entropy considerations in numerical simulations of non equilibrium rarefied flows abstract non equilibrium rarefied flows are encountered frequently in supersonic flight at high altitudes vacuum technology and in microscale devices prediction of the onset of non equilibrium is important for accurate numerical simulation of such flows we formulate and apply the discrete version of boltzmann s h theorem for analysis of non equilibrium onset and accuracy of numerical modeling of rarefied gas flows the numerical modeling approach is based on the deterministic solution of kinetic model equations the numerical solution approach comprises the discrete velocity method in the velocity space and the finite volume method in the physical space with different numerical flux schemes the first order the second order minmod flux limiter and a third order weno schemes the use of entropy considerations in rarefied flow simulations is illustrated for the normal shock the riemann and the two dimensional shock tube problems the entropy generation rate based on kinetic theory is shown to be a powerful indicator of the onset of non equilibrium accuracy of numerical solution as well as the compatibility of boundary conditions for both steady and unsteady problems

calculating functions of fuzzy numbers abstract this paper discusses the calculation of functions of fuzzy numbers our general approach follows that of the fuzzy weighted algorithm fwa of wong and dong however by developing a classification scheme for functions and their arguments we show that there are many cases for which the fwa and all other published methods will give wrong results we suggest an alternative approach which will work in all cases for certain restricted classes of functions we develop methods which require less computation than the fwa these algorithms have been implemented in a constraint based logic programming language examples are given to illustrate the method and its relation to other discrete methods and the exact nonlinear programming approach

parametric representation of fuzzy numbers and application to fuzzy calculus we present several models to obtain simple parametric representations of the fuzzy numbers or intervals based on the use of piecewise monotonic functions of different forms the representations have the advantage of allowing flexible and easy to control shapes of the fuzzy numbers we use the standard cuts setting but also the membership functions are obtained immediately and can be used directly to obtain error controlled approximations of the fuzzy calculus in terms of a finite set of parameters the general setting is the hermite type interpolation where the values and the slopes of the monotonic interpolators are given by appropriate parameters and the overall errors of the fuzzy computations can be controlled within a prefixed tolerance by eventually augmenting the total number of pieces and of the parameters by which the results are obtained the representations are designed to model the lower and the upper extremal values of each cut compact intervals of the fuzzy numbers and are able to produce almost any possible configuration differentiable continuous or with a finite number of discontinuity points by using parametric monotonic functions of different types we show applications in the standard fuzzy calculus and we stress the generality and the applicability of the proposed representation to a large class of problems including the numerical solution of fuzzy differential equations the fuzzy linear regression and the stochastic extensions of the fuzzy mathematics the proposed model is called the lower upper representation and we denote the associated fuzzy numbers or intervals as lu fuzzy

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

existence and multiplicity of solutions of a kind of fourth order boundary value problem abstract in this paper the existence and multiplicity results of the solutions are obtained for the fourth order boundary value problem u 4 t f t u t for all t 0 1 subject to u 0 u 1 u 0 u 1 0 where f is continuous the monotone operator theory and critical point theory are employed to discuss this problem respectively

necessary and sufficient conditions for the equality of interactive and non interactive extensions of continuous functions abstract in this contribution we find the class of n dimensional joint possibility distributions with the property that the interactive extension principle coincides with the non interactive extension principle as long as the interactive operations are determined by continuous functions strictly increasing in each argument this result completes recent studies by the authors where the particular case of interactive additions and multiplications versus non interactive additions and multiplications were investigated in addition this time we propose results that also cover the cases when we know the fuzzy numbers only from their membership functions it means that we eliminated the limitations that appear when we cannot pass from membership function representation to parametric representation of fuzzy numbers as important new applications we mention the study on the completely correlated fuzzy numbers also of note is that we propose two simple methods to extend bidimensional joint possibility distributions to n dimensional joint possibility distributions one method is based on an inductive construction while the other one is based on a pairwise construction

a generalization of hukuhara difference and division for interval and fuzzy arithmetic we propose a generalization of the hukuhara difference first the case of compact convex sets is examined then the results are applied to generalize the hukuhara difference of fuzzy numbers using their compact and convex level cuts finally a similar approach is suggested to attempt a generalization of division for real intervals and fuzzy numbers applications to solving interval and fuzzy linear equations and fuzzy differential equations are shown

simulation of discrete linear time invariant fuzzy dynamic systems in this paper a new approach to simulation of discrete linear time invariant dynamic systems with parameters and initial conditions described by fuzzy numbers is proposed this approach solves the problem of fuzzy simulation by determining at each iteration a nested family of intervals corresponding to different levels of certainty a cuts that enclose the possible system states using optimisation the originality of this approach consists in considering uncertain parameters as time invariant during the simulation process the optimisation problems associated with the approach proposed must be solved globally in order to guarantee that the minimum volume box enclosing the region of system states for each level of certainty has been derived then a global optimisation algorithm based on the theory of moments and linear matrix inequality relaxations suitable for the type of problems polynomial programs that appear in the fuzzy simulation algorithm is used finally two real engineering application examples allow assessing the performance of the approach proposed

equivalent conditions for laws of large numbers for t related l r fuzzy numbers in this paper we give some necessary and sufficient conditions for laws of large numbers for sequence of mutually t related l r fuzzy numbers when t is an continuous archimedean t norm and diameter of support of the fuzzy numbers is not uniformly bounded we also consider some necessary and sufficient conditions for laws of large numbers for l r fuzzy random numbers and generalize earlier results of fuller fuzzy sets and systems 45 1992 299 303 triesch fuzzy sets and systems 58 1993 339 342 and hong and lee fuzzy sets and systems 121 2001 537 543

fuzzy portfolio allocation models through a new risk measure and fuzzy sharpe ratio a new portfolio risk measure that is the uncertainty of portfolio fuzzy return is introduced in this paper beyond the well known sharpe ratio i e the reward to variability ratio in modern portfolio theory we initiate the so called fuzzy sharpe ratio in the fuzzy modeling context in addition to the introduction of the new risk measure we also put forward the reward to uncertainty ratio to assess the portfolio performance in fuzzy modeling corresponding to two approaches based on rm t rm m and rm t rm w fuzzy arithmetic two portfolio optimization models are formulated in which the uncertainty of portfolio fuzzy returns is minimized while the fuzzy sharpe ratio is maximized these models are solved by the fuzzy approach or by the genetic algorithm ga solutions of the two proposed models are shown to be dominant in terms of portfolio return uncertainty compared with those of the conventional mean variance optimization mvo model used prevalently in the financial literature in terms of portfolio performance evaluated by the fuzzy sharpe ratio and the reward to uncertainty ratio the model using rm t rm w fuzzy arithmetic results in higher performance portfolios than those obtained by both the mvo and the fuzzy model which employs rm t rm m fuzzy arithmetic we also find that using the fuzzy approach for solving multiobjective problems appears to achieve more optimal solutions than using ga although ga can offer a series of well diversified portfolio solutions diagrammed in a pareto frontier

on the algebra of interactive fuzzy numbers the continuous case abstract in 1 we discussed default methods involving an approximate joint distribution or the extension principle to calculated m n m n and evaluate m n m n etc for positively and negatively associated discrete fuzzy numbers m and n this paper continues that research where now m and n are continuous fuzzy numbers we also discuss default procedures for extended max and min

the use of fuzzy variables in pert abstract in this paper we present a method called fpert for estimating a project completion time in the situation when activity duration times in the project network model are given in the form of fuzzy variables fuzzy sets on time space theoretical foundations of the method as well as results of calculations derived from a simple example are included

general lidstone problems multiplicity and symmetry of solutions abstract for the 2mth order lidstone boundary value problem y 2 m t f y t y t y 2 i t y 2 m 1 t t 0 1 y 2 i 0 y 2 i 1 0 0 i m 1 where 1 mf r m 0 thinsp is continuous growth conditions are imposed on f which yield the existence of at least three symmetric positive solutions this generalizes earlier papers which have applied avery s generalization of the leggett williams theorem to lidstone problems we then prove the analogous result for difference equations

solving fuzzy equations we start in the next section 10 2 with looking at possible solutions to the simple fuzzy linear equation overline a cdot overline x overline b overline c we discuss three different types of solution which we have studied before in solving fuzzy equations then we present a fourth type of solution based on our fuzzy monte carlo method in section 10 2 2 this new solution is based on random fuzzy numbers in section 10 3 we look at only classical solutions to the fuzzy quadratic equation and apply our fuzzy monte carlo method to obtain new solutions then in section 10 4 we consider the fuzzy matrix equation overline a cdot overline x overline b and a number of solution types for overline x and then another solution based on fuzzy monte carlo techniques the last section contains a brief summary and our conclusions

positive solutions of a 2nth order boundary value problem involving all derivatives via the order reduction method this paper is mainly concerned with the existence multiplicity and uniqueness of positive solutions for the 2nth order boundary value problem 1 nu 2 n f t u u 1 i 2 u i 1 n 1u 2 n 1 u 2 i 0 u 2 i 1 1 0 i 0 1 n 1 where n 2 and f c 0 1 xr 2 n r r 0 we first use the method of order reduction to transform the above problem into an equivalent initial value problem for a first order integro differential equation and then use the fixed point index theory to prove the existence multiplicity and uniqueness of positive solutions for the resulting problem based on a priori estimates achieved by developing spectral properties of associated parameterized linear integral operators finally as a byproduct our main results are applied for establishing the existence multiplicity and uniqueness of symmetric positive solutions for the lidstone problem involving all derivatives

bifurcation techniques for lidstone boundary value problems abstract by employing bifurcation techniques this paper investigates the existence of nontrivial solutions single and multiple for lidstone boundary value problems depending on higher order derivatives our results improve on those in the literature

spl alpha cut fuzzy arithmetic simplifying rules and a fuzzy function optimization with a decision variable the problems of alpha cut fuzzy arithmetic have been shown like in interval arithmetic that distinct states of fuzzy parameters or fuzzy variable values may be chosen and produce an overestimated fuzziness meanwhile local extrema of a function may exist inside the support of fuzzy parameters and cause an underestimation of fuzziness and an illegal fuzzy number s result previous approaches to overcoming these problems have appeared in literature yet the computational burden of these approaches became even heavier thus this paper is based on the vertex method in literature and extensively proposes newly devised rules observed greatly useful for simplifying the vertex method these rules are devised through a function partitioned into subfunctions distinguishing the types of fuzzy parameter variable occurrences and types of subfunctions or functions with the various observations the improved efficiency has been found able to significantly reduce the combination vertex test of the vertex method for the fuzzy parameters alpha cut endpoints possibly to only a few fuzzy parameters endpoint combinations also as related a procedure for the fuzzy optimization of fuzzy functions with a fuzzy blurred argument a single variable is examined with the vertex method as well a proper and useful preliminary algorithm is proposed numerical examples with results are provided

real time constrained fuzzy arithmetic klir introduced constrained fuzzy arithmetic cfa as a solution to the unnecessary precision loss when dealing with fuzzy quantities that represent linguistic variables since then some attempts have been made to make cfa efficient but none of these solutions is suitable for real time applications in this paper we will propose a new cfa algorithm that can be used in such environments

falscal a fuzzy multidimensional scaling algorithm the conventionally adopted alternating least squares scaling alscal procedure of multidimensional scaling mds is a valuable mathematical scheme for analyzing data in areas where organized concepts and underlying dimensions are inadequately defined or developed fuzzy set theory fst attempts to formulate human reasoning and perceptions therefore targeting problems in areas where human factors significantly impact the result of decision making to our knowledge the fst and alscal approaches have not yet been integrated this study integrates and modifies the fst and alscal procedures fuzzy data collected from fuzzy questionnaires are adopted as the input of the mds ensuring that the uncertainty of input data can be incorporated into the analysis the conventionally adopted alscal procedure is then modified to cope with fuzzy input data by adopting the notion of fuzzy distances fuzzy disparities and fuzzy ranking to represent the similarities between fuzzy data related approximation operations of the triangular fuzzy number are also introduced to facilitate computation in fuzzy alscal

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

existence and multiplicity of solutions to even order ordinary differential equations via variational methods abstract in this paper we study the existence and multiplicity of solutions for two point boundary value problems for even order ordinary differential equation by using the critical point theory and the property of projective operator we establish some conditions on the nonlinearity which guarantee that the nonlinear hammerstein integral equation with quasi positive definite kernel has at least one solution two nontrivial solutions and finite pairs of solutions respectively as an application we get the corresponding results for two point boundary value problems for even order ordinary differential equation moreover an example is presented to illustrate our main results

solving singular boundary value problems of higher even order in this paper we present a new algorithm to solve a kind of singular boundary value problems of higher even order based on the theory of numerical analysis in the reproducing kernel space w 2 2 m 3 0 1 singular boundary value problem of sixth order has been discussed for the problem u 6 x a x u 4 x b x u x c x u x f x 0

analysis of linear systems with fuzzy parametric uncertainty in this paper the problem of determining the stability margin of a linear system with fuzzy parametric uncertainty is addressed this kind of systems arise when the exact value of a process parameter is not available but an estimation of the value is given by an expert more or less 5 as the cut representation of a fuzzy number is a functional interval interval analysis shows to be a suitable tool to deal with fuzziness based on the argoun s stability test for interval systems a new methodology with a natural graphical interpretation is presented an independent confidence degree for every coefficient is considered allowing a clear understanding on how each coefficient affects stability an analysis on how precise one should be on some coefficients to allow more uncertainty on others can be carried out a comparison with tsypkin polyak plot is also presented

multiple positive solutions of a boundary value problem for a class of 2 nth order m point singular integro differential equations in banach spaces in this paper we obtain the existence of multiple positive solutions of a boundary value problem for 2nth order singular nonlinear integro differential equations in a banach space by means of fixed point index theory of completely continuous operators

the reduction of interval type 2 lr fuzzy sets type reduction of interval type 2 it2 fuzzy sets is essential in conducting the type 2 fuzzy sets expressed with the resolution forms of it2 fuzzy sets several type reduction methods such as km ekm and centroid flow have been proposed these methods are relatively easy to implement but still computation intensive because they need to invoke an iterative switching point finding procedure this study derives a theorem and proposes a heuristic algorithm which can fast and accurately identify the minimum and maximum switching points of a piecewise smooth it2 fuzzy set it also demonstrates that it is easy to derive the close form expressions of the switching points of a piecewise smooth it2 fuzzy set if both of its upper and lower membership functions can be parameterized as lr fuzzy sets which are defined in this paper then the type reduction of piecewise smooth it2 fuzzy sets can be simplified to solve the close form expressions of their switching points in terms of lr parameters experiments on it2 fuzzy sets with various piecewise smooth membership functions including linear gaussian and hybrid shaped ones were made the results showed that the proposed type reduction method can obtain solutions which accurately approximate to the desired switching points with much lower computational overhead than the karnik mondel km and enhanced km ekm methods

fuzzy quantities in networks one of possible applications of the fuzzy sets theory in network analysis is described and discussed it is shown that the uncertainty connected with the length of activities forming the network can be represented by fuzzy quantities summing these fuzzy quantities it is possible to derive the uncertainty connected with more complex paths and reserves an illustrative numerical example is presented at the end of the paper

positive solutions for a class of 2nth order singular boundary value problems abstract we establish sufficient and necessary conditions for guaranteeing the existence of c 2 n 2 0 1 as well as c 2 n 1 0 1 positive solutions for 2 n th order singular boundary value problems the arguments are based upon the fixed point theorem an the maximal principle as well as constructing lower and upper solutions our nonlinearity f t x 1 x 2 x n may be singular at x i 0 i 1 2 n t 0 and or t 1 meanwhile an example is worked out to demonstrate the main results

positive solutions for 2nth order singular sub linear m point boundary value problems abstract this paper investigates the existence of positive solutions for 2nth order singular sub linear m point boundary value problems firstly we establish a comparison theorem then we define a partial ordering in c2n 2 a b c2n a b and construct lower and upper solutions to give a necessary and sufficient condition for the existence of c2n 2 0 1 as well as c2n 1 0 1 positive solutions our nonlinearity f t x1 x2 xn may be singular at xi 0 i 1 2 n t 0 and or t 1

a class of 2nth order singular boundary value problems abstract this paper investigates the existence of positive solutions for 2 n th order n 1 singular sub linear boundary value problems first of all we establish the maximal principle and some important lemmas then we define a partial ordering in c 2 n 2 0 1 c 2 n 0 1 and construct lower and upper solutions to give a necessary and sufficient condition for the existence of c 2 n 2 0 1 as well as c 2 n 1 0 1 positive solutions our nonlinearity f t x 1 x 2 x n may be singular at x i 0 i 1 2 n t 0 and or t 1

type 1 owa operators for aggregating uncertain information with uncertain weights induced by type 2 linguistic quantifiers the owa operator proposed by yager has been widely used to aggregate experts opinions or preferences in human decision making yager s traditional owa operator focuses exclusively on the aggregation of crisp numbers however experts usually tend to express their opinions or preferences in a very natural way via linguistic terms these linguistic terms can be modelled or expressed by type 1 fuzzy sets in this paper we define a new type of owa operator the type 1 owa operator that works as an uncertain owa operator to aggregate type 1 fuzzy sets with type 1 fuzzy weights which can be used to aggregate the linguistic opinions or preferences in human decision making with linguistic weights the procedure for performing type 1 owa operations is analysed in order to identify the linguistic weights associated to the type 1 owa operator type 2 linguistic quantifiers are proposed the problem of how to derive linguistic weights used in type 1 owa aggregation given such type of quantifier is solved examples are provided to illustrate the proposed concepts

solutions of two point boundary value problems for even order differential equations abstract the existence of solutions of the two point boundary value problems consisting of the even order differential equations x 2 n t f t x t x t x 2 n 2 t r t 0 t 1 and the boundary value conditions i x 2 i 0 i x 2 i 1 0 0 i x 2 i 1 i x 2 i 1 1 0 i 0 1 n 1 is studied sufficient conditions for the existence of at least one solution of above bvps are established it is interesting that the nonlinearity f in the equation depends on all lower derivatives especially odd order derivatives and the growth conditions imposed on f are allowed to be super linear the degrees of phases variables are allowed to be greater than 1 if it is a polynomial the results are different from known ones since we do not apply the green s functions of the corresponding problem and the method to obtain a priori bounds of solutions is different from known ones examples that cannot be solved by known results are given to illustrate our theorems

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

existence of positive solutions for 2nth order singular sublinear boundary value problems abstract this paper investigates the existence of positive solutions for 2nth order n 1 singular sub linear boundary value problems a necessary and sufficient condition for the existence of c 2 n 2 0 1 as well as c 2 n 1 0 1 positive solutions is given by constructing lower and upper solutions and with the maximal theorem our nonlinearity f t x 1 x 2 x n may be singular at x i 0 i 1 2 n t 0 and or t 1

min and max operators for fuzzy intervals and their potential use in aggregation operators this paper aims at expressing min and max operations when triangular fuzzy intervals are taken as inputs that is when zadeh s extension principle is considered first presented approach consists in representing fuzzy intervals by means of their profiles in this context a computation algorithm can be easily derived for implementing the min and max operations another methodology based on interval relations is then proposed for determining a general analytical expression of the min and max operations the potential use of these expressions in the framework of uncertain aggregation operators is illustrated with the two additive choquet integral

t sum of l r fuzzy numbers abstract the sum of l r fuzzy numbers based on a given archimedian continuous t norm t is studied a series of improving sufficient conditions of several authors ensuring an analytical form of the output sum is completed by a sufficient and necessary condition i e the best sufficient condition some applications are indicated

existence and multiplicity of solutions to 2mth order ordinary differential equations abstract in this paper the existence and multiplicity of solutions are obtained for the 2mth order ordinary differential equation two point boundary value problems 1 m u 2 m t i 1 m 1 m i a i u 2 m i t f t u t for all t 0 1 subject to dirichlet neumann mixed and periodic boundary value conditions respectively where f is continuous a i r for all i 1 2 m since these four boundary value problems have some common properties and they can be transformed into the integral equation of form u i 1 m a i t i u t m f u we firstly deal with this nonlinear integral equation by using the strongly monotone operator principle and the critical point theory we establish some conditions on f which are able to guarantee that the integral equation has a unique solution at least one nonzero solution and infinitely many solutions furthermore we apply the abstract results on the integral equation to the above four 2mth order two point boundary problems and successfully resolve the existence and multiplicity of their solutions

fuzzy cardinality and the modeling of imprecise quantification abstract the concept of cardinality of a fuzzy set has received attention from several researchers and has been defined in several apparently independent manners a systematic investigation of this notion is performed which unifies and improves previous attempts the cardinality of a fuzzy set viewed as a fuzzy integer is related to scalar cardinality indices the closely related question of the probability of a fuzzy event is dealt with lastly the usefulness of fuzzy cardinality for meaning representation of statements or queries involving fuzzy linguistic quantifiers is emphasized

development of a particle particle hybrid scheme to simulate multiscale transitional flows in the present work a new particle particle hybrid method that combines the statistical ellipsoidal statistical bhatnagar gross krook and direct simulation monte carlo methods is developed the switching criterion between the two methods is based on the deviation of the cumulative velocity distribution function from maxwellian as measured by the kolmogorov smirnov statistical test unlike other hybrid approaches that use switching criteria based on macroscopic properties the selection of a particular particle method in a cell is determined by the local kolmogorov smirnov parameter value with respect to a preset global switching criterion a numerically efficient technique to compute the kolmogorov smirnov parameter was developed to enable the efficient calculation of the degree of nonequilibrium two well known fluid flow problems expanding argon flow through a nozzle and hypersonic flow over a blunt body were studied the kolmogorov smirnov parameter is shown to demarcate the regions of nonequilibriu

t sum of bell shaped fuzzy intervals the usual arithmetic operations on real numbers can be extended to arithmetical operations on fuzzy intervals by means of zadeh s extension principle based on a t norm t a t norm is called consistent with respect to a class of fuzzy intervals for some arithmetic operation if this arithmetic operation is closed for this class it is important to know which t norms are consistent with a particular type of fuzzy intervals recently dombi and gyorbiro proved that addition is closed if the dombi t norm is used with two bell shaped fuzzy intervals a result proved by mesiar on a strict t norm based shape preserving additions of lr fuzzy intervals with unbounded support is recalled as applications we define a broader class of bell shaped fuzzy intervals then we study t norms which are consistent with these particular types of fuzzy intervals dombi and gyorbiro s results are special cases of the results described in this paper

a necessary and sufficient condition for 2nth order singular super linear m point boundary value problems abstract in this paper we study the existence of positive solutions for singular super linear m point boundary value problems of 2nth order ordinary differential equations a necessary and sufficient condition for the existence of c 2 n 2 0 1 positive solutions as well as c 2 n 1 0 1 positive solutions is given by means of the fixed point theorems on cones

a note to the addition of fuzzy numbers based on a continuous archimedean t norm abstract a simple method of computing the t sum of special types of fuzzy numbers is introduced for the lukasiewicz t norm tl the proposed method can be applied to the addition of fuzzy numbers with convex membership functions the classes of fuzzy numbers leading to t idempotents are studied for continuous t norms

analytical and numerical solutions of fuzzy differential equations in this paper we study analytical and numerical solutions of fuzzy differential equations based on the extension principle for linear fuzzy differential equations we state some results on the behaviour of the solutions and study their relationship with the generalised hukuhara derivative in order to approximate the solutions of linear and non linear fuzzy differential equations we propose a new fuzzification of the classical euler method and then incorporate an unconstrained optimisation technique this combination offers a powerful tool to tackle uncertainty in any numerical method an efficient computational algorithm is also provided to guarantee the convexity of fuzzy solutions on the time domain several illustrative examples are given

single level constraint interval arithmetic in the present paper we propose a variant of constraint interval arithmetic that operates with a single parameter level in each interval operand of an expression this results in a computationally simple interval arithmetic that has several desirable properties not shared in general by interval operations previously defined in the literature the actualization and usefulness of this approach to fuzzy interval arithmetic is accomplished via alpha levels the evaluation of arithmetic expressions using the proposed single level constrained arithmetic is discussed in detail and we extend this theory to the fuzzy context

unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement this paper describes a unified flow solver ufs for rarefied and continuum gas flows the ufs separates the rarefied and continuum flow domains and selects appropriate solvers to combine the efficiency of continuum models with the accuracy of kinetic models the direct numerical solution of the boltzmann transport equation is used in rarefied regions while kinetic schemes of continuum fluid dynamics are used elsewhere using similar computational techniques for the kinetic and continuum solvers and employing intelligent domain decomposition algorithms attain the efficiency and numerical stability of the ufs solutions of test problems are presented to illustrate the capabilities of the ufs for high and low speed flows it is shown that the ufs can dynamically adapt the computational mesh and automatically introduce and remove kinetic patches to provide significant savings by limiting molecular scale solutions only to the regions where they are needed

the iterative solutions of 2nth order nonlinear multi point boundary value problems abstract the aim of this paper is to investigate the existence of iterative solutions for a class of 2 n th order nonlinear multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence problem is based on the method of upper and lower solutions and its associated monotone iterative technique a monotone iteration is developed so that the iterative sequence converges monotonically to a maximal solution or a minimal solution depending on whether the initial iteration is an upper solution or a lower solution two examples are presented to illustrate the results

some results on the addition of fuzzy intervals abstract a simple new method of computing t sum of fuzzy intervals having the same results as the sum of fuzzy intervals based on the weakest t norm t w is introduced this work extends that of mesiar fuzzy sets and systems 91 1997 231 237 and markova fuzzy sets and systems 91 1997 253 258

inverse controller design for fuzzy interval systems this paper aims at designing and analyzing an inverse controller for stable inversible minimum phase fuzzy interval linear and or multilinear systems the controller is designed from the fuzzy interval ranges of the system parameters using an spl alpha cut methodology indeed for a given spl alpha cut of the fuzzy system parameters representing an uncertainty level the control objective can be viewed as maintaining the system output within a tolerance envelope around the exact trajectory specified by the degree of preference spl alpha on the fuzzy trajectory the stability is ensured in the way that the controller restricts the system output divergence within the tolerance envelope the validity of the proposed method is illustrated by simulation examples

quotient mi groups a many identities group mi group for short is a special algebraic structure in which identity like elements called pseudoidentities are specified and collected into a monoidal substructure in this way many algebraic structures such as monoids of fuzzy intervals numbers or convex bodies possessing behavior very similar to that of a group structure may be well described and investigated using a new approach which seems to be superfluous for the classical structures the concept of mi groups was recently introduced by holcapek and st pnicka in the paper mi algebras a new framework for arithmetics of extensional fuzzy numbers to demonstrate how a standard structure can be generalized in terms of mi algebras this paper is a continuation of the development of mi group theory and is focused on the construction of quotient mi groups and a specification of the conditions under which the isomorphism theorems for groups are fulfilled for mi groups

cpk index estimation using fuzzy numbers process capacity indices pcis were developed and have been successfully used by companies to compete in and dominate the high profit markets by improving the quality and the productivity since the past two decades there is an essential assumption in the conventional application wherein the output process measurements are precise and distributed as normal random variables since the assumption of normal distribution is untenable errors can occur if the cpk index is computed using non normal data in the present study we address the situation that the output of data from measurement of the quality of a product is insufficiently precise or scarce this is possible when the quality measurement refers to the decision maker s subjective determination in such a situation the linguistic variable that is easier to capture the decision maker s subjective perception is applied to construct the pci cpk the present approach can mitigate the effect when the normal assumption is inappropriate and extends the application of cpk index

on generalized quotient mi groups abstract many identities group mi group for short is an algebraic structure generalizing the group structure where an involutive anti automorphism satisfying certain properties is used instead of the standard group inversion the concept of mi group in a more general form than in this article has been introduced by holcapek and st pnicka in the paper mi algebras a new frame work for arithmetics of extensional fuzzy numbers to describe properties of different approaches to arithmetics of vaguely specified quantities e g stochastic or fuzzy quantities in a unified way this article is a continuation of the effort to develop the theory of mi groups and is focused on a generalization of the construction of quotient mi groups induced by so called normal full mi subgroups which has been introduced by holcapek et al recently in the paper quotient mi groups besides a more general definition of quotient mi groups we prove three isomorphism theorems for mi groups in this new framework

a fourth order compact finite difference method for nonlinear higher order multi point boundary value problems a fourth order compact finite difference method is proposed for a class of nonlinear 2nth order multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as the three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term the convergence and the fourth order accuracy of the method are proved an efficient monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems various sufficient conditions for the construction of upper and lower solutions are obtained some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach

higher order lidstone boundary value problems for elliptic partial differential equations the aim of this paper is to show the existence and uniqueness of a solution for a class of 2nth order elliptic lidstone boundary value problems where the nonlinear functions depend on the higher order derivatives sufficient conditions are given for the existence and uniqueness of a solution it is also shown that there exist two sequences which converge monotonically from above and below respectively to the unique solution the approach to the problem is by the method of upper and lower solutions together with monotone iterative technique for nonquasimonotone functions all the results are directly applicable to 2nth order two point lidstone boundary value problems

on differential equations with interactive fuzzy parameter via t norms abstract in this paper we study fuzzy differential equations with uncertain parameters modeled by interactive fuzzy numbers the interactivity is formalized with the help of upper semicontinuous t norms to obtain solutions to fuzzy differential equations we use two different approaches the first uses a family of differential inclusions while the second is the fuzzification of the deterministic problem using the zadeh extension principle we show that the solutions obtained by the two methods are equal interactivity among the parameters and variable of the problem are studies with respect two methods we show that a hierarchy in uncertainties of the solutions arises when we chose one of the basic t norm the minimum the t norm of the product the t norm of lukasiewicz and t norm of the drastic product to model the interaction between parameters finally to illustrate the concepts introduced in the paper we study the malthusian model with interactive parameters

triple positive solutions and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem y 2 m t f y t y 2 j t y 2 m 1 t 0 t 1 y 2 i 0 0 y 2 i 1 0 i m 1 where 1 m f 0 growth conditions are imposed on f and inequalities involving an associated green s function are employed which enable us to apply the leggett williams fixed point theorem to cones in ordered banach spaces this in turn yields the existence of at least three positive symmetric concave solutions the emphasis here is that f depends on higher order derivatives

multiple positive solutions for nth order m point boundary value problems with all derivatives abstract we investigate the existence of positive solutions for the boundary value problem u n t f t u t u t u n 2 t u n 1 t 0 0 t 1 u 0 u 0 u n 2 0 0 u n 1 1 i 1 m 2 k i u n 1 i where f 0 1 r n r is continuous k i 0 i 1 2 m 2 0 i 1 m 2 k i 1 we give at first the associated green s function and some of its properties then imposing growth conditions on f we obtain the existence of at least three positive solutions for the boundary value problem by using the five functional fixed point theorem finally we give an example to demonstrate our result

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

on product sum of triangular fuzzy numbers abstract we study the problem if a i i n are fuzzy numbers of triangular form then what is the membership function of the infinite or finite sum a 1 a 2 defined via the sup product norm convolution

existence of multiple symmetric positive solutions of higher order lidstone problems this paper gives the sufficient conditions of the existence of at least three symmetric positive solutions for one type of higher order autonomous lidstone problem by applying the five functionals fixed point theorem the analogous result for higher order nonautonomous singular lidstone problem is also proved here

an approximation to the extension principle using decomposition of fuzzy intervals this article presents a proposal for the decomposition of large ranges of uncertainty associated with the cartesian product of two fuzzy intervals we compute an approximation of the fuzzy set obtained by applying extension principle to a real function by means of this decomposition and piecewise linearization of the function it is proved the efficiency of the method via convergence in d metric we also extend the method to the case when the function has more than two arguments some examples are included to illustrate the proposed method and to compare it with others

fuzzy strategic replacement analysis abstract this paper presents a fuzzy methodology for replacement of equipment issues such as fuzzy modeling of degradation parameters and determining fuzzy strategic replacement and economic lives are extensively discussed for the strategic purpose addible market and cost effects from the replacements against the counterpart i e the existing equipment cost and market obsolescence are modeled fuzzily and interactively in addition to the equipment deterioration both the standard fuzzy arithmetic and here re termed requisite constraint vertex fuzzy arithmetic or the vertex method are applied and investigated two numerical examples and the results are provided

the extension principle and a decomposition of fuzzy sets we give an algorithm to decompose a fuzzy interval u using this decomposition and the multilinearization of a univariate function f we obtain an approximation of the fuzzy interval where is obtained from f by applying the extension principle we provide approximation bounds some numeric illustration is provided

modeling vague beliefs using fuzzy valued belief structures abstract this paper presents a rational approach to the representation and manipulation of imprecise degrees of belief in the framework of evidence theory we adopt as a starting point the non probabilistic interpretation of belief functions provided by smets transferable belief model as well as previous generalizations of evidence theory allowing to deal with fuzzy propositions we then introduce the concepts of interval valued and fuzzy valued belief structures defined respectively as crisp and fuzzy sets of belief structures verifying hard or elastic constraints we then proceed with a generalization of various concepts of dempster shafer theory including those of belief and plausibility functions combination rules and normalization procedures most calculations implied by the manipulation of these concepts are based on simple forms of linear programming problems for which analytical solutions exist making the whole scheme computationally tractable we discuss the application of this framework in the areas of decision making under uncertainty and classification of fuzzy data

positive solutions of 2nth order ordinary differential equations with multi point boundary conditions in this paper we study the existence of positive solutions for 2nth order n 1 boundary value problems 1 nu 2n t f t u 0 t 1 subject to the 2 n m point boundary value conditions u 2j 0 u 2j 1 0 j 0 1 2 m m n 2 u 2j 0 bju 2j j 1 0 u 2j 1 dju 2j j 2 0 j m 1 n 1 where bj 0 dj 0 0 j 1 j 2 1 j m 1 n 1 by means of the fixed point theorems and the lower and upper solution method some results on the existence of positive solutions to the above 2 n m point boundary value problems are obtained

computing expensive multivariate functions of fuzzy numbers using sparse grids fuzzy arithmetic provides a powerful tool to introduce uncertainty into mathematical models with zadeh s extension principle one can obtain a fuzzy extension of any objective function computing expensive multivariate functions of fuzzy numbers however often poses a difficult problem due to non applicability of common fuzzy arithmetic algorithms severe overestimation or very high computational complexity this paper proposes a new approach based on sparse grids consisting of two parts first we compute a surrogate function using sparse grid interpolation second we perform the fuzzy valued evaluation of the surrogate function by a suitable implementation of the extension principle based on real or interval arithmetic the new approach gives accurate results and requires only few function evaluations

a probabilistic approach to the arithmetics of fuzzy numbers an alternative look on fuzzy numbers based on two random variables and their summation and multiplication is introduced this approach to summation covers the standard zadeh s extension principle and triangular norm based approach however it is more general we illustrate it on some examples moreover copula based summation preserving the class of triangular trapezoidal fuzzy numbers is discussed

constrained fuzzy hierarchical analysis for portfolio selection under higher moments marginal impacts of assets on portfolio higher moments are characterized by triangular fuzzy numbers and then evaluated by fuzzy ranking procedures in order to assemble fuzzy reciprocal matrices that are needed for the constrained fuzzy analytic hierarchy process ahp methods the proposed methodology increases the scope for emphasizing objective quantitative measures thus alleviating the influence of subjective qualitative factors employing constrained fuzzy arithmetic during fuzzy ahp application produces greater precision and reliability compared with applications of the standard fuzzy arithmetic by reference to higher moments the investor is able to strategize portfolios so that there is a reduction not only in exposure to normal risk i e volatility but also to risk of asymmetry skewness and the risk of fat tails kurtosis the efficiency of the proposed approach is highlighted in the ability to handle investor preferences regarding higher moment risks more optimal values for a particular risk can be obtained if that risk is of more concern moreover in terms of obtaining portfolio diversification the method is shown to be more useful than conventional optimization models

interval and fuzzy average internal rate of return for investment appraisal in investment appraisal uncertainty can be managed through intervals or fuzzy numbers because the arithmetical properties and the extension principle are well established and can be successfully applied in a rigorous way we apply interval and fuzzy numbers to the average internal rate of return airr recently introduced for overcoming the problems of the traditional internal rate of return irr in the setting of interval and fuzzy arithmetic we establish relations between the interim capitals invested the profits and the cash flows which are the ingredients of the airr and shed lights on the different ways uncertainty propagates depending on which variable is known and which one is derived the relations between fuzzy airr and fuzzy net present value are also investigated

regranulation a granular algorithm enabling communication between granular worlds abstract in this paper we describe a granular algorithm for translating information between two granular worlds represented as fuzzy rulebases these granular worlds are defined on the same universe of discourse but employ different granulations of this universe in order to translate information from one granular world to the other we must regranulate the information so that it matches the information granularity of the target world this is accomplished through the use of a first order interpolation algorithm implemented using linguistic arithmetic a set of elementary granular computing operations we first demonstrate this algorithm by studying the common fuzzy pd rulebase at several different granularities and conclude that the 3 3 granulation may be too coarse for this objective we then examine the question of what the natural granularity of a system might be this is studied through a 10 fold cross validation experiment involving three different granulations of the same underlying mapping for the problem under consideration we find that a 7 7 granulation appears to be the minimum necessary precision

a two stage logarithmic goal programming method for generating weights from interval comparison matrices a two stage logarithmic goal programming tlgp method is proposed to generate weights from interval comparison matrices which can be either consistent or inconsistent the first stage is devised to minimize the inconsistency of interval comparison matrices and the second stage is developed to generate priorities under the condition of minimal inconsistency the weights are assumed to be multiplicative rather than additive in the case of hierarchical structures a nonlinear programming method is used to aggregate local interval weights into global interval weights a simple yet practical preference ranking method is investigated to compare the interval weights of criteria or rank alternatives in a multiplicative aggregation process the proposed tlgp is also applicable to fuzzy comparison matrices when they are transformed into interval comparison matrices using a level sets and the extension principle six numerical examples including a group decision analysis problem with a group of comparison matrices a hierarchical decision problem and a fuzzy decision problem using fuzzy comparison matrix are examined to show the applications of the proposed methods comparisons with other existing procedures are made whenever possible

approximate fuzzy arithmetic operations using monotonic interpolations abstract we suggest the use of piecewise monotonic interpolations to approximate and represent a fuzzy number or interval and to derive a procedure to control the absolute error associated to the arithmetic operations between fuzzy numbers in order to reduce the distance between the true result of the operation and its approximation the monotonic functions are then used to define a parametric representation of a large class of fuzzy numbers having general shapes of the membership function and a simple and accurate procedure is introduced for the fuzzy arithmetic several computational experiments are given to show the good performance of the proposed procedure

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

a convexity problem and a new proof for linearity preserving additions of fuzzy intervals in this paper we propose a new sufficient condition for convexity of function this result provides a simple new proof for linearity preserving additions of fuzzy intervals obtained by the author in 2002 which was a conjecture suggested by mesiar in 1997

mi algebras the existing arithmetics of fuzzy numbers usually stemming from the cut arithmetic do not preserve some of the important properties of the standard arithmetics of real numbers although one cannot expect a generalization of standard arithmetic to perfectly preserve all its properties the most important properties should be preserved at least in a weakened form we present a novel framework for arithmetics of extensional fuzzy numbers that more or less preserves all the important algebraic properties of the arithmetic of real numbers and thus seems to be an important seed for further investigations on this topic the suggested approach to arithmetics of extensional fuzzy numbers is demonstrated by many examples and besides its algebraic properties it is also shown to carry some desirable practical properties the investigation leads to novel algebraic structures mi algebras mi monoids mi groups mi fields that abstract the discussed properties the main idea of these structures is based on a set of pseudoidentities that complements the only commonly used identity element in a monoid group structure these pseudoidentities are elements that generalize the identity like behavior and allow us to weaken the standard forms of algebraic properties that are highly desirable to preserve at least in a weakened form appropriate properties of suggested mi structures are introduced and demonstrated by a number of practical examples

symmetric positive solutions of nonlinear boundary value problems abstract we study the nonlinear boundary value problem u 2 m f t u u u 2 m 2 t 0 1 u 2 i 0 u 2 i 1 0 i 0 m 1 the existence of symmetric positive solutions of the above problem is discussed sufficient conditions are obtained for the problem to have one any finite number and a countably infinite number of such solutions our results extend some recent work in the literature on boundary value problems of ordinary differential equations we illustrate our results by two examples none of which can be handled using the existing results

discrete velocity models and numerical schemes for the boltzmann bgk equation in plane and axisymmetric geometries we present new numerical models for computing transitional or rarefied gas flows as described by the boltzmann bgk and bgk es equations we first propose a new discrete velocity model based on the entropy minimization principle this model satisfies the conservation laws and the entropy dissipation moreover the problem of conservation and entropy for axisymmetric flows is investigated we find algebraic relations that must be satisfied by the discretization of the velocity derivative appearing in the transport operator then we propose some models that satisfy these constraints owing to these properties we obtain numerical schemes that are economic in terms of discretization and robust in particular we develop a linearized implicit scheme for computing stationary solutions of the discrete velocity bgk and bgk es models this scheme is the basis of a code which can compute high altitude hypersonic flows in 2d plane and axisymmetric geometries our results are analyzed and compared to other methods

multiple solutions of some fourth order boundary value problems abstract for some fourth order boundary value problems several new existence theorems on multiple positive negative and sign changing solutions are obtained the critical point theory and the supersolution and subsolution method are employed to discuss this problem

the transformation method for the simulation and analysis of systems with uncertain parameters in this paper the transformation method is introduced as a powerful approach for both the simulation and the analysis of systems with uncertain model parameters based on the concept of cuts the method represents a special implementation of fuzzy arithmetic that avoids the well known effect of overestimation which usually arises when fuzzy arithmetic is reduced to interval computation systems with uncertain model parameters can thus be simulated without any artificial widening of the simulation results as a by product of the implementation scheme the transformation method also provides a measure of influence to quantitatively analyze the uncertain system with respect to the effect of each uncertain model parameter on the overall uncertainty of the model output by this a special kind of sensitivity analysis can be defined on the basis of fuzzy arithmetic finally to show the efficiency of the transformation method the method is applied to the simulation and analysis of a model for the friction interface between the sliding surfaces of a bolted joint connection

cardinality approach to fuzzy number arithmetic to solve problems from the area of computing with words arithmetic operations often have to be carried out on fuzzy numbers the author proposes to call fuzzy numbers fuzzy sets of numbers fsofn to emphasize the fact that they are sets of many numbers in the continuous case of infinitely many numbers and not one single number as the name fuzzy number suggests because it occasionally leads to incorrect interpretations of their concept to realize arithmetic operations on fsofn the standard extension principle is used this principle was intended for possibilistic fsofn however in practical tasks many fsofn are of probabilistic character if for such fsofn the standard extension principle is used the results will be incorrect in this paper a cardinality extension principle is proposed it realizes arithmetic operations on probabilistic fsofn this principle is proposed in 2 versions as a normal and as a generalized version that enables context dependent constraints to be taken into account in addition a new form of the possibilistic constraint extension principle of klir is presented in this paper many examples are given that illustrate computations with both extension principles to make them less abstract and more user friendly