super pseudo differential operators the elementary dbts are triggered by the gauge operators constructed from the wave functions and adjoint wave functions of the hierarchy iterating these elementary dbts we obtain not only wronskian type but also binary type superdeterminant representations of the solutions

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

technical communique an improvement on multivariable pid controller design via iterative lmi approach this note improves the results of zheng wang and lee automatica 38 2002 517 on the design of multivariable pid controllers via lmi approach the idea of the improvement is to transform the problem of pid controller design to that of static output feedback sof controller design for a system in descriptor form the merits of the present result are 1 the present condition removes the invertibility constraint on the controllers in zheng et al 2002 and incorporates it to the matrix inequality to be solved 2 once the sof matrix is obtained the pid gains are ready in hand and no additional computation is needed 3 the present approach can deal with the case of some system parameters subject to perturbations

robust h control for descriptor systems with non linear uncertainties this paper considers robust performance analysis and robust h x control for a class of descriptor systems with time varying and non linear uncertainties these uncertainties exist not only in the state the control input and exogenous input but also in the derivative of the state a sufficient condition is firstly proposed to analyse the robust h x performance problem of the unforced systems feasible design procedures are presented to construct both the robust state feedback and output feedback controllers such that the closed loop system has a unique non impulsive and asymptotically stable solution and satisfies a robust h x performance criterion a numerical example is provided to show the efficiency of the approach

affine lie algebraic origin of constrained kp hierarchies an affine sl n 1 algebraic construction of the basic constrained kp hierarchy is presented this hierarchy is analyzed using two approaches namely linear matrix eigenvalue problem on hermitian symmetric space and constrained kp lax formulation and it is shown that these approaches are equivalent the model is recognized to be the generalized non linear schrodinger gnls hierarchy and it is used as a building block for a new class of constrained kp hierarchies these constrained kp hierarchies are connected via similarity backlund transformations and interpolate between gnls and multi boson kp toda hierarchies our construction uncovers the origin of the toda lattice structure behind the latter hierarchy

state space h controller design for descriptor systems this paper proposes a new linear matrix inequality lmi method to design state space h controllers for linear time invariant descriptor systems unlike preceding studies where descriptor type controllers are first computed and then numerically transformed to state space controllers the proposed method carries out the transformation analytically in the parameter domain we derive a necessary and sufficient lmi condition for the existence of a state space controller with the same dynamic order of the descriptor system to be controlled which makes the closed loop system regular impulse free stable and guarantees the h norm bound imposed on the closed loop transfer function furthermore we present parameterization of all such state space controllers by variables satisfying the lmi condition and an arbitrary nonsingular matrix the lmis utilized in this paper are strict ones that is those containing no equality while lmis with equality constraints have been extensively used in the analysis and design for descriptor systems the strict lmis play key roles in deriving the results of this paper

kp constraints from reduced multi component hierarchies the m vector k constrained kadomtsev petviashvili kp hierarchy is shown to be a pseudo reduction of the m 1 component kp hierarchy to facilitate the implementation of this reduction on the level of the solutions the typical multi component kp solutions are mapped onto solutions of a toda molecule type equation from which wronskian and grammian solutions for the constrained kp hierarchy follow the reduction of the associated linear systems is discussed and its importance for the choice of bilinear representation of the reduced systems is explained

robust h 2 control of descriptor system with time varying uncertainty for a descriptor system with time varying structured uncertainty this paper considers the robust h 2 control problem of finding a controller satisfying the prescribed h 2 performance level against all possible uncertainty under the constraint of robust stability a sufficient condition for the robust h2 performance is derived in terms of a linear matrix inequality lmi based on this lmi condition we develop a design method of a descriptor variable feedback controller achieving the robust performance the present design method can solve the robust h 2 control problem for a wider class of plants than the state space approach a numerical example is also included to show the applicability of the present method

admissibility and stabilization of stochastic singular markovian jump systems with time delays abstract this paper deals with the admissibility analysis and stabilization of stochastic singular markovian jump systems ssmjss with time delays more general conditions for the existence and uniqueness of the impulse free solution to delayed ssmjss are presented consequently by constructing stochastic lyapunov krasovskii functional and applying generalized ito formula new sufficient conditions for the stability of ssmjss and delayed ssmjss are obtained in terms of strict linear matrix inequalities state feedback controller is designed to ensure stabilization of the delayed ssmjss three illustrative examples including an r l c circuit network and an oil catalytic cracking process are employed to verify the effectiveness and usefulness of the obtained results

categorial studies of fuzzy sets abstract this paper presents some recent studies on the categorial aspects of fuzzy sets and sets with a valued equality in order to suitably develop a theory fuzzy sets it seems useful to construct a category such an approach is very helpful when a classical concept has to be generalized for performing a choice between all the apparently possible definitions such a category has besides to look as much as possible like the usual category of sets one can wish it would be a topos or at least something similar

constraints of the kadomtsev petviashvili hierarchy for the kadomtsev petviashvili kp hierarchy constructed in terms of the famous sato theory a k constraint is proposed that leads the hierarchy to the nonlinear system involving a finite number of dynamical coordinates the eigenvalue problem of the k constrained system is naturally obtained from the linear system of the kp hierarchy which takes the form of kth order polynomial coupled with a first order one thus we are able to derive the correspondent lax pair recursion operator bi hamiltonian structures and conserved quantities the constraints for the bkp hierarchy are also sketched

classical extended conformal algebras associated with constrained kp hierarchy the conformal property of the second hamiltonian structure of constrained kp hierarchy derived by oevel and strampp is examined it is found that it naturally gives a family of nonlocal extended conformal algebras two examples of such algebras are given and it is found that they are similar to bilal s v algebra by taking a gauge transformation one can map the constrained kp hierarchy to kupershmidt s nonstandard lax hierarchy the second hamiltonian structure in this representation is considered it is shown that after mapping the lax operator to a pure differential operator the second structure becomes the sum of the second and the third gelfand dickey brackets defined by this differential operator the corresponding conformally covariant form of this differential operator is also given

modifying the kp the n th constrained kp hierarchies and their hamiltonian structures the kadomtsev petviashvili kp hierarchy has infinitely many hamiltonian pairs then th pair of them is associated withl n wherel is the pseudodifferential operator pdo 3 4 in this paper by the factorizationl n l n l 1 withl j j 1 n being the independent pdos we construct the miura transformation for the kp which leads to a decomposition of the second hamiltonian structure in then th pair to a direct sum each term in the sum is the second structure in the initial pair associated withl j when we impose a constraint 1 9 i e a new type of reduction to the kp hierarchy we obtain the similar results for the constrained kp hierarchy in particular the second hamiltonian structure for this hierarchy is transformed to a vastly simpler one

when does a category built on a lattice with a monoidal structure have a monoidal structure in a word sometimes and it gets harder if the structure on l is not commutative in this paper we consider the question of what properties are needed on the lattice l equipped with an operation for several different kinds of categories built using sets and l to have monoidal and monoidal closed structures this works best for the goguen category set l in which membership but not equality is made fuzzy and maps respect membership commutativity becomes critical if we make the equality fuzzy as well this can be done several ways so a progression of categories is considered using sets with an l valued equality and functions which respect that equality gives a monoidal category which is closed if we use a strong form of the transitive law if we use strict extensional total relations and a strong transitive law and is commutative and nearly idempotent we get a monoidal structure we also recall some constructions by mulvey nawaz and hohle on quantales with properties making them commutative enough to have non symmetric monoidal structures

miura and auto backlund transformations for the ckp and cmkp hierarchies we construct the miura and auto backlund transformations for the ckp and cmkp hierarchies both the eigenfunctions and the adjoint eigenfunctions of the hierarchies are used to trigger the transformations the canonical properties of the constructed miura and auto backlund transformations are also investigated

foundations of fuzzy sets abstract this paper gives an overview of the origins of fuzzy set theory and the problems for the foundations of fuzzy sets arising from those origins and current practice it then gives detailed accounts of categorical approaches using a closed structure to capture the fuzzy and connective within these categories weak forms of subobject representations provide an internal second order logic an approach to fuzzy real numbers and fuzzy topology is included to illustrate the use of this internal second order theory

categories of fuzzy sets and weak topos in this paper definitions of middle object in a category and concept of weak topos are given a weak topos is a cartesian closed category with a middle object it is proved that category of functors from a small category to category fuz of fuzzy sets is a weak topos

topoi and categories of fuzzy sets abstract let h be a completely distributive lattice and hence a heyting algebra goguen s category of fuzzy sets set h eytan s logos fuz h and the topos of sheaves on h sh h are interconnected by pairs of adjoint functors between them each of the categories has a predicate calculus these predicate calculi are related through the functors between the categories change of base lattice gives rise to several functors which preserve or reflect specific kinds of statements in the predicate calculi this paper gives details of the categories the predicate calculi attached to them the functors between the categories and the preservation properties of those functors