are grades of membership probabilities abstract a small sample is given of the difficulties with present day fuzzy set theory with respect to the requirements of a scientific theory and a summary of some probabilistic interpretations of grades of membership to be found in the literature the tee model for grades of membership is then suggested instead of starting out from mathematical postulates such as the assumed max and min operations for or and and the tee model uses a semantic physio logical psycho logical starting point by investigating the possibilities which a person has for assigning linguistic labels and partial grades of membership in a meaningful way the assignment of partial grades of membership is explained by the assumption that humans have the ability to take errors of observation and intersubject differences into consideration a membership value uex is identified with the subject s estimate of the probability that the label e g tall would be assigned to an object of exact attribute value uex in a natural language situation this situation being a labeling or a yes no one in which at least one of three possible sources of uncertainty is present almost all the accepted formulas of fuzzy set theory as well as the shapes of the grade of membership curves are then derived instead of having to be postulated however the derived formulas for or and and reduce to the max and min operations only in a number of special cases in another special case they reduce to the summation and the zero operation respectively the generally valid formulas are presented

comments on exact solutions of electromagnetic fields in both near and far zones radiated by thin circular loop antennas a general representation with reply this paper presents an alternative vector analysis of the electromagnetic em fields radiated from thin circular loop antennas of arbitrary radius a this method which employs the dyadic green s function in the derivation of the em radiated fields makes the analysis more general compact and straightforward than those two methods published recently by werner 1996 and overfelt 1996 both near and far zones are considered so that the em radiated fields are expressed in terms of the vector wave eigenfunctions not only the exact solution of the em fields in the near and far zones outside the region where r a is derived by the use of the spherical hankel function of the first kind but also the closed series form of the em fields radiated in the near zone inside the region 0 spl les r a is obtained in series of the spherical bessel functions of the first kind as an example a fourier cosine series is used to expand an arbitrary current distribution along the loop and the exact representations of the em radiated fields due to the loop everywhere are obtained in closed form the closed form reduces to those for the sinusoidal current loop and further for the uniform current loop validity of the approximate formulas is discussed and clarified error analysis based on numerical computations of the radiated fields is also given to show the accuracy of the limiting cases

a note on fuzzy relational equations with min implication composition the fuzzy relational equations with min implication composition are considered where the implication operation involved is the residuum with respect to the minimum operation the solution set of such a system of equations whenever nonempty can be characterized by a minimum solution and finitely many maximal solutions it is shown that a system of fuzzy relational equations with min implication composition can be reformulated in polynomial time into a system of 0 1 mixed integer linear inequalities and consequently the structure of its solution set can be revealed in a succinct manner besides it is shown that the determination of all maximal solutions is polynomially reducible to the transversal hypergraph generation problem of which some practically well performed algorithms are available

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

numerical and applicational aspects of fuzzy relational equations we discuss numerical methods leading to a resolution of fuzzy relational equations which create a formal description of ill defined systems an applicability of the numerical approach is shown in solving some problems in fuzzy systems such as identification and control an idea of fuzzy discretization fuzzy chopping as a meaningful tool for handling various problems is also investigated

the concept of grade of membership abstract the concept of grade of membership is bound up with the general problem of fuzzy reasoning in this paper an approach to the latter problem is outlined and the consequences for the notion of grade of membership are described particular care is taken from the beginning to base the theory on concepts which have an immediate practical meaning for this reason assertions rather than sentences are taken as fundamental an assertion being characterized in terms of utility and decision theory fuzzy sentences and degrees of truth and then fuzzy sets and grades of membership are defined in terms of assertions the definitions being illustrated with examples the problem of defining unary and binary operations connectives for assertions sentences and fuzzy sets is discussed finally some other aspects of the logic of assertions are outlined and the extent to which the concepts of fuzzy set and grade of membership can be incorporated into the theory is discussed

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

exact field expressions for circular loop antennas using spherical functions expansion a new technique is introduced for evaluating the vector potential of circular loop antennas clas with arbitrary current distribution and dimensions which can be used to obtain the exact electromagnetic fields in near and far regions of the antenna the technique is based on the spherical bessel functions sbfs and associated legendre polynomials alps expansion this expansion is used to decouple the azimuth variables of the integrand of the vector potential integrals from the other variables in a separate functions that depend exponentially on azimuth variables this technique reduces greatly the mathematical operations for evaluation of the vector potential to only one simple integration the technique is used to obtain exact electromagnetic fields expressions for a thin wire cla of arbitrary current distribution and dimension which is valid everywhere except on the loop the well known far fields for the cla of arbitrary current distribution as well as near and far fields for uniform current loop and small loop are obtained from the general field expressions presented in this paper as an special cases

scaling the membership function on propose une base theorique pour construire une fonction de qualite de membre au moyen de comparaisons par paire

on several definitions of the differential of a fuzzy mapping abstract several definitions of the differential of a fuzzy function are discussed in the restricted framework of fuzzy mappings from r to r

new exact solution procedure for the near fields of the general thin circular loop antenna an analytical method is presented which gives the various fields including the scalar potential of a general circular loop antenna in cylindrical coordinates in terms of six separate series three of the series are elementary and the other three involve legendre functions of the second kind and half integral order the results obtained are compared with direct numerical integration and with two far field formulae the standard textbook expression and what appears to be a new formula which gives more accurate results the series solution is more robust than numerical integration for the higher fourier terms of the general solution close to the loop the solution has also been compared with the solution of werner and it agrees exactly the solution presented here gives better convergence close to the loop whereas werner s solution is better far from the loop sample calculations are given for the axisymmetric case

a model for the measurement of membership and the consequences of its empirical implementation representation and uniqueness theorems are presented for the fundamental measurement of the membership of a fuzzy set when the domain of discourse is order dense the conclusion that membership is on an interval scale is justified by the inapplicability of extensive measurement to fuzziness and the lack of a natural origin for membership some descriptive statistics are also presented from an empirical study currently in progress to construct membership functions they suggest that subjects differ in their perception of fuzziness and that the models in the literature for certain linguistic operators such as not very and antonym are not sufficiently general to account for the range of individual perceptions of vague attributes a more general mathematical model is proposed the question of the meaningfulness of statements about membership functions is also discussed in the light of the assertion that membership can be fundamentally measured at best on an interval scale some of the difficulties which arise in fuzzy set theory due to this property are illustrated and a solution is suggested involving the use of a function which is derived from membership but is on an absolute scale

fuzzy arithmetic with requisite constraints abstract the purpose of this paper is to show that the standard fuzzy arithmetic does not take into account known constraints when applied to states of linguistic variables these constraints referred to as requisite constraints represent additional information when they are ignored as in the standard fuzzy arithmetic the obtained results exhibit in general information deficiency it is thus important as argued in the paper to revise fuzzy arithmetic to take relevant requisite constraints into account some typical requisite constraints are overviewed with a more extensive examination of the requisite equality constraint crisp or fuzzy

weighted norm inequalities for the continuous square function we prove new weighted norm inequalities for real variable analogues of the lusin area function we apply our results to obtain new i weighted norm inequalities for singular integral operators ii weighted sobolev inequalities iii eigenvalue estimates for degenerate schrodinger operators

infinite fuzzy relational equations on a complete brouwerian lattice abstract let i be a finite and j an infinite index sets a a ij i j and b b i i i a given matrix and a column vector with elements from a complete brouwerian lattice in this paper the problem of a minimal solution of a fuzzy relational equation a x b where is the sup inf composite operation is investigated

solution of a system of linear equations with fuzzy numbers in this paper the interval nature of fuzzy numbers is revealed by showing that many interesting results from classical interval analysis transfer also into the fuzzy case the paper deals with a solution of a fuzzy interval system of linear equations i e a system in which fuzzy intervals numbers appear instead of crisp numbers

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

applications of fuzzy relational equations for methods of reasoning in presence of fuzzy data the paper deals with implementation of a general scheme for fuzzy logic inference with the aid of fuzzy relational equations it is shown that antecedents and consequents are related in terms of a fuzzy relation various implementation methods of generalized modus ponens and modus tollens are put into extensive study it is pointed out that fuzzy relational equations form an effective and plausible tool for implementations of reasoning methods with fuzzy premises two examples of fuzzy controllers being a direct application of schemes of approximate reasoning are also discussed

on the set of solutions of composite fuzzy relation equations in this paper we introduce some algorithms which have minimization properties about the fuzziness of solutions in the maxmin fuzzy relation equations

on the convergence exponent of decomposable relations this paper deals with an decomposable fuzzy relation on 0 1 the decomposition problem of a fuzzy relation can be stated as follows given a fuzzy relation r determine whether there exist two fuzzy sets a and b such that r a b where the operator is nothing but godel implication at first two necessary and sufficient conditions that a fuzzy relation r is decomposable are given then the solution set of a and b with a b r is obtained in the end the convergence of the decomposable fuzzy relation is investigated and an algorithm is shown to calculate the convergence exponent for a given decomposable fuzzy relation

the exact analytical solution for large circular loops radiating around a dielectric coated conducting sphere in this paper the electromagnetic field of electrically large circular loop antennas with different radii and different nonuniform current values around a dielectric coated conducting sphere is considered one or more loop antennas are located on the outer surface of a spherical dielectric shell covering a conducting sphere eigenfunction series solutions for the field are assumed in two regions the current distribution on the wire loop driven by a voltage source is determined by fourier series expansion and all necessary harmonics are taken into account exact analytical field expressions in closed forms are derived and field patterns are plotted the antenna model and formulation presented in this paper offer exact analytical solutions to several loop antenna problems

an exact integration procedure for vector potentials of thin circular loop antennas a direct integration procedure for far zone vector potentials of thin circular loop antennas has been known for many years this method is general in the sense that it leads to simple integrals which have closed form solutions for most commonly assumed loop current distributions however a comparable integration technique has not been available for evaluating the more complicated near zone vector potentials this paper introduces a systematic approach for the exact integration of general near zone vector potentials associated with current carrying circular loop antennas a particular example is considered where this new integration technique is used to find exact solutions to the vector potential and electromagnetic field integrals for loops with a fourier cosine series expansion of the current the observation is made that degenerate forms of these exact representations lead to simplified expressions for the important special cases of a uniform and cosinusoidal current loop two equivalent forms of exact series expansions are derived for the uniform current vector potential and field integrals it is shown that the familiar small loop approximations as well as the classical far field expressions may be obtained as limiting cases of the more general exact series representations for the uniform current loop obtained in this paper convenient asymptotic far field expansions are derived for the loop with a cosinusoidal current distribution finally the far field analysis for the cosinusoidal loop is generalized to loops having an arbitrary current represented by a fourier cosine series

near fields of the constant current thin circular loop antenna of arbitrary radius assuming a known constant current distribution on the thin circular loop antenna of arbitrary radius in free space an exact integration of the vector potential is performed without recourse to approximations the only restrictions on the solution variables are that the observation point distance must be greater than the loop radius and that the polar angle must run between 0 and spl pi the resulting vector potential infinite series solution possesses a real part composed of linear combinations of complete elliptic integrals of the first and second kind and an imaginary part composed of elementary functions thus it is possible to obtain an exact solution which is valid everywhere that r a and 0 spl les spl theta spl les spl pi the electromagnetic field components of the constant current circular loop antenna are then determined by direct series differentiation these solutions are valid in the near and induction fields converging rapidly there and are also valid in the far field although many terms of the series are needed for convergence

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

edge effects in axial arrays of circular rings a boundary value problem is solved for the current induced by any source in particular for an incident plane wave on any number of identical rings that are arranged to form an axisymmetric linear array the net aperture distribution along the finite array is approximated by a combination of three physically identifiable components a term from the related infinite array plus contributions from two independent edge effect waves that originate from each of the two ends however if individually resonant loops are closer than a critical spacing the entire structure exhibits a wild resonance and the simple approximation becomes invalid

a new look at solving a system of fuzzy relational equations abstract solving a system of fuzzy relational equations exactly has been studied for many years a usual understanding on the solvable conditions is that the input fuzzy sets must be normal and pairwise disjoint such understanding is reexamined in this paper we show that the usual pairwise disjoint condition is too conservative and a system of max t fuzzy relational equations can be solved exactly when the input fuzzy sets are semi overlapped a condition commonly found in most rule based system applications in addition the boundary condition of solvability with respect to the compactness of the input fuzzy sets is derived if it cannot be satisfied we show that the system of equations could still be solved almost exactly by specifying the t norm as drastic product the results have been applied to study the capacity of fuzzy relations

entire domain mom analysis of an array of arbitrarily oriented circular loop antennas a general formulation this paper presents a general formulation for the entire domain method of moments analysis of an n element array of arbitrarily oriented coupled circular loop antennas the orientations of the loops with respect to each other are defined through the use of eulerian angles the positions of the loops in space are defined by translation vectors from the center of a loop to another by making use of coordinate transformations and vector algebra a concise set of linear equations for solving the currents on the loops is obtained the formulation is sufficiently general to be used for all the previously reported coupled loop orientations it is also ideal for the study of skewed circular loops which may be oriented in innumerable ways and in turn demand a general formulation for their analysis the use of the general formulation is illustrated by analyzing the coupling currents between collocated skewed loops the formulation is also applied to study the effects on the transfer impedance between a pair of noncollocated skewed loops as one of the loops is rotated about various axes

comment on analytical results for a bessel function times legendre polynomials class integrals a result is obtained stemming from gegenbauer where the products of certain bessel functions and exponentials are expressed in terms of an infinite series of spherical bessel functions and products of associated legendre functions closed form solutions for integrals involving bessel functions times associated legendre functions times exponentials recently elucidated by neves et al j phys a math gen 39 l293 are then shown to result directly from the orthogonality properties of the associated legendre functions this result offers greater flexibility in the treatment of classical heisenberg chains and may do so in other problems such as occur in electromagnetic diffraction theory

a model for the prediction of diseases of firms by means of fuzzy relations the core of our proposal is the determination of a matrix of economic financial knowledge which states fuzzy relations between symptoms and causes in this case the symptoms of a firm can be represented by the diverse ratios which are in the end the indicators the analysts look at to make their predictions and suggestions on the other hand the diseases of the firm will be the causes to be determined which generate anomalies at the firm the ability to create such a matrix of economic financial knowledge implies that given the level of intensity of the chosen symptoms we can determine the extent to which the main problems or diseases affecting a firm s cost effectiveness and efficiency are present at the firm in order to act upon them with the utilization of the same one can be complemented the activity of the analyst of the business

method of solution to fuzzy equations in a complete brouwerian lattice abstract up to now how to solve a fuzzy relation equation in a complete brouwerian lattice is still an open problem as di nola et al point out to this problem the key problem is whether there exists a minimal element in the solution set when a fuzzy relation equation is solvable in this paper we first show that there is a minimal element in the solution set of a fuzzy relation equation a x b where a a 1 a 2 a n and b are known and x x 1 x 2 x n t is unknown when its solution set is nonempty and b has an irredundant finite join decomposition further we give the method to solve a x b in a complete brouwerian lattice under the same conditions finally a method to solve a more general fuzzy relation equation in a complete brouwerian lattice when its solution set is nonempty is also given under similar conditions

some remarks on the identification problem in fuzzy systems abstract several identification problems in fuzzy systems are considered which are described by means of fuzzy relational equations the determination of a family of fuzzy relations of the system is described in detail

structural and functional quantization of vagueness abstract this paper proposes two methods of quantizing vague categories in terms of quantities whose identities have been either widely recognized in the fuzzy set theory literature or established by bayesian decision theory the methods are called structural quantization and functional quantization respectively the idea behind structural quantization is that for a comparative vague category like good scores there exists a preference ordering among the possible scores and how good a particular score is depends on how large the portion of examinees whose scores are lower than that score is functional quantization adopts the conclusion of bayesian decision theory that a rational agent s belief can be represented by a probability to quantize the term good score the method first assesses the two probabilities that respectively represent a rational agent s beliefs about john s score before and after learning that john has got a good score it then derives a membership function by comparing those two probabilities the equivalence of the two quantization methods is hypothesized the logic connectives not and and or are discussed in the setting of structural quantization while aggregation of information is addressed in the setting of functional quantization

on the existence of solutions of systems of fuzzy equations abstract a necessary and sufficient condition for the existence of solutions of a fuzzy equation given by e sanchez is extended to systems of fuzzy equations

fuzzy relational structures the state of art fuzzy relational structures built with the aid of fuzzy sets and fuzzy relations constitute a fundamental class of constructs arising in fuzzy set theory they give rise to fuzzy relation equations in this paper we will develop the concepts of single and multilevel relational structures different logic operations will be discussed in detail aspects of the knowledge representation will be studied and illustrated with several schemes of applications

the general α decomposition problem of fuzzy relations abstract this paper deals with a general decomposition problem of fuzzy relations which can be stated as follows given a fuzzy relation r f x y determine two fuzzy relations q f x z and t f z y such that r q t where x resp y is a finite set firstly we point out that every fuzzy relation r is always generally decomposable and give an algorithm to construct q and t with r q t for a given r secondly we show that the general content r with r min z r q t q f x z t f z y is equal to the chromatic number of the simple graph f r generated by r therefore finding an exact algorithm for calculating r is an np complete problem

synthesis of wire antennas using the multipole expansion method a rigorous mathematical solution to the problem of synthesizing wire antennas using the multipole expansion method mem is presented the input current and required size of the antenna are estimated using the knowledge of the required field pattern this solution is based on three steps first obtaining the expansion s weighting coefficients that carry both the signature of the current and size of the antenna in essence the analysis of the direct modeling needs to be addressed first in order to define these coefficients second extracting the coefficients from the required field pattern using the natural orthogonality property of the spherical harmonic functions finally performing a numerical search for the argument of the spherical bessel functions to satisfy the ratio between two successive moments once the size of the antenna is found a recursive expression is used to obtain the input current for the current distributions assumed herein the odd order coefficients of the mem are shown to be the only necessary coefficients to construct the radiation far field also it is found that the estimated model parameters are in excellent agreement with the analytical ones

erratum to weighted norm inequalities for the continuous square function trans amer math soc 314 1989 no 2 661 692 mr0972707 91e 42025

probability of a fuzzy event an axiomatic approach abstract axioms are proposed that could justify the natural definition of the probability of a fuzzy event initially given by zadeh they are based 1 on the postulate that the sum of the conditional probability of a fuzzy event and of its complement given any fuzzy event adds to one or 2 on soft independence for orthogonal sets with independent constitutive elements a general postulate is also required concerning the complement of a fuzzy set the classical definition of the operator representing the complement can also be deduced

necessary and sufficient conditions for the existence of solution of generalized fuzzy relation equations a x b abstract in 2013 li and jin studied a particular type of fuzzy relational equations on finite sets where the introduced min bi implication composition is based on ukasiewicz equivalence in this paper such fuzzy relation equations are studied on a more general level namely complete residuated lattice valued fuzzy relation equations of type y y a x y x y b x are analyzed and the existence of solutions s is studied first a necessary condition for the existence of solution is established then conditions for lower and upper limits of solutions are given and finally sufficient conditions for the existence of the smallest and largest solutions respectively are characterized if such general or global solutions do not exist there might still be partial or point wise solutions this is a novel way to study fuzzy relation equations such point wise solutions are studied on ukasiewicz product and godel t norm based residuated lattices on the real unit interval

novel neural algorithms based on fuzzy d rules for solving fuzzy relation equations part iii abstract in this paper we first design a fuzzy neuron which possesses some generality this fuzzy neuron is founded by replacing the operators of the traditional neuron with a pair of abstract fuzzy operators as which we call fuzzy neuron operators for example it may be or etc it is an extended fuzzy neuron and a network composed o such neurons is an extended fuzzy neural network then we discuss the relationship between the fuzzy neuron operators and t norm and t conorm and point out fuzzy neuron operators are based on t norm but much wider than t norm in this paper we will emphatically discuss a two layered network and its training algorithm which will have to satisfy a set of various operators this work is very related to solving fuzzy relation equations so it can be used to resolve fuzzy relation equations furthermore the new fuzzy neural algorithm is found to be stronger than other existing methods to some degree some simulation results will be reported in detail

additions of interactive fuzzy numbers this paper provides an account of an approach to modeling unknown data by means of fuzzy set theory and addresses the problem of deriving the uncertainty on a sum of variables whose values lie within fuzzy intervals the first part is an extensive presentation of the theoretical background of the approach the extension principle is stated in terms of possibility of an event the concept of variable interaction is investigated at length section ii gives new results regarding the effective practical computation of additions of fuzzy numbers its originality lies in the introduction of interaction which enables to control the growth of uncertainty in calculations moreover the problem of computing mathematical expectations with fuzzy probabilities is solved the results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities more generally fuzzy arithmetic can be an important tool for sophisticated computationally tractable sensitivity analysis in systems modeling computer aided design and operations research

identification of fuzzy relational equations by fuzzy neural networks abstract although the identification of a fuzzy system by a feedforward neural network is obviously possible whenever the system was continuous however when we use fuzzy max min neural network some problems on the learning of the network arise in this paper we present a method to identify a fuzzy relation by a fuzzy max min neural network we adapt the learning algorithm backpropagation to learning of max min neural network by using one which we will denote as smooth derivative finally we present some examples and a comparison with a similar method

comments on exact field expressions for circular loop antennas using spherical functions expansion the author would like to thank the commentators dr zheng and dr li for reviewing and commenting on the work reported in 1 the author disagrees with their conclusion that the author in 1 is criticizing the merits of their published papers the author is merely explaining the differences between the author s work in 1 and the commentators work reported in 2 this should not detract from the contribution they made in 2 as indicated in their comments

sharp weighted inequalities for the vector valued maximal function we prove in this paper some sharp weighted inequalities for the vector valued maximal function mq of fefferman and stein defined by 00 llq mqf x mfi o q where m is the hardy littlewood maximal function as a consequence we derive the main result establishing that in the range 1 a jf x jq mw x dx a rn where c is a constant independent of a 1 motivation and description of the main results the purpose of this paper is to obtain some sharp weighted inequalities for the vector valued maximal function mq which are not within the scope of the standard ap theory for vector valued singular integrals as can be found in rrt we start with a review of some of the classical estimates and then we shall state the main results 1 1 background let m be the hardy littlewood maximal function and let mq be the vector valued maximal operator defined by mq f x e m fi x q this nonlinear operator was introduced by c fefferman and e m stein in fs as a generalization of both the scalar maximal function m and the classical integral of marcinkiewicz and since then it has played an important role in the development of modern harmonic analysis we recall the two basic estimates obtained in fs for 1 q oo received by the editors may 19 1997 1991 mathematics subject classification primary 42b20 42b25 42b15 this work was partially supported by dgicyt grant pb940192 spain 2000 american mathematical society 3265 this content downloaded from 207 46 13 102 on sat 30 jul 2016 04 48 12 utc all use subject to http about jstor org terms

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

powers of matrices over distributive lattices a review in this paper we review the results on powers of matrices over a distributive lattice we stress the significance of the graph theoretical approach and show how several previous results for matrices of special types can be obtained in a unified way

efficient modeling of radiation and scattering for a large array of loops a computationally efficient technique based on the method of moments mom formulation is invoked in the characterization of radiation and scattering properties of an array of coaxial circular non identical loops a set of pocklington type integral equations for the loop currents is formulated and subsequently discretized by a standard procedure thanks to a suitable choice of the basis functions the resulting matrix corresponding to the pertinent linear system is forced to consist of circulant blocks this type of system is solvable by an innovative recursive algorithm featuring several important advantages such as lower memory and execution time consumption over standard purely numerical inversion the overall procedure is simpler in implementation than already existing methods based on fourier analysis the procedure invokes almost exclusively elementary functions and is applicable to large arrays with respect to diameter or number of loops data for such configurations are presented for the first time in literature

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

decisions and evaluations by hierarchical aggregation of information usually in preparing a decision a hierarchy of criteria for evaluating alternatives can he agreed upon by the decision makers the problem addressed in this paper is the appropriate modelling of such evaluations and their aggregation in order to arrive at a proper overall evaluation function by which the decision alternatives can be ordered easily representing the criteria subjective categories by fuzzy sets it is possible to use aggregation operators as models for the amalgamation of those criteria in the framework of an empirical research project a hierarchy of criteria was developed for the evaluation of credit worthiness of customers 50 credit applications were evaluated by a number of credit managers of banks and their judgement could be appropriately modelled by a four level hierarchy of criteria

further relation equations with equality and difference composition operators in this ch we will introduce other composition operators which have a plausible logical interpretation they will be called equality and difference operator respectively we first define a notion of equality and difference of any two grades of membership of a fuzzy set and of a fuzzy relation in the sequel this concept will be utilized to form the respective composition operators and the related fuzzy equations afterwards we provide a method of resolution of these equations characterizing completely the set of the solutions all the fuzzy sets involved are defined on finite sets and with membership values in 0 1

approximately solving fuzzy relation equations some mathematical results and some heuristic proposals abstract fuzzy relation equations are intimately connected with the approach toward fuzzy control unfortunately systems of such equations are solvable only under quite restrictive conditions therefore a notion of approximate solution is discussed which applied also in cases where such systems do not have solutions for some types of fuzzy equations best possible approximate solutions are known for other types some strategies are considered to suitably guess approximate solutions furthermore that approach is applied to some types of cascaded fuzzy relation equations which are connected with ideas to join neural network structures with fuzzy control

fuzzy relation equations on a finite set abstract the family of all the solutions of a fuzzy relation equation on a finite set is considered it is characterized by the set of all the lower solutions which can be obtained by a combinatorial algorithm

response to comments on exact field expressions for circular loop antennas using spherical functions expansion the author would like to thank the commentators dr zheng and dr li for reviewing and commenting on the work reported in the author disagrees with their conclusion that the author in is criticizing the merits of their published papers the author is merely explaining the differences between the author s work in and the commentators work reported in this should not detract from the contribution they made in as indicated in their comments