jump discontinuities of semilinear strictly hyperbolic systems in two variables creation and propagation the creation and propagation of jump discontinuities in the solutions of semilinear strictly hyperbolic systems is studied in the case where the initial data has a discrete set xi i 1n of jump discontinuities lets be the smallest closed set which satisfies i s is a union of forward characteristics ii s contains all the forward characteristics from the points xi i 1n iii if two forward characteristics ins intersect then all forward characteristics from the point of intersection lie ins

a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

reasonable properties for the ordering of fuzzy quantities ii as a continuation of the first part related to the first and second class of ordering approaches this paper deals with the fulfilment of reasonable properties in the third class of ordering approaches to do so we briefly introduce fuzzy relations on which the third class of approaches is based then we recall some transitivity related concepts and an ordering procedure based on a acyclic fuzzy relation acyclicity is a very weak restriction on a fuzzy relation we prove that many fuzzy relations used for the comparison of fuzzy quantities satisfy some conditions stronger than acyclicity so we give a widely applicable formulation to derive a total ranking order from a fuzzy relation with our formulation we examine all the ordering indices in the third class with respect to the proposed axioms in part i

a simplification of the zoeppritz equations the compressional wave reflection coefficient r given by the zoeppritz equations is simplified to the following r r0 a0r0 1 2 sin2 1 2 vpvp tan2 sin2 the first term gives the amplitude at normal incidence 0 the second term characterizes r at intermediate angles and the third term describes the approach to critical angle the coefficient of the second term is that combination of elastic properties which can be determined by analyzing the offset dependence of event amplitude in conventional multichannel reflection data if the event amplitude is normalized to its value for normal incidence then the quantity determined is a a0 1 1 2 r0 a0 specifies the normal gradual decrease of amplitude with offset its value is constrained well enough that the main information conveyed is r0 where is the contrast in poisson s ratio at the reflecting interface and r0 is the amplitude at normal incidence this simplified formula for r accounts for all of the relations between r

ranking fuzzy numbers with integral value abstract ranking fuzzy numbers is important in decision making since very often the alternatives are evaluated by fuzzy numbers in a vague environment a comparison between these fuzzy numbers is indeed a comparison between alternatives this paper proposes a method of ranking fuzzy numbers with integral value the method which is independent of the type of membership functions used and the normality of the functions can rank more than two fuzzy numbers simultaneously it is relatively simple in computation especially in ranking triangular and trapezoidal fuzzy numbers further an index of optimism is used to reflect the decision maker s optimistic attitude discussion on comparative advantages is included

a multiobjective fuzzy stopping in a stochastic and fuzzy environment abstract in a stochastic and fuzzy environment a multiobjective fuzzy stopping problem is discussed the randomness and fuzziness are evaluated by probabilistic expectations and linear ranking functions respectively pareto optimal fuzzy stopping times are given under the assumption of regularity for stopping rules by using optimal stopping times

fuzzy versions of the covering circle problem abstract in this paper we consider two fuzzy versions of the well known problem of determining the smallest circle center and radius that would cover a given finite set of points in the plane when the locations of points are not precise but fuzzy the first is modeled as a possibility constrained mathematical program while the second is modeled as a necessity constrained one polynomial algorithms are presented for both the versions also a numerical example is included for one case these models and solutions are of interest in both theoretical and practical contexts

a probability based approach to comparison of fuzzy numbers and applications to target oriented decision making in this paper we introduce a new comparison relation on fuzzy numbers based on their alpha cut representation and comparison probabilities of interval values basically this comparison process combines a widely accepted interpretation of fuzzy sets together with the uncertain characteristics inherent in the representation of fuzzy numbers the proposed comparison relation is then applied to the issue of ranking fuzzy numbers using fuzzy targets in terms of target based evaluations some numerical examples are used to illuminate the proposed ranking technique as well as to compare with previous methods more interestingly according to the interpretation of the new comparison relation on fuzzy numbers we provide a fuzzy target based decision model as a solution to the problem of decision making under uncertainty with which an interesting link between the decision maker s different attitudes about target and different risk attitudes in terms of utility functions can be established moreover an application of the proposed comparison relation to the fuzzy target based decision model for the problem of fuzzy decision making with uncertainty is provided numerical examples are also given for illustration

fast model based migration velocity analysis and reflector shape estimation migration velocity analysis can be made more efficient by preselecting the traces that contribute to a series of common reflection point crp gathers and migrating only those traces the data traces that contribute to a crp for one reflection point on one layer are defined in a two step procedure first poststack parsimonious kirchhoff depth migration of zero offset or stacked traces defines approximate reflector positions and orientations then ray tracing from the reflection points for nonzero reflection angles defines the source and receiver locations of the data traces in the crp gather these traces are then prestack depth migrated and the interval velocity model adjustment is obtained by fitting the velocity that maximizes the stack amplitude over the predicted nonhyperbolic moveout a small number 2 3 of iterations converge to a 2d model of layer shape and interval velocity further efficiency is obtained by implementing layer stripping the computation time is greatly reduced by combinin

criteria for evaluating fuzzy ranking methods abstract in this paper four criteria for evaluating fuzzy ranking methods are investigated fuzzy preference representation rationality of fuzzy ordering distinguishability and robustness based on these criteria two existing ranking methods baas and kwakernaak s and nakamura s are evaluated and a new improved ranking method is suggested

a method for ranking fuzzy numbers and its application to decision making since fuzzy numbers represent uncertain numeric values it is difficult to rank them according to their magnitude in the paper a method for ranking fuzzy numbers is proposed the method considers the overall possibility distributions of fuzzy numbers in their evaluations for ranking and provides users with a method of changing viewpoints for evaluations users represent their viewpoints with fuzzy sets the method evaluates fuzzy numbers with a satisfaction function and the viewpoint given by users and then ranks the numbers according to their evaluation values the satisfaction function is a measure of comparisons between fuzzy numbers in order to illustrate the ranking method two numeric examples are shown and for the comparative study our method is compared with four existing ranking methods through eight examples as an example of potential applications the proposed method is applied to a decision making problem a two person game with fuzzy profit and loss the ranking method is used to analyze player choices

the revised method of ranking fuzzy numbers with an area between the centroid and original points in 2002 chu and tsao proposed a method to rank fuzzy numbers they employed an area between the centroid and original points to rank fuzzy numbers however there were some problems with the ranking method in this paper we want to indicate these problems of chu and tsao s method and then propose a revised method which can avoid these problems for ranking fuzzy numbers since the revised method is based on the chu and tsao s method it is easy to rank fuzzy numbers in a way similar to the original method

ranking fuzzy numbers with an area method using radius of gyration ranking fuzzy numbers plays an very important role in linguistic decision making and some other fuzzy application systems many methods have been proposed to deal with ranking fuzzy numbers chu pointed out some shortcomings of the existing distance method and proposed to rank the fuzzy numbers with the area between the centroid point and original point however drawbacks are also found in the area method for example it cannot rank fuzzy numbers when some fuzzy numbers have the same centroid point in this paper we propose a modified area method to rank fuzzy numbers the modified method can effectively rank various fuzzy numbers and their images we also used some comparative examples to illustrate the advantage of the proposed method

duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables linear programming problems with trapezoidal fuzzy variables fvlp have recently attracted some interest some methods have been developed for solving these problems by introducing and solving certain auxiliary problems here we apply a linear ranking function to order trapezoidal fuzzy numbers then we establish the dual problem of the linear programming problem with trapezoidal fuzzy variables and hence deduce some duality results in particular we prove that the auxiliary problem is indeed the dual of the fvlp problem having established the dual problem the results will then follow as natural extensions of duality results for linear programming problems with crisp data finally using the results we develop a new dual algorithm for solving the fvlp problem directly making use of the primal simplex tableau this algorithm will be useful for sensitivity or post optimality analysis when using primal simplex tableaus

duality in fuzzy number linear programming by use of a certain linear ranking function we explore some duality properties in fuzzy number linear programming problems by use of a linear ranking function we introduce the dual of fuzzy number linear programming primal problems we then present several duality results

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

solving linear programming problems under fuzziness and randomness environment using attainment values in this paper the author presents a model to measure attainment values of fuzzy numbers fuzzy stochastic variables these new measures are then used to convert the fuzzy linear programming problem or the fuzzy stochastic linear programming problem into the corresponding deterministic linear programming problem numerical comparisons are provided to illustrate the effectiveness of the proposed method

ranking fuzzy numbers based on decomposition principle and signed distance by using the decomposition principle and the crisp ranking system on r we construct a new ranking system for fuzzy numbers which is very realistic and also matching our intuition as in r

ranking fuzzy numbers with preference weighting function expectations based on the recent research developments of fuzzy number ranking the paper extends the centroid expectation approach and proposes a preference weighting function expectation method to rank fuzzy numbers the weighting function gives different emphasis on the possible values of the fuzzy number which can be used to represent the decision maker s pessimistic neutral or optimistic attitude an optimistic measure to represent this preference information is proposed some properties of the weighting function and optimistic measure are systematically analyzed then the paper proposes a special class weighting function with maximum entropy principle and analyzes some of its properties with this parameterized maximum entropy weighting function we can get preference expectation of any fuzzy number with given optimistic level next the weighting function is extended to represent the confidential attitude of the decision maker by giving different emphasis on the cut level set of the fuzzy number some results about confidential attitude that parallel to the valuation attitude are attained we especially extended the badd basic defuzzification distribution approach with preference expectation of fuzzy numbers and another confidential measure with badd operator is proposed finally the expected valuations of fuzzy numbers are used to ranking the fuzzy numbers compared with other fuzzy number ranking methods the method of this paper is easily interpreted robust flexible and invariant with translating and scaling for real numbers

ranking alternatives using fuzzy numbers a computational approach abstract the method presented by buckley fuzzy sets and systems 15 1985 21 31 for ranking alternatives by using fuzzy numbers is discussed as far as computational aspects are concerned an alternative procedure is presented which allows for the use of any convex and normalized fuzzy set to assess the contribution of each alternative to some overall objective according to a set of criteria the procedure also allows the criteria to be structured in n level hierarchies an example is presented to illustrate the algorithm

characterization for the solvability of nonlinear partial differential equations within the nonlinear theory of generalized functions introduced earlier by the author a number of existence and regularity results have been ob tained one of them has been the first global version of the cauchy kovalevskaia theorem which proves the existence of generalized solutions on the whole of the domain of analyticity of arbitrary analytic nonlinear pdes these generalized solutions are analytic everywhere except for closed nowhere dense subsets which can be chosen to have zero lebesgue measure this paper gives a certain extension of that result by establishing an algebraic necessary and sufficient condition for the existence of generalized solutions for arbitrary polynomial nonlinear pdes with continuous coefficients this alge braic characterization given by the so called neutrix or off diagonal condition is proved to be equivalent to certain densely vanishing conditions useful in the study of the solutions of general nonlinear pdes

the preference order of fuzzy numbers abstract many fuzzy number ranking approaches are developed in the literature for multiattribute decision making problems almost all of the existing approaches focus on quantity measurement of fuzzy numbers for ranking purpose in this paper we consider the ranking process to determine a decision maker s preference order of fuzzy numbers a new ranking index is proposed to not only take quantity measurement but incorporate quality factor into consideration for the need of general decision making problems for measuring quantity several cuts of fuzzy numbers are used a signal noise ratio is defined to evaluate quality of a fuzzy number this ratio considers the middle point and spread of each cut of fuzzy numbers as the signal and noise respectively a fuzzy number with the stronger signal and the weaker noise is considered better moreover the associated levels are treated as the degree of belief about the cut and used as weights in the index for strengthening the influence of cut with higher levels the membership functions of fuzzy numbers are not necessarily to be known beforehand while applying this index only a few left and right boundary values of cuts of fuzzy numbers are required we have proved the feature of the proposed index in a particular case several examples are also used to illustrate the feature and applicability in ranking fuzzy numbers

ranking and defuzzification methods based on area compensation abstract we present some interesting properties related to the area compensation procedure to compare fuzzy numbers it has been proved that this method produces more than a fuzzy interval order it induces a ranking of fuzzy numbers some further results are given about the transitivity property and about computational aspects extensions to non normal fuzzy numbers and fuzzy quantities are also proposed

ranking alternatives using fuzzy numbers we investigate the problem of employing expert opinion to rank alternatives across a set of criteria the experts use fuzzy numbers to express their preferences and we employ fuzzy arithmetic to compute an issue s fuzzy ranking this leads to a partition of the alternatives into sets h1 h2 where h1 contains the highest ranked issues h2 has all the second highest ranked alternatives etc the total ranking process is shown to possess a number of important properties an example is presented to illustrate the method

ranking fuzzy numbers with index of optimism abstract fuzzy numbers are convenient for representing imprecise numerical quantities in a vague environment and their comparison or ranking is very important for application purposes despite many methods suggested in the literature there is no single measure that is universally applicable to a wide variety of situations this paper suggests a new method for comparing fuzzy numbers based on the combination of maximizing possibility and minimizing possibility using an index of optimism in 0 1 reflecting the decision maker s risk taking attitude the method is simple but has many comparative advantages

a new approach for ranking fuzzy numbers based on fuzzy simulation analysis method a new approach for ranking fuzzy numbers based on a fuzzy simulation analysis fsa method is proposed a combination method including computer and math application is developed first the method is intuitive and can be used to order fuzzy numbers easily the approach is illustrated by numerical examples showing that it overcomes the shortcoming existing fuzzy ranking approaches especially when it is difficult to use other methods to solve ranking fuzzy problems

optimal stopping problems in a stochastic and fuzzy system in a stochastic and fuzzy environment two kinds of stopping models are discussed and compared the optimal fuzzy stopping times are given under the assumptions of monotonicity and regularity for stopping rules also we find that fuzzy stopping times are favored in a comparison between fuzzy and classical stopping models

on the role of quadratic oscillations in nonlinear schrodinger equations we consider a nonlinear semi classical schrodinger equation for which it is known that quadratic oscillations lead to focusing at one point described by a nonlinear scattering operator if the initial data is an energy bounded sequence we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations the proof relies on a linearizability condition which can be expressed in terms of wigner measures when the initial data is a sum of such quadratic oscillations we prove that the associate solution is the superposition of the nonlinear evolution of each of them up to a small remainder term in an appendix we transpose those results to the case of the nonlinear schrodinger equation with harmonic potential

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

an approximate approach for ranking fuzzy numbers based on left and right dominance this study presents an approximate approach for ranking fuzzy numbers based on the left and right dominance the proposed approach only requires a few left and right spreads at some levels of fuzzy numbers to determine the respective dominance of one fuzzy number over the other the total dominance is then determined by combining the left and right dominance based on a decision maker s optimistic perspectives such a dominance is useful in ranking the fuzzy numbers when membership functions cannot be acquired the approach proposed herein is relatively simple in terms of computational efforts and is efficient when ranking a large quantity of fuzzy numbers by using a few left and right spreads two groups of examples demonstrate the accuracy and applicability of the proposed approach

ranking l r fuzzy number based on deviation degree this paper proposed a novel approach to ranking fuzzy numbers based on the left and right deviation degree l r deviation degree in the approach the maximal and minimal reference sets are defined to measure l r deviation degree of fuzzy number and then the transfer coefficient is defined to measure the relative variation of l r deviation degree of fuzzy number furthermore the ranking index value is obtained based on the l r deviation degree and relative variation of fuzzy numbers additionally to compare the proposed approach with the existing approaches five numerical examples are used the comparative results illustrate that the approach proposed in this paper is simpler and better

velocities from amplitude variations with offset the sensitivity of amplitude variations with offset avo to normal moveout nmo velocity errors has usually been taken to be a significant limitation of the method however this sensitivity can in most cases be exploited to obtain more accurate velocities from a fully automated procedure the error due to velocity in the avo gradient is in phase quadrature with the avo zero offset intercept furthermore if the majority of reflectors within a suitable data window contain a similar gradient quadrature component then it may be inferred that this component is due to an nmo velocity error both the intercept and gradient traces may be made analytic complex by combining them with i times their hilbert transforms the imaginary part of their joint correlation coefficient quantifies the gradient quadrature component and is proportional to the average of the fractional nmo velocity error over the data window unlike semblance which is always positive the sign of the imaginary correlation coefficient ind

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

reflection of delta waves for nonlinear wave equations in one space variable 0 the functions f g are assumed to be smooth and globally lipschitz continuous f 0 g 0 0 other boundary conditions can be treated as well we focus on a nonlinear neumann condition which is the most relevant among nonlinear boundary conditions from the viewpoint of applications see for example the discussion of boundary conditions in graff l the initial data are the sum of a singular part a a consisting of distributions with discrete support in 0 oo and a regular part go g the boundary data g t are also assumed to be sufficiently regular let u be the classical solution to

optimal fuzzy counterparts of scheduling rules the optimality of a fuzzy logic alternative to the usual treatment of uncertainties in a scheduling system using probability theory is examined formally fuzzy scheduling techniques proposed in the literature either fuzzify directly the existing scheduling rules or solve mathematical programming problems to determine the optimal schedules in the former method the fuzzy optimality for the optimal scheduling rules is usually not justified but still assumed in this paper the necessary conditions for fuzzy optimality are defined and fuzzy counterparts of some of the well known scheduling rules such as shortest processing time spt and earliest due date edd are developed

weak limits of solutions to semilinear hyperbolic systems on considere un systeme d ordre 1 semi lineaire strictement hyperbolique sous forme diagonale t x t x u x t f x t u x t x t r 2 u x 0 a x x r ou u r 2 r n f r n 2 r n est lisse et une matrice n n lisse a valeur reelle diagonale on etudie le cas ou les donnees initiales sont des distributions a support en un nombre fini de points on montre que sous des hypotheses additionnelles sur la non linearite f les solutions du systeme avec donnees initiales regularisees convergent dans 7b d r 2 et qu il est possible de calculer la limite

an index for ordering fuzzy numbers abstract a new index that is useful for ordering ranking of fuzzy numbers is proposed to use this index the membership functions of fuzzy numbers need not be normal or convex properties of this index and its relationship to the 1981 yager s index are explained a numerical comparison between the proposed and four other ordering procedures is presented

the mean value of a fuzzy number abstract the notion of expectation is defined for fuzzy numbers of intervals viewing them as consonant random sets this expectation is interval valued and remains additive in the sense of fuzzy interval addition as a by product some light is shed on the debate between probability and possibility theory and the problem of how to carry measure theoretic notions into the field of possibility theory

an approach to stopping problems of a dynamic fuzzy system we formulate a stopping problem for dynamic fuzzy systems concerning with fuzzy decision environment it could be regarded as a natural fuzzification of non fuzzy stopping problem with a deterministic dynamic system the validity of the approach by cuts of fuzzy sets will be discussed in constructing one step look ahead policy of an optimal fuzzy stopping time a numerical example is given to illustrate the theoretical results and to show a merit to use fuzzy stopping times in dynamic fuzzy systems

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

a procedure for ranking fuzzy numbers using fuzzy relations abstract this paper presents a method to give fuzzy order relations between fuzzy numbers these are founded on the concept of comparison function defined in the paper and the use of fuzzy measures related with the same numbers the link of such relations with fuzzy operations is investigated too

ranking fuzzy numbers with maximizing set and minimizing set abstract up to now these are five methods of ranking n fuzzy numbers in order but these methods contain some confusions and occasionally conflict with intuition this paper introduces the concept of maximizing set and minimizing set to decide the ordering value of each fuzzy number and uses these values to determine the order of the n fuzzy numbers in addition we give a method for calculating the ordering value of each fuzzy number with triangular trapezoidal and two sided drum like shaped membership functions

nonlinear superposition and absorption of delta waves in one space dimension in the study of the initial value problem for linear hyperbolic partial differential equations it is usually possible to extend the solution operator from classical to distribution initial data this extension is useful for several reasons first the existence of fundamental solutions allows one to construct other solutions by the superposition principle second distribution data are idealizations of certain kinds of classical initial data the delta functon is an idealization of a high localized peak the derivative of a delta function is an idealization of a single sharp localized oscillation and so forth if one can understand how the idealized data are propagated then with suitable continuity one has understood some of the essential features of the classical solutions with data which are close to the distribution data in the linear theory there are many theorems which give precise analytical content to this idea in the nonlinear case the first reason for studying distribution data is not operative since there is no general superposition principal for nonlinear equations the second reason is certainly appropriate however since one is very interested in the propagation of high peaks sharp jumps and

a new criterion of choice between generalized triangular fuzzy numbers in this paper we consider a new criterion of choice between generalized or rounded triangular fuzzy numbers based on the concept of the weighted possibilistic mean introduced by fullr and majlender first we introduce a preference relation which establishes a partial order on the family of generalized triangular fuzzy numbers second we present a weak preference relation which leads to a total order on this family then we consider the special case of triangular fuzzy numbers and rephrase the preceding results in terms of selected parameters characterizing a triangular fuzzy number finally we compare our results with a number of new approaches recently discussed in the literature