knot concordance whitney towers and l 2 signatures we construct many examples of nonslice knots in 3 space that cannot be distinguished from slice knots by previously known invariants using whitney towers in place of embedded disks we define a geometric filtration of the 3 dimensional topological knot concordance group the bottom part of the filtration exhibits all classical concordance invariants including the cassongordon invariants as a first step we construct an infinite sequence of new obstructions that vanish on slice knots these take values in the l theory of skew fields associated to certain universal groups finally we use the dimension theory of von neumann algebras to define an l 2 signature and use this to detect the first unknown step in our obstruction theory

describing function method for stability analysis of pd and pi fuzzy controllers a fuzzy logic controller saturates outside the universe of discourse region this saturation gives rise to a universal nonlinearity for this type of controller in this paper the effects of this nonlinearity are analyzed for the most usual fuzzy controllers pd and pi fuzzy controllers it is shown that due to this saturation some pathological behaviors such as oscillations or unstable limit cycles can occur the methodology employed in the analysis is based on the describing function which is determined for pd and pi fuzzy controllers one example illustrates the relevance of the proposed methodology

fuzzy piecewise multilinear and piecewise linear systems as universal approximators in sobolev norms a piecewise multilinear and a piecewise linear model for fuzzy systems are introduced and their approximation capabilities investigated classical results on the approximation of regular functions are improved showing that fuzzy systems may keep their semantic structure while approximating to any degree of accuracy not only sufficiently regular functions but also their derivatives

whitney towers and abelian invariants of knots we relate certain abelian invariants of a knot namely the alexander polynomial the blanchfield form and the arf invariant to intersection data of a whitney tower in the 4 ball bounded by the knot we also give a new 3 dimensional algorithm for computing these invariants

on the notion of concept ii it is well known that classical set theory is not expressive enough to adequately model categorization and prototype theory recent work on compositionality and concept determination showed that the quantitative solution initially offered by classical fuzzy logic also led to important drawbacks several qualitative approaches were thereafter tempted that aimed at modeling membership through ordinal scales or lattice fuzzy sets most of the solutions obtained by these theoretical constructions however are of difficult use in categorization theory we propose a simple qualitative model in which membership relative to a given concept f is represented by a function that takes its value in a finite abstract set a f equipped with a total order this function is recursively built through a stratification of the set of concepts at hand based on a notion of complexity similarly the typicality associated with a concept f will be described using an ordering that takes into account the characteristic features of f once the basic notions of membership and typicality are set the study of compound concepts is possible and leads to interesting results in particular we investigate the internal structure of concepts and obtain the characterization of all smooth subconcepts of a given concept

on stability of fuzzy systems expressed by fuzzy rules with singleton consequents addresses the issue of stability of a fuzzy system described by fuzzy rules with singleton consequents it first presents two canonical forms of a fuzzy system a parametric expression and a state space expression a fuzzy system with singleton consequents is found to be a piecewise polytopic affine system then the paper gives stability theorems for a discrete time system as well as for a continuous time system it also gives a brief survey on the stability issues of fuzzy control systems

a novel fuzzy framework for nonlinear system control nonlinear pid and gain scheduling controls have attracted much research interests in recent years these control strategies can accommodate some nonlinear characteristics by allowing the gains varied or rescheduled as a function of system states in this paper a novel fuzzy framework is developed to transfer this type of nonlinear controller to its fuzzy domain representation it is proved that the resulting fuzzy controller is functionally exactly identical to the original control system one of the benefits of the suggested approach is that it provides a well defined prototype for the design of fuzzy control system the resulting linguistic representation can facilitate investigation in linguistic terms into how the controller operates whereas expert knowledge can be effectively implemented to improve control performance the viability of the proposed mapping technique is demonstrated by using simulations corresponding to a flexible link robot

fuzzy control rules and their natural laws abstract we consider the convergence of fuzzy controllers as the number of fuzzy rules tends to infinity we assume that the fuzzy rules are regular and show the existence of limit laws for five different fuzzy control rules

reformulation of the theory of conceptual spaces this paper presents a conceptual system based on two independently developed extensions of gardenfors formulation of conceptual spaces the new approach continues to emphasize the role of properties and their associations in conceptual representation and to recognize the importance of similarity judgments in reasoning tasks in the new theory domains are sets equipped with a measure and a property is a measurable membership function on a domain concepts are sets of properties and their binary associations a new notion of the degree of ambiguity of properties for a given reasoning task is used to focus attention upon properties having the greatest discrimination power the theory is demonstrated on a maritime application involving the categorization of vessel routes

fuzzy controller with stability and performance rules for nonlinear systems this paper presents a fuzzy controller which can be designed through two steps for nonlinear systems the proposed fuzzy controller consists of two groups of fuzzy rules namely the stability and performance rules stability conditions which involve the stability rules only will be derived to guarantee the system stability by properly designing the shapes of the membership functions of the fuzzy controller the stability conditions can be relaxed the proposed fuzzy controller offers a property that the performance rules will not alter the system stability this favorable property enables the performance rules to be designed freely to achieve the system performance without affecting the system stability consequently the system stability and performance can be designed in two separate processes an application example will be presented to show the merits of the proposed approach

constructing nonlinear variable gain controllers via the takagi sugeno fuzzy control we investigated the analytical structure of the takagi sugeno ts type of fuzzy controllers which was unavailable in the literature the ts fuzzy controllers we studied employ a new and simplified ts control rule scheme in which all the rule consequent use a common function and are proportional to one another greatly reducing the number of parameters needed in the rules other components of the fuzzy controllers are general arbitrary input fuzzy sets any type of fuzzy logic and the generalized defuzzifier which contains the popular centroid defuzzifier as a special case we proved that all these ts fuzzy controllers were nonlinear variable gain controllers and characteristics of the gain variation were parametrized and governed by the rule proportionality we conducted an in depth analysis on a class of nonlinear variable gain proportional derivative pd controllers we present the results to show 1 how to analyze the characteristics of the variable gains in the context of control 2 why the nonlinear variable gain pd controllers can outperform their linear counterpart and 3 how to generate various gain variation characteristics through the manipulation of the rule proportionality

homotopy invariants of links this paper answers a question originally asked by john milnor in 1957 it offers significant progress toward another question from that paper in m 1 an m 2 john milnor defined a collection of isotopy invariants of links stallings showed these were actually concordance i variants of the ink stallings work suggested that milnor s invariants were related to other algebraically defined invariants the massey products in the exterior of the link a connection made precise in tu and p thus the results in this paper can be reinterpreted as results about massey products in the link exterior recently mike freedman has suggested that some of these invariants may be the obstructions to the topological surgery theorem in dimension four f i if2 fl in this paper the approach to studying higher dimensional link concordance presented in o 1 is extended to the study of links in the 3 sphere a collection of new link invariants is defined and related to milnor s invariants one fundamental advantage of this approach is that these new invariants make explicit use of the three sphere containing the link the link exterior contains all the relations among milnor s invariants by accounting for the ambient sphere our invariants eliminate redundancy see theorem 15 we answer this question originally asked by milnor 1 to what extent are milnor s invariants independent of each other theorem 15 milnor also asked this question 2 can one extract invariants from the group rc n where 7r is the link group and n is the intersection of the lower central series of the link group theorem 15 answers question 1 partial results can be found in c 1 ma i on and t the only previously known results for two component links were k 2 8 where the number of linearly independent invariants are 0 1 0 5 0 6 and 4 respectively for links with three or more components tess was known

structural analysis of fuzzy controllers with nonlinear input fuzzy sets in relation to nonlinear pid control with variable gains the popular linear pid controller is mostly effective for linear or nearly linear control problems nonlinear pid controllers however are needed in order to satisfactorily control highly nonlinear plants time varying plants or plants with significant time delay this paper extends our previous papers in which we show rigorously that some fuzzy controllers are actually nonlinear pi pd and pid controllers with variable gains that can outperform their linear counterparts in the present paper we study the analytical structure of an important class of two and three dimensional fuzzy controllers we link the entire class as opposed to one controller at a time to nonlinear pi pd and pid controllers with variable gains by establishing the conditions for the former to structurally become the latter unlike the results in the literature which are exclusively for the fuzzy controllers using linear fuzzy sets for the input variables this class of fuzzy controllers employs nonlinear input fuzzy sets of arbitrary types our structural results are thus more general and contain the existing ones as special cases two concrete examples are provided to illustrate the usefulness of the new results

a general formulation of conceptual spaces as a meso level representation representing cognitive processes remains one of the great research challenges many important application areas such as clinical diagnosis operate in an environment of relative magnitudes counts shapes colours etc which are not well captured by current representational approaches this paper presents conceptual spaces as a meso level representation for cognitive systems between the high level symbolic representations and the subconceptual connectionist representations which have dominated ai conceptual spaces emphasize orders and measures and therefore naturally represent counts magnitudes and volumes taking grdenfors decade long investigation of conceptual spaces grdenfors conceptual spaces the geometry of thought mit press 2000 as start point the paper presents a formal foundation for conceptual spaces shows how they are theoretically and practically linked to higher and lower representational levels and develops dynamics which allow the orbits of states in the space to solve appropriate meso level reasoning tasks interpretations of conceptual spaces are given to illustrate the formal definitions and show the flexibility of the representation

fuzzy control theory the linear case abstract a linear fuzzy proportional integral pi controller with one input and one output is defined in terms of piecewise linear membership functions for fuzzification control rules and defuzzification algorithm it is shown that a linear fuzzy controller is not equivalent to a linear non fuzzy pi controller if the rules are evaluated using the zadeh probability or lukasiewicz fuzzy logic alone however the linear fuzzy controller is precisely equivalent to a linear non fuzzy pi controller if mixed fuzzy logic is used to evaluate the control rules when the fuzzy logics used are selected with due regard to prior associations implied by the control rule operands themselves this finding has relevance to the selection of fuzzy logics in fuzzy expert systems and for fuzzy logics in general

hierarchical conceptual spaces for concept combination we introduce a hierarchical framework for conjunctive concept combination based on conceptual spaces and random set theory the model has the flexibility to account for composition of concepts at various levels of complexity we show that the conjunctive model includes linear combination as a special case and that the more general model can account for non compositional behaviours such as overextension non commutativity preservation of necessity and impossibility of attributes and to some extent attribute loss or emergence we investigate two further aspects of human concept use the conjunction fallacy and the guppy effect

4 manifold topology ii dwyer s filtration and surgery kernels even when the fundamental group is intractable i e not good many interesting 4 dimensional surgery problems have topological solutions we unify and extend the known examples and show how they compare to the presumed counterexamples by reference to dwyer s filtration on second homology the development brings together many basic results on the nilpotent theory of links as a special case a class of links only slightly smaller than homotopically trivial links is shown to have free slices on their whitehead doubles

studies on the output of fuzzy controller with multiple inputs abstract in this paper we discuss the output of a fuzzy controller in which an arbitrary number of fuzzy control inputs and an arbitrary number of fuzzy sets employed to fuzzify every linguistic variable are considered in the fuzzy control rules a rule base with a rule mapping concept is considered a defuzzifier with the general t norm and t conorm operators is formulated a triangular family is adopted so that the defuzzifier of the complete rule mapping is represented by a formula consisting of only some rules using a specific logic pair to evaluate values we derive that the output of the fuzzy controller can be simply represented by a linear parametric function of the inputs to the fuzzy controller

higher order intersections in low dimensional topology we show how to measure the failure of the whitney move in dimension 4 by constructing higher order intersection invariants of whitney towers built from iterated whitney disks on immersed surfaces in 4 manifolds for whitney towers on immersed disks in the 4 ball we identify some of these new invariants with previously known link invariants such as milnor sato levine and arf invariants we also define higher order sato levine and arf invariants and show that these invariants detect the obstructions to framing a twisted whitney tower together with milnor invariants these higher order invariants are shown to classify the existence of twisted whitney towers of increasing order in the 4 ball a conjecture regarding the nontriviality of the higher order arf invariants is formulated and related implications for filtrations of string links and 3 dimensional homology cylinders are described

stability analysis and design of fuzzy control systems abstract the stability analysis and the design technique of fuzzy control systems using fuzzy block diagrams are discussed first we show the concept of fuzzy blocks and consider the connection problems of fuzzy blocks diagrams we derive some theorems and corollaries with respect to two basic types of connections of fuzzy blocks in order to preserve some properties in a connection of fuzzy blocks continuous piecewise polynomial membership functions are defined secondly a sufficient condition which guarantees the stability of fuzzy systems is obtained in terms of lyapunov s direct method we give an important fact based on this condition thirdly we propose a new design technique of a fuzzy controller the fuzzy block diagrams and the stability analysis are applied to the design problems of a model based fuzzy controller

fuzzy controllers synthesis and equivalences it has been proved that fuzzy controllers are capable of approximating any real continuous control function on a compact set to arbitrary accuracy in particular any given linear control can be achieved with a fuzzy controller for a given accuracy the aim of this paper is to show how to automatically build this fuzzy controller the proposed design methodology is detailed for the synthesis of a sugeno or mamdani type fuzzy controller precisely equivalent to a given pi controller the main idea is to equate the output of the fuzzy controller with the output of the pi controller at some particular input values called modal values the rule base and the distribution of the membership functions can thus be deduced the analytic expression of the output of the generated fuzzy controller is then established for sugeno type fuzzy controllers precise equivalence is directly obtained for mamdani type fuzzy controllers the defuzzification strategy and the inference operators have to be correctly chosen to provide linear interpolation between modal values the usual inference operators satisfying the linearity requirement when using the center of gravity defuzzification method are proposed

whitney towers and gropes in 4 manifolds many open problems and important theorems in low dimensional topology have been formulated as statements about certain 2 complexes called gropes this paper describes a precise correspondence between embedded gropes in 4 manifolds and the failure of the whitney move in terms of iterated towers of whitney disks the flexibility of these whitney towers is used to demonstrate some geometric consequences for knot and link concordance connected to n solvability k cobordism and grope concordance the key observation is that the essential structure of gropes and whitney towers can be described by embedded unitrivalent trees which can be controlled during surgeries and whitney moves it is shown that a whitney move in a whitney tower induces an ihx jacobi relation on the embedded trees

uncertainty modelling for vague concepts a prototype theory approach an epistemic model of the uncertainty associated with vague concepts is introduced label semantics theory is proposed as a framework for quantifying an agent s uncertainty concerning what labels are appropriate to describe a given example an interpretation of label semantics is then proposed which incorporates prototype theory by introducing uncertain thresholds on the distance between elements and prototypes for description labels this interpretation naturally generates a functional calculus for appropriateness measures a more general model with distinct threshold variables for different labels is discussed and we show how different kinds of semantic dependence can be captured in this model

cobordism for links in we give an explicit finite set of based links which generates under connected sum the k cobordism classes of links we show that the union of these generating sets 2 1 and in this case c m does not appear to have a natural group structure moreover we consider only linking properties as opposed to knotting of the individual components of the link the invariants we consider are independent of the concordance class of the component knots our links are oriented ordered and tame weaker equivalence relations have recently been considered by n sato k orr and the author sa 02 c2 c3 let us immediately generalize these in order to state our main results definition 1 1 a link l l1 l 1 in 53 is b1 bm cobordant to l l1 l4 where bi are ordinal numbers if there are disjointly embedded compact connected oriented surfaces vi in 53 x 0 1 such that i vi n s3 x o 1 a vi a vi ii a vil where l vio l and d vil received by the editors august 10 1989 1980 mathematics subject classification 1985 revision primary 57m25 secondary 55s30

a new fuzzy lyapunov function approach for a takagi sugeno fuzzy control system design in this paper a new fuzzy lyapunov function approach is presented for a class of continuous time takagi sugeno fuzzy control system the proposed fuzzy lyapunov function is formulated as a line integral of a fuzzy vector which is a function of the state and it can be regarded as the work done from the origin to the current state in the fuzzy vector field unlike the approaches using a fuzzy blending of multiple quadratic lyapunov functions the time derivatives of membership functions do not appear in the proposed approach the effectiveness of the proposed approach is shown through numerical examples

nonmonotonic reasoning preferential models and cumulative logics abstract many systems that exhibit nonmonotonic behavior have been described and studied already in the literature the general notion of nonmonotonic reasoning though has almost always been described only negatively by the property it does not enjoy i e monotonicity we study here general patterns of nonmonotonic reasoning and try to isolate properties that could help us map the field of nonmonotonic reasoning by reference to positive properties we concentrate on a number of families of nonmonotonic consequence relations defined in the style of gentzen 13 both proof theoretic and semantic points of view are developed in parallel the former point of view was pioneered by gabbay 10 while the latter has been advocated by shoham 38 five such families are defined and characterized by representation theorems relating the two points of view one of the families of interest that of preferential relations turns out to have been studied by adams 2 the preferential models proposed here are a much stronger tool than adams probabilistic semantics the basic language used in this paper is that of propositional logic the extension of our results to first order predicate calculi and the study of the computational complexity of the decision problems described in this paper will be treated in another paper

rational whitney tower filtration of links we present complete classifications of links in the 3 sphere modulo framed and twisted whitney towers in a rational homology 4 ball this provides a geometric characterization of the vanishing of the milnor invariants of links in terms of whitney towers our result also says that the higher order arf invariants which are conjectured to be nontrivial measure the potential difference between the whitney tower theory in rational homology 4 balls and that in the 4 ball extensively developed by conant schneiderman and teichner

robust stability analysis of fuzzy control systems abstract robust stability of fuzzy control systems is studied through the popov lyapunov approach first we transform a fuzzy control system into a lur e system with uncertainties or nonlinearities second the lyapunov s direct method is used to guarantee the stability of the perturbed lur e system and then a robustness measurement which gives the bound on allowable uncertainties or nonlinearities is derived also if the bound of the uncertainties nonlinearities of the system is known we can find an estimated region of asymptotic stability in phase space a nonlinear numerical example and a magnetic bearing experiment demonstrating the feasibility of the method are illustrated

fuzzy controllers lifting the linear nonlinear frontier abstract this paper brings back some ideas related with the linear analysis of fuzzy controllers a fuzzy control algorithm does not go further than a non linear function described by its inference map firstly it is possible to obtain the inference map corresponding to a classical pid controller by choosing adequately the linguistic terms the membership functions and the table of rules then the inverse problem is presented to obtain the closer pid to a given fuzzy controller from these ideas the paper focuses its attention on the obtainment of a useful tool to design fuzzy controllers at least as good as the pid that allows the system to follow a specified behaviour the next step is to improve the fuzzy controller parameters finally an industrial application of these ideas is shown the design of a fuzzy controller over a clinker cooler with grill improving p and pi controllers already functioning is discussed

a linear output structure for fuzzy logic controllers in this paper the output structure of multiple input single output fuzzy logic controller flcr is studied the input and output variables of the flcr are all characterized by normal triangular shaped membership functions and fuzzy partitions are considered for corresponding universes of discourse a special mapping linear rule mapping fuzzy sets and systems 57 1993 149 which describes the relationship between the input variables and the output variables of one rule is used the conclusion proposed is that the output of multiple input single output flcr can be represented by the convex linear combination of its crisp input variables a precise mathematical model of flcr which is helpful to analyze fuzzy control and tune the fuzzy logic controller is given

explicit analytical formulation and exact inversion of decomposable fuzzy systems with singleton consequents abstract this paper is devoted to the inversion of fuzzy systems expressed by fuzzy rules with singleton consequents if input variables are described using strong triangular partitions as pointed out in recent works such fuzzy systems can be decomposed into collections of multi linear subsystems in this paper an analytical formulation of the system output is explicitly developed and directly used in order to determine solutions to the inversion problem based on this analytical methodology an algorithm is proposed for computing inverse solutions as the inversion is handled analytically the exactness of the obtained solutions is guaranteed furthermore according to the decomposability of the studied fuzzy systems all inverse solutions are found finally whatever the fuzzy system under consideration there is no need to study its invertibility beforehand since the algorithm is able to handle all possible situations no solution one unique solution multiple solutions an infinity of solutions the proposed approach can be easily extended to other types of fuzzy systems provided that decomposability is preserved in other words with regard to exact inversion which often plays a key role in engineering applications such as control or diagnosis decomposability is probably the first criterion that should be considered when choosing a specific fuzzy system structure

the three semantics of fuzzy sets abstract three main semantics for membership functions seem to exist in the literature similarity preference and uncertainty each semantics underlies a particular class of applications similarity notions are exploited in clustering analysis and fuzzy controllers uncertainty is captured by fuzzy sets in the framework of possibility theory the membership function of a fuzzy set is also sometimes a kind of utility function that represents flexible constraints in decision problems this paper advocates the claim that progress in operational semantics of membership functions presupposes that these distinct semantics be acknowledged and related to more basic measurement issues in terms of distance cost and frequency on which scientific traditions exist

grope cobordism and feynman diagrams we explain how the usual algebras of feynman diagrams behave under the grope degree introduced in ct we show that the kontsevich integral rationally classifies grope cobordisms of knots in 3 space when the class is used to organize gropes this implies that the grope cobordism equivalence relations are highly nontrivial in dimension 3 we also show that the class is not a useful organizing complexity in 4 dimensions since only the arf invari ant survives in contrast measuring gropes according to height does lead to very interesting 4 dimensional information cot finally several low degree calculations are explained in particular we show that s equivalence is the same relation as grope cobordism based on the smallest tree with an internal vertex

stable and optimal fuzzy control of linear systems a number of stable and optimal fuzzy controllers are developed for linear systems based on some classical results in control theory we design the structure and parameters of fuzzy controllers such that the closed loop fuzzy control systems are stable provided that the process under control is linear and satisfies certain conditions it turns out that if stability is the only requirement there is much freedom in choosing the fuzzy controller parameters therefore a performance criterion is set to optimalize the parameters using the pontryagin minimum principle we design an optimal fuzzy controller for linear systems with quadratic cost function finally the optimal fuzzy controller is applied to a ball and beam system