spectral hp least squares finite element formulation for the navier stokes equations we consider the application of least squares finite element models combined with spectral hp methods for the numerical solution of viscous flow problems the paper presents the formulation validation and application of a spectral hp algorithm to the numerical solution of the navier stokes equations governing two and three dimensional stationary incompressible and low speed compressible flows the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity or velocity gradients as additional independent variables and the least squares method is used to develop the finite element model high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method spectral convergence of the l2 least squares functional and l2 error norms is verified using smooth solutions to the two dimensional stationary poisson and incompressible navier stokes equations numerical results for flow over a backward facing step steady flow past a circular cylinder three dimensional lid driven cavity flow and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation

hierarchical pinning models quadratic maps and quenched disorder we consider a hierarchical model of polymer pinning in presence of quenched disorder introduced by derrida et al j stat phys 66 1189 1213 1992 which can be re interpreted as an infinite dimensional dynamical system with random initial condition the disorder it is defined through a recurrence relation for the law of a random variable r n n 1 2 which in absence of disorder i e when the initial condition is degenerate reduces to a particular case of the well known logistic map the large n limit of the sequence of random variables 2 n log r n a non random quantity which is naturally interpreted as a free energy plays a central role in our analysis the model depends on a parameter 0 1 related to the geometry of the hierarchical lattice and has a phase transition in the sense that the free energy is positive if the expectation of r 0 is larger than a certain threshold value and it is zero otherwise it was conjectured in derrida et al j stat phys 66 1189 1213 1992 that disorder is relevant respectively irrelevant or marginally relevant if 1 2 1 2 we find the correct scaling form for weak disorder of the critical point shift

a comparison of discrete algorithms for fuzzy weighted average fuzzy weighted average fwa which can be applied to various fields such as engineering design decision analysis etc and as function of fuzzy numbers is suitable for the problem of multiple occurrences of fuzzy parameters additional fuzziness may be introduced in the alpha cut arithmetic this paper reviews and compares discrete algorithms for the fwas in both theoretical comparison and numerical comparisons as opposed to the linear programming algorithms that may be efficient but require the help of linear programming software an alternative efficient algorithm is also proposed the algorithm introduces an all candidate criteria ratings weights replaced benchmark adjusting procedure other than a binary dichotomy search in the existing methods the theoretical worst case comparison shows that the algorithms of guu and guh and our alternative algorithm require the same order of calculation complexity o n where n is the number of fwa terms yet the guh algorithm always requires 2 n 1 calculations to be performed the algorithm of guu requires the elemental comparison complexity o n and is superior to the other algorithms in the worst case on the other hand our alternative algorithm and lee and park algorithm have the same comparison complexity o n log n yet our alternative algorithm requires o n calculation complexity which is better than o n log n of the lee and park algorithm the numerical experiments show that the algorithm of guu and our alternative algorithm generally perform and converge faster by requiring fewer calculations than that of lee and park s algorithm the alternative algorithm requires a slightly smaller number of calculations than that of the guu algorithm due to the use of the convergent benchmark adjustment procedure rather than a somewhat fixed binary search conversely in the number of element comparisons lee and park s algorithm is shown numerically generally slightly better than the alternative algorithm due to the simple binary search scheme used the alternative algorithm also requires fewer comparisons than that of the guu algorithm furthermore in comparison of the average cpu time requirements the ranks of the results of these algorithms are consistent with those of the algorithms for the average numbers of calculations in general the alternative algorithm may provide an efficient alternative algorithm to the fwas in the worse case situation the guu algorithm still may be considered as an efficient alternative

a new approach to normalization of interval and fuzzy weights a new approach to normalization of interval and fuzzy weights based on the so called interval extended zero method is proposed the three desirable intuitively obvious properties of the normalization procedure are defined the main of them is based on the assumption that the sum of normalized interval or fuzzy weights should be an interval or a fuzzy value centered around 1 with a minimal width the advantages of a new approach are illustrated with the use of numerical examples it is shown that a new approach performs better than known methods for normalization of interval and fuzzy weights as it provides the results with the properties which are close to the desirable ones

global convergence of karnik mendel algorithms karnik mendel km algorithms are the most commonly used iterative type reduction methods in interval type 2 fuzzy sets and systems as well as new techniques for computing the fuzzy weighted average fwa various extensions and improvements have been proposed however no proof has been provided for the convergence of these extensions it is necessary to provide the proof because many of the iterative algorithms may have divergence cases in the present study we provide a theoretical proof that km algorithms exhibit global convergence different initialization methods and iteration formats can always obtain the same unique optimal solution thus there are no concerns about the possibility of divergence in extensions of km algorithms our proof provides theoretical support for the applications of km algorithms especially the type reduction designs used in type 2 fuzzy systems and fwa computations because of the important roles of km algorithms in these methods

existence of a non averaging regime for the self avoiding walk on a high dimensional infinite percolation cluster let z n be the number of self avoiding paths of length n starting from the origin on the infinite cluster obtained after performing bernoulli percolation on mathbb z d with parameter p p c mathbb z d the object of this paper is to study the connective constant of the dilute lattice limsup n rightarrow infty z n 1 n which is a non random quantity we want to investigate if the inequality limsup n rightarrow infty z n 1 n le lim n rightarrow infty mathbb e z n 1 n obtained with the borel cantelli lemma is strict or not in other words we want to know if the quenched and annealed versions of the connective constant are equal on a heuristic level this indicates whether or not localization of the trajectories occurs we prove that when d is sufficiently large there exists p 2 c p c such that the inequality is strict for p in p c p 2 c

least squares finite element methods for optimization and control problems for the stokes equations the approximate solution of optimization and control problems for systems governedby the stokes equations is considered modern computational techniques for such problems are predominantly based on the application of the lagrange multiplier rule while penalty formulations even though widely used in other settings have not enjoyed the same level of popularity for this class of problems a discussion is provided that explains why naively defined penalty methods may not be practical then practical penalty methods are defined using methodologies associated with modern least squares finite element methods the advantages with respect to efficiency of penalty leasts squares methods for optimal control problems compared to methods based on lagrange multipliers are highlighted a tracking problem for the stokes system is used for illustrative purposes

on the partition function of a directed polymer in a gaussian random environment the purpose of this work is the study of the partition function of a dimensional lattice directed polymer in a gaussian random environment being the inverse of temperature in the low dimensional cases we prove that for all the renormalized partition function converges to 0 and the correlation of two independent configurations does not converge to 0 in the high dimensional case a lower tail of has been obtained for small furthermore we express some thermodynamic quantities in terms of the path measure alone

self avoiding walks in quenched random environments the self avoiding walk in a quenched random environment is studied using realspace and field theoretic renormalization and flory arguments these methods indicate that the system is described for d de by a strong disorder fixed point corresponding to a glass state in which the polymer is confined to the lowest energy path this fixed point is characterized by scaling laws for the size of the walk l n with n the number of steps and the fluctuations in the free energy af l the bound 1 co pore and suggests that this inequality holds for d 2 and 3 although pure cannot be excluded particularly for d 2 for d d c there is a transition between strong and weak disorder phases at which pure the strong disorder fixed point for saws on percolation clusters is discussed the analogy with directed walks is emphasized

the fuzzy weighted average within a generalized means function the fuzzy weighted average is widely used to solve hierarchical evaluation problems including fuzzy consideration for the operations of scoring weighting and aggregating previous works considered the fuzziness of score and weight and used the additive function to aggregate these weighted scores this study considers the aggregation operator also as a fuzzy variable and uses a generalized means function to fuzzify the aggregation operator within a fuzzy weighted average in practice the proposed model not only considers both the relative important of the criteria and its achieved performance but also conveys the influence of the dm s decision maker s evaluation attitude thus the proposed model can flexibly reflect any dm s evaluation attitude such as open neutral or rigorous thereby the proposed model can make an objective evaluation that approaches a real decision making situation and thus has the potential to be a useful management tool for improved resolution of fuzzy hierarchical evaluation problems

large scale computation of incompressible viscous flow by least squares finite element method abstract the least square finite element method lsfem based on the velocity pressure vorticity formulation is applied to large scale three dimensional steady incompressible navier stokes problems this method can accommodate equal order interpolations and results in a symmetric positive definite algebraic system which can be solved effectively by simple iterative methods the first order velocity bernoulli pressure vorticity formulation for incompressible viscous flows is also tested the first order velocity pressure stress formulation is not elliptic in the ordinary sense so we do not recommend its use for newtonian flows for three dimensional flows a compatibility equation i e zero divergence of vorticity vector is included to make the first order system elliptic as a by product of proving the ellipticity of first order systems a rigorous mathematical technique has been developed to justify the number of permissible boundary conditions for the navier stokes equations the simple substitution or newton s method is employed to linearize the partial differential equations the lsfem is used to obtain discretized equations and the system of algebraic equations is solved using the jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices matrix free to achieve high efficiency to show the validity of this method for large scale computation we give numerical results for the 2d driven cavity problem at re 10000 with 408 400 bilinear elements the flow in a 3d cavity is calculated at re 100 400 and 1000 with 50 52 50 trilinear elements the taylor gortler like vortices are observed for re 1000

the number of open paths in oriented percolation we study the number n n of open paths of length n in supercritical oriented percolation on zd times n with d ge 1 we prove that on the percolation event inf n n 0 n n 1 n almost surely converges to a positive deterministic constant we also study the existence of directional limits the proof relies on the introduction of adapted sequences of regenerating times on subadditive arguments and on the properties of the coupled zone in supercritical oriented percolation

marginal relevance of disorder for pinning models the effect of disorder on pinning and wetting models has attracted much attention in theoretical physics in particular it has been predicted on the basis of the harris criterion that disorder is relevant annealed and quenched model have different critical points and critical exponents if the return probability exponent alpha a positive number that characterizes the model is larger than 1 2 weak disorder has been predicted to be irrelevant i e coinciding critical points and exponents if alpha 1 2 recent mathematical work has put these predictions on firm grounds in renormalization group terms the case alpha 1 2 is a marginal case and there is no agreement in the literature as to whether one should expect disorder relevance or irrelevance at marginality the question is particularly intriguing also because the case alpha 1 2 includes the classical models of two dimensional wetting of a rough substrate of pinning of directed polymers on a defect line in dimension 3 1 or 1 1 and of pinning of an heteropolymer by a point potential in three dimensional space here we prove disorder relevance both for the general alpha 1 2 pinning model and for the hierarchical version of the model proposed by b derrida v hakim and j vannimenus jsp 1992 in the sense that we prove a shift of the quenched critical point with respect to the annealed one in both cases we work with gaussian disorder and we show that the shift is at least of order exp 1 beta 4 for beta small if beta is the standard deviation of the disorder

applying the fuzzy weighted average approach to evaluate network security systems imprecise or vague importance as well as satisfaction levels of criteria may characterize the evaluation of alternative network security systems and may be treated by the fuzzy set theory as fuzzy numbers are adopted the fuzzy weighted averages fwas may become a suitable computation for the fuzzy criteria ratings and fuzzy weightings in this paper we propose to use the fwa approach that operates on the fuzzy numbers and obtains the fuzzy weighted averages during the evaluations of network security systems the algorithm constructs a benchmark adjustment solution approach for fwa a case study of the evaluation of network security systems is provided and demonstrates the usefulness of this study

self avoiding walk in 5 or more dimensions using an expansion based on the renormalization group philosophy we prove that for a t step weakly self avoiding random walk in five or more dimensions the variance of the endpoint is of order t and the scaling limit is gaussian as t

fuzzy weighted averages and implementation of the extension principle abstract this paper addresses the computational aspect of the extension principle when the principle is applied to algebraic mappings and in particular to weighted average operations in risk and decision analysis a computational algorithm based on the cut representation of fuzzy sets and interval analysis is described the method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner examples are given to illustrate the method and its relation to other discrete methods and the exact approach by non linear programming the algorithm has been implemented in a computer program which can handle very general extended algebraic operations on fuzzy numbers

fractional programming approach to fuzzy weighted average this paper proposes a fractional programming approach to construct the membership function for fuzzy weighted average based on the cut representation of fuzzy sets and the extension principle a pair of fractional programs is formulated to find the cut of fuzzy weighted average owing to the special structure of the fractional programs in most cases the optimal solution can be found analytically consequently the exact form of the membership function can be derived by taking the inverse function of the cut for other cases a discrete but exact solution to fuzzy weighted average is provided via an efficient solution method examples are given for illustration

centroids of type 1 and type 2 fuzzy sets when membership functions have spikes centroids are practically important in type 1 and type 2 fuzzy logic systems as a method of defuzzification and type reduction however computational problems arise when membership functions mf have singleton spikes the novel thresholding aggregation operators that were described in our companion paper new classes of threshold aggregation functions based upon the tsallis q exponential with applications to perceptual computing produce such mfs with spikes such spikes may occur when modeling concepts defined on a real valued domain and they are also formed in unions of fuzzy sets in which some have mfs with discrete support and others have support defined on an interval this paper presents a modified definition of the centroid of a fuzzy set that avoids the computational problems associated with the usual definition and reduces to this definition when mfs are continuous and normal i e of unit height on some interval we also present an enhanced karnik mendel type algorithm to compute the modified centroid of interval type 2 fuzzy sets whose mfs have spikes

non coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster in this paper we study the abundance of self avoiding paths of a given length on a supercritical percolation cluster on mathbb z d more precisely we count z n the number of self avoiding paths of length n on the infinite cluster starting from the origin which we condition to be in the cluster we are interested in estimating the upper growth rate of z n limsup n rightarrow infty z n 1 n which we call the connective constant of the dilute lattice after proving that this connective constant is a s non random we focus on the two dimensional case and show that for every percolation parameter p in 1 2 1 almost surely z n grows exponentially slower than its expected value in other words we prove that limsup n rightarrow infty z n 1 n lim n rightarrow infty mathbb e z n 1 n where the expectation is taken with respect to the percolation process this result can be considered as a first mathematical attempt to understand the influence of disorder for self avoiding walks on a quenched dilute lattice our method which combines change of measure and coarse graining arguments does not rely on the specifics of percolation on mathbb z 2 so our result can be extended to a large family of two dimensional models including general self avoiding walks in a random environment

fuzzy measures for profit maximization with fuzzy parameters profit maximization is an important issue to the firms that pursue the largest economic profit possible this paper extends the situation from the deterministic to uncertain where the coefficients are represented by fuzzy numbers intuitively when the problem has fuzzy parameters the derived profit value should be a fuzzy number as well the extension principle is utilized to develop a pair of two level mathematical programs to calculate the upper and lower bounds of the profit value at cuts following the duality theorem and a variable separation technique the two level mathematical programs are transformed into a class of one level signomial geometric programs to solve an example is given to illustrate the idea proposed in this paper

forecasting methods using fuzzy concepts in this paper fuzzy concepts are applied in forecasting product price and sales in the semiconductor industry which is often conceived as a highly dynamic environment first two fuzzy forecasting methods including fuzzy interpolation fi and fuzzy linear regression flr are developed and discussed forecasts generated by these methods are fuzzy valued next the subjective beliefs about whether the industry is booming or slumping and the speed at which this change in prosperity takes place during a given period are also considered two subjective functions are defined and used to adjust fuzzy forecasts practically fuzzy forecasts are incorporated with fuzzy programming like fuzzy linear programming flp or fuzzy nonlinear programming fnp for mid term or long term planning advantages over traditional methods are shown in our examples

modeling uncertain variables of the weighted average operation by fuzzy vectors the paper deals with the fuzzy extension of the weighted average operation first we study the convenient ways how uncertain weights and weighted values can be modeled by fuzzy vectors we show that in comparison to a tuple of fuzzy numbers that have been used for modeling uncertain values of particular weights and weighted values up to now fuzzy vectors extend the possibilities of utilizing the vague expert information concerning the weights and weighted values next we focus on computation of a fuzzy weighted average of a fuzzy vector of weighted values with a fuzzy vector of weights we derive a general formula and we study its special forms the advantage of the approach presented in the paper is that the resulting fuzzy weighted average is not overly imprecise since every available information about its variables is involved in computation this fact is illustrated by several examples finally we briefly discuss the problem of defuzzification of the resulting fuzzy weighted average

multivariate modeling and type 2 fuzzy sets this paper explores the link between type 2 fuzzy sets and multivariate modeling elements of a space x are treated as observations fuzzily associated with values in a multivariate feature space a category or class is likewise treated as a fuzzy allocation of feature values possibly dependent on values in x we observe that a type 2 fuzzy set on x generated by these two fuzzy allocations captures imprecision in the class definition and imprecision in the observations in practice many type 2 fuzzy sets are in fact generated in this way and can therefore be interpreted as the output of a classification task we then show that an arbitrary type 2 fuzzy set can be so constructed by taking as a feature space a set of membership functions on x this construction presents a new perspective on the representation theorem of mendel and john the multivariate modeling underpinning the type 2 fuzzy sets can also constrain realizable forms of membership functions because averaging operators such as centroid and subsethood on type 2 fuzzy sets involve a search for optima over membership functions constraining this search can make computation easier and tighten the results we demonstrate how the construction can be used to combine representations of concepts and how it therefore provides an additional tool alongside standard operations such as intersection and subsethood for concept fusion and computing with words

directed polymers in random environment are diffusive at weak disorder in this paper we consider directed polymers in random environment with discrete space and time for transverse dimension at least equal to 3 we prove that diffusivity holds for the path in the full weak disorder region i e where the partition function differs from its annealed value only by a non vanishing factor deep inside this region we also show that the quenched averaged energy has fluctuations of order 1 in complete generality arbitrary dimension and temperature we prove monotonicity of the phase diagram in the temperature

a spectral multidomain penalty method model for the simulation of high reynolds number localized incompressible stratified turbulence a spectral multidomain penalty method model has been developed for the simulation of high reynolds number localized stratified turbulence this is the first time that a penalty method with a particular focus on subdomain interface treatment has been used with a multidomain scheme to simulate incompressible flows the temporal discretization ensures maximum temporal accuracy by combining third order stiffly stable and backward differentiation schemes with a high order boundary condition for the pressure in the non periodic vertical direction a spectral multidomain discretization is used and its stability for under resolved simulations at high reynolds numbers is ensured through use of penalty techniques spectral filtering and strong adaptive interfacial averaging the penalty method is implemented in different formulations for both the explicit non linear term advancement and the implicit treatment of the viscous terms the multidomain model is validated by comparing results of simulations of the mid to late time stratified turbulent wake with non zero net momentum to the corresponding laboratory data for a towed sphere the model replicates correctly the characteristic vorticity and internal wave structure of the stratified wake and exhibits robust agreement with experiments in terms of the temporal power laws in the evolution of mean profile characteristic velocity and lengthscales

fuzzy weighted average an improved algorithm abstract this paper suggests a modification on the fuzzy weighted average fwa method developed by dong and wong 1987 a steepest descend ascend method is proposed to facilitate steps 3 and 4 of the fwa to find the minimum maximum of f x1 x2 xn w1 w2 wn i 1n wixi i 1n wi an example used by dong and wong is again employed to illustrate the new method the same results are obtained but the new method requires much less evaluations and computations the advantages and disadvantages of the two algorithms are discussed also several relevant theorems are stated and discussed in this paper and the proofs of these theorems are listed in the appendix

a space time coupled p version least squares finite element formulation for unsteady two dimensional navier stokes equations this paper presents a p version least squares finite element formulation for two dimensional unsteady fluid flow described by navier stokes equations where the effects of space and time are coupled the dimensionless form of the navier stokes equations are first cast into a set of first order differential equations by introducing auxiliary variables this permits the use of c0 element approximation the element properties are derived by utilizing the p version approximation functions in both space and time and then minimizing the error functional given by the space time integral of the sum of squares of the errors resulting from the set of first order differential equations this results in a true space time coupled least squares minimization procedure the application of least squares minimization to the set of coupled first order partial differential equations results in finding a solution vector which makes gradient of error functional with respect to a null vector this is accomplished by using newton s method with a line search a time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance the space time coupled p version approximation functions provide the ability to control truncation error which in turn permits very large time steps what literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space time coupled approach the generality success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity the results are compared with the analytical solutions and those reported in the literature the formulation presented here is ideally suited for space time adaptive procedures the element error functional values provide a mechanism for adaptive h p or hp refinements

aggregation using the linguistic weighted average and interval type 2 fuzzy sets the focus of this paper is the linguistic weighted average lwa where the weights are always words modeled as interval type 2 fuzzy sets it2 fss and the attributes may also but do not have to be words modeled as it2 fss consequently the output of the lwa is an it2 fs the lwa can be viewed as a generalization of the fuzzy weighted average fwa where the type 1 fuzzy inputs are replaced by it2 fss this paper presents the theory algorithms and an application of the lwa it is shown that finding the lwa can be decomposed into finding two fwas since the lwa can model more uncertainties it should have wide applications in distributed and hierarchical decision making

space time coupled spectral hp least squares finite element formulation for the incompressible navier stokes equations we consider least squares finite element models for the numerical solution of the non stationary navier stokes equations governing viscous incompressible fluid flows the paper presents a formulation where the effects of space and time are coupled resulting in a true space time least squares minimization procedure as opposed to a space time decoupled formulation where a least squares minimization procedure is performed in space at each time step the formulation is first presented for the linear advection diffusion equation and then extended to the navier stokes equations the formulation has no time step stability restrictions and is spectrally accurate in both space and time to allow the use of practical c0 element expansions in the resulting finite element model the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity as an additional independent variable and the least squares method is used to develop the finite element model of the governing equations high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method in matrix free form spectral convergence of the l2 least squares functional and l2 error norms in space time is verified using a smooth solution to the two dimensional non stationary incompressible navier stokes equations numerical results are presented for impulsively started lid driven cavity flow oscillatory lid driven cavity flow transient flow over a backward facing step and flow around a circular cylinder the results demonstrate the predictive capability and robustness of the proposed formulation even though the space time coupled formulation is emphasized we also present the formulation and numerical results for least squares space time decoupled finite element models the numerical results show that the space time coupled formulation has superior predictive capabilities for flows demanding high space time resolution exemplified here by the transient flow over a backward facing step

on various approaches to normalization of interval and fuzzy weights abstract the paper deals with the problem of fuzzification of the procedure of normalization of weights first the existing methods for normalization of interval and fuzzy weights are reviewed second we study the problem of normalization of a fuzzy vector of weights that expresses the joint possibility distribution of initial weights we show that a correct way is to apply the extension principle proposed by zadeh since the result of such normalization is the fuzzy vector of normalized weights that expresses the true joint possibility distribution of normalized weights further we establish some properties of this approach to normalization that are important from the point of view of real applications finally since an n tuple of non interactive interval or fuzzy weights can be viewed as a fuzzy vector of weights of a special kind we investigate normalization of such kind of fuzzy vectors of weights according to the extension principle we show that from the point of view of the way of modelling uncertain normalized weights the result of this approach can be directly compared only with the result of normalization proposed by wang and elhag 2006 35 we find out that it is not sufficient to express the result of normalization only by an n tuple of normalized interval or fuzzy weights together with the constraint that the sum of the weights is equal to 1 since it can cause a false increase of uncertainty in the model this fact is illustrated by an example

fuzzy weighted averages revisited the fuzzy weighted average operations are common operations in risk and decision analysis based on zadeh s extension principle dong and wang fuzzy sets and systems 21 1987 183 proposed a fwa method a method based on the cut representation of fuzzy sets and interval analysis the fwa method provides a discrete approximation of the cuts of fuzzy weighted average the fwa method has an exponential complexity subsequently liou and wang fuzzy sets and systems 49 1992 307 proved that indeed the fwa method can be simplified to be a nonconstrainted linear fractional programming problem with lower and upper bounds for each of the variables they proposed an improved algorithm called ifwa which is of o n2 complexity where n is the number of decision variables in ifwa guh et al comput math appl 32 8 1996 115 proposed a max min paired elimination method pfwa they claimed their pfwa was of order n in complexity yet the pfwa is a heuristic method lacking any proof of convergence in 1997 lee and park proposed an algorithm efwa which is of o nlogn complexity most recently kao and liu fuzzy sets and systems 120 2001 435 solved the fwa by transforming the fractional program into a linear program in this note we shall relate the ifwa framework with the nonconstrainted 0 1 integer linear fractional programming problem in the mathematical programming literature several o n time algorithms have been existing for instance robillard naval res logist quart 18 1971 47 and hansen et al math programming 52 1991 255 the algorithm of hansen et al will be presented for easy reference

rating technical attributes in fuzzy qfd by integrating fuzzy weighted average method and fuzzy expected value operator quality function deployment qfd is a planning and problem solving tool that is gaining acceptance for translating customer requirements into the technical attributes of a product deriving the rating order of technical attributes from input variables is a crucial step in applying qfd when the relative weights of customer requirements and the relationship measures between customer requirements and technical attributes are expressed as fuzzy numbers calculating the importance of each technical attribute falls into the category of fuzzy weighted average in which the derived membership function of the fuzzy importance of each technical attribute is not explicitly known thus most ranking methods are not suitable under these circumstances a method is proposed in this paper using fuzzy weighted average method in the fuzzy expected value operator in order to rank technical attributes in fuzzy qfd an example of a flexible manufacturing system design is cited to demonstrate the application of the proposed approach

fuzzy weighted average the linear programming approach via charnes and cooper s rule abstract to improve the method of dong and wong fuzzy sets and systems 21 1987 183 199 for obtaining the fuzzy weighted average a steepest descent ascent method was proposed by liou and wang fuzzy sets and systems 49 1992 307 315 in this paper we propose to replace the steepest descent ascent method by linear programming which is a much more powerful approach for large problems the reduction of the original nonlinear programming problem into a linear one is achieved by the use of the charnes and cooper s linear transformation method the example used by dong and wong is again used to illustrate the approach

least squares finite element method for the stokes problem with zero residual of mass conservation in this paper the simulation of incompressible flow in two dimensions by the least squares finite element method lsfem in the vorticity velocity pressure version is studied in the lsfem the equations for continuity of mass and momentum and a vorticity equation are minimized on a discretization of the domain of interest a problem is these equations are minimized in a global sense thus this method may not enforce that div underline u 0 at every point of the discretization in this paper a modified lsfem is developed which enforces near zero residual of mass conservation i e div underline u is nearly zero at every point of the discretization this is accomplished by adding an extra restriction in the divergence free equation through the lagrange multiplier strategy in this numerical method the inf sup or say lbb condition is not necessary and the matrix resulting from applying the method on a discretization is symmetric the uniqueness of the solution and the application of the conjugate gradient method are also valid numerical experience is given in simulating the flow of a cylinder with diameter 1 moving in a narrow channel of width 1 5 results obtained by the lsfem show that mass is created or destroyed at different points in the interior of discretization the results obtained by the modified lsfem show the mass is nearly conserved everywhere

super exponential convergence of the karnik mendel algorithms for computing the centroid of an interval type 2 fuzzy set computing the centroid of an interval t2 fs is an important operation in a type 2 fuzzy logic system where it is called type reduction but it is also a potentially time consuming operation the karnik mendel km iterative algorithms are widely used for doing this in this paper we prove that these algorithms converge monotonically and super exponentially fast both properties are highly desirable for iterative algorithms and explain why in practice the km algorithms have been observed to converge very fast thereby making them very practical to use

a collocation penalty least squares finite element formulation for incompressible flows abstract we present a penalty least squares finite element formulation for incompressible flows the pressure degree of freedom is eliminated using the classical penalty approach and the least squares model is formed in terms of velocity vorticity and dilatation an iterative penalization is implemented which allows the use of low penalty parameters and retains a manageable conditioning number of the global coefficient matrix the h convergence is verified using the exact solution of kovasznay s flow numerical results are presented for a number of benchmark problems e g steady flow past a large circular cylinder in a channel transient flow over a backward facing step unsteady flow past a circular cylinder and buoyancy driven natural convection in a square cavity for all numerical examples the effect of penalty parameter on the accuracy is investigated thoroughly and it is concluded that the present model produces accurate results for low penalty parameters 10 100 these problems are also solved on coarse meshes to show that good mass conservation is achieved

a fast schur complement method for the spectral element discretization of the incompressible navier stokes equations the weak formulation of the incompressible navier stokes equations in three space dimensions is discretized with spectral element approximations and gauss lobatto legendre quadratures the uzawa algorithm is applied to decouple the velocities from the pressure the equation that results for the pressure is solved by an iterative method within each pressure iteration a helmholtz operator has to be inverted this can efficiently be done by separating the equations for the interior nodes from the equations at the interfaces according to the schur method fast diagonalization techniques are applied to the interior variables of the spectral elements several ways to deal with the resulting interface problem are discussed finally a comparison is made with a more classical method c 1995 academic press inc

spectral hp penalty least squares finite element formulation for the steady incompressible navier stokes equations in this paper we present spectral hp penalty least squares finite element formulation for the numerical solution of steady incompressible navier stokes equations the continuity equation is treated as a constraint on velocity field and the constraint is enforced using the penalty method which is equivalent to replacing the pressure variable p with p n p n email protected u where u is the velocity field c is the penalty parameter and superscript n is non linear iteration number the modified navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity dilatation or stresses as additional independent variables and the least squares method is used to develop the finite element model high order basis functions are used to construct the discrete model unlike other penalty finite element formulations equal order integration is used for all terms of the coefficient matrix schur complement method is used and all interior element degrees of freedom are condensed out and interface degrees of freedom are solved either by direct solver banded cholesky factorization or iterative solver preconditioned conjugate gradient present formulation produces very accurate results for even very low penalty parameters 10 40 in this way present formulation and implementation fixes perennial problem of bad conditioning of coefficient matrix associated with penalty formulations proposed formulation is verified using the exact solution of kovasznay flow numerical results are also presented for a number of two dimensional benchmark problems e g 2d lid driven cavity flow flow over a backward facing step steady flow past a circular cylinder and 2d flow past a large circular cylinder in a channel finally the proposed formulation is extended to coupled velocity temperature problems and buoyant flow inside a square enclosure is solved to test the penalty least squares finite element model developed herein

an efficient algorithm for fuzzy weighted average in multisensor intelligent systems the information fusion plays an important role several algorithms have been proposed for the purpose of aggregating imprecise sensory information represented by fuzzy numbers this paper proposes an efficient algorithm to compute fuzzy weighted average which turned out to be superior to the previous works by reducing the number of comparisons and arithmetic operations to o n log n

self avoiding walk connectivity constant and theta point on percolating lattices the average connectivity constant mu of self avoiding walks saws is obtained from exact enumeration of saws on monte carlo generated percolating clusters in a randomly diluted square lattice for averages over the infinite percolating cluster mu decreases almost linearly with bond dilution 1 p where p is the bond occupation concentration the authors find mu pc 1 31 or 0 03 at the percolation threshold pc and could not detect any significant difference between mu pc and pc mu 1 the variation of theta point for saws on the same lattice with dilution is also estimated analysing the partition function zeros within the limited accuracy of their analysis its variation with dilution is observed as being quite weak and the theta point increases somewhat compared to pure lattice value near pc they find a non vanishing theta point ktheta pc equivalent to 0 59 where k0 j k theta on the square lattice percolation cluster at pc

on the roles of minimization and linearization in least squares finite element models of nonlinear boundary value problems in this paper we examine the roles of minimization and linearization in the least squares finite element formulations of nonlinear boundary values problems the least squares principle is based upon the minimization of the least squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations in the present case we consider l 2 norms since the least squares method is independent of the discretization procedure and the solution scheme the least squares principle suggests that minimization should be performed prior to linearization where linearization is employed in the context of either the picard or newton iterative solution procedures however in the least squares finite element analysis of nonlinear boundary value problems it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations the main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged the assessment is performed mathematically through an examination of the variational setting for the least squares formulation of an abstract nonlinear boundary value problem and also computationally through the numerical simulation of the least squares finite element solutions of both a nonlinear form of the poisson equation and also the incompressible navier stokes equations the assessment suggests that although the least squares principle indicates that minimization should be performed prior to linearization such an approach is often impractical and not necessary

fuzzy weighted average a max min paired elimination method by the use of max min paired elimination an efficient method for obtaining the fuzzy weighted average fwa was developed this approach is much more efficient than either the approach developed by dong and wong 1 or the improved approach developed by liou and wang 2 the computational requirements for the method of dong and wong are 22n and the largest computational requirements for the improved method of liou and wang are 2 n n 1 the computational requirements for the proposed approach are proportional to the problem size n and the total computational requirements are 2 n 1 the same example used by dong and wong is used to illustrate the approach in addition a more complicated problem is also solved

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

new results about the centroid of an interval type 2 fuzzy set including the centroid of a fuzzy granule the centroid of an interval type 2 fuzzy set it2 fs provides a measure of the uncertainty of such a fs its calculation is very widely used in interval type 2 fuzzy logic systems in this paper we present properties about the centroid of an it2 fs we also illustrate many of the general results for a t2 fuzzy granule fg in order to develop some understanding about the uncertainty of the fg in terms of its vertical and horizontal dimensions at present the t2 fg is the only it2 fs for which it is possible to obtain closed form formulas for the centroid and those formulas are in this paper

subjective assessment of mental workload a fuzzy linguistic multi criteria approach abstract in this paper a fuzzy linguistic multi criteria method for subjective assessment of mental workload is proposed the criteria ratings and the corresponding importance weights are assessed in linguistic terms which are described by fuzzy numbers with triangular and trapezoidal membership function a fuzzy weighted average algorithm based on the extension principle is employed to aggregate these fuzzy numbers the aggregated fuzzy number is then translated back to linguistics and the overall mental workload is expressed in linguistic terms

least squares spectral element solution of incompressible navier stokes equations with adaptive refinement least squares spectral element solution of steady two dimensional incompressible flows are obtained by approximating velocity pressure and vorticity variable set on gauss lobatto legendre nodes constrained approximation method is used for h and p type nonconforming interfaces of quadrilateral elements adaptive solutions are obtained using a posteriori error estimates based on least squares functional and spectral coefficient effective use of p refinement to overcome poor mass conservation drawback of least squares formulation and successful use of h and p refinement together to solve problems with geometric singularities are demonstrated capabilities and limitations of the developed code are presented using kovasznay flow flow past a circular cylinder in a channel and backward facing step flow

a discontinuous petrov galerkin methodology for adaptive solutions to the incompressible navier stokes equations the discontinuous petrov galerkin methodology with optimal test functions dpg of demkowicz and gopalakrishnan 18 20 guarantees the optimality of the solution in an energy norm and provides several features facilitating adaptive schemes whereas bubnov galerkin methods use identical trial and test spaces petrov galerkin methods allow these function spaces to differ in dpg test functions are computed on the fly and are chosen to realize the supremum in the inf sup condition the method is equivalent to a minimum residual method for well posed problems with sufficiently regular solutions dpg can be shown to converge at optimal rates the inf sup constants governing the convergence are mesh independent and of the same order as those governing the continuous problem 48 dpg also provides an accurate mechanism for measuring the error and this can be used to drive adaptive mesh refinements we employ dpg to solve the steady incompressible navier stokes equations in two dimensions building on previous work on the stokes equations and focusing particularly on the usefulness of the approach for automatic adaptivity starting from a coarse mesh we apply our approach to a manufactured solution due to kovasznay as well as the lid driven cavity flow backward facing step and flow past a cylinder problems

gradual numbers and their application to fuzzy interval analysis in this paper we introduce a new way of looking at fuzzy intervals instead of considering them as fuzzy sets we see them as crisp sets of entities we call gradual real numbers they are a gradual extension of real numbers not of intervals such a concept is apparently missing in fuzzy set theory gradual numbers basically have the same algebraic properties as real numbers but they are functions a fuzzy interval is then viewed as a pair of fuzzy thresholds which are monotonic gradual real numbers this view enables interval analysis to be directly extended to fuzzy intervals without resorting to alpha cuts in agreement with zadeh s extension principle several results show that interval analysis methods can be directly adapted to fuzzy interval computation where end points of intervals are changed into left and right fuzzy bounds our approach is illustrated on two known problems computing fuzzy weighted averages and determining fuzzy floats and latest starting times in activity network scheduling