generalized means as model of compensative connectives abstract empirical investigations have shown that instead of strong logical and and or connectives there is often some degree of compensation in between in this paper the generalized means are proposed as connective operators for fuzzy set theory which easily allow for modelling the degree of compensation in a natural manner including the arithmetic and geometric means as well as the maximum and minimum operators as special cases numerical examples with empirical data form an illustration of this concept some aspects of applications to decision making are discussed

double sweep preconditioner for optimized schwarz methods applied to the helmholtz problem this paper presents a preconditioner for non overlapping schwarz methods applied to the helmholtz problem starting from a simple analytic example we show how such a preconditioner can be designed by approximating the inverse of the iteration operator for a layered partitioning of the domain the preconditioner works by propagating information globally by concurrently sweeping in both directions over the subdomains and can be interpreted as a coarse grid for the domain decomposition method the resulting algorithm is shown to converge very fast independently of the number of subdomains and frequency the preconditioner has the advantage that like the original schwarz algorithm it can be implemented as a matrix free routine with no additional preprocessing

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

applying gmres to the helmholtz equation with shifted laplacian preconditioning what is the largest shift for which wavenumber independent convergence is guaranteed there has been much recent research on preconditioning discretisations of the helmholtz operator delta k 2 k2 subject to suitable boundary conditions using a discrete version of the so called shifted laplacian delta k 2 mathrm i varepsilon k2 i for some varepsilon 0 0 this is motivated by the fact that as varepsilon increases the shifted problem becomes easier to solve iteratively despite many numerical investigations there has been no rigorous analysis of how to chose the shift in this paper we focus on the question of how large varepsilon can be so that the shifted problem provides a preconditioner that leads to k k independent convergence of gmres and our main result is a sufficient condition on varepsilon for this property to hold this result holds for finite element discretisations of both the interior impedance problem and the sound soft scattering problem with the radiation condition in the latter problem imposed as a far field impedance boundary condition note that we do not address the important question of how large varepsilon should be so that the preconditioner can easily be inverted by standard iterative methods

using fuzzy sets in operational research abstract during the last four decades a large number of mainly mathematical tools have been developed in and for operational research they are primarily devices to find optimal solutions after a problem has been modelled formally hardly any new tools have been developed for proper modelling of problems which are not well structured and which can not easily be cast into classical mathematical models fuzzy set theory as proposed by l zadeh in 1965 promises to bridge part of this gap it is much more adaptable to different problem structures and better suited to model human evaluation and decision making processes than classical mathematics this paper will primarily focus attention on four questions why fuzzy sets in or what are fuzzy sets where can they be used and how are they being used

matrix reordering using multilevel graph coarsening for ilu preconditioning incomplete lu factorization ilu techniques are a well known class of preconditioners often used in conjunction with krylov accelerators for the iterative solution of linear systems of equations however for certain problems ilu factorizations can yield factors that are unstable and in some cases quite dense reordering techniques based on permuting the matrix prior to performing the factorization have been shown to improve the quality of the factorization and the resulting preconditioner in this paper we examine the effect of reordering techniques based on multilevel graph coarsening ideas on one level ilu factorizations such as the level based ilu k or the dual threshold ilut algorithms we consider an aggregation based coarsening idea that implements two main coarsening frameworks a top down approach and a bottom up approach each utilizing one of two different strategies to select the next level coarse graph numerical results are presented to support our findings

low rank damping modifications and defective systems the structural modification of dynamical systems is an important issue in a wide range of applications for example in vibration suppression or in active and passive control it is well known that for a proportionally or classically damped system there always exists a real matrix of eigenvectors which simultaneously diagonalizes the three system matrices of inertia damping and stiffness even if the system possesses repeated eigenvalues for general viscously damped systems the eigenvalue analysis must be performed in state space and for systems with distinct eigenvalues the corresponding eigenvectors diagonalize the state space matrices however with general viscous damping systems with repeated complex eigenvalues may have insufficient linearly independent complex eigenvectors these systems are termed defective in contrast to non defective or simple systems for defective systems only a jordan decomposition exists in this paper conditions on rank 1 rank 2 and higher rank modifications are derived which ensure that the modified system is simple if none of the eigenvalues of the unmodified system is an eigenvalue of the modified system then every rank 1 modification that produces a pair of repeated complex eigenvalues leads to a defective system under the same assumptions there exist higher rank modifications which lead to simple systems either these modifications produce a proportionally damped system or the restrictions on these modifications are rather strict which suggests that in practical cases every rank 2 or higher modification that produces repeated pairs of complex eigenvalues will lead to a defective system the findings are demonstrated by simulated examples

optimized schwarz methods without overlap for the helmholtz equation the classical schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel convergence is achieved through overlap of the subdomains we study in this paper a variant of the schwarz method which converges without overlap for the helmholtz equation we show that the key ingredients for such an algorithm are the transmission conditions we derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps these conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the schwarz method this leads to an algorithm in the class of optimized schwarz methods we present an asymptotic analysis of the optimized schwarz method for two types of transmission conditions robin conditions and transmission conditions with second order tangential derivatives numerical results illustrate the effectiveness of the optimized schwarz method on a model problem and on a problem from industry

membership function as an evaluation abstract after giving an overview of the different kinds of mathematical forms of the membership functions we extract the different demands and determine the rational class of the membership functions from the generalisation of our result we can determine it giving four parameters the internal a b the sharpness and the decision level we show the correspondence of this membership function to the zimmermann zysno model and using their empirical data we also test our mathematical model we also show the connections between the operators of the evaluation and membership functions which gives the generalisation of this concept in a more general form as a special case we can get a linear form

the decoupling of damped linear systems in free or forced vibration abstract the purpose of this paper is to extend classical modal analysis to decouple any viscously damped linear system in non oscillatory free vibration or in forced vibration based upon an exposition of how exponential decay in a system can be regarded as imaginary oscillations the concept of damped modes of imaginary vibration is introduced by phase synchronization of these real and physically excitable modes a time varying transformation is constructed to decouple non oscillatory free vibration when time drifts caused by viscous damping and by external excitation are both accounted for a time varying decoupling transformation for forced vibration is derived the decoupling procedure devised herein reduces to classical modal analysis for systems that are undamped or classically damped this paper constitutes the second and final part of a solution to the classical decoupling problem together with an earlier paper a general methodology that requires only the solution of a quadratic eigenvalue problem is developed to decouple any damped linear system

aggregation operators ordering and bounds standard ordering of aggregation operators in several distinguished classes is characterized lower and upper bounds of finite systems in those classes of aggregation operators are discussed several examples are given

spectral analysis of the discrete helmholtz operator preconditioned with a shifted laplacian shifted laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the helmholtz equation in this paper we present a comprehensive spectral analysis of the helmholtz operator preconditioned with a shifted laplacian our analysis is valid under general conditions the propagating medium can be heterogeneous and the analysis also holds for different types of damping including a radiation condition for the boundary of the computational domain by combining the results of the spectral analysis of the preconditioned helmholtz operator with an upper bound on the gmres residual norm we are able to provide an optimal value for the shift and to explain the mesh dependency of the convergence of gmres preconditioned with a shifted laplacian we illustrate our results with a seismic test problem

the transformation of second order linear systems into independent equations the class of second order linear dynamical systems is considered a method and algorithm are presented to transform any system with n degrees of freedom into n independent second order equations the conversion utilizes a real invertible but nonlinear mapping and is applicable to practically every linear system two examples from earthquake engineering are provided to indicate the utility of this approach

the distributivity property of increasing binary operations abstract this paper is mainly devoted to solving the functional equations of distributivity and conditional distributivity of increasing binary operations with the unit our investigations are motivated by distributive logical connectives and their generalizations used in fuzzy set theory in particular some assumptions namely associativity and commutativity and results about conditional distributivity of uninorms and triangular norms and triangular conorms were simplified moreover some other properties e g componentwise convexity concavity for special operations from the class of quasi copulas is considered

short communication an impossibility theorem for indicators aggregation using the fuzzy set theory we prove that there is no indicator at the same time aggregated and sensitive anticatastrophic and noncompensatory such a result theoretically justifies the weaknesses of certain indicators used at present as well as the noncomparability of certain social issues such as the quality of life etc

pictorial representations of fuzzy connectives part i cases of t norms t conorms and averaging operators part i of this paper shows a number of examples of existing t norms and t conorms which are known as good models of fuzzy set theoretic intersection and union and then the pictorial representations of t norms and t conorms are made with the aid of a computer several new examples of t norms and t conorms are also given averaging operators between min and max are summarized and their pictorial representations are also made

additively decomposed quasiconvex functions letf be a real valued function defined on the product ofm finite dimensional open convex setsx1 xm assume thatf is quasiconvex and is the sum of nonconstant functionsf1 fm defined on the respective factor sets then everyfi is continuous with at most one exception every functionfi is convex if the exception arises all the other functions have a strict convexity property and the nonconvex function has several of the differentiability properties of a convex function we define the convexity index of a functionfi appearing as a term in an additive decomposition of a quasiconvex function and we study the properties of that index in particular in the case of two one dimensional factor sets we characterize the quasiconvexity of an additively decomposed functionf either in terms of the nonnegativity of the sum of the convexity indices off1 andf2 or equivalently in terms of the separation of the graphs off1 andf2 by means of a logarithmic function we investigate the extension of these results to the case ofm factor sets of arbitrary finite dimensions the introduction discusses applications to economic theory

extended neuro fuzzy models of multilayer perceptrons in this paper the famous neural model the multilayer perceptron is extended to a new neural model that is called the additive takagi sugeno type multilayer perceptron the present study proves that this new model can also act as a universal approximator and thus it can be used in many fields such as system modeling and identification the concept of f duality and the fuzzy operator interactive or are used to prove that the proposed neural model is functionally equal to a kind of fuzzy inference system further this paper presents another new neuro fuzzy model that is called the sigmoid adaptive network based fuzzy inference system simulation studies show that our proposed models both have stronger approximation capability than multilayer perceptrons

an extended rule based inference for general decision making problems complex decision making problems are usually modeled in terms of a number of decisive variables that are related hierarchically pieces of evidence are aggregated in a bottom up way to determine the final decision goal for dealing with such problems ishizuka et al 5 6 developed a rule based inference procedure by using fuzzy sets to represent imprecise data and employed the dempster shafer theory to aggregate evidence for structural damage assessment this paper highlights the necessity of applying the rules with all possible consequent clauses in the procedure with the considerations of information quality and quantity when applying the procedure to more general decision making problems the use of such rules can provide high quality information which contains much less ignorance in the resultant information also it can afford more information due to the decrease in total ignorance a simple example with two cases is used to compare the results based on the two different presentations of rules in the inference procedure for illustrating the importance of considering the rules proposed in this paper

the self dual core and the anti self dual remainder of an aggregation operator in most decisional models based on pairwise comparison between alternatives the reciprocity of the individual preference representations expresses a natural assumption of rationality in those models self dual aggregation operators play a central role in so far as they preserve the reciprocity of the preference representations in the aggregation mechanism from individual to collective preferences in this paper we propose a simple method by which one can associate a self dual aggregation operator to any aggregation operator on the unit interval the resulting aggregation operator is said to be the self dual core of the original one and inherits most of its properties our method constitutes thus a new characterization of self duality with some technical advantages relatively to the traditional symmetric sums method due to silvert in our framework moreover every aggregation operator can be written as a sum of a self dual core and an anti self dual remainder which in some cases seems to give some indication on the dispersion of the variables in order to illustrate the method proposed we apply it to two important classes of continuous aggregation operators with the properties of idempotency symmetry and stability for translations the owa operators and the exponential quasiarithmetic means

basic concepts for a theory of evaluation hierarchical aggregation via autodistributive connectives in fuzzy set theory abstract the aggregative operator introduced by combi 1982 in his analysis of basic concepts for a theory of evaluation in this journal is based on the axioms of associativity continuity and monotonicity in this paper an alternative approach is given where associativity is replaced by another axiom viz autodistributivity implying a kind of hierarchical aggregation instead of the former horizontal one the new set of properties yields a general class of operators for connecting fuzzy sets which are formally similar to the aggregative operators but are purely compensative in a strong sense moreover this modified approach generalizes a recent approach based on generalized means as compensative connectives it appears to be similar to expected utility approaches in decision making under uncertainty and seems to be a very promising tool to handle representation problems as they grow from other fields of human decision making e g of multicriteria analysis and of welfare theory

tackling an mcdm problem with the help of some results from fuzzy set theory abstract the problem to be addressed and tackled in this paper arose as a byproduct from some efforts at solving problems involving multiple goals by linking linear and goal programming models the critical issue was that some forms for interdependence among the goals could not be handled in the programming models here we will deal with a set of goals with realistic counterparts in a finnish plywood industry in which a subset of the goals are i conflicting another subset ii unilaterally supporting and a third subset iii mutually supporting it is furthermore observed that the elements of a studied set of goals may be partly independent and partly interdependent which makes the context a fullfledged mcdm problem it is tackled with a technique which is based on the theory of fuzzy sets the conceptual framework for fuzzy decisions and the algorithms developed for fuzzy mathematical programming the resulting fuzzy multiobjective programming model is simplified and tested with the help of a fairly complex numerical example

preasymptotic error analysis of cip fem and fem for helmholtz equation with high wave number part ii hp version in this paper which is the second in a series of two the preasymptotic error analysis of the continuous interior penalty finite element method cip fem and the fem for the helmholtz equation in two and three dimensions is continued while part i contained results on the linear cip fem and fem the present part deals with approximation spaces of order p ge 1 by using a modified duality argument preasymptotic error estimates are derived for both methods under the condition of frac kh p le c 0 frac p k frac 1 p 1 where k is the wave number h is the mesh size and c 0 is a constant independent of k h p and the penalty parameters it is shown that the pollution errors of both methods in h 1 norm are o k 2p 1 h 2p if p o 1 and are o frac k p 2 frac kh sigma p 2p if the exact solution u in h 2 omega which coincide with existent dispersion analyses for the fem on cartesian grids here sigma is a constant independent of k h p and the penalty par

aggregation functions construction methods conjunctive disjunctive and mixed classes in this second part of our state of the art overview on aggregation theory based again on our recent monograph on aggregation functions we focus on several construction methods for aggregation functions and on special classes of aggregation functions covering the well known conjunctive disjunctive and mixed aggregation functions some fields of applications are included

the decoupling of defective linear dynamical systems in free motion abstract it was demonstrated in two earlier papers that there exists a real linear time varying transformation that decouples any non defective linear dynamical system in free vibration in the configuration space as an extension of this work the present paper represents the first systematic effort to decouple defective systems it is shown that the decoupling of defective systems is a rather delicate procedure that depends on the multiplicities of the system eigenvalues while any defective system can be decoupled with the eigenvalues kept invariant the geometric multiplicities of these eigenvalues may not be preserved several numerical examples are provided to illustrate the theoretical developments

multilevel preconditioner with stable coarse grid corrections for the helmholtz equation in this paper we consider a class of robust multilevel preconditioners for the helmholtz equation with high wave number the key idea in this work is to use the continuous interior penalty finite element methods studied in h wu ima j numer anal 34 2014 pp 1266 1288 l zhu and h wu siam j numer anal 51 2013 pp 1828 1852 to construct the stable coarse grid correction problems the multilevel methods based on gmres smoothing on coarse grids are then served as a preconditioner in the outer gmres iteration in the one dimensional case the convergence property of the modified multilevel methods is analyzed by the local fourier analysis from our numerical results we find that the proposed methods are efficient for a reasonable range of frequencies the performance of the algorithms depends relatively mildly on wave number

aggregation operators and commuting commuting is an important property in any two step information merging procedure where the results should not depend on the order in which the single steps are performed we investigate the property of commuting for aggregation operators in connection with their relationship to bisymmetry in case of bisymmetric aggregation operators we show a sufficient condition ensuring that two operators commute while for bisymmetric aggregation operators with neutral element we even provide a full characterization of commuting n ary operators by means of unary distributive functions the case of associative operations especially uninorms is considered in detail

social choice axioms for fuzzy set aggregation the problem of aggregating n fuzzy sets f1 f2 fn on a set is viewed as one of merging the opinions of n individuals e g experts that rate objects belonging to this approach contrasts with the pure set theoretic point of view and leads to interpreting already known axioms underlying fuzzy connectives in a way different from that of multiple criteria aggregation various natural properties of a voting procedure including the ones proposed by arrow are expressed in the fuzzy set setting a number of conditions limiting the choice of fuzzy set operations are proposed and classified according to whether they are imperative mainly technical or facultative families of solutions are obtained that include those proposed in earlier works the case of non homogeneous groups is briefly examined lastly the application of the voting paradigm to the management of antagonistic local decision rules in knowledge based systems is outlined

compensatory fuzzy multiple level decision making abstract fuzzy set theory has been shown to be an effective tool to overcome the computational difficulties encountered in solving large multiple level programming problems shih et al 1996 in this paper compensatory operators are introduced for adjusting the decision making process between the different levels and also between the decision makers of the same level after a brief consideration of the bi level and three level systems the large decentralized organizations with both equal and unequal goals are investigated various numerical examples are given to compare the influences of compensation and to illustrate the approaches

gpu implementation of a helmholtz krylov solver preconditioned by a shifted laplace multigrid method a helmholtz equation in two dimensions discretized by a second order finite difference scheme is considered krylov methods such as bi cgstab and idr s have been chosen as solvers since the convergence of the krylov solvers deteriorates with increasing wave number a shifted laplace multigrid preconditioner is used to improve the convergence the implementation of the preconditioned solver on cpu central processing unit is compared to an implementation on gpu graphics processing units or graphics card using cuda compute unified device architecture the results show that preconditioned bi cgstab on gpu as well as preconditioned idr s on gpu is about 30 times faster than on cpu for the same stopping criterion

uniform convergence of multigrid v cycle iterations for indefinite and nonsymmetric problems in this paper an analysis of a multigrid method for nonsymmetric and or indefinite elliptic problems is presented in this multigrid method various types of smoothers may be used one type of smoother considered is defined in terms of an associated symmetric problem and includes point and line jacobi and gauss seidel iterations smoothers based entirely on the original operator are also considered one smoother is based on the normal form that is the product of the operator and its transpose other smoothers studied include point and line jacobi and gauss seidel it is shown that the uniform estimates of j h bramble and j e pasciak math comp 60 1993 pp 447 471 for symmetric positive definite problems carry over to these algorithms more precisely the multigrid iteration for the nonsymmetric and or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine but not dependent on the number of multigrid levels

basic concepts for a theory of evaluation the aggregative operator abstract starting with an explication of the aggregative concept and deducing a general structure which satisfies a number of minimal requirements properties of clustering the main features of a new mathematical theory called theory of evaluation are developed the theory sheds new light on such well known concepts as membership conjunction and disjunction and seems to be a very promising tool to handle representation problems as they grow from the fields of theory of fuzzy set and its many applications of human decision making and of multicriteria analysis

a multigrid method enhanced by krylov subspace iteration for discrete helmholtz equations standard multigrid algorithms have proven ineffective for the solution of discretizations of helmholtz equations in this work we modify the standard algorithm by adding gmres iterations at coarse levels and as an outer iteration we demonstrate the algorithm s effectiveness through theoretical analysis of a model problem and experimental results in particular we show that the combined use of gmres as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200 for fixed wave numbers it displays grid independent convergence rates and has costs proportional to the number of unknowns

different interpretations of triangular norms and related operations triangular norms and triangular conorms and uni norms can be treated as semigroup two place function also as commutative semiring multiplications it can also be derived from the difference operation in a difference poset we discuss here these different interpretations and their consequences

iterative solution of helmholtz problem with high order isogeometric analysis and finite element method at mid range frequencies abstract solving wave problems with isogeometric analysis has attracted a significant attention in the past few years it is well known that keeping a fixed number of degrees of freedom per wavelength leads to an increased error as higher wavenumbers are considered this behaviour often cited as the pollution error improves significantly with isogeometric analysis when compared to the conventional finite element method the improvement in handling pollution along with the ability to represent exact geometries has been the main reasons behind the attention that isogeometric analysis has received furthermore using high order elements also presents major advantages over low order elements for this range of frequencies however it remains to be studied how iterative linear solvers often necessary for solving high frequency wave problems perform when using isogeometric analysis compared to the finite element method especially at high polynomial orders this paper is one of the first studies in this direction in this work we investigate the generalised minimal residual method a standard krylov subspace iterative technique for solving the linear system resulting form isogeometric analysis furthermore we look into the use of some recently proposed preconditioners for helmholtz problem such as shifted laplace or ilu with a complex shift preconditioners and how they perform with high order isogeometric analysis and finite element method in general the results show improvement when using isogeometric analysis in terms of the number of iterations required for convergence compared to the finite element method for both preconditioned and non preconditioned linear systems we use eigenvalue spectra to understand this improvement

compensatory operators in fuzzy linear programming with multiple objectives previously h j zimmermann 11 presented a fuzzy approach to solve the linear vectormaximum problem in the mentioned approach the author uses the minimum or the product operators to translate the semantic meaning of the and the aim of this paper is to reconsider this approach by using operators which allow some degree of compensation between aggregated membership functions firstly the c operator already defined in 12 is used to combine our fuzzy objectives it is shown that solutions obtained by this way are always efficient unfortunately the substitute problem is not generally computationally feasible we suggest then the min bounded sum operator which is a compensatory one has some desired properties and is supported by empirical arguments it allows a considerable decrease of the computational burden in the substitute problem and leads to a solution which is attractive from the stand point of efficiency

analysis of errors and approximations in the use of modal co ordinates abstract a basic operation in several updating procedures and other modal analysis applications concerns the transformation of the complex modes identified from experimental tests into the real ones typical of theoretical models in this transformation several approximations are normally adopted this paper presents a discussion of two of these approximations the first refers to a series expansion of the complex modes into the subspace of real modes commonly used for normalization of complex modes the second used in advanced modal analysis applications relates the physical damped co ordinates to the modal ones through the real modal matrix an explanation of the limitations of these assumptions is given through analytical developments and numerical results obtained from a discrete model

how large a shift is needed in the shifted helmholtz preconditioner for its effective inversion by multigrid the shifted helmholtz operator has received a lot of attention over the past decade as a preconditioner for the iterative solution of the helmholtz equation the idea is that if one uses a small complex shift the shifted helmholtz operator is still close to the original helmholtz operator and could thus be an effective preconditioner it was shown in m j gander i g graham and e a spence numer math 53 2015 pp 573 579 that the shift can be at most o k to prove rigorously wave number independent convergence of the preconditioned system solved with gmres provided the preconditioner is inverted exactly in practice however the preconditioner is inverted only approximately and if one shifts enough this can be done effectively by standard multigrid methods we show in this paper that for a finite element discretization the shift has to be at least o k 2 to be able to invert the shifted helmholtz preconditioner using multigrid there is therefore a gap between being a good precondi

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

field of values analysis of a two level preconditioner for the helmholtz equation in this paper we study the convergence of a two level preconditioned gmres for linear systems related to first order finite element discretizations of helmholtz equation in a lossy media due to losses the finite element system matrix is nonnormal to handle this nonnormality we use a field of values based convergence criterion for gmres the focus is on a priori analysis to study the dependency between gmres convergence and wave number losses and coarse as well as fine grid mesh sizes before any actual computations are done the analysis indicates that the coarse grid mesh size h should satisfy the constraint kappa 3 h ll 1 to guarantee wave number and mesh size independent convergence of the preconditioned iteration the obtained theoretical results are illustrated in two and three dimensional numerical examples

a canonical form of the equation of motion of linear dynamical systems the equation of motion of a discrete linear system has the form of a second order ordinary differential equation with three real and square coefficient matrices it is shown that for almost all linear systems such an equation can always be converted by an invertible transformation into a canonical form specified by two diagonal coefficient matrices associated with the generalized acceleration and displacement this canonical form of the equation of motion is unique up to an equivalence class for non defective systems as an important by product a damped linear system that possesses three symmetric and positive definite coefficients can always be recast as an undamped and decoupled system

an approach to generalization of fuzzy topsis method the topsis method is a technique for establishing order preference by similarity to the ideal solution and was primarily developed for dealing with real valued data this technique is currently one of most popular methods for multiple criteria decision making mcdm in many cases it is hard to present precisely exact ratings of alternatives with respect to local criteria and as a result these ratings are seen as fuzzy values a number of papers have been devoted to fuzzy extensions of the topsis method in the literature but these extensions are not complete since the ideal solutions are usually presented as real values not by fuzzy values or as fuzzy values which are not attainable in the decision matrix in most of these papers a defuzzification of elements of the fuzzy decision matrix is used which leads inevitably to a loss of important information and may even produce the wrong results in this paper we propose a new direct approach to the fuzzy extension of the topsis method which is free of the limitations of other known approaches we show that the distances of the alternatives from the ideal solutions may be treated in some sense as modified weighted sums of local criteria it is known that using weighted sums is not the best approach to the aggregation of local criteria in many real world situations therefore here we propose the use in addition to weighted sums some other types of local criteria aggregation in the topsis method and we develop a method for the generalization of different aggregation modes providing compromised final results