least squares splines with free knots global optimization approach splines with free knots have been extensively studied in regard to calculating the optimal knot positions the dependence of the accuracy of approximation on the knot distribution is highly nonlinear and optimisation techniques face a difficult problem of multiple local minima the domain of the problem is a simplex which adds to the complexity we have applied a recently developed cutting angle method of deterministic global optimisation which allows one to solve a wide class of optimisation problems on a simplex the results of the cutting angle method are subsequently improved by local discrete gradient method the resulting algorithm is sufficiently fast and guarantees that the global minimum has been reached the results of numerical experiments are presented

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

multidimensional bisection a dual viewpoint abstract this paper provides an alternative viewpoint of multidimensional bisection global optimization methods of wood a dual coordinate representation of convex bodies is introduced which leads to an easy implementation and eliminates the need to see the geometry of intersecting simplexes although developed in the context of global optimization the techniques deal more generally with regions represented as the union of convex bodies with this dual framework the algorithm can be implemented efficiently using any multiattribute index data structure that allows for quick range queries a c version using a multi key double linked skip list based on pugh s skip list has been implemented

rewriting systems over similarity and generalized pseudometric spaces and their properties we present a study of confluence and related properties of fuzzy relations defined over similarity spaces the ordinary confluence is an essential property of relations connected to the idea of rewriting and substituting which appears in abstract rewriting systems this paper is a continuation of our previous paper where we have introduced analogous notions related to substitutability in graded setting using residuated lattices as structures of truth degrees leaving the ordinary notions a particular case when the underlying structure is the two valued boolean algebra in this paper we further extend our previous results by developing the notions of confluence and related properties respecting a given similarity relation we also present definitions of confluence and related properties of relations on a generalized pseudometric spaces using the well known link between generalized pseudometrics and similarities we describe the connection of the notions defined on generalized pseudometric spaces to the corresponding notions on similarity spaces the introduced notions are further illustrated by example of rewriting based on graded if then rules

random number generators in genetic algorithms for unconstrained and constrained optimization abstract presented here is a genetic algorithm that computes an approximate solution to constrained and unconstrained global optimization problems this technique has been implemented using several pseudo and quasi random number generators and the results of several test examples are presented the performance of this technique is based on a ranked comparison of relative error

fuzzy equivalence relations and their equivalence classes in this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice we give certain characterizations of fuzzy semi partitions and fuzzy partitions over a complete residuated lattice as well as over a linearly ordered complete heyting algebra in the latter case for a fuzzy equivalence relation over a linearly ordered complete heyting algebra we construct an algorithm for calculation of a minimal family of its equivalence classes which generates it most of the presented results are new but some of them are generalizations of known results given in a way which simplifies and clarifies them

a new filled function for unconstrained global optimization abstract seeking global optima of an unconstrained and multi modal global optimization problem min x f x by constructing a filled function is concerned in this paper on the basis of analyzing filled functions presented before a new filled function is proposed and it is proved to meet the properties of a filled function moreover solutions of numerical experiments show that the function is quite effective

fuzzy implication operators for difference operations for fuzzy sets and cardinality based measures of comparison in this paper we determine by means of fuzzy implication operators two classes of difference operations for fuzzy sets and two classes of symmetric difference operations for fuzzy sets which preserve properties of the classical difference operation for crisp sets and the classical symmetric difference operation for crisp sets respectively the obtained operations allow us to construct as in b de baets h de meyer transitivity preserving fuzzification schemes for cardinality based similarity measures european journal of operational research 160 2005 726 740 cardinality based similarity measures which are reflexive symmetric and transitive fuzzy relations and to propose two classes of distances metrics which are fuzzy versions of the well known distance of cardinality of the symmetric difference of crisp sets

convergence of line search methods for unconstrained optimization line search methods are traditional and successful methods for solving unconstrained optimization problems its convergence has attracted more attention in recent years in this paper we analyze the general results on convergence of line search methods with seven line search rules it is clarified that the search direction plays a main role in these methods and that step size guarantees the global convergence in some cases it is also proved that many line search methods have same convergence property these convergence results can enable us to design powerful effective and stable algorithms in practice finally a class of special line search methods is investigated

a note on approximate equality versus the poincare paradox the purpose of this note is to comment on a paper by m de cock and e e kerre that was recently accepted for publication in fuzzy sets and systems the paper in contrast to the view of several other researchers states that fuzzy equivalence relations are not always able to resolve the poincare paradox this brief paper discusses this controversial contribution from the present author s subjective point of view

transitivity bounds in additive fuzzy preference structures transitivity plays a crucial role in preference modeling and related fields in this paper we discuss this property in the general context of additive fuzzy preference structures of particular interest is the decomposition of a large preference relation r in its symmetric part i indifference relation and its asymmetric part p strict preference relation by means of a so called indifference generator i given the type of transitivity of a large preference relation r w r t a conjunctor and a generator we establish basic lower bounds and general upper bounds on the transitivity of p and i these bounds are due to the careful design of generic counterexamples moreover we identify the situations in which these bounds are effectively reached thereby establishing connections with interesting properties such as dominance bisymmetry the 1 lipschitz property and rotation invariance

t partitions of the real line generated by idempotent shapes abstract the idea of generating fuzzy numbers as equivalence classes of particular t equivalences on the real line r is fully exploited scales or generators are used to define certain pseudo metrics on r by means of a shape function these pseudo metrics are then transformed into binary fuzzy relations on r shapes leading to t equivalences and hence to a class of fuzzy numbers forming a t partition of r are completely characterized in the case of a continuous generator this characterization problem is shown to be closely related to determining the idempotents w r t the t addition of fuzzy numbers

global optimization of univariate lipschitz functions ii new algorithms and computational comparison we consider the following global optimization problems for a lipschitz functionf implicitly defined on an interval a b problem p find a globallye optimal value off and a corresponding point problem q find a set of disjoint subintervals of a b containing only points with a globallye optimal value and the union of which contains all globally optimal points a two phase algorithm is proposed for problem p in phase i this algorithm obtains rapidly a solution which is often globallye optimal moreover a sufficient condition onf for this to be the case is given in phase ii the algorithm proves thee optimality of the solution obtained in phase i or finds a sequence of points of increasing value containing one with a globallye optimal value the new algorithm is empirically compared on twenty problems from the literature with a best possible algorithm for which the optimal value is assumed to be known with a passive algorithm and with the algorithms of evtushenko galperin shen and zhu piyavskii timonov and schoen for smalle the new algorithm requires only a few percent more function evaluations than the best possible one an extended version of piyavskii s algorithm is proposed for problem q a sufficient condition onf is given for the globally optimal points to be in one to one correspondance with the obtained intervals this result is achieved for all twenty test problems

deriving calculus with cotriples we construct a taylor tower for functors from pointed categories to abelian categories via cotriples associated to cross effect functors the tower was inspired by goodwillie s taylor tower for functors of spaces and is related to dold and puppe s stable derived functors and mac lane s q construction we study the layers d n f fiber p n f p n 1 f and the limit of the tower for the latter we determine a condition on the cross effects that guarantees convergence we define differentials for functors and establish chain and product rules for them we conclude by studying exponential functors in this setting and describing their taylor towers

i fuzzy equivalence relations and i fuzzy partitions a t fuzzy equivalence relation is a fuzzy binary relation on a set x which is reflexive symmetric and t transitive for a t norm t recently mesiar et al r mesiar b reusch h thiele fuzzy equivalence relations and fuzzy partitions j multi valued logic soft comput 12 2006 167 181 have generalised the t norm t to any general conjunctor c and investigated the minimal assumptions required on such operations called duality fitting conjunctors such that the fuzzification of the equivalence relation admits any value from the unit interval and also the one one correspondence between the fuzzy equivalence relations and fuzzy partitions is preserved in this work we conduct a similar study by employing a related form of c transitivity viz i transitivity where i is an implicator we show that although every i fuzzy equivalence relation can be shown to be a c fuzzy equivalence relation there exist c fuzzy equivalence relations that are not i fuzzy equivalence relations and hence these concepts are not equivalent most importantly we show that the class of duality fitting implicators i is much richer than the residuals of the duality fitting conjunctors in the study of mesiar et al we also show that the i fuzzy partitions have a constant wise structure

a deterministic algorithm for global optimization we present an algorithm for finding the global maximum of a multimodal multivariate function for which derivatives are available the algorithm assumes a bound on the second derivatives of the function and uses this to construct an upper envelope successive function evaluations lower this envelope until the value of the global maximum is known to the required degree of accuracy the algorithm has been implemented in ratfor and execution times for standard test functions are presented at the end of the paper

splitting criteria for homotopy functors of spectra we explore the interaction between the taylor tower and cotower as defined in deriving calculus with cotriples and dual calculus for functors to spectra of functors of spectra this leads to new splitting criteria which generalize the results in dual calculus for functors to spectra

on the construction of fuzzy betweenness relations from metrics abstract we consider the problem of constructing a fuzzy betweenness relation from a metric more precisely given a continuous archimedean triangular norm we present two construction methods for a fuzzy betweenness relation from a metric by making use of the pseudo inverse of either a continuous additive generator or a continuous multiplicative generator of the triangular norm in case the metric is bounded and given a 1 lipschitz continuous triangular norm we present a third construction method for a fuzzy betweenness relation from a metric by making use of the residual implication of the triangular norm since the ukasiewicz and product triangular norms are both continuous archimedean and 1 lipschitz continuous all three construction methods may be used interestingly the construction method based on the residual implication is proved to coincide with that based on a continuous additive generator for the ukasiewicz triangular norm and with that based on a continuous multiplicative generator for the product triangular norm we end by noting that all three construction methods result in a fuzzy prebetweenness relation when considering a pseudometric instead of a metric

approximate solutions of fuzzy relational equations and a characterization of t norms that define metrics for fuzzy sets abstract if one considers the membership degrees of fuzzy sets as truth values of a many valued logic there is a canonical way to define a fuzzified equality for fuzzy sets this generalized equality relation can be used to define a degree to which some value of the unknown variable is a solution of the fuzzy equation as well as a degree of solvability of the whole equation furthermore there is an intimate relationship between such a degree to which some fuzzy set is a solution and the quality to which it is an approximate solution the background is that the negation of the fuzzified equality measures a kind of distinctness of fuzzy sets formally the definition of this fuzzified equality contains as parameter some t norm which acts as truth functions of a conjunction connective for some such t norms this negation of the fuzzified equality really has the properties of a metric in the mathematical sense of the word the paper gives a necessary and sufficient condition which characterizes all those t norms which in that way yield a metric for fuzzy sets

products of elements in vague semigroups and their implementations in vague arithmetic vague arithmetic different from the present literature of fuzzy arithmetic has been proposed in demirci internat j uncertainty fuzziness and knowledge based systems 10 1 2002 25 internat j general systems 32 2 2003 157 177 to model vaguely defined arithmetic operations resulting from the indistinguishability of real numbers the main motivating problem of this paper is to introduce the notion of vague product sum of a finite number of real numbers in vague arithmetic and to point out their fundamental properties from a more abstract mathematical point of view the vague product sum of a finite number of real numbers in vague arithmetic and their properties can be considered as the vague product of a finite number of elements in vague semigroups and their relevant properties for this reason a large part of this paper is devoted to the vague product of a finite number of elements in vague semigroups and their elementary properties as a direct implementation of the present results it is shown that the vague product sum of a finite number of real numbers in vague arithmetic can be easily evaluated in terms of the underlying many valued equivalence relations furthermore various non trivial examples for the vague product sum of a finite number of real numbers in vague arithmetic are designed and a simple technique for the construction of such non trivial examples is stated

metrics and t equalities abstract the relationship between metrics and t equalities is investigated the latter are a special case of t equivalences a natural generalization of the classical concept of an equivalence relation it is shown that in the construction of metrics from t equalities triangular norms with an additive generator play a key role conversely in the construction of t equalities from metrics this role is played by triangular norms with a continuous additive generator or equivalently by continuous archimedean triangular norms these results are then applied to the biresidual operator e t of a triangular norm t it is shown that e t is a t equality on 0 1 if and only if t is left continuous furthermore it is shown that to any left continuous triangular norm t there correspond two particular t equalities on f x the class of fuzzy sets in a given universe x one of these t equalities is obtained from the biresidual operator e t t by means of a natural extension procedure these t equalities then give rise to interesting metrics on f x

the goodwillie tower of the identity functor and the unstable periodic homotopy of spheres we investigate goodwillie s taylor tower of the identity functor from spaces to spaces more specifically we reformulate johnson s description of the goodwillie derivatives of the identity and prove that in the case of an odd dimensional sphere the only layers in the tower that are not contractible are those indexed by powers of a prime moreover in the case of a sphere the tower is finite in v k periodic homotopy

the generalized associative law in vague groups and its applications ii as a continuation of part i vague integral powers of elements in vague groups and their representation properties are introduced in this paper thereafter some rudimentary algebraic properties of vague integral powers of elements obtained from the generalized vague associative law formulated in part i are established the present paper particularly provides the abstract foundations of integral powers and multiples of real numbers in vague arithmetic for this reason special attention is also paid to the calculation of integral powers and multiples of real numbers in vague arithmetic and some practical applications related to the discrete structure of measurement instruments are also given

convergence rates of a global optimization algorithm this paper presents a best and worst case analysis of convergence rates for a deterministic global optimization algorithm superlinear convergence is proved for lipschitz functions which are convex in the direction of the global maximum concave in the direction of the global minimum computer results are given which confirm the theoretical convergence rates

the generalized associative law in smooth groups in this paper assuming a multiplicative symbol for smooth groups successive products in smooth groups are introduced as a counterpart to the products of a finite number of elements in classical group theory and then a generalized associative law in smooth groups is formulated by means of successive products various elementary properties of successive products are established and some applications of the generalized associative law in smooth groups are given

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

on the number of iterations of piyavskii s global optimization algorithm piyavskii s algorithm maximizes a univariate function f satisfying a lipschitz condition we compare the numbers of iterations needed by piyavskir s algorithm np to obtain a bounding function whose maximum value is within e of the unknown globally optimal value with that required by the best possible algorithm nb the main result is that np 2nb 1 and that this bound is sharp we also show that the number of iterations needed by piyavskii s algorithm to obtain a globally e optimal value together with a corresponding point npy satisfies npy 4nb 1 and that this bound is again sharp lower and upper bounds on nb are obtained as functions of f x e l0 and l1 where l0 is the smallest valid lipschitz constant and l1 the lipschitz constant used several properties of nb and of the distribution of evaluation points are deduced from these bounds

inclusion functions and global optimization 11 inclusion functions combined with special subdivision strategies are an effective means of solving the global unconstrained optimization problem although these techniques were determined and numerically tested about ten years ago they are nearly unknown and scarcely used in order to make the role of inclusion functions and subdivision strategies more widespread and transparent we will discuss a related simplified basic algorithm it computes approximations of the global minimum and at the same time bounds the absolute approximation error we will show that the algorithm works and converges under more general assumptions than it has been known hitherto that is only appropriate inclusion functions are expected to exist the number of minimal points finite or infinite is not of importance lipschitz conditions or continuity are not assumed as shown in the appendix the required inclusion functions can be constructed and programmed without difficulty in a natural way using interval analysis

iterative methods for the localization of the global maximum methods for the search of a maximum utilizing a sequence of majorants to generate a sequence of localizations are analyzed necessary and sufficient conditions for the convergence of this sequence are given the case of lipschitz continuous functions is studied in detail and an analogy with iterative and relaxation methods for the solution of a system of linear equations is made methods of jacobi gauss seidel and chaotic type are proposed

t evaluators and s evaluators the scalar evaluation of the aggregation of elements from a complete lattice via the operation meet join is compared with the aggregation of scalar evaluations of these elements via a t norm t conorm based on this comparison the notion of a t evaluator s evaluator is introduced and its basic properties are studied the relationship between t s evaluators fuzzy measures and generalized filters is investigated an application of t evaluators in fuzzy preference modelling is presented

quotients with respect to similarity relations abstract the problem of the quotient w r t a similarity relation is closely related to poincare s paradox and central problems in cluster analysis amazingly this problem remains open for more than 30 years as a generalization of strong m categories we introduce the concept of m valued sets which leads to the definition of the category m set in m set we give a complete solution of the quotient problem

similarity based fuzzy limits in this paper the concept of similarity based fuzzy limits in metric spaces is introduced and it is shown that it is an application of limit maps of topologies on 0 1 valued sets introduced by hohle and sostak furthermore it is observed that the similarity based setting introduced is a more general approach to fuzzy limits than the one proposed by burgin

functional equations and their related operads using functional equations we define functors that generalize standard examples from calculus of one variable examples of such functors are discussed and their taylor towers are computed we also show that these functors factor through objects enriched over the homology of little n cubes operads and discuss the relationship between functors defined via functional equations and operads in addition we compute the differentials of the forgetful functor from the category of n poisson algebras in terms of the homology of configuration spaces

global search based on efficient diagonal partitions and a set of lipschitz constants in the paper the global optimization problem of a multidimensional black box function satisfying the lipschitz condition over a hyperinterval with an unknown lipschitz constant is considered a new efficient algorithm for solving this problem is presented at each iteration of the method a number of possible lipschitz constants are chosen from a set of values varying from zero to infinity this idea is unified with an efficient diagonal partition strategy a novel technique balancing usage of local and global information during partitioning is proposed a new procedure for finding lower bounds of the objective function over hyperintervals is also considered it is demonstrated by extensive numerical experiments performed on more than 1600 multidimensional test functions that the new algorithm shows a very promising performance

some results on the degree of symmetry of fuzzy relations abstract in this paper we investigate the degree of the symmetry of fuzzy relations on a set x based on the fuzzy e equality derived from uninorms with unit element e the degree of the symmetry of fuzzy relations are defined by two different approaches the type i degree of symmetry and the type ii degree of symmetry the main work of this paper include first we discuss some basic properties of those two degrees especially for the relationship between them in particular for continuous t norms we give a necessary and sufficient condition such that the type i degree of symmetry and the type ii degree of symmetry are equal and then we obtain that the type ii degree of symmetry is more appropriate to character the degree of the symmetry of fuzzy relations than the type i degree of symmetry with respect to whether they preserve the fuzzy e equalities or not in the meantime for a special case with continuous t norms we provide a necessary and sufficient condition such that the type i degree of symmetry preserves the fuzzy equalities finally for conjunctive left continuous idempotent uninorms with neutral element e 0 1 we find out a symmetric fuzzy relation s which is close enough to the given fuzzy relation r with respect to fuzzy e equality such that the degree of fuzzy e equality of s and r is the type ii degree of symmetry of r in particular we obtain analogous results for continuous t norms and one class of left continuous t norms

the bisection method in higher dimensions is the familiar bisection method part of some larger scheme the aim of this paper is to present a natural and useful generalisation of the bisection method to higher dimensions the algorithm preserves the salient features of the bisection method it is simple convergence is assured and linear and it proceeds via a sequence of brackets whose infinite intersection is the set of points desired brackets are unions of similar simplexes an immediate application is to the global minimisation of a lipschitz continuous function defined on a compact subset of euclidean space

efficiently computing many roots of a function we present a new bisection based method for counting and computing roots of a function in a given interval our method is focused on very large problems i e instances with the number of roots of the order of hundreds the method draws its power from the fact that the roots are expected to be many and is able to discover a large percentage of them very efficiently its main advantage apart from its efficiency is the fact that it requires only the sign of the function at a certain point and not its actual value also its simplicity makes it a suitable preprocessing step for reducing the size of the problem prior to more robust but also more demanding methods the algorithm is accompanied by a probabilistic analysis of its behavior which shows that a simple existence criterion like bolzano s rule can be a powerful tool in the zerofinding process

multidimensional bisection applied to global optimisation abstract a family of deterministic algorithms is introduced designed to solve the global optimisation problem for lipschitz continuous functions of many variables all the algorithms can be considered as generalisations of the bisection method they proceed via a sequence of brackets whose infinite intersection is the set of global optima brackets are unions of similar simplexes acceleration methods convergence properties and optimality questions are considered

on approximations of zadeh s extension principle in this paper we provide several results concerning the approximation of zadeh s extension of a given function we provide several simple observations due to which we provide a general procedure allowing to approximate zadeh s extension of any continuous map among other things we demonstrated that the technique of f transform can also be used to approximate this extension further we studied how the choice of the metric on the space of fuzzy sets resp fuzzy numbers can affect the quality of the approximation and we clarified that in comparison to recent habits the choice of the levelwise metric need not be the most appropriate one and that the approximation process could be simplified our results are general in the sense that we mostly require nothing but the continuity of the original map

a hybrid global optimization method the multi dimensional case we extend the hybrid global optimization method proposed by xu j comput appl math 147 2002 301 314 for the one dimensional case to the multi dimensional case the method consists of two basic components local optimizers and feasible point finders local optimizers guarantee efficiency and speed of producing a local optimal solution in the neighbourhood of a feasible point feasible point finders provide the theoretical guarantee for the new method to always produce the global optimal solution s correctly if a nonlinear nonconvex inverse problem has multiple global optimal solutions our algorithm is capable of finding all of them correctly three synthetic examples which have failed simulated annealing and genetic algorithms are used to demonstrate the proposed method

local tuning and partition strategies for diagonal go methods in this paper global optimization go lipschitz problems are considered where the multi dimensional multiextremal objective function is determined over a hyperinterval an efficient one dimensional go method using local tuning on the behavior of the objective function is generalized to the multi dimensional case by the diagonal approach using two partition strategies global convergence conditions are established for the obtained diagonal geometric methods results of a wide numerical comparison show a strong acceleration reached by the new methods working with estimates of the local lipschitz constants over different subregions of the search domain in comparison with the traditional approach

a distance metric for a space of linguistic summaries producing linguistic summaries of large databases or temporal sequences of measurements is an endeavor that is receiving increasing attention these summaries can be used in a continuous monitoring situation like eldercare where it is important to ascertain if the current summaries represent an abnormal condition it is therefore necessary to compute the distance between summaries as a basis for such a determination in this paper we propose a dissimilarity measure between summaries based on fuzzy protoforms and prove that this measure is a metric we take into account not only the linguistic meaning of the summaries but also two quality evaluations namely the truth values and the degrees of focus we present examples of how the distance metric behaves and show that it corresponds with intuition

global optimization of univariate lipschitz functions i survey and properties we consider the following global optimization problems for a univariate lipschitz functionf defined on an interval a b problem p find a globally optimal value off and a corresponding point problem p find a globally optimal value off and a corresponding point problem q localize all globally optimal points problem q find a set of disjoint subintervals of small length whose union contains all globally optimal points problem q find a set of disjoint subintervals containing only points with a globally optimal value and whose union contains all globally optimal points we present necessary conditions onf for finite convergence in problem p and problem q recall the concepts necessary for a worst case and an empirical study of algorithms i e those ofpassive and ofbest possible algorithms summarize and discuss algorithms of evtushenko piyavskii shubert timonov schoen galperin shen and zhu presenting them in a simplified and uniform way in a high level computer language we address in particular the problems of using an approximation for the lipschitz constant reducing as much as possible the expected length of the region of indeterminacy which contains all globally optimal points and avoiding remaining subintervals without points with a globally optimal value new algorithms for problems p and q and an extensive computational comparison of algorithms are presented in a companion paper

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

l approximation spaces and l fuzzy quasi uniform spaces we introduce the notion of generalized l upper and generalized l lower approximation spaces under a family of information systems we define l upper and l lower quasi approximation operators in fuzzy rough sets and l fuzzy quasi uniform spaces as a view point of the entourage approach in a strictly two sided commutative bi quantale we investigate the relations between l quasi approximation spaces and l fuzzy quasi uniform spaces

a filled function method with one parameter for box constrained global optimization in this paper a new auxiliary function with one parameter on box constrained for escaping the current local minimizer of global optimization problem is proposed first a new definition of the filled function for box constrained minimization problem is given and under mild assumptions this new auxiliary function is really a filled function then a new solution algorithm is proposed according to the theoretical analysis and some numerical results demonstrate the efficiency of this method for box constrained global optimization

vague groups and generalized vague subgroups on the basis of 0 1 le in this paper various elementary properties of vague groups and some properties of vague binary operations related with their associativity aspects are obtained furthermore the concept of vague isomorphism is defined and some basic properties of this concept are studied the concept of external direct product of vague groups is established later the definition of generalized vague subgroup which is a generalization of the vague subgroup defined by demirci is introduced and the validity of some classical results in this setting is investigated on the basis of the particular integral commutative complete quasi monoidal lattice 0 1

global root bracketing method with adaptive mesh refinement an efficient method for finding all real roots of a univariate function in a given bounded domain is formulated the proposed method uses adaptive mesh refinement to locate bracketing intervals based on bisection criterion for root finding each bracketing interval encloses one root an adaptive form of error is introduced to enclose roots in a desired tolerance based on how close the roots are detecting roots with even multiplicity which is regarded as beyond the realm of bracketing methods becomes possible with the method proposed in this paper also strategies for finding odd multiple roots with the least number of function evaluations are proposed adaptive mesh refinement lead to considerable reduction in function evaluations in comparison to previous global root bracketing methods the reliability of the proposed method is illustrated by several examples

fuzzy preorder and fuzzy topology this paper presents by means of alexandrov topology for fuzzy preordered sets and specialization order for fuzzy topological spaces a systematic investigation of the interrelationship between fuzzy preordered sets topological spaces and fuzzy topological spaces

on the compactness of admissible transformations of fuzzy partitions in terms of t equivalence relations this paper analyzes constraints in terms of fuzzy logical concepts for transforming families of fuzzy sets in order to guarantee linguistically interpretable solutions of fine tuned or optimized fuzzy systems and further to make the whole process of fine tuning a well posed problem for this purpose transformations for fuzzy sets are introduced which are constructed by means of the compositional rule of inference it is investigated under which conditions on these transformations various characteristics of fuzzy partitions and fuzzy sets like redundancy convexity and topological aspects remain invariant it is argued that further restrictions like the compactness of the set of admissible transformations are required in order to make the process of fine tuning well posed as main result it is demonstrated that compactness of this function space endowed with the uniform norm can be characterized in terms of purely fuzzy logical concepts that is by means of fuzzy equivalence relations

acceleration of univariate global optimization algorithms working with lipschitz functions and lipschitz first derivatives this paper deals with two kinds of the one dimensional global optimization problem over a closed finite interval i the objective function f x satisfies the lipschitz condition with a constant l ii the first derivative of f x satisfies the lipschitz condition with a constant m in the paper six algorithms are presented for the case i and six algorithms for the case ii in both cases auxiliary functions are constructed and adaptively improved during the search in the case i piecewise linear functions are constructed and in the case ii smooth piecewise quadratic functions are used the constants l and m either are taken as values known a priori or are dynamically estimated during the search a recent technique that adaptively estimates the local lipschitz constants over different zones of the search region is used to accelerate the search a new technique called the local improvement is introduced in order to accelerate the search in both cases i and ii the algorithms are

algebraic goodwillie calculus and a cotriple model for the remainder goodwillie has defined a tower of approximations for a functor from spaces to spaces that is analogous to the taylor series of a function his n th order approximation p n f at a space x depends on the values of f on coproducts of large suspensions of the space f m x we define an algebraic version of the goodwillie tower p alg n f x that depends only on the behavior of f on coproducts of x when f is a functor to connected spaces or grouplike h spaces the functor p alg n f is the base of a fibration 1 f f p alg n f whose fiber is the simplicial space associated to a cotriple built from the n 1 st cross effect of the functor f in a range in which f commutes with realizations for instance when f is the identity functor of spaces the algebraic goodwillie tower agrees with the ordinary topological goodwillie tower so this theory gives a way of studying the goodwillie approximation to a functor f in many interesting cases

convergence qualification of adaptive partition algorithms in global optimization following the presentation of a general partition algorithm scheme for seeking the globally best solution in multiextremal optimization problems necessary and sufficient convergence conditions are formulated in terms of respectively implied or postulated properties of the partition operator the convergence results obtained are pertinent to a number of deterministic algorithms in global optimization permitting their diverse modifications and generalizations

relations in fuzzy class theory this paper studies fuzzy relations in the graded framework of fuzzy class theory fct this includes i rephrasing existing work on graded properties of binary fuzzy relations in the framework of fuzzy class theory and ii generalizing existing crisp results on fuzzy relations to the graded framework our particular aim is to demonstrate that fuzzy class theory is a powerful and easy to use instrument for handling fuzzified properties of fuzzy relations this paper does not rephrase the whole theory of fuzzy relations instead it provides an illustrative introduction showing some representative results with a strong emphasis on fuzzy preorders and fuzzy equivalence relations

an algorithm for finding the global maximum of a multimodal multivariate function this algorithm for global optimization uses an arbitrary starting point requires no derivatives uses comparatively few function evaluations and is not side tracked by nearby relative optima the algorithm builds a gradually closer piecewise differentiable approximation to the objective function the computer program exhibits a theoretically expected strong tendency to cluster around relative optima close to the global results of testing with several standard functions are given

a non gradient approach to global extremum seeking an adaptation of the shubert algorithm the main purpose of this paper is to adapt the so called shubert algorithm for extremum seeking control of general dynamic plants this algorithm is a good representative of the sampling optimization methods that achieve global extremum seeking on compact sets in the presence of local extrema the algorithm applies to lipschitz mappings the model of the system is assumed unknown but the knowledge of its lipschitz constant is assumed the controller depends on a design parameter the waiting time and tuning guidelines that relate the design parameter and the region of convergence and accuracy of the algorithm are presented the analysis shows that semi global practical convergence in the initial states to the global extremum can be achieved in presence of local extrema if compact sets of inputs are considered numerical simulations for global optimization in the presence of local extrema are provided to demonstrate the proposed approach

several filled functions with mitigators several filled functions were reported to seek the global minimum of multimodal functions of multiple variables this paper proposes an alternative formulation that may reduce the negative definite effect of the hessian of a filled function proposed before furthermore a class of mitigators is defined and applied to improve the computational characteristics of filled functions results of numerical experiments on typical testing functions are also reported

similarity relations and bk relational products bk relational products and their role in the analysis and synthesis of complex systems has been discussed by bandler and kohout in a series of papers we show that bk products preserve similarity of the relations in that if r1 r2 and s1 s2 are similar then r1 s1 and r2 s2 are also similar for any type of bk products applications of the results are discussed

an introduction to vague complemented ordered sets abstract the purpose of this paper is to introduce a theory of fuzzily defined complement operations on nonempty sets equipped with fuzzily defined ordering relations many valued equivalence relation based fuzzy ordering relations also called vague ordering relations provide a powerful and a comprehensive mathematical modelling of fuzzily defined partial ordering relations for this reason starting with a nonempty set x equipped with a many valued equivalence relation and a vague ordering relation a fuzzily defined complement operation called a vague complement operation on x will be formulated by means of the underling many valued equivalence relation and vague ordering relation because of the fact that the practical implementations of vague complement operations basically depend on their representation properties a considerable part of this paper is devoted to the representations of vague complement operations in addition to this the present paper provides various nontrivial examples for vague complements and introduces a many valued logical interpretation of quantum logic as a real application of vague complements

dominance in the family of sugeno weber t norms the dominance relationship between two members of the family of sugeno weber t norms is proven by using a quantifier elimination algorithm further it is shown that dominance is a transitive and therefore also an order relation on this family of t norms

a generalization of the mulholland inequality for continuous archimedean t norms it is well known that dominance between strict t norms is closely related to the mulholland inequality which can be seen as a generalization of the minkowski inequality however strict t norms constitute only one part of the class of continuous archimedean t norms the basic elements from which all continuous t norms are composed in this paper dominance between continuous archimedean t norms is shown to be related to a generalization of the mulholland inequality we provide sufficient and necessary conditions for its fulfillment

derived koszul duality and tq homology completion of structured ring spectra abstract working in the context of symmetric spectra we consider algebraic structures that can be described as algebras over an operad o topological quillen homology or tq homology is the spectral algebra analog of quillen homology and stabilization working in a framework for homotopical descent the associated tq homology completion map appears as the unit map of an adjunction comparing the homotopy categories of o algebra spectra with coalgebra spectra over the associated comonad via tq homology we prove that this adjunction between homotopy categories can be turned into an equivalence by replacing o algebras with the full subcategory of 0 connected o algebras we also construct the spectral algebra analog of the unstable adams spectral sequence that starts from the tq homology groups tq x of an o algebra x and prove that it converges strongly to x when x is 0 connected

subdivision sampling and initialization strategies for simplical branch and bound in global optimization abstract we consider the problem of optimizing a lipshitzian function the branch and bound technique is a well known solution method and the key components for this are the subdivision scheme the bound calculation scheme and the initialization for lipschitzian optimization the bound calculations are based on the sampling of function values we propose a branch and bound algorithm based on regular simplexes initially the domain in question is covered with regular simplexes and our subdivision scheme maintains this property the bound calculation becomes both simple and efficient and we describe two schemes for sampling points of the function midpoint sampling and vertex sampling the convergence of the algorithm is proved and numerical results are presented for the two dimensional case for which also a special initial covering is presented

the semi continuous quadratic mixture design problem description and branch and bound approach the semi continuous quadratic mixture design problem scqmdp is described as a problem with linear quadratic and semi continuity constraints moreover a linear cost objective and an integer valued objective are introduced the goal is to deal with the scqmd problem from a branch and bound perspective generating robust solutions therefore an algorithm is outlined which identifies instances where decision makers tighten requirements such that no e robust solution exists the algorithm is tested on several cases derived from industry

measures of specificity over continuous spaces under similarity relations we introduce the concept of specificity and indicate its importance as a measure of uncertainty for information represented using fuzzy sets or possibility distributions we provide formal measures of specificity for variables whose domain is an interval of the real line similarity relations are discussed and formalized the class of width based similarity relations is introduced we extend the measure of specificity to allow for the effect of an underlying similarity relationship the manifestation of this extension for a number of different similarity relations is investigated particularly notable results are obtained for width based similarity relationships more generally we note the connection between the size of a granule of information and its uncertainty motivated by this we investigate the effect of a similarity relationship on the perception of distance in the underlying space and use this to develop an extension of the specificity measure