anisotropic elastic plates with holes cracks inclusions subjected to out of plane bending moments the problems of anisotropic plates containing holes cracks and or inclusions have been studied extensively for two dimensional deformations although the correspondence between the two dimensional problems and the plate bending problems has been observed long time ago without clarifying the involved mathematical details one still cannot get the solutions of the plate bending problems directly from the solutions of the corresponding two dimensional problems based upon the correspondence relation recently we developed a stroh like formalism for the bending theory of anisotropic plates by this newly developed formalism most of the relations for bending problems can be organized into the forms for two dimensional problems thus by using the stroh like formalism the analytical solutions for problems of anisotropic plates with holes cracks inclusions subjected to out of plane bending moments can be obtained directly from the solutions of the corresponding two dimensional problems the stresses or resultant bending moments around the hole crack inclusion boundaries are also given explicitly in this paper since the explicit closed form solutions for the present cases have not been found in the literature comparison is made with some special cases of which the analytical solutions exist which shows that our solutions are exact simple and general

some results on the resolution of fuzzy relation equations abstract in this paper two special cases of the solution of the inverse problem are investigated the necessary and sufficient conditions for the existence of a solution such that the upper bound is the unit vector and for a unique solution of the inverse problem equal to the unit vector are presented an algorithmic approach which derives the output vector and the state matrix in each one of these cases is also proposed

some theoretical aspects of fuzzy relation equations describing fuzzy systems abstract this paper contains some theoretical results on fuzzy relation equations and applications to analysis of fuzzy systems as well as identification and control problems if ai 1 i k are fuzzy inputs b is a fuzzy output and r is a fuzzy relation describing the system the above problems are illustrated by means of a max min multidimensional equation of type b a 1 a k r furthermore we investigate some properties of fuzzy systems described by fuzzy relation equations focusing our attention on energy and entropy measures of fuzziness of solutions of the class of equations under discussion

resolution of fuzzy relational equations method algorithm and software with applications analytical methods and algorithms for inverse problem resolution of fuzzy linear systems of equations in some bl algebras godel algebra in case of max min and min max compositions and goguen algebra in case of max product composition are presented algorithms with software realization for solving fuzzy linear systems of equations are proposed applications include fuzzy optimization with fuzzy linear systems of equation constraint fuzzy machines and covering problem

self organizing maps for imprecise data self organizing maps soms consist of a set of neurons arranged in such a way that there are neighbourhood relationships among neurons following an unsupervised learning procedure the input space is divided into regions with common nearest neuron vector quantization allowing clustering of the input vectors in this paper we propose an extension of the soms for data imprecisely observed self organizing maps for imprecise data soms id the learning algorithm is based on two distances for imprecise data in order to illustrate the main features and to compare the performances of the proposed method we provide a simulation study and different substantive applications

on solution of fuzzy functional equations abstract the idea of a fuzzy functional equation of the type y f x1 x2 xn a1 a2 as where x1 x2 xn a1 a2 as are treated as fuzzy numbers and the methods of its resolution are dealt with it is shown that in the case of existence of a1 a2 as satisfying the equation given above the resolution problem is equivalent to the resolution of fuzzy relational equation of the type y a 1 a 2 a s r the idea of fuzzy functional equations generated by fuzzy relation is presented and the algorithm using the notion of probabilistic sets and leading to the resolution of fuzzy functional equations in the case of noises is given as well some numerical examples form an illustration of presented algorithms pointing out there the most characteristic features

deconstructing plane anisotropic elasticity part ii stroh s formalism sans frills abstract eigensolutions for all types of anisotropic elastic materials are obtained in terms of the eigenvalues and the anisotropic elastic stiffness the generalized eigenvectors and eigensolutions in the degenerate and extra degenerate cases are obtained by the derivative rule a complete set of unnormlized eigenvectors now given in terms of the elastic moduli define the barnett lothe tensors by the same expressions irrespective of material degeneracy explicit expressions of the barnett lothe tensors are obtained in various forms depending on the multiplicity of eigenvalues these expressions complement the alternative expressions of part i in terms of the elastic compliances a new family of extra degenerate materials is found suggesting the superabundance of such materials a concise proof of the equivalence of the eigensystems of the compliance based and elasticity based formalisms is given eigenrelations applicable to all cases of material degeneracy are presented in both three dimensional and six dimensional matrix formalisms

new solutions to pressuring shearing torsion and extension of a cylindrically anisotropic elastic circular tube or bar the problem of a cylindrically anisotropic elastic circular tube or bar subjected to a uniform pressure shearing torsion or extension has been studied by ting the stroh formalism for twodimensio

a unified formalism for elastostatics or steady state motion of compressible or incompressible anisotropic elastic materials abstract for a two dimensional deformation of an anisotropic elastic material the lekhnitski formalism applies to elastostatics only it does not apply to steady state motion such as surface waves or a moving line dislocation the stroh formalism applies to elastostatics as well as steady state motion however it does not apply to an incompressible material because some of the elastic stiffnesses become unbounded in the incompressible limit modifying a new formalism studied recently a more explicit and remarkably simple expression of the matrices in the new formalism is given and a unified formalism that is applicable to elastostatics as well as steady state motion of a compressible or incompressible anisotropic elastic material is obtained as an application the secular equation is presented for surface waves propagating in a monoclinic material with the symmetry plane at x 3 0 that has a more explicit expression than what is available in the literature for a general anisotropic elastic material explicit expressions are obtained for the sextic equation for the eigenvalues p and the eigenvectors b and a for any steady state wave speed the explicit expressions recover results related to one component surface waves when the material is incompressible it is shown that the eigenvalues p remain complex for elastostatics we also show that when the material is incompressible a monoclinic material with the symmetry plane at x 3 0 behaves like that of an orthotropic material

resolution of the cartesian product of fuzzy sets abstract the problem given a in v n b in v m and b in v m find all a in v n such that a a b b where denotes max min composition x denotes cartesian product and v n denotes the set of all fuzzy vectors with n members is investigated the condition for the existence of a solution is shown and an analytical solution is given

the complete resolution of cartesian products of fuzzy sets abstract a complete solution is given to the cartesian problem given the fuzzy sets a u b v and b v find all fuzzy sets a u such that a a b b where denotes the max min composition this work is an extension of the work by pappis fuzzy sets and systems 26 1988 387 391 and kim et al fuzzy sets and systems 41 1991 293 297

similarity and dissimilarity measures between fuzzy sets a formal relational study the paper deals with the well known notion of dis similarity measures between fuzzy sets we provide three separate lists of axioms that fit with the respective notions of general comparison measure similarity measure and dissimilarity measure then we review some of the most important axiomatic definitions of dis similarity measures in the literature by referring to the axioms in those lists satisfied by each specific definition this common framework will make our study about the formal relationships among different axiomatic definitions much easier some of them which are apparently different do in fact share many commonalities we provide a self contained picture of these relationships by providing formal results and counterexamples that reflect which of the dis similarity definitions in the literature are connected by implication relations and which of them are not we finalize the paper with an in depth study about the notion of duality between similarity and dissimilarity measures as well as with some concluding remarks

an inference network for bidirectional approximate reasoning based on an equality measure an inference network is proposed as a tool for bidirectional approximate reasoning the inference network can be designed directly from the given fuzzy data knowledge if a fuzzy input is given for the inference network then the network renders a reasonable fuzzy output after performing approximate reasoning based on an equality measure conversely due to the bidirectional structure the network can yield its corresponding reasonable fuzzy input for a given fuzzy output this property makes it possible to perform forward and backward reasoning in the knowledge base system

determination of means by invariance abstract if m is a mean on r n and m f x 1 f x 2 f x n f m x 1 x 2 x n then we say that m is invariant under f the problem is to find a class of functions that by invariance determines a mean uniquely we focus on the geometric mean which can be transformed to obtain results for other means

handling interpretability issues in anfis using rule base simplification and constrained learning abstract adaptive neuro fuzzy inference system anfis is a well known neuro fuzzy model for approximating highly complex non linear systems anfis uses precise fuzzy modelling concept that aims at accuracy of a fuzzy model being designed than on interpretability but interpretability of a fuzzy system is an equally important aspect of fuzzy modelling as accuracy so far the research based on anfis has been mostly application based due to which the various issues related to the interpretability of anfis have not been dealt with the rule base of anfis is typically obtained using data driven clustering algorithms but this process introduces redundancy in the system in terms of similar fuzzy sets and redundant fuzzy rules which unnecessarily increases system complexity this in turn reduces both the interpretability as well as generalization capability of anfis additionally in case of anfis unconstrained gradient descent based learning algorithms are used to fine tune the membership function parameters which usually result in a rule base with inconsistent excessively overlapping and indistinguishable membership functions for input variables and thus the interpretability of the final optimized system is not guaranteed this paper is based on addressing the issue of rule base redundancy in anfis to reduce complexity and enforcing constraints during learning phase to ensure interpretability of the final optimized system rule base redundancy is removed using similarity analysis based rule base simplification in which similar fuzzy sets are merged and subsequently resulting fuzzy rules with equal premises are combined hybrid learning technique which is an efficient parameter tuning method for anfis is constrained to prevent inconsistency excessive overlapping and inclusion of membership functions so that the final fuzzy partitions of inputs stay interpretable the empirical analysis of the impact of rule base simplification and constrained learning on anfis is done by application to two well known benchmark problems and a real world stock price prediction problem the introduction of rule base simplification and constrained learning in anfis modeling has shown better results in terms of obtaining a desired accuracy interpretability tradeoff than conventional anfis

finite fuzzy relation equations with a unique solution in linear lattices abstract we characterize fuzzy relation equations possessing a unique solution the equations presented here involve fuzzy sets and fuzzy relations assuming values in a linear lattice and defined on finite sets a suitable example also shows that our main result generally does not hold if the fuzzy equations are assigned on complete brouwerian lattices not necessarily linear

on solving relational equations in brouwerian lattices abstract the problems of finding generic and minimal solutions of a fuzzy relational equations in brouwerian lattices are considered

fuzzy relation equations and causal reasoning abstract fuzzy relation based models for handling uncertainty in a non probabilistic way in diagnosis problems have been introduced by sanchez about 20 years ago and since then applied and developed by several researchers the paper first reviews the existing proposals also including the association based abductive model proposed by reggia and his co workers which uses non fuzzy relations a new model is presented in order to have a more expensive representation capable of handling uncertainty and also of distinguishing i between manifestations which are certainly absent from those which are not yet observed and ii between manifestations which cannot be caused by a given disorder and manifestations for which we do not know if they can or cannot be caused by this disorder this new model is based on possibility theory and the so called twofold fuzzy sets previously introduced by the authors

convergence eigen fuzzy sets and stability analysis of relational matrices abstract if stable control applications are formulated with a max product composition then instability in the relational matrix under any conditions is undesirable this paper reviews the stability analysis of relational matrices combined with the max min composition 19 and then presents an analysis of the stability of relational matrices combined with the max product composition this analysis includes results defining the convergence properties of the relational matrix determination of the eigen fuzzy sets of the stable matrices and algorithmic solutions for several important cases a method is also presented to stabilize unstable relational matrices for some control applications

a rule based method to calculate exactly the widest solution sets of a max min fuzzy relational inequality abstract in this paper we propose a rule based method to calculate exactly all the widest solution sets of a fuzzy relational inequality viewed as an extension of fuzzy relational equations fuzzy relational inequalities can be applied to many fields of fuzzy logic yet they provide with more flexibility as they enable to get rid of the sensitivity of fuzzy relational equations to numerical values up to now however fuzzy relational inequalities have received few attention from researchers and few methods have been proposed to solve these inequalities the proposed analytical methods furthermore do not calculate exactly all the widest solutions sets of a fuzzy relational inequality and often lead to redundant solution sets the rule based method we propose in this paper determines easily exactly all the widest solution sets of a fuzzy relational inequality

sextic formalism in anisotropic elasticity for almost non semisimple matrix n abstract the sextic formalism of stroh for anisotropic elasticity leads to the eigen relation n p in which n is a 6 6 real matrix the orthogonality and closure relations as well as many other relations involving the eigenvalues p and the eigenvectors x are based on the assumption that n is simple or semisimple so that the six eigenvectors span a six dimensional space problems arise when n is non semisimple in fact there are problems even when n is almost non semisimple we present a modified formalism which is valid regardless of whether n is simple almost non semisimple or non semisimple the modified formalism does not apply when n is semisimple

a similarity based bidirectional approximate reasoning method for decision making systems this paper presents a similarity based bidirectional approximate reasoning method which can express the decision maker s disposition for this we shall propose a new type of similarity measure based on the ordered weighted aggregation owa operator the proposed similarity measure has a structure to involve the disposition of a decision maker by choosing a suitable weighting vector of the owa operator from this property we derive a simple and flexible type of similarity based bidirectional approximate reasoning method which can support the making of a decision under more complex situations such as industrial inspection systems for example

on the existence of minimal solutions for fuzzy equations with tolerances abstract we investigate the existence of minimal elements of solution sets for fuzzy equation problems with tolerances where the cardinality of the basic spaces of fuzzy inputs and outputs of an object need not be finite it turns out the semi continuity of applied membership functions plays a central role crisp models with fuzzy parameters are considered as well

calculation and application of a minimum fuzzy relational matrix abstract this paper compares the maximum and minimum inverse formulations for fuzzy identification of a relational matrix given a series of non normal input output data baboshin and naryshkin 1990 showed for the case of normal input data that identification of the relational matrix using an estimate of the minimum inverse resulted in a smaller hamming distance between actual and predicted output values than the mamdani identification method this paper extends their work to include unrestricted or non normal input data and confirms that the estimate of the minimum inverse for this case is again better than mamdani s method it is then proven that the relative rank of identification algorithms for minimization of the hamming distance between the actual and predicted output values is the maximum inverse calculation of sanchez 1976 the minimum inverse formulation initial formulation by sanchez 1977 and extended to a series of input output data by sessa 1984 the estimate of the minimum inverse and then the mamdani method however this ranking applies only in the fuzzy domain and not to the discrete or defuzzified domain relevant to many practical applications

minimizing a linear function under a fuzzy max min relational equation constraint in this paper we investigate the problem of minimizing a linear objective function subject to a fuzzy relational equation constraint a necessary condition for optimal solution is proposed based on this necessary condition we propose three rules to simplify the work of computing an optimal solution numerical examples are provided to illustrate the procedure experimental results are reported showing that our new procedure systematically outperforms our previous work

solution algorithms for fuzzy relational equations with max product composition the conditions for the existence of an inverse solution to the max min composition of fuzzy relational equations have been well documented since the original work by sanchez 30 31 these same existence theorems have been extended to the t norm composition of relational equations in which the max product composition is a member 5 13 26 several studies 8 15 24 33 34 38 have shown that the max min operator may not always be the most desirable fuzzy relational composition and in fact the max product operator was superior in these instances this paper reviews the algorithms necessary to determine the complete solution of the inverse for fuzzy relational equations with max product composition

surface waves in anisotropic elastic materials for which the matrix n υ is extraordinary degenerate degenerate or semisimple the 6 x 6 real matrix n for anisotropic elastic materials under a two dimensional steady state motion with speed is extraordinary degenerate when n has three identical complex eigenvalues p but has only one associated eigenvector it has been an open question if such an n exists for surface waves in this paper we first modify the solution for ordinary surface waves to the case when n is extraordinary degenerate the displacement consists of three waves one of which decays exponentially as in the ordinary surface waves according to e ndash 2 where is a positive constant the other two waves decay in proportion to 2endash 2 and x 22endash 2 we then consider a class of materials that has seven independent elastic constants for two dimensional deformations for this class of materials it is shown that there is a five parameter family of extraordinary degenerate n for surface waves thus not only an extraordinary degenerate n for surface waves exists its existence is not as rare as one might have expected we also study surface waves for which n is degenerate or semisimple in the former n has two or three identical complex eigenvalues but has two independent eigenvectors in the latter two of the three eigenvalues are identical but n has three independent eigenvectors it is shown that there are families of degenerate and semisimple n for surface waves

multi input multi output fuzzy systems and the inverse problem abstract for the inverse problem given a fuzzy relation r u v and a fuzzy subset b v find all a u such that a r b an analytical solution has already been given this solution is applicable to one input one output fuzzy systems it is shown that it is also applicable to multi input multi output fuzzy systems

from classic statistical characterization to fuzzy windowing based characterization for the exploratory analysis of miscellaneous time variables example in the field of car driving studies the problem of data characterization of quantitative and qualitative measurement scales is stated in the context of an exploratory multivariate statistical analysis an example from a car driving study is considered where the quantitative data correspond to the car and head movements while the qualitative data correspond to objects being viewed road bridge sign post etc for each of these two sets the literature is analyzed first in terms of data characterizing methods and relationship obtaining methods then we propose to evaluate and compare nine quantitative data characteriing methods five corresponding to classic statistical indicators two to crisp space windowing with either two or three windows and two on fuzzy windowing with either two or three windows logically the last method appears as the best according to our evaluation procedure then we propose a bidimensional fuzzy windowing instead of a crisp one to characterize the gaze positions finally the multiple correspondence analysis is used to investigate the membership value averages obtained from the characterization stage

resolution of cartesian products of fuzzy sets abstract the problem given the fuzzy subsets a u b v and b v find all a u such that a a b b where denotes max min composition and x denotes cartesian product is investigated the condition for the existence of a solution is shown and an analytical solution is given

similarity based inverse approximate reasoning this paper investigates the inverse problem of ap proximate reasoning to conclude a from b and a b from different angles we develop a method to and a solution to the inverse problem by the use of the law of contrapositive symme try in the process generalized modus ponens generalized modus tollens and their generating functions are studied extensively an attempt is made here to establish that the concept of similarity plays an important role in inverse approximate reasoning we apply the concept of similarity to model inverse approximate reasoning and develop a new methodology that is named similarity based inverse approximate reasoning for a solution to the problem algorithms are presented and illustrated with simple yet concrete examples we then use zadeh s compositional rule of inference cri as a combi nation of cylindrical extension and projection and thereby model generalized modus tollens in a different way we thus develop a third way to find a solution to the inverse problem of approximate reasoning a comparison between the result of application of our technique and the real static characteristic of a dc shunt motor is presented

adaptive fuzzy interpolation fuzzy interpolative reasoning strengthens the power of fuzzy inference by the enhancement of the robustness of fuzzy systems and the reduction of the systems complexity however after a series of interpolations it is possible that multiple object values for a common variable are inferred leading to inconsistency in interpolated results such inconsistencies may result from defective interpolated rules or incorrect interpolative transformations this paper presents a novel approach for identification and correction of defective rules in interpolative transformations thereby removing the inconsistencies in particular an assumption based truth maintenance system atms is used to record dependences between interpolations and the underlying technique that the classical general diagnostic engine gde employs for fault localization is adapted to isolate possible faulty interpolated rules and their associated interpolative transformations from this an algorithm is introduced to allow for the modification of the original linear interpolation to become first order piecewise linear the approach is applied to a realistic problem which predicates the diarrheal disease rates in remote villages to demonstrate the potential of this study

inverse problem in fuzzy relational equations abstract the paper deals with the problem of handling fuzzy relational equations of the form x r y in the case of an empty solution set with respect to x x r y discussing the essence of the structure of the fuzzy relation r with respect to provided y algorithms have been proposed which enable us to modify the original equation i e the fuzzy set y in a way leading to its genuine solution or at least an approximate one it is explained how these modifications relax a set of constraints needed to be satisfied for any solvable equation a grade of the modification of the original problem is expressed in terms of distortion of the fuzzy set y which forms a cornerstone of the proposed procedures numerical examples are provided some aspects related to the area of diagnostic models in which these equations are studied are also treated

improvements and critique on sugeno s and yasukawa s qualitative modeling investigates sugeno s and yasukawa s 1993 qualitative fuzzy modeling approach we propose some easily implementable solutions for the unclear details of the original paper such as trapezoid approximation of membership functions rule creation from sample data points and selection of important variables we further suggest an improved parameter identification algorithm to be applied instead of the original one these details are crucial concerning the method s performance as it is shown in a comparative analysis and helps to improve the accuracy of the built up model finally we propose a possible further rule base reduction which can be applied successfully in certain cases this improvement reduces the time requirement of the method by up to 16 in our experiments

rough set analysis of a general type of fuzzy data using transitive aggregations of fuzzy similarity relations abstract in this paper we initially present and compile the tools and theories that are necessary for a rough set analysis of a general type of fuzzy data from this study we have developed two original contributions that can enhance the application of fuzzy rough sets techniques to data that may be affected by several sources of uncertainty in its measurement process for this type of analysis we have first considered the construction of homogeneous granules in which reflexivity symmetry and a certain transitivity property are satisfied this has encouraged us to study the application of t similarity relations in this framework and to examine the available theorems that permit us to aggregate this type of relations as a part of this study we have also developed a new theorem that allows us to use the dual of the generalized means as an aggregation operator in the fuzzy rough sets context another theoretical development that tries to respond to a practical problem is the concept of precision aggregation we use this new notion to export ziarko s variable precision rough set model to the analysis of fuzzy data a generalization that becomes a necessity for the fuzzy rough sets analysis of large databases the utility of this approach is demonstrated with an application example in the vehicle database donated by the turing institute

axiomatic characterization of nonlinear homomorphic means one important problem in mathematics and statistics is to formulate a mean from a set of data various definitions of mean have been studied to represent in some sense an average of data the applications of these means involve all the fields of science and technology with nonlinear filtering being a primary example in the area of image processing 1 3 the goal of this work is to axiomatically investigate the rationale for a large class of means which is referred to as quasi arithmetic mean 4 nonlinear homomorphic mean or nonlinear homomorphic filter 3

neurocomputational approach to solve a convexly combined fuzzy relational equation with generalized connectives abstract in this paper we present a method to solve a convexly combined fuzzy relational equation with generalized connectives for this we propose a neural network whose structure represents the fuzzy relational equation then we derive a learning algorithm by using the concept of back propagation learning since the proposed method can be used for a general form of fuzzy relational equations such fuzzy max min or min max relational equations can be treated as its special cases

review measures of divergence on credal sets when there is uncertainty about the values of a variable it is possible to have different pieces of information about it this can be due to the fact that they represent the points of view of different agents and these agents have followed different criteria when codify the pieces of information by means of numbers the aim of this paper is to provide tools for the comparison of these representations this is done in the framework of imprecise probabilities more concretely the credal set theory is used the reason is that credal sets are very general including as particular cases possibility theory probability theory and belief functions so it can serve as basis to compare the representations of information obtained by following these formalisms the measures used for comparison are inconsistency measure inclusion index and informative distance

further contributions to the study of finite fuzzy relations equations abstract an interesting property of the schein rank of a fuzzy matrix stressed by k h kim and f w roush is formalized as definition of the basis of a set of fuzzy vectors defined over a commutative semiring with respect to another set of fuzzy vectors we apply this definition and another related concept in the setting of finite fuzzy relation equations in particular we give a sufficient condition for the existence of the smallest solution of a max min fuzzy equation

new methods for students evaluation using fuzzy sets in an earlier work biswas 1995 presented two methods for the application of fuzzy sets in students answerscripts evaluation in this paper we extend his work to propose two new methods for evaluating students answerscripts using fuzzy sets the proposed methods can overcome the drawbacks in biswas 1995 due to the fact that they do not need to perform the complicated matching operations and they can evaluate students answerscripts in a more fair manner

a motion compression reconstruction method based on max t norm composite fuzzy relational equations a motion compression reconstruction method based on max t norm composite fuzzy relational equations mcf is proposed where into intra pictures i pictures and predictive pictures p pictures of the original motion are compressed by uniform and non uniform coders respectively the non uniform coders of the proposed method can preserve edge information of p pictures on the compressed image to perform an effective compression reconstruction of the p pictures a design method of non uniform coders is proposed based on an overlap level of fuzzy sets and a fuzzy equalization an experiment using 10 p pictures confirms that the root mean square error of the reconstructed images obtained by the proposed non uniform coders is decreased to 89 4 of that one of the uniform coders under the condition that compression rate the ratio between the file size of compressed image and original one is 0 0057 two test motions tennis and woman 100 frames are compressed and reconstructed by the proposed mcf

a new algorithm for minimizing a linear objective function with fuzzy relation equation constraints in this paper we study the problem of minimizing a linear objective function subject to a fuzzy system constraint by utilizing the fuzzy system s compact system we define k form chained solutions and spaces of k form chained solutions then we prove that any minimal solution of the fuzzy system is a k form chained solution by using an operation method expressed in tables we sieve out basic solutions from the spaces of k form chained solutions finally we obtain optimal solutions of the studied problem from the obtained basic solutions examples are provided to show that our algorithm is simple and convenient

granular counting of uncertain data abstract we propose a definition of granular count realized in the presence of uncertain data modeled through possibility distributions we show that the resulting counts are fuzzy intervals in the domain of natural numbers based on this result we devise two algorithms for granular counting an exact counting algorithm with quadratic time complexity and an approximate counting algorithm with linear time complexity we compare the two algorithms on synthetic data and show their application to a bioinformatics scenario concerning the assessment of gene expressions in cells

unique real variable expressions of displacement and traction fundamental solutions covering all transversely isotropic elastic materials for 3d bem a general efficient and robust boundary element method bem formulation for the numerical solution of three dimensional linear elastic problems in transversely isotropic solids is developed in the present work the bem formulation is based on the closed form real variable expressions of the fundamental solution in displacements uik and in tractions tik originated by a unit point force valid for any combination of material properties and for any orientation of the radius vector between the source and field points a compact expression of this kind for uik was introduced by ting and lee q j mech appl math 1997 50 407 426 in terms of the stroh eigenvalues on the oblique plane normal to the radius vector working from this expression of uik and after a revision of their final formula a new approach based on the application of the rotational symmetry of the material for deducing the derivative kernel uik j and the corresponding stress kernel ijk and traction kernel tik has been developed in the present work these expressions of uik uik j ijk and tik do not suffer from the difficulties of some previous expressions obtained by other authors in different ways with complex valued functions appearing for some combinations of material parameters and or with division by zero for the radius vector at the rotational symmetry axis the expressions of uik uik j ijk and tik have been presented in a form suitable for an efficient computational implementation the correctness of these expressions and of their implementation in a three dimensional collocational bem code has been tested numerically by solving problems with known analytical solutions for different classes of transversely isotropic materials copyright 2007 john wiley sons ltd

the inverse problem for distributive lattices abstract the inverse problem givena in vl n b in vl m andb in vl m find all a in vl n such that a a b b where l is distributive lattice denotes max min composition vl n denotes the set of all vector with n member on l and x a1 a2 an b1 b2 bn ai bj n m is investigated the necessary and sufficient condition for the existence of a solution is shown and an analytical solution is given

fuzzy linear systems abstract fuzzy linear systems of equations and inequalities over a bounded chain are studied both from theoretical and computational points of view a unified approach is presented for solving such systems completed with polynomial time algorithms the main results are concerned with establishing the consistency of the system computing all kinds of solutions or marking the contradictory equations respectively inequalities if the system is inconsistent applications in fuzzy linear programming multivalued logic fuzzy matrix equations or inequalities fuzzy relation equations or inclusions fuzzy automata and medicine are presented

fuzzy relational equations and the inverse problem abstract the inverse problem concerned with fuzzy relations is investigated the conditions for the existence of a solution are shown and an analytical solution is given a method for the improvement of the solution is proposed

deconstructing plane anisotropic elasticity part i the latent structure of lekhnitskii s formalism abstract general solutions of the stress and displacements in two dimensional anisotropic elasticity may be represented by eigenvectors and analytic functions of the complex variables x iy but the representation takes different forms for five distinct types of materials as determined by the elastic compliance matrix in this paper explicit expressions of the general solutions are derived for each type of anisotropic materials in terms of the eigenvalues i and the elements of it is shown that for degenerate and extra degenerate materials the generalized eigenvectors and associated eigensolutions may be obtained by the derivative rule the barnett lothe tensors are defined in terms of unnormalized eigenvectors by the same set of relations regardless of material degeneracy explicit expressions of these tensors are given in concise forms depending only on the multiplicity of the eigenvalues the six dimensional matrix formalism and normalization of the eigenvectors are found to be neither essential nor expedient for the analysis except as a device for abridged expressions of matrix identities

classification and coding approaches to part family formation under a fuzzy environment classification and coding is a major approach to solve the part family formation problem in group technology applications traditionally all features in a classification and coding system are considered crisp it has been recognized that more realistically some features should be fuzzy this paper addresses issues involved in using the classification and coding approach for part family formation under a fuzzy environment methods for dealing with major issues such as part coding computation of similarity between parts and part family formation are proposed and discussed in detail examples were given to show how these methods could be applied

the stroh formalism for anisotropic materials that possess an almost extraordinary degenerate matrix n abstract it has been discovered recently that the 6 x 6 elasticity matrix n for an anisotropic elastic material under a two dimensional deformation can be extraordinary degenerate i e n can have three identical pairs of complex conjugate eigenvalues but has only one pair of complex conjugate eigenvector the stroh formalism and the modified formalism for a simple degenerate n available in the literature need to be revised what is more likely to happen in applications is that the matrix n is almost extraordinary degenerate in that case the orthonormalized eigenvectors can be very large causing certain numerical problems we therefore present in this paper a modified formalism that is applicable to the case when n can be extraordinary degenerate or almost extraordinary degenerate

two new algorithms for solving optimization problems with one linear objective function and finitely many constraints of fuzzy relation inequalities this paper studies the optimization model of a linear objective function subject to a system of fuzzy relation inequalities fri with the max einstein composition operator if its feasible domain is non empty then we show that its feasible solution set is completely determined by a maximum solution and a finite number of minimal solutions also an efficient algorithm is proposed to solve the model based on the structure of fri path the concept of partial solution and the branch and bound approach the algorithm finds an optimal solution of the model without explicitly generating all the minimal solutions some sufficient conditions are given that under them some of the optimal components of the model are directly determined some procedures are presented to reduce the search domain of an optimal solution of the original problem based on the conditions then the reduced domain is decomposed if possible into several sub domains with smaller dimensions that finding the components of the optimal solution in each sub domain is very easy in order to obtain an optimal solution of the original problem we propose another more efficient algorithm which combines the first algorithm these procedures and the decomposition method furthermore sufficient conditions are suggested that under them the problem has a unique optimal solution also a comparison between the recently proposed algorithm and the known ones will be made

generalized fuzzy indices for similarity matching the purpose of this paper is to introduce a new family of fuzzy similarity indices referred to as the generalised tversky index gti the development of gti is based on the theoretical findings by amos tversky regarding the human perception of similarity between different objects as formulated by the features contrast model fc although gti was developed starting from tversky s fc it is shown that it provides a fuzzy extension and generalization of several widely used similarity indices like the jaccard and dice coefficients the mathematical properties of two members of the gti family namely tim and tip are studied and their interpretation of similarity is explained by comparison to other conventional indices

nonlinear mean filters in image processing the use of nonlinear means in image processing is introduced the properties of these means in the presence of different types of noise are investigated it is shown that nonlinear filters based on these means behave well for both additive and impulse noise their performance in the presence of signal dependent noise is satisfactory they preserve the edges better than linear filters and they reject the noise better than median filters

general solution for eshelby s problem of 2d arbitrarily shaped piezoelectric inclusions abstract eshelby s problem of piezoelectric inclusions arises sometimes in exploiting the electromechanical coupling effect in piezoelectric media for example it intervenes in the nanostructure design of strained semiconductor devices involving strain induced quantum dot qd and quantum wire qwr growth using the extended stroh formalism the present work gives a general analytical solution for eshelby s problem of two dimensional arbitrarily shaped piezoelectric inclusions the key step toward obtaining this general solution is the derivation of a simple and compact boundary integral expression for the eigenfunctions in the extended stroh formalism applied to eshelby s problem the simplicity and compactness of the boundary integral expression derived make it much less difficult to analytically tackle eshelby s piezoelectric problem for a large variety of non elliptical inclusions in the present work explicit analytical solutions are obtained and detailed for all polygonal inclusions and for the inclusions characterized by jordan s curves and laurent s polynomials by considering the piezoelectric material gaas 110 the analytical solutions provided are illustrated numerically to verify the coincidence between different expressions and to clarify the jump across the boundary of the inclusion and the singularity around the corner of the inclusion

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

ranking type 2 fuzzy numbers type 2 fuzzy sets are a generalization of the ordinary fuzzy sets in which each type 2 fuzzy set is characterized by a fuzzy membership function in this paper we consider the problem of ranking a set of type 2 fuzzy numbers we adopt a statistical viewpoint and interpret each type 2 fuzzy number as an ensemble of ordinary fuzzy numbers this enables us to define a type 2 fuzzy rank and a type 2 rank uncertainty for each intuitionistic fuzzy number we show the reasonableness of the results obtained by examining several test cases

interval valued fuzzy backward reasoning the importance and efficiency of backward reasoning in nonfuzzy reasoning has been stressed for a long time especially in the case of expert systems and decision support systems the extension of this reasoning method to fuzzy theory however has never been considered in this paper the authors propose a definition of fuzzy backward reasoning based on the generalized modus ponens and show the necessity of considering interval valued fuzzy backward reasoning then the authors propose solving methods for fuzzy backward reasoning in the case of a rule with one or several conditions as well as in the case of several rules

a modified lekhnitskii formalism a la stroh for anisotropic elasticity and classifications of the 6 6 matrix n the stroh formalism for two dimensional deformations of anisotropic elastic materials computes the eigenvalue p and the eigenvector a from an eigenrelation the vector b is then determined from a depending on the number of repeated eigenvalues p and the number of independent eigenvectors a b the system has it can be classified into six groups the lekhnitskii formalism has no eigenrelation to speak of we present a modified lekhnitskii formalism that computes the eigenvalue p and the eigenvector b from an eigenrelation the vector a is then determined from b thus the modified lekhnitskii formalism is a dual to the stroh formalism not only does the modified formalism enable us to do the classifications it is much simpler than using the stroh formalism the six groups are the sp ss d1 d2 ed and es groups the es group does not exist for a real material the ss group that has p 1 p 2 p 3 and the d2 group that has p 1 p 2 p 3 can be identified without computing the eigenvalues p and the eigenvectors a b we show that the repeated eigenvalue p 1 p 2 in the ss and d2 groups is simply a root of the quadratic equation l 2 0 we present an explicit expression of p 3 for the ss group the ed group that has three identical p can also be identified without computing p and a b however we do present an explicit expression of p we show that monoclinic materials with the symmetry plane at x 3 0 cannot belong to the ed group the identification of the sp and d1 groups is the only one that requires computation of p but not a b for special classes of materials however they can be identified without computing p in all cases the eigenvectors and the generalized eigenvectors are obtained explicitly

a collaborative recommender system based on probabilistic inference from fuzzy observations the problem of building recommender systems has attracted considerable attention in recent years the objective of this paper is to automatically suggest and rank a list of new items to a user based on the past voting patterns of other users with similar tastes the proposed model can be considered as a soft computing based collaborative recommender system the combination of bayesian networks which enables an intuitive representation of the mechanisms that govern the relationships between the users and the fuzzy set theory enabling us to represent ambiguity or vagueness in the description of the ratings improves the accuracy of the system

an algorithmic approach to some special cases of the generalized inverse problem abstract in this paper two special cases of the generalized inverse problem namely given r and b find a such that a r b where r b and a are fuzzy matrices is considered in the first a is supposed to be unique and equal to the unit matrix in the second the upper bound of the set of all possible a s is the unit matrix the necessary and sufficient conditions for the existence of such types of solution are determined in the case of a and b being matrices in both cases an algorithmic analysis is adopted

equations in classes of fuzzy relations abstract particular solutions of fuzzy relation equations are examined in diverse classes of fuzzy relations the main results concern the existence of solutions and the description of the family of all desired solutions

on the relation between equations with max product composition and the covering problem systems of equations with max product composition are considered it is shown that solving these equations is closely related with the covering problem which belongs to the category of np hard problems it is proved that minimal solutions of equations correspond to irredundant coverings in terms of the covering problem the conditions of compatibility of equations of redundancy of equations of uniqueness of solution of uniqueness of minimal solution are determined concepts of essential non essential semi essential and super essential variables are suggested ways of simplification of a covering problem and methods of its solving are considered

fuzzy clustering of fuzzy data based on robust loss functions and ordered weighted averaging abstract in many real cases the data are not expressed in term of single values but are imprecise in all these cases standard clustering methods for single valued data are unable to properly take into account the imprecise nature of the data in this paper by considering the partitioning around medoids pam approach in a fuzzy framework we propose a fuzzy clustering method for imprecise data formalized in a fuzzy manner in particular in order to neutralize the negative effects of possible outlier fuzzy data in the clustering process we proposed a robust fuzzy c medoids clustering method for fuzzy data based on the combination of huber s m estimators and yager s owa ordered weighted averaging operators the proposed method is able to smooth the influence of anomalous data by means of a suitable parameter the so called typicality parameter capable to tune the influence of the outliers the performance of the proposed method has been shown by means of a simulation study composed of experiments on i simple two dimensional dataset ii benchmark datasets and iii the fuzzy art outliers dataset the comparison made with the robust clustering methods known from the literature indicates the competitiveness of the introduced method to others an application of the suggested method to a real dataset is also provided and the results of the method has been compared with other clustering methods suggested in the literature in the application the comparative assessment has shown the informational gain in term of additional information of the proposed method vs the other robust methods

resolution of a system of the max product fuzzy relation equations using lou factorization in this paper the lu factorization is extended to the fuzzy square matrix with respect to the max product composition operator called l u factorization equivalently we will find two fuzzy lower and upper triangular matrices l and u for a fuzzy square matrix a such that a l u where is the max product composition an algorithm is presented to find the matrices l and u furthermore some necessary and sufficient conditions are proposed for the existence and uniqueness of the l u factorization for a given fuzzy square matrix a an algorithm is also proposed to find the solution set of a square system of fuzzy relation equations fre using the l u factorization the algorithm finds the solution set without finding its minimal solutions and maximum solution it is shown that the two algorithms have a polynomial time complexity as o n 3 since the determination of the minimal solutions is an np hard problem the algorithm can be very important from the practical point of view

convexly combined fuzzy relational equations and several aspects of their application to fuzzy information processing abstract this paper presents some theoretical results on a set of the convexly combined fuzzy relational equations represented in the form x k 2 x k r y k k 1 2 k where x k and y k denote fuzzy sets r and 2 fuzzy relations the sup min compositional operator the inf max one and a fuzzy convex combinator whose complement is furthermore we propose a method to formulate the fuzzy information processing approximately by means of the convexly combined fuzzy relational equations focusing our attention on the modeling of fuzzy input output systems normality of fuzzy sets characterizing fuzzy input fuzzy logical aspects of systems and association between fuzzy information finally it is suggested that the proposed method is suitable for the modelling of several types of fuzzy systems

study on stability of fuzzy closed loop control systems abstract the problem of stability of fuzzy closed loop control systems has long been unsolved through analyzing the max min composition operator in fuzzy systems this paper presents a reasonable analytical model of fuzzy closed loop systems and then proposes a method to analyze the stability of control systems by the relational matrix of fuzzy systems an example is given to show the method available

solution of fuzzy equations with extended operations abstract assuming that is any operation defined on a product set x y and taking values on a set z it can be extended to fuzzy sets by means of zadeh s extension principle given a fuzzy subset c of z it is here shown how to solve the equation a b c or a b c when a fuzzy subset a of x or a fuzzy subset b of y is given the methodology we provide includes as a special case the resolution of fuzzy arithmetical operations i e when stands for or extended to fuzzy numbers fuzzy subsets of the real line the paper is illustrated with several examples in fuzzy arithmetic

sets of solution set invariant coefficient matrices of simple fuzzy relation equations abstract a set of coefficient matrices which are invariant to a solution set of a simple fuzzy relation equation is considered the equipollency of the cardinal numbers of solution sets of simple fuzzy relation equations for two constant vectors is also shown

system of fuzzy relation equations with inf composition complete set of solutions the problem of solvability of a system of equations with inf composition is considered on finite universes equations are expressed using operations of a bl algebra we study complete set of solutions of the respective system in the particular case one equation and in the general case in both cases various conditions of solvability are found and proved we characterize all maximal solutions and prove that under certain conditions each solution of the system an equation is less than or equal to a respective maximal one as a result we are able to characterize the complete set of solutions examples of single equations and their systems are considered and complete sets of solutions are found for each

lower solutions of systems of fuzzy equations lower solutions of a system of fuzzy equations are determined by lower solutions of each equation the result is applied to fuzzy relation equations so that they are decomposed to simple fuzzy relation equations

adjointness of fuzzy systems this paper deals with two adjoint fuzzy systems based on and compositions which correspond to the normal use of and in l fuzzy logic respectively the resolution of composite fuzzy relational inequations for these adjoint fuzzy systems can be skillfully treated by the gentzen type sequent calculus as well as by the lattice theoretical way owing to the adjointness of and compositions

a software routine to solve the generalized inverse problem of fuzzy systems abstract a software routine to produce the solution of the generalized inverse problem of fuzzy relational equations is presented given the fuzzy matrices r and b defined on u v and w v respectively the routine produces all fuzzy matrices a defined on the product space w u which satisfy a r b where denotes maxmin composition

finite fuzzy relation equations with unique solution in complete brouwerian lattices abstract we characterize finite fuzzy relation equations having a unique solution for sake of completeness we first recall some results already established in foregoing papers in the context of the linear lattices new results are established in the general setting of the complete brouwerian lattices suitable examples illustrate the theorems obtained

fault tree based diagnostics using fuzzy logic fuzzy set theory is investigated as a tool for the diagnostics of systems described by means of a fault tree the objective is to diagnose component failures from the observation of fuzzy symptoms using the information contained in a fault tree a two step procedure is used to solve the problem in this first step causal reasoning is used to diagnose failure modes consisting of minimal cut sets of basic events from the observation of triggered gates treated as symptoms in the second step the authors identify the particular components which have failed based on the diagnosed failure modes to perform this second step the solution of a fuzzy relational equation a v product s sup t alpha x connecting failure mode a to basic events x is derived with this method the diagnostics equations can be symmetrically generated and solved in terms of the tree s basic events the systematic nature with which a diagnosis can be generated from a fault tree lends this method to potential application of object based programming techniques

a statistical comparative study of different similarity measures of consensus in group decision making an essential aim in group decision making gdm problems is to achieve a high level of consensus among experts consensus is defined as general or widespread agreement and it is usually modelled mathematically via a similarity function measuring how close experts opinions or preferences are similarity functions are defined based on the use of a metric describing the distance between experts opinions or preferences in the literature different metrics or distance functions have been proposed to implement in consensus models but no study has been carried out to analyse the influence the use of different distance functions can have in the gdm process this paper presents a comparative study of the effect of the application of some different distance functions for measuring consensus in gdm by using the nonparametric wilcoxon matched pairs signed ranks test it is concluded that different distance functions can produce significantly different results moreover it is also shown that their application also has a significant effect on the speed of achieving consensus finally these results are analysed and used to derive decision support rules based on a convergent criterion that can be used to control the convergence speed of the consensus process using the compared distance functions

a computer algorithm for the solution of the inverse problem of fuzzy systems abstract a computer algorithm based on a method for dealing with the so called inverse problem is presented the algorithm produces the solution semi lattice formed by the fuzzy subsets a u satisfying the fuzzy relational equation a r b where denotes maxmin composition given a fuzzy relation r u v and a fuzzy subset b v

some results in the setting of fuzzy relation equations theory abstract in this paper we point out certain theoretical aspects of fuzzy relation equations connecting some known results between composite and o fuzzy equations several theorems are obtained and some examples are provided too

a comparison of similarity measures of fuzzy values abstract this paper extends the work of pappis and karacapilidis 1993 to present and compare the properities of several measures of similarity of fuzzy values the measures examined in this paper are based on the geometric model the set theoretic approach and the matching function s we presented in chen 1988 it is shown that several properties are common to all measures and some properties do not hold for all of them