an exposition of the ahp in reply to the paper remarks on the analytic hierarchy process it is a fact that people make decisions and have been making decisions for a very long time contrary to what some of us who are interested in decision making may like to believe most people do not take seriously the existence of theories which purport to set their thinking and feeling right they claim to know their own value system and what they want they may wonder how anyone else can know well enough to tell them how best to organize their thinking in order to make better choices yet research has shown that complex decisions are beyond the capacity of the brain to synthesize intuitively and efficiently since decision making is a natural characteristic of people how do we describe what they do so that an ordinary mortal can understand what we are saying we do not wish to legislate the method with which people should make decisions but only to describe it even when it is prescribed by some method in the process we may learn things that can help people make better decisions how the analytic hierarchy process ahp forman et al harker 1986 harker and vargas 1987 saaty 1986 1988a b saaty and vargas 1987 xu 1988 golden et al 1989 saaty and alexander 1989 is a theory of measurement when applied in decision making it assists one to describe the general decision operation by decomposing a complex problem into a multi level hierarchic structure of objectives criteria subcriteria and alternatives the ahp provides a fundamental scale of relative magnitudes expressed in dominance units to represent judgments in the form of paired comparisons a ratio scale of relative magnitudes expressed in priority units is then derived from each set of comparisons an overall rcatio scale of priorities is then synthesized to obtain a ranking of the alternatives from its axioms to its procedures the ahp has turned out to be historically and theoretically a different and independent theory of decision making from utility theory much as a dialogue evolved in mathematics around the consistency of different geometries and around absolute and relative space and time in physics both to dispel absolute notions those who believe that only utility theory can tell us the absolute truth about man s decision making might take a close look at the ahp it has found varied and serious applications it also has a particular way of generating ratio scales and dealing with inconsistency in judgment that have contributed to its effectiveness in resource allocation and in the setting of priorities by a group of decision makers utility theory is a normative process the ahp as a descriptive theory encompasses procedures leading to outcomes as would be ranked by a normative theory but it must go beyond to deal with outcomes not accounted for by the demanding assumptions of a normative theory we must briefly describe the ahp to enable the reader to see that a practicable theory based on ratio scales need not dilute itself to satisfy expectations of people who derive their understanding from a theory based on interval scales this is particularly true if the rival theory in aspiring for generality also makes unrealistic assumptions for example about the transitivity and consistency of preferences and the difficult use of lotteries

on stability of fuzzy systems expressed by fuzzy rules with singleton consequents addresses the issue of stability of a fuzzy system described by fuzzy rules with singleton consequents it first presents two canonical forms of a fuzzy system a parametric expression and a state space expression a fuzzy system with singleton consequents is found to be a piecewise polytopic affine system then the paper gives stability theorems for a discrete time system as well as for a continuous time system it also gives a brief survey on the stability issues of fuzzy control systems

a clarification of remarks on the analytic hierarchy process the replies by saaty 1990 and by harker and vargas 1990 to my remarks on the analytic hierarchy process dyer 1990 are useful in focusing attention on areas of disagreement regarding the ahp and provide readers with sufficient information to form their own opinions the purpose of this response is not to extend the debate but to clarify the two methods proposed in my paper to resolve the problem of rank reversal in the ahp harker and vargas 1989 misinterpreted my first proposed method method 1 in their reply the key observation they missed is that both the weights on the criteria and the scores of the alternatives on the criteria must be normalized with respect to the same range of alternative values in method 1 the decision maker must specify the ranges over which he assumed the alternatives might vary on each criterion when he determined the weights an example may be useful in clarifying the key ideas suppose each alternative in the harker and vargas 1990 example represents the choice of a decision maker among sums of money deposited in two different bank accounts and each criterion is simply the amount of money deposited in a single bank account obviously the decision maker would use the sum of the accounts as the basis for ranking the alternatives and would select b from the three alternatives a b and c let us see how the appropriate use of method 1 would lead to this result without recourse to the supermatrix technique or to the concept of dependence of the criteria on the alternatives the decision maker would consider the deposits in each bank account to be equally important if he assumed that the ranges of the possible deposits were equal for example suppose the decision maker assumes that the values for each bank account criterion vary over the range 1 80 he responds that changes from 1 to 80 on each criterion are equally important so the normalized eigenvector for the ahp criterion weights is 1 2 1 2 using method 1 since the range of values on criterion c1 is 75 80 and the decision maker assumed a range of 1 80 in determining the equal criterion weights we must introduce a dummy alternative with the value of 1 on criterion c1 this dummy alternative does not need to be explicitly introduced with values on the other criteria but it could be and would not affect the results as long as its assumed values on the other criteria are in the range 1 80 similarly a dummy alternative with a value of 80 must be introduced for criterion c2 the normalized eigenvector for criterion c1 would become 80 235 79 235 75 235 1 235 for example where the last component 1 235 is the score for the dummy alternative we normalize each resulting eigenvector by subtracting its smallest component and then dividing by the largest component the appropriate scores for the three alternatives are

brief paper an analytical study on structure stability and design of general nonlinear takagi sugeno fuzzy control systems in this paper we first study analytical structure of general nonlinear takagi sugeno ts for short fuzzy controllers then establish a condition for analytically determining asymptotic stability of the fuzzy control systems at the equilibrium point and finally use the stability condition in design of the control systems that are at least locally stable the general ts fuzzy controllers use arbitrary input fuzzy sets any types of fuzzy logic and ts fuzzy rules with linear consequent and the generalized defuzzifier which contains the popular centroid defuzzifier as a special case we have mathematically proved that the general ts fuzzy controllers are nonlinear controllers with variable gains continuously changing with controllers input variables using lyapunov s linearization method we have established a necessary and sufficient condition for analytically determining local asymptotic stability of ts fuzzy control systems each of which is made up of a fuzzy controller of the general class and a nonlinear plant we show that the condition can be used in practice even when the plant model is not explicitly known we have utilized the stability condition to design with or without plant model general ts fuzzy control systems that are at least locally stable three numerical examples are given to illustrate in detail how to use our new results our results offer four important practical advantages 1 our stability condition being a necessary and sufficient one is the tightest possible stability condition 2 the condition is simple and easy to use partially because it only needs the fuzzy controller structure around the equilibrium point 3 the condition can be used for determining system local stability and designing fuzzy control systems that are stable at least around the equilibrium point even when the explicit plant models are unavailable and 4 the condition covers a very broad range of nonlinear ts fuzzy control systems for which a meaningful global stability condition seems impossible to establish

existence regularity and properties of generalized apparent horizons we prove a conjecture of tom ilmanen and hubert bray on the existence of the outermost generalized apparent horizon in an initial data set and that it is outer area minimizing

analytical structure of a fuzzy controller with linear control rules abstract we prove that a fuzzy controller with linear control rules and mamdani minimum inference is the sum of a global two dimensional multilevel relay and a local nonlinear proportional integral pi controller the properties of the nonlinear gains of the local pi controller are investigated moreover it is proven that as the number of input fuzzy sets approaches the global multilevel relay approaches a global linear pi controller while the local nonlinear pi controller disappears

realization of linear outputs by using mixed fuzzy logics abstract the realization of the linear defuzzified output of the fuzzy controller with two inputs is discussed in this paper using the mixed fuzzy logics to evaluate the fuzzy control rule we show that the defuzzified output by an appropriate choice of each component can be precisely equivalent to a linear function of the inputs to the fuzzy controller

the existence of least area surfaces in 3 manifolds this paper presents a new and unified approach to the existence theorems for least area surfaces in 3 manifolds introduction a surface f smoothly embedded or immersed in a riemannian manifold m is minimal if it has mean curvature zero at all points it is a least area surface in a class of surfaces if it has finite area which realizes the infimum of all possible areas for surfaces in this class the connection between these two ideas is that a surface which is of least area in a reasonable class of surfaces must be minimal the converse is false minimal surfaces are in general only critical points for the area function there are close analogies between these two concepts and the theory of geodesics in a riemannian manifold minimal surfaces correspond to geodesics and least area surfaces correspond to geodesic arcs or closed geodesics which have shortest length in some class of paths any geodesic a in a riemannian manifold m has the property that it is locally shortest i e if p and q are nearby points on a then the subarc of a which joins p and q is the shortest path in m from p to q it can also be proved that minimal surfaces are locally of least area but the proof is difficult and involves substantial knowledge of the theory of partial differential equations there are now a large number of theorems asserting the existence of surfaces of least area in various classes surfaces of this type have become an important tool in 3 dimensional topology in this paper we present a new approach to the proofs of these existence theorems this yields a simplified and unified method for the proof of the existence of minimal surfaces in 3 dimensional riemannian manifolds in 1930 douglas do and rado ra independently showed that a simple closed curve in ri which bounds a disk of finite area bounds a disk of least possible area this result was extended by morrey moi in 1948 to a general class of riemannian manifolds the homogeneously regular manifolds a class which includes all closed manifolds work of osserman 0 and gulliver g later showed that least area disks in 3 manifolds were immersed in their interiors more recently there has been a series of new existence results for closed surfaces of least area in manifolds of any dimension it follows from work of sacks and uhlenbeck s ui that if m is closed and 7r2 m is nonzero then there is an essential map of the 2 sphere into m which has least area among all essential maps sacks and uhlenbeck s uii and schoen and yau s y independently showed that if f f m is a map of a closed orientable surface f not the 2 sphere into a closed riemannian manifold m such received by the editors june 23 1986 1980 mathematics subject classification 1985 revision primary 53a10 secondary 57m35 the first author was supported by nsf postdoctoral fellowship dms84 14097 1988 american mathematical society 0002 9947 88 1 00 25 per page

a unified approach to ahp with linking pins abstract recent work demonstrates the need for paired comparison estimates of criteria in the analytic hierarchy process ahp to follow prescribed rules and for the subsequent analysis to be contingent on these rules many different approaches are valid each reflecting a different rule for criteria comparison a generalized formula from which valid approaches may be generated is presented and the connection between these methods and saaty s supermatrix approach is explored

topological censorship from the initial data point of view we introduce a natural generalization of marginally outer trapped surfaces called immersed marginally outer trapped surfaces and prove that three dimensional asymptotically flat initial data sets either contain such surfaces or are diffeomorphic to mathbb r 3 we establish a generalization of the penrose singularity theorem which shows that the presence of an immersed marginally outer trapped surface generically implies the null geodesic incompleteness of any spacetime that satisfies the null energy condition and which admits a non compact cauchy surface taken together these results can be viewed as an initial data version of the gannon lee singularity theorem the first result is a non time symmetric version of a theorem of meeks simon yau which implies that every asymptotically flat riemannian 3 manifold that is not diffeomorphic to mathbb r 3 contains an embedded stable minimal surface we also obtain an initial data version of the spacetime principle of topological censorship under physically natural assumptions a 3 dimensional asymptotically flat initial data set with marginally outer trapped boundary and no immersed marginally outer trapped surfaces in its interior is diffeomorphic to mathbb r 3 minus a finite number of open balls an extension to higher dimensions is also discussed

pad analysis of fuzzy control stability abstract by means of an index represented rule mapping the response g of a fuzzy controller is decomposed into two parts f the classical component and g f the nonlinear component called pad we intend to analyse the pad in order to know what the relation is between fuzzy and classical pid controllers fuzzy controllers mentioned here are defuzzified by the typical center of gravity method in the 1 input case we get that f g 0 which means that the fuzzy controller is exactly the same as the classical p controller whenever the rule mapping is simple definition 1 2 and the linguistic values are a binary family definition 4 1 in the 2 input case of error and error change for simple rule mapping and for simple linguistic values family 4 7 and perfectly for the wxyz linguistic values family definition 6 1 the fuzzy controller can be viewed as a series connection of a classical pd controller and a 1 pad element the describing function of a 1 pad element is very simple and fine and ensures that such fuzzy controller will be stable if the pd controller is the similar consequences are also obtained for the case of three inputs

constructing nonlinear variable gain controllers via the takagi sugeno fuzzy control we investigated the analytical structure of the takagi sugeno ts type of fuzzy controllers which was unavailable in the literature the ts fuzzy controllers we studied employ a new and simplified ts control rule scheme in which all the rule consequent use a common function and are proportional to one another greatly reducing the number of parameters needed in the rules other components of the fuzzy controllers are general arbitrary input fuzzy sets any type of fuzzy logic and the generalized defuzzifier which contains the popular centroid defuzzifier as a special case we proved that all these ts fuzzy controllers were nonlinear variable gain controllers and characteristics of the gain variation were parametrized and governed by the rule proportionality we conducted an in depth analysis on a class of nonlinear variable gain proportional derivative pd controllers we present the results to show 1 how to analyze the characteristics of the variable gains in the context of control 2 why the nonlinear variable gain pd controllers can outperform their linear counterpart and 3 how to generate various gain variation characteristics through the manipulation of the rule proportionality

on the evolution of hypersurfaces by their inverse null mean curvature we introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets that unites the theory of marginally outer trapped surfaces mots with the study of inverse mean curvature flow a theory of weak solutions is developed using level set methods and an appropriate variational principle this new flow has a natural application as a variational type approach to constructing mots and this work also gives new insights into the theory of weak solutions of the inverse mean curvature flow

rank from comparisons and from ratings in the analytic hierarchy network processes rank preservation and reversal are important subjects in multi criteria decision making particularly if a theory uses only one of two ways of creating priorities rating alternatives one at a time with respect to an ideal or standard or comparing them in pairs it is known that our minds can do both naturally and without being tutored when rating alternatives they must be assumed to be independent and rank should be preserved when comparing alternatives they must be assumed to be dependent and rank may not always be preserved however even in making comparisons rank can be preserved if one uses idealization instead of normalization with the original set of alternatives and preserves that ideal from then on unless that ideal itself is deleted for some reason so often it is a matter of judgment as to whether it is desirable to force rank preservation or allow rank to adjust as necessary examples are given to illustrate the foregoing ideas

on the topology of initial data sets with higher genus ends in this note we study the topology of 3 dimensional initial data sets with horizons of a sort associated with asymptotically locally anti de sitter spacetimes we show that within this class those initial data sets that contain no immersed marginally outer trapped surfaces in their interior must have simple topology they are a product of a surface and an interval or a mild variation thereof depending on the connectedness of the horizon and on its genus relative to that of the end the results obtained here extend results in eichmair et al j differ geom 95 389 405 2013 to the case of higher genus ends

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

remarks on the analytic hierarchy process the analytic hierarchy process ahp is flawed as a procedure for ranking alternatives in that the rankings produced by this procedure are arbitrary this paper provides a brief review of several areas of operational difficulty with the ahp and then focuses on the arbitrary rankings that occur when the principle of hierarchic composition is assumed this principle requires that the weights on the higher levels of a hierarchy can be determined independently of the weights on the lower levels virtually all of the published examples of the use of the ahp to evaluate alternatives relative to a set of criteria have assumed this principle the key to correcting this flaw is the synthesis of the ahp with the concepts of multiattribute utility theory

decision making with the ahp why is the principal eigenvector necessary in this paper it is shown that the principal eigenvector is a necessary representation of the priorities derived from a positive reciprocal pairwise comparison judgment matrix a aij when a is a small perturbation of a consistent matrix when providing numerical judgments an individual attempts to estimate sequentially an underlying ratio scale and its equivalent consistent matrix of ratios near consistent matrices are essential because when dealing with intangibles human judgment is of necessity inconsistent and if with new information one is able to improve inconsistency to near consistency then that could improve the validity of the priorities of a decision in addition judgment is much more sensitive and responsive to large rather than to small perturbations and hence once near consistency is attained it becomes uncertain which coefficients should be perturbed by small amounts to transform a near consistent matrix to a consistent one if such perturbations were forced they could be arbitrary and thus distort the validity of the derived priority vector in representing the underlying decision

pid type fuzzy controller and parameters adaptive method abstract the authors of this paper try to analyze the dynamic behavior of the product sum crisp type fuzzy controller revealing that this type of fuzzy controller behaves approximately like a pd controller that may yield steady state error for the control system by relating to the conventional pid control theory we propose a new fuzzy controller structure namely pid type fuzzy controller which retains the characteristics similar to the conventional pid controller in order to improve further the performance of the fuzzy controller we work out a method to tune the parameters of the pid type fuzzy controller on line producing a parameter adaptive fuzzy controller simulation experiments are made to demonstrate the fine performance of these novel fuzzy controller structures

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

infinite boundary value problems for constant mean curvature graphs in h2 r and s2 r we extend the classical existence and uniqueness theory of jenkins serrin h 0 and spruck h 0 for the constant mean curvature equation over a domain in r 2 to domains in h 2 or s 2 this theory allows prescribed boundary data including plus or minus infinity on entire arcs of the boundary necessarily these arc must have curvature 2h or 2h with respect to the domain we give necessary and sufficient conditions for existence in terms of so called admissible polygons the key idea as in previous proofs is to study the flux of monotone increasing and decreasing sequences of solutions

lower bounds on volumes of hyperbolic haken 3 manifolds we prove a volume inequality for 3 manifolds having c o metrics bent along a surface and satisfying certain curvature conditions the result makes use of perelman s work on the ricci flow and geometrization of closed 3 manifolds corollaries include a new proof of a conjecture of bonahon about volumes of convex cores of kleinian groups improved volume estimates for certain haken hyperbolic 3 manifolds and a lower bound on the minimal volume of orientable hyperbolic 3 manifolds an appendix compares estimates of volumes of hyperbolic 3 manifolds drilled along a closed embedded geodesic with experimental data

analytical structure of the typical fuzzy controllers employing trapezoidal input fuzzy sets and nonlinear control rules fuzzy controllers are generally nonlinear controllers and as such it is important toreveal analytical structure of fuzzy controllers relative to conventional nonlinear controllers so that the well developed nonlinear control theory can be utilized to effectively analyze and design fuzzy control systems in our previous papers we have shown that the analytical structure of the fuzzy controllers using triangular input fuzzy sets and linear fuzzy control rules is the sum of a global two dimensional multilevel relay and a local nonlinear pi controller in this paper we present the analytical structure of much more general and typical fuzzy controllers the ones that employ trapezoidal input fuzzy sets and arbitrary nonlinear control rules we analytically prove that the structure of such controllers is the sum of a global nonlinear controller and a local nonlinear pi like controller whose proportional gain and integral gain vary both globally and locally with change of the controllers inputs this result significantly generalizes our previous ones and contains them only as special cases analytical results are presented which reveal the characteristics and properties of the fuzzy controllers we show that the global nonlinear controller is characterized solely by fuzzy control rules that is different nonlinear control rules result in different types of global nonlinear controllers when the number of input fuzzy sets approaches oo the global nonlinear controller approaches a control rules dependent nonlinear controller whereas the local nonlinear pi like controller disappears the results presented in this paper can directly be used to generate analytical structure of any specific fuzzy controllers and we demonstrate how by illustrative examples

a nonlinear fuzzy controller with linear control rules is the sum of a global two dimensional multilevel relay and a local nonlinear proportional integral controller abstract the author analytically proves that a nonlinear fuzzy controller with linear control rules and n members for input fuzzy sets is the sum of a global two dimensional multilevel relay and a local nonlinear proportional integral pi controller which adjusts the control action generated by the global multilevel relay as n increases the resolution of the global multilevel relay is enhanced but the role of the local nonlinear pi controller in total control action is decreased as n approaches the global multilevel relay approaches a regular linear pi controller while the control action from the local nonlinear pi controller approaches zero the role of the global multilevel relay and the local nonlinear pi controller in total control action is quantitatively described as is the degree of nonlinearity of the fuzzy controllers with different n

studies on the output of fuzzy controller with multiple inputs abstract in this paper we discuss the output of a fuzzy controller in which an arbitrary number of fuzzy control inputs and an arbitrary number of fuzzy sets employed to fuzzify every linguistic variable are considered in the fuzzy control rules a rule base with a rule mapping concept is considered a defuzzifier with the general t norm and t conorm operators is formulated a triangular family is adopted so that the defuzzifier of the complete rule mapping is represented by a formula consisting of only some rules using a specific logic pair to evaluate values we derive that the output of the fuzzy controller can be simply represented by a linear parametric function of the inputs to the fuzzy controller

regular paper the takagi sugeno fuzzy controllers using the simplified linear control rules are nonlinear variable gain controllers takagi sugeno ts for short fuzzy controllers have been used and treated as black box controllers and there exists no explicit structure of any ts fuzzy controller in the literature in this paper we investigate the analytical structure of the ts fuzzy controllers that use our newly introduced simplified ts control rule scheme our scheme requires that all the rule consequent employ a common linear function and be proportional to one another the scheme drastically reduces the number of adjustable parameters required by the original ts rule scheme product fuzzy logic and was employed in the rule antecedents we theoretically proved that the ts controllers using the simplified rule scheme were nonlinear variable gain controllers the proportionality of the rule consequent parameterized the gain variation characteristics as special cases the ts controllers with three or two input variables and the simplified linear rule consequent were nonlinear variable gain pid pi or pd controllers as an illustration we mathematically analyzed the nonlinear variable gain pi controllers and revealed that the gain variation empowered the controllers to outperform their linear counterparts stability of the nonlinear pi control systems was also addressed a practical example is given to show how to achieve the desired characteristics of the gain variation by properly designing the simplified rules

stability analysis and stabilization of fuzzy state space models abstract a new stability test method for linguistic fuzzy state space models is proposed in this paper this method allows the determination of stability of fuzzy models by applying the lyapunov function to each rule a gradient based algorithm is suggested to simplify the stability tests using this algorithm the lyapunov equation can be updated towards the direction where the stability can be determined more correctly an additional gradient based stabilizing algorithm is also derived which allows a systematic approach to find stabilizing controls by updating the lyapunov equations towards the direction where unstable eigenvalues are decreased examples demonstrate the convenience of the stability test and the two suggested gradient based algorithms

jenkins serrin type results for the jang equation let m g k be an initial data set for the einstein equations of general relativity we prove that there exist solutions of the plateau problem for marginally outer trapped surfaces motss that are stable in the sense of motss this answers a question of g galloway and n o murchadha raised in some remarks on the size of bodies and black holes classical quantum gravity 25 2008 no 10 105009 9 mr 2416045 and is an ingredient in the proof of the spacetime positive mass theorem given by l h huang d lee r schoen and the first named author we show that a canonical solution of the jang equation exists in the complement of the union of all weakly future outer trapped regions in the initial data set with respect to a given end provided that this complement contains no weakly past outer trapped regions the graph of this solution relates the area of the horizon to the global geometry of the initial data set in a non trivial way we prove the existence of a scherk type solution of the jang equation outside the union of all weakly future or past outer trapped regions in the initial data set this result is a natural exterior analogue for the jang equation of the classical jenkins serrin theory we extend and complement existence theorems for scherk type constant mean curvature graphs over polygonal domains in m g where m g is a complete riemannian surface we can dispense with the a priori assumptions that a sub solution exists and that m g has particular symmetries also our method generalizes to higher dimensions

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

the existence of embedded minimal surfaces and the problem of uniqueness it was in the early nineteen thirties that douglas and radd solved the plateau problem for a rectifiable jordan curve in euclidean space the minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk in 1948 morrey generalized the theorem of douglas and radd to minimal surfaces in a homogeneously regular riemannian manifold in both cases the solutions are defective as they may possess branch points it was not until 1968 that osserman 24 was able to prove that interior branch points do not exist on the douglas radd morrey solution osserman s theorem in this full generality was proved by alt 2 3 and gulliver 8 if the jordan curve is real analytic gulliver and lesley 9 showed that the solution surface is free of branch points even on its boundary hence at least in this case we know that the douglas radd morrey solution is an immersion and represents a classical surface however the douglas radd morrey solution may in general have selfintersections it was then a classical problem to find conditions on the jordan curve to guarantee that the solution is an embedded disk in 1932 radd was the first to produce a general condition of this type he proved that if the curve can be projected onto a convex curve in the plane then it bounds a unique embedded minimal disk which is a graph over this plane it was not until the seventies that there was further progress on the problem of embeddedness osserman had conjectured that if a jordan curve is extreme that is lies on the boundary of its convex hull then the douglas rad6 solution is embedded gulliver and spruck 10 proved this conjecture under the additional assumption that the absolute total curvature of the jordan curve is less than 4 z right after this work tomi and tromba 32 showed that an extreme curve must bound an embedded minimal disk which need not be stable in 1976 almgren and simon 1 established the existence for such a curve of an embedded stable minimal disk which need not be a douglas radd solution finally in the same year the authors 18 proved that any douglas rad6 solution must be embedded

continuity estimates for solutions to the prescribed curvature dirichlet problem on etudie des solutions au probleme de dirichlet pour l equation de courbure moyenne prescrite ainsi que pour des equations de weingarten plus generales on etablit des modules d estimation de continuite dependant seulement des valeurs limites

isometric embeddings into the minkowski space and new quasi local mass the definition of quasi local mass for a bounded space like region in space time is essential in several major unsettled problems in general relativity the quasi local mass is expected to be a type of flux integral on the boundary two surface sigma partial omega and should be independent of whichever space like region sigma bounds an important idea which is related to the hamiltonian formulation of general relativity is to consider a reference surface in a flat ambient space with the same first fundamental form and derive the quasi local mass from the difference of the extrinsic geometries this approach has been taken by brown york 4 5 and liu yau 16 17 see also related works 3 6 9 12 14 15 28 32 to define such notions using the isometric embedding theorem into the euclidean three space however there exist surfaces in the minkowski space whose quasilocal mass is strictly positive 19 it appears that the momentum information needs to be accounted for to reconcile the difference in order to fully capture this information we use isometric embeddings into the minkowski space as references in this article we first prove an existence and uniqueness theorem for such isometric embeddings we then solve the boundary value problem for jang s 13 equation as a procedure to recognize such a surface in the minkowski space in doing so we discover a new expression of quasi local mass for a large class of admissible surfaces see theorem a and remark 1 1 the new mass is positive when the ambient space time satisfies the dominant energy condition and vanishes on surfaces in the minkowski space it also has the nice asymptotic behavior at spatial infinity and null infinity some of these results were announced in 29

the plateau problem for marginally outer trapped surfaces we solve the plateau problem for marginally outer trapped surfaces in general cauchy data sets we employ the perron method and tools from geometric measure theory to force and control a blow up of jang s equation substantial new geometric insights regarding the lower order properties of marginally outer trapped surfaces are gained in the process the techniques developed in this paper are flexible and can be adapted to other non variational existence problems

theory of the fuzzy controller abstract we first introduce notation in an attempt to standardize the description of a basic fuzzy controller if r 1 r 2 r n are the real number inputs to the fuzzy controller and is its defuzzified output then f r 1 r n for some function f by the theory of the fuzzy controller we mean in this paper the study of the structure of f for elementary fuzzy controllers using this standardized notation we first survey the known results on the theory of the fuzzy controller and then discuss important future research topics in this area

on the geometry and topology of initial data sets with horizons we study the relationship between initial data sets with horizons and the existence of metrics of positive scalar cur vature we dene a cauchy domain of outer communications cdoc to be an asymptotically at initial set m g k such that the boundary m of m is a collection of marginally outer or inner trapped surfaces motss and or mitss and such that mn m contains no motss or mitss this denition is meant

on the relativity of relative measures accommodating both rank preservation and rank reversals in the ahp abstract the analytic hierarchy process ahp has been criticized for allowing the introduction of new alternatives to cause rank reversals among existing alternatives while in many cases this is a perfectly valid phenomenon there are also many cases where rank should be preserved to address both contingencies the ahp now supports two types of synthesis procedures one that preserves rank and one that does not this paper proposes guidelines for the selection of one synthesis mode over the other it also proposes several modifications to software implementations of the ahp to better support both synthesis modes and to reduce user stress over the choice

blowup of jang s equation at outermost marginally trapped surfaces the aim of this paper is to accurately describe the blowup of jang s equation first we discuss how to construct solutions that blow up at an outermost mots second we exclude the possibility that there are extra blowup surfaces in data sets with non positive mean curvature then we investigate the rate of convergence of the blowup to a cylinder near a strictly stable mots and show exponential convergence with an identifiable rate near a strictly stable mots

group decision making based on multiplicative consistent reciprocal preference relations the main aim of this paper is to investigate the consistency and consensus of multiplicative consistent reciprocal preference relations based on the multiplicative consistency property a straightforward method to obtain the priority vector for multiplicative consistent reciprocal preference relation is introduced together with some of its desired properties in addition some interesting properties on consistency and consensus for reciprocal preference relations are further investigated a consensus optimization model for group decision making with reciprocal preference relations is proposed which enables the decision makers to reach consensus with the highest overall consensus level numerical examples are provided to make a discussion and comparison with some existing similar methods

a linear output structure for fuzzy logic controllers in this paper the output structure of multiple input single output fuzzy logic controller flcr is studied the input and output variables of the flcr are all characterized by normal triangular shaped membership functions and fuzzy partitions are considered for corresponding universes of discourse a special mapping linear rule mapping fuzzy sets and systems 57 1993 149 which describes the relationship between the input variables and the output variables of one rule is used the conclusion proposed is that the output of multiple input single output flcr can be represented by the convex linear combination of its crisp input variables a precise mathematical model of flcr which is helpful to analyze fuzzy control and tune the fuzzy logic controller is given

the jang equation reduction of the spacetime positive energy theorem in dimensions less than eight we extend the jang equation proof of the positive energy theorem due to schoen and yau commun math phys 79 2 231 260 1981 from dimension n 3 to dimensions 3 n 8 this requires us to address several technical difficulties that are not present when n 3 the regularity and decay assumptions for the initial data sets to which our argument applies are weaker than those in schoen and yau commun math phys 79 2 231 260 1981

a self generating method for fuzzy system design abstract in this paper a self generating method based on genetic algorithms is proposed to automatically construct a fuzzy system a parameter structure of fuzzy system is presented along with a fuzzy set elimination method which enables the self generating method not only to determine an appropriate fuzzy system with a high performance but also to favor the selected system with fewer fuzzy rules an inverted pendulum control and a system modeling problem are utilized respective to illustrate the efficiency of the proposed method in the fuzzy system design

geometric inequalities for quasi local masses in this paper lower bounds are obtained for quasi local masses in terms of charge angular momentum and horizon area in particular we treat three quasi local masses based on a hamiltonian approach namely the brown york liu yau and wang yau masses the geometric inequalities are motivated by analogous results for the adm mass they may be interpreted as localized versions of these inequalities and are also closely tied to the conjectured bekenstein bounds for entropy of macroscopic bodies in addition we give a new proof of the positivity property for the wang yau mass which is used to remove the spin condition in higher dimensions furthermore we generalize a recent result of lu and miao to obtain a localized version of the penrose inequality for the static wang yau mass

least area incompressible surfaces in 3 manifolds let m be a riemannian manifold and let f be a closed surface a map f f m is called least area if the area of f is less than the area of any homotopic map from f to m note that least area maps are always minimal surfaces but that in general minimal surfaces are not least area as they represent only local stationary points for the area function the existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group 18 17 in this paper we shall consider the possible singularities of such immersions our results show that the general philosophy is that least area surfaces intersect least meaning that the intersections and self intersections of least area immersions are as small as their homotopy classes allow when measured correctly one should note that evidence supporting this view had been found by meeks yau in their embedding theorems for minimal disks and 2 spheres 13 143 our main result asserts that if a least area immersion is homotopic to an embedding then it has no self intersections which clearly exemplifies the above philosophy the precise result is the following

curvature estimates for stable marginally trapped surfaces we derive local integral and sup estimates for the curvature of stable marginally outer trapped surfaces in a sliced space time the estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature of a slice containing the surface these estimates are well adapted to situations of physical interest such as dynamical horizons