a cosine maximization method for the priority vector derivation in ahp abstract the derivation of a priority vector from a pair wise comparison matrix pcm is an important issue in the analytic hierarchy process ahp the existing methods for the priority vector derivation from pcm include eigenvector method ev weighted least squares method wls additive normalization method an logarithmic least squares method lls etc the derived priority vector should be as similar to each column vector of the pcm as possible if a pair wise comparison matrix pcm is not perfectly consistent therefore a cosine maximization method cm based on similarity measure is proposed which maximizes the sum of the cosine of the angle between the priority vector and each column vector of a pcm an optimization model for the cm is proposed to derive the reliable priority vector using three numerical examples the cm is compared with the other prioritization methods based on two performance evaluation criteria euclidean distance and minimum violation the results show that the cm is flexible and efficient

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

polynomially riesz perturbations in this paper we investigate perturbations by holomorphically and polynomially riesz operators concerning the sets of upper and lower semi fredholm operators the sets of left and right fredholm operators as well as the sets of upper and lower semi browder operators and the sets of left and right browder operators we consider in particular perturbations of shifts by polynomially riesz operators 1 introduction and preliminaries let c denote the set of all complex numbers and let x be an infinite dimensional banach space we denote by b x the set of all linear bounded operators on x for a2 b x let r a and n a denote the range and the null space of a respectively and let a dimn a and a dimx r a codim r a the sets of upper and lower semi fredholm operators respectively are defined as x fa2 b x a 1 and r a is closedg and x fa 2 b x a 1g set x x x for the set of semi fredholm operators and x x x for the set of fredholm operators for a2 x the index is defined by i a a a the calkin algebra over x is the quotient algebra c x b x k x and let b x c x denote the natural homomorphism it is well known that a2 x if and only if a is invertible in c x 5 theorem 3 2 8 an operator a2 b x is left fredholm if a 1 and r a is a closed and complemented subspace of x while a is right fredholm if a 1 and n a is a complemented subspace of x set l x for the set of all left fredholm operators on x and r x for the set of all right fredholm operators on x for a 2 b x denote by asc a dsc a the ascent the descent of a 2 b x i e the smallest non negative integer n such that n a n n a n 1 r a n r a n 1 if such n does not exist then asc a 1 dsc a 1 an operator a2 b x is upper semi browder if it is upper semi fredholm with finite ascent while a is lower semi browder if it is lower semi fredholm with finite descent letb x b x denote the set of all upper lower semi browder operators we shall say that an operator a2 b x is left browder if it is left fredholm with finite ascent while a is right browder if it is right fredholm with finite descent letbl x br x denote the set of all left right browder operators an operator a2 b x is browder if it is fredholm with finite ascent and finite descent the set of all browder operators on x is denoted byb x clearly b x b x b x bl x br x

the singular set of mean curvature flow with generic singularities a mean curvature flow starting from a closed embedded hypersurface in mathbf r n 1 must develop singularities we show that if the flow has only generic singularities then the space time singular set is contained in finitely many compact embedded n 1 dimensional lipschitz submanifolds plus a set of dimension at most n 2 if the initial hypersurface is mean convex then all singularities are generic and the results apply in mathbf r 3 and mathbf r 4 we show that for almost all times the evolving hypersurface is completely smooth and any connected component of the singular set is entirely contained in a time slice for 2 or 3 convex hypersurfaces in all dimensions the same arguments lead to the same conclusion the flow is completely smooth at almost all times and connected components of the singular set are contained in time slices a key technical point is a strong parabolic reifenberg property that we show in all dimensions and for all flows with only generic singularities we also show that the entire flow clears out very rapidly after a generic singularity these results are essentially optimal

a semiring based study of judgment matrices properties and models in decision making and group decision making multiplicative reciprocal judgment matrices and additive reciprocal judgment matrices are used as two kinds of important preference information in this paper semirings are applied to the discussion of judgment matrix properties first two special semirings are constructed second the properties of the consistent judgment matrices are given as a set of equations all in the semiring sense which include idempotency equations and fixed point equations we find that there exists one and only one specially constrained fixed point as the priority vector of a consistent judgment matrix third optimization models for inconsistent judgment matrices are presented finally we offer some simple illustrative examples

group decision making with a fuzzy linguistic majority abstract the determination of solutions in group decision making is considered the point of departure is a collection of individual fuzzy preference relations a solution is derived either directly from the individual fuzzy preference relations or by constructing first a social fuzzy preference relation as opposed to conventional approaches in which a crisp threshold type majority rule is used we employ a fuzzy majority rule specified by a fuzzy linguistic quantifier e g most much more than 50 etc a calculus of linguistically quantified propositions is applied using the fuzzy majority various solution concepts are derived mainly of the type of core minimax opposition set and consensus winner

partial regularity at the first singular time for hypersurfaces evolving by mean curvature in this paper we consider smooth properly immersed hypersurfaces evolving by mean curvature in some open subset of mathbb r n 1 on a time interval 0 t 0 we prove that p integrability with p ge 2 for the second fundamental form of these hypersurfaces in some space time region b r y times 0 t 0 implies that the mathcal h n 2 p measure of the first singular set vanishes inside b r y for p 2 and n 2 this was established by han and sun our result furthermore generalizes previous work of xu ye and zhao and of le and sesum for p ge n 2 in which case the singular set was shown to be empty by a theorem of ilmanen our integrability condition is satisfied for p 2 and n 2 if the initial surface has finite genus thus the first singular set has zero mathcal h 2 measure in this case this is the conclusion of brakke s main regularity theorem for the special case of surfaces but derived without having to impose the area continuity and unit density hypothesis it follows from recent work of head and of huisken and sinestrari that for the flow of closed k convex hypersurfaces that is hypersurfaces whose sum of the smallest k principal curvatures is positive our integrability criterion holds with exponent p n 3 k alpha for all small alpha 0 as long as 1 le k le n 1 therefore the first singular set of such solutions is at most k 1 dimensional which is an optimal estimate in view of some explicit examples

aggregation of monotone reciprocal relations with application to group decision making this paper is composed of two complementary parts the first part is a formal investigation into the interplay of properties of reciprocal relations how monotonicity relates to some natural and intuitive properties including stochastic transitivity the goal is to aggregate monotone reciprocal relations on a given set of alternatives monotonicity is expressed w r t a linear order on the set of alternatives the second part is a practical protocol to both determine the best fitting linear order underlying the alternatives and construct a reciprocal relation monotone w r t it we formulate the problem as an optimization problem where the aggregated linear order is that for which the implied stochastic monotonicity conditions are closest to being satisfied by the distribution of the input monotone reciprocal relations we show that if stochastic monotonicity conditions are satisfied a monotone reciprocal relation is easily found on the basis of the possibly constructed stochastically monotone reciprocal distributional relation

fuzzy sets and decision analysis abstract this survey points out recent advances in multiple attribute decision making methods dealing with fuzzy or ill defined information fuzzy maut as well as fuzzy outranking methods are reviewed aggregation procedures choice problems and treatment of interactive attributes are covered trends in research and open problems are indicated

the geometric consistency index approximated thresholds abstract crawford and williams journal of mathematical psychology 29 1985 387 suggested for the row geometric mean method rgmm one of the most extended ahp s priorization procedure a measure of the inconsistency based on stochastic properties of a subjacent model in this paper we formalize this inconsistency measure hereafter called the geometric consistency index gci and provide the thresholds associated with it these thresholds allow us an interpretation of the inconsistency tolerance level analogous to that proposed by saaty multicriteria decision making the analytic hierarchy process new york 1980 for the consistency ratio cr used with the right eigenvector method in conventional ahp

topological change in mean convex mean curvature flow consider the mean curvature flow of an n 1 dimensional compact mean convex region in euclidean space or if n 7 in a riemannian manifold we prove that elements of the mth homotopy group of the complementary region can die only if there is a shrinking sk rn k singularity for some k m we also prove that for each m with 1 m n there is a nonempty open set of compact mean convex regions k in rn 1 with smooth boundary k for which the resulting mean curvature flow has a shrinking sm rn m singularity

the extension problem of the mean curvature flow i we show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in mathbb r 3

uniqueness of convex ancient solutions to mean curvature flow in mathbb r 3 r 3 a well known question of perelman concerns the classification of noncompact ancient solutions to the ricci flow in dimension 3 which have positive sectional curvature and are kappa noncollapsed in this paper we solve the analogous problem for mean curvature flow in mathbb r 3 and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in mathbb r 3 which is strictly convex and noncollapsed

a compendium of fuzzy weak orders representations and constructions the present paper gives a state of the art overview of representation and construction results for fuzzy weak orders we do not assume that the underlying domain is finite instead we concentrate on results that hold in the most general case that the underlying domain is possibly infinite this paper presents three fundamental representation results each of which also provides a construction method i score function based representations ii inclusion based representations iii representations by decomposition into crisp linear orders and fuzzy equivalence relations which also facilitates a pseudo metric based construction

convex solutions to the mean curvature flow in this paper we study the classication of ancient convex solutions to the mean curvature ow in r n 1 an open problem related to the classication of type ii singularities is whether a convex translating solution is k rotationally symmetric for some integer 2 k n namely whether its level set is a sphere or cylinder s k 1 r n k in this paper we give an afrmative answer for entire solutions in dimension 2 in high dimensions we prove that there exist nonrotationally symmetric entire convex translating solutions but the blow down in space of any entire convex translating solution is k rotationally symmetric we also prove that the blow down in space time of an ancient convex solution which sweeps the whole space r n 1 is a shrinking sphere or cylinder

stability of mean convex cones under mean curvature flow we consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature if another solution is initially close to the cone at infinity then the difference to the homothetically expanding solution becomes small for large times the proof involves the construction of appropriate barriers

computing random consistency indices and assessing priority vectors reliability abstract the paper deals with two crucial steps in multi criteria decision analysis that are consistency of the judgments and priority vectors for alternatives criteria from a side several consistency indices are proposed for measuring the consistency of a pairwise comparison matrix from another one conditions weaker than consistency such as transitivity and weak consistency are proposed for representing further levels of coherence of a decision maker when he she expresses his her preferences by means of reciprocal pairwise comparison matrix firstly in this paper a simulation is performed in order to establish a relation between random consistency index and coherence level then since weak consistency ensures reliability to priority vectors proposed in literature a second simulation is performed in order to measure in case of no weak consistency the reliability of these priority vectors

singularities of mean convex level set flow in general ambient manifolds abstract we prove two new estimates for the level set flow of mean convex domains in riemannian manifolds our estimates give control exponential in time for the infimum of the mean curvature and the ratio between the norm of the second fundamental form and the mean curvature in particular the estimates remove a stumbling block that has been left after the work of white 16 17 20 and haslhofer kleiner 9 and thus allow us to extend the structure theory for mean convex level set flow to general ambient manifolds of arbitrary dimension

the size of the singular set in mean curvature flow of mean convex sets in this paper we study the singularities that form when a hypersurface of positive mean curvature moves with a velocity that is equal at each point to the mean curvature of the surface at that point it is most convenient to describe the results in terms of the level set flow also called biggest flow i2 of chen giga goto cgg and evans spruck es under the level set flow any closed set k in r generates a one parameter family of closed sets ft k t 0 with f0 k k if the boundary of k is a smooth compact hypersurface then so is the boundary of ft k for t in some interval 0 and for such t s the evolution coincides with motion by mean curvature as defined classically by partial differential equations and differential geometry as in h1 however if k is compact then the boundary of ft k will necessarily become singular at some finite time our goal is to show that the singular sets are necessarily quite small this we do provided the initial set k is compact and mean convex in the sense that ft k interior k for all t 0 in case m k is a smooth hypersurface we have the following equivalent characterization of mean convexity k is mean convex if and only if the mean curvature of m with respect to the inward unit normal is everywhere non negative given a compact set k let k be the region in spacetime swept out by the ft k k x t r r t 0 x ft k a point x x t in the boundary of k is called a regular point if 1 x has a neighborhood in which k is a smooth manifold with boundary and 2 the tangent plane to k at x is not horizontal i e is not r 0 note that if x x t is a regular point then in a neighborhood of x ft k is a smooth manifold with boundary in r a point x x t in k with t 0 that is not a regular point is called a singular point

building consensus in group decision making with an allocation of information granularity abstract consensus is defined as a cooperative process in which a group of decision makers develops and agrees to support a decision in the best interest of the whole it is a questioning process more than an affirming process in which the group members usually modify their choices until a high level of agreement within the group is achieved given the importance of forming an accepted decision by the entire group the consensus problem has attained a great attention as it is a major goal in group decision making in this study we propose the concept of the information granularity being regarded as an important and useful asset supporting the goal to reach consensus in group decision making by using fuzzy preference relations to represent the opinions of the decision makers we develop a concept of a granular fuzzy preference relation where each pairwise comparison is formed as a certain information granule say an interval fuzzy set rough set and alike instead of a single numeric value as being more abstract the granular format of the preference model offers the required flexibility to increase the level of agreement within the group using the fact that we select the most suitable numeric representative of the fuzzy preference relation

incomplete pairwise comparison and consistency optimization this paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem the matrix is completed by minimizing a measure of global inconsistency thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements the optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions some other methods proposed in the literature are discussed and a numerical example is presented

fuzzy rationality measures fuzzy preference relations formalize intensity of individual preferences over fixed sets of alternatives it is therefore natural to extend to fuzzy preferences the notion of rationality or consistency with this goal in mind we address in this paper the problem of giving an axiomatic basis for defining the concept of fuzzy rationality specifically we establish a collection of conditions that any fuzzy rationality measure must satisfy some examples of fuzzy rationality measures are then given and analyzed

erratum to partial regularity at the first singular time for hypersurfaces evolving by mean curvature the purpose of this note is to inform the reader of a major error in the original article namely in the proof of theorem 1 8 i would like to point out that under the additional assumption of embeddedness of the solution the dimension estimates for k convex hypersurfaces in corollary 1 7 have been established by white 2 for k n and in 1 for all 1 k n i therefore believe that this statement is also correct in the case of immersed solutions but the proof may require different techniques to ours i am also confident that the statement of theorem 1 1 is correct for immersed hypersurfaces the error in the proof of theorem 1 8 can be found in inequality 11 on page 13 of our paper the centre of the ball should be y j j rather than y j the resulting ball may not contain the unit ball around the origin therefore the validity of inequality 13 and all subsequent statements in the proof are unclear since the proof of theorem 1 1 uses theorem 1 8 the validity of theorem 1 1 is also unclear the same holds for corollary 1 2 remark 1 3 as well as corollaries 1 4 and 1 5 the curvature integral estimate for k convex hypersurfaces with 1 k n 1 is however correct corollary 1 7 relies on corollary 1 2 though so its validity is again not clear i thank my colleague dr theodora bourni for pointing out this error to me

an index formula for the product of linear relations abstract let x y and z be linear spaces and let a and b be linear relations from x to y and from y to z respectively the main result of this note is a formula which relates the nullities and the defects of the relations a and b with those of the product relation ba

a rational consensus model in group decision making using linguistic assessments abstract a new consensus model for the consensus reaching process in a linguistic framework is presented in heterogeneous group decision making problems called rational consensus model it is guided by some linguistic consensus and linguistic consistency meaures all the measures are calculated from a set of linguistic preference relations used to provide experts opinions this consensus model allows more rational consensus solutions to be obtained and thus more human consistency to be incorporated in decision support systems

group decision making using bilateral agreement matrix abstract group decision making based on a bilateral agreement matrix is investigated in this paper to derive a group decision from the opinions of individual decision makers pairs of decision makers are allowed to negotiate with one another and provide their bilateral negotiation values two algorithms are provided to help pairs of decision makers reach bilateral agreements which enables the construction of the bilateral agreement matrix the definition of a consistent bilateral agreement matrix is introduced to measure the credibility of the final result and its properties are investigated based on which a method is provided to construct a consistent bilateral agreement matrix from an inconsistent one models are then given to derive the weight vector from the bilateral agreement matrix based on which a consistency index of the matrix is defined two algorithms are then given to improve the level of consistency of the bilateral agreement matrix the final group decision is obtained by aggregating individual opinions using the weight vector the proposed methods are further extended to deal with the problem of incomplete bilateral agreement matrices several examples are also given to illustrate the proposed approach

a note on the proportionality between some consistency indices in the ahp analyzing the consistency of preferences is an important step in decision making with pairwise comparison matrices and several indices have been proposed in order to estimate it in this paper we prove the proportionality between some consistency indices in the framework of the analytic hierarchy process knowing such equivalences eliminates redundancy in the consideration of evidence for consistent preferences

inscribed radius estimates for inverse curvature flow in sphere and hyperbolic space abstract in this paper we study a fully nonlinear inverse curvature flow in sphere and hyperbolic space and prove a non collapsing property for this flow using maximum principle precisely when ambient space is sphere we show that upon some conditions on speed function the radius of the largest touching interior ball is bounded below by a multiple of the reciprocal of the speed where ambient space is hyperbolic we obtain an upper bound of the curvature of the largest touching interior ball for star shaped inverse mean curvature flow

embedded minimal tori in s3 and the lawson conjecture we show that any embedded minimal torus in s 3 is congruent to the clifford torus this answers a question posed by h b lawson jr in 1970

regularity of the level set flow we showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative and satisfies the equation classically we show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closed c1 manifold with cylindrical singularities 2017 wiley periodicals inc

ascent descent nullity defect and related notions for linear relations in linear spaces for a linear relation in a linear space the concepts of ascent descent nullity and defect are introduced and studied it is shown that the results of a e taylor and m a kaashoek concerning the relationship between ascent descent nullity and defect for the case of linear operators remain valid in the context of linear relations sometimes under the additional condition that the linear relation does not have any nontrivial singular chains in particular it is shown for a linear relation a with a trivial singular chain manifold whose ascent p is finite and whose nullity and defect are equal and finite that the linear space h is a direct sum of ker a p and ran a p furthermore it is shown that the various results which require the absence of singular chains are not valid when such chains are present c 2007 elsevier inc all rights reserved

on essentially semi regular linear relations abstract the characterization of bounded essentially semi regular operators in terms of kato decomposition of finite type was studied by several authors in this paper we extend this characterization to the case of essentially semi regular linear relations we also give other characterization of such linear relations further we apply the obtained results to analyse the stability of the class of essentially semi regular linear relations under additional operator perturbations finally as an application we get some useful connections between the fredholm spectrum and the essentially semi regular spectrum of linear relations

a further discussion of a semiring based study of judgment matrices properties and models information sciences 181 2011 2166 2176 abstract in the paper a semiring based study of judgment matrices properties and models information sciences 181 2011 2166 2176 the semiring is applied to the discussion of judgment matrix properties this paper shows that the suggested semirings could be replaced by a more convenient algebraic structure

on the cycle transitivity of the mutual rank probability relation of a poset the mutual rank probability relation associated with a finite poset is a reciprocal relation expressing the probability that a given element succeeds another one in a random linear extension of that poset we contribute to the characterization of the transitivity of this mutual rank probability relation also known as proportional probabilistic transitivity by situating it between strong stochastic transitivity and moderate product transitivity the methodology used draws upon the cycle transitivity framework which is tailor made for describing the transitivity of reciprocal relations

singularity of mean curvature flow of lagrangian submanifolds in this article we study the tangent cones at first time singularity of a lagrangian mean curvature flow if the initial compact submanifold 0 is lagrangian and almost calibrated by re in a calabi yau n fold m and t 0 is the first blow up time of the mean curvature flow then the tangent cone of the mean curvature flow at a singular point x 0 t is a stationary lagrangian integer multiplicity current in r 2n with volume density greater than one at x 0 when n 2 the tangent cone is a finite union of at least two 2 planes in r 4 which are complex in a complex structure on r 4

minimal 2 spheres in 3 spheres we prove that any manifold diffeomorphic to s3 and endowed with a generic metric contains at least two embedded minimal 2 spheres the existence of at least one minimal 2 sphere was obtained by simon and smith in 1983 our approach combines ideas from min max theory and mean curvature flow we also establish the existence of smooth mean convex foliations in 3 manifolds we apply our methods to solve a problem posed by s t yau in 1987 on whether the planar 2 spheres are the only minimal spheres in ellipsoids centered about the origin in r4 finally considering the example of degenerating ellipsoids we show that the assumptions in the multiplicity 1 conjecture and the equidistribution of widths conjecture are in a certain sense sharp

granular representation and granular computing with fuzzy sets in this study we introduce a concept of a granular representation of numeric membership functions of fuzzy sets which offers a synthetic and qualitative view at fuzzy sets and their ensuing processing the notion of consistency of the granular representation is formed which helps regard the problem as a certain optimization task more specifically the consistency is referred to a certain operation f which gives rise to the concept of f consistency likewise introduced is a concept of granular consistency with regard to a collection of several operations given the essential role played by logic operators in computing with fuzzy sets detailed investigations include and and or consistency as well as and or consistency of granular representations of membership functions with the logic operators implemented in the form of various t norms and t conorms the optimization framework supporting the realization of the f consistent optimization process is provided through particle swarm optimization further conceptual and representation issues impacted processing fuzzy sets are discussed as well

a statistical comparative study of different similarity measures of consensus in group decision making an essential aim in group decision making gdm problems is to achieve a high level of consensus among experts consensus is defined as general or widespread agreement and it is usually modelled mathematically via a similarity function measuring how close experts opinions or preferences are similarity functions are defined based on the use of a metric describing the distance between experts opinions or preferences in the literature different metrics or distance functions have been proposed to implement in consensus models but no study has been carried out to analyse the influence the use of different distance functions can have in the gdm process this paper presents a comparative study of the effect of the application of some different distance functions for measuring consensus in gdm by using the nonparametric wilcoxon matched pairs signed ranks test it is concluded that different distance functions can produce significantly different results moreover it is also shown that their application also has a significant effect on the speed of achieving consensus finally these results are analysed and used to derive decision support rules based on a convergent criterion that can be used to control the convergence speed of the consensus process using the compared distance functions

sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature in this paper we consider complete noncompact riemannian manifolds m g with nonnegative ricci curvature and euclidean volume growth of dimension n ge 3 for every bounded open subset omega subset m with smooth boundary we prove that begin aligned int limits partial omega left frac mathrm h n 1 right n 1 mathrm d sigma ge mathrm avr g big mathbb s n 1 big end aligned where mathrm h is the mean curvature of partial omega and mathrm avr g is the asymptotic volume ratio of m g moreover the equality holds true if and only if m setminus omega g is isometric to a truncated cone over partial omega an optimal version of huisken s isoperimetric inequality for 3 manifolds is obtained using this result finally exploiting a natural extension of our techniques to the case of parabolic manifolds we also deduce an enhanced version of kasue s non existence result for closed minimal hypersurfaces in manifolds with nonnegative ricci curvature

the nature of singularities in mean curvature flow of mean convex sets let k be a compact subset of r or more generally of an n 1 dimensional riemannian manifold we suppose that k is mean convex if the boundary of k is smooth and connected this means that the mean curvature of k is everywhere nonnegative with respect to the inward unit normal and is not identically 0 more generally it means that ft k is contained in the interior of k for t 0 where ft k is the set obtained by letting k evolve for time t under the level set mean curvature flow as k evolves it traces out a closed set k of spacetime k x t r r x ft k also there is associated to k a brakke flow m t 7 mt of rectifiable varifolds we call the pair m k a mean convex flow let x x t be a point in spacetime with t 0 suppose xi ti is a sequence of points converging to x and i is a sequence of positive numbers tending to infinity translate the pair m and k in spacetime by y 7 y xi ti and then dilate parabolically by y 7 iy i to get new flows mi and ki the sequence mi ki is called a blow up sequence at x general compactness theorems guarantee that this sequence will have subsequential limits a subsequential limit m k is called a limit flow here m t 7 m t

the harmonic consistency index for the analytic hierarchy process a new consistency measure the harmonic consistency index is obtained for any positive reciprocal matrix in the analytic hierarchy process we show how this index varies with changes in any matrix element a tight upper bound is provided for this new consistency measure when the entries of matrix are at most 9 as is often recommended using simulation the harmonic consistency index is shown to give numerical values similar to the standard consistency index but it is easier to compute and interpret in addition new properties of the column sums of reciprocal matrices are obtained

a goal programming approach to deriving interval weights in analytic form from interval fuzzy preference relations based on multiplicative consistency abstract this paper focuses on how to find an analytic solution of optimal interval weights from consistent interval fuzzy preference relations ifprs and obtain approximate solution based interval weights in analytic form from inconsistent ifprs the paper first analyzes the popularly used interval weight additive normalization model and illustrates its drawbacks on the existence and uniqueness for characterizing 0 1 valued interval weights obtained from ifprs by examining equivalency of 0 1 valued interval weight vectors a novel framework of multiplicatively normalized interval fuzzy weights mnifws is then proposed and used to define multiplicatively consistent ifprs the paper presents significant properties for multiplicatively consistent ifprs and their associated mnifws these properties are subsequently used to establish two goal programming gp models for obtaining optimal mnifws from consistent ifprs by the lagrangian multiplier method analytic solutions of the two gp models are found for consistent ifprs the paper further devises a two step procedure for deriving approximate solution based mnifws in analytic form from inconsistent ifprs two visualized computation formulas are developed to determine the left and right bounds of approximate solution based mnifws of any ifpr the paper shows that this approximate solution is an optimal solution if an ifpr is multiplicatively consistent three numerical examples including three ifprs and comparative analyses are offered to demonstrate rationality and validity of the developed model

mean curvature flow with surgery of mean convex surfaces in mathbb r 3 we define a notion of mean curvature flow with surgery for two dimensional surfaces in mathbb r 3 with positive mean curvature our construction relies on the earlier work of huisken and sinestrari in the higher dimensional case one of the main ingredients in the proof is a new estimate for the inscribed radius established by the first author invent math 2015

finite topology self translating surfaces for the mean curvature flow in r3 abstract finite topology self translating surfaces for the mean curvature flow constitute a key element in the analysis of type ii singularities from a compact surface because they arise as limits after suitable blow up scalings around the singularity we prove the existence of such a surface m r 3 that is orientable embedded complete and with three ends asymptotically paraboloidal the fact that m is self translating means that the moving surface s t m t e z evolves by mean curvature flow or equivalently that m satisfies the equation h m e z where h m denotes mean curvature is a choice of unit normal to m and e z is a unit vector along the z axis this surface m is in correspondence with the classical three end costa hoffman meeks minimal surface with large genus which has two asymptotically catenoidal ends and one planar end and a long array of small tunnels in the intersection region resembling a periodic scherk surface this example is the first non trivial one of its kind and it suggests a strong connection between this problem and the theory of embedded complete minimal surfaces with finite total curvature

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

a fully nonlinear flow for two convex hypersurfaces in riemannian manifolds we consider a one parameter family of closed embedded hypersurfaces moving with normal velocity g kappa big sum i j frac 1 lambda i lambda j 2 kappa big 1 where lambda 1 le cdots le lambda n denote the curvature eigenvalues and kappa is a nonnegative constant this defines a fully nonlinear parabolic equation provided that lambda 1 lambda 2 2 kappa in contrast to mean curvature flow this flow preserves the condition lambda 1 lambda 2 2 kappa in a general ambient manifold our main goal in this paper is to extend the surgery algorithm of huisken sinestrari to this fully nonlinear flow this is the first construction of this kind for a fully nonlinear flow as a corollary we show that a compact riemannian manifold satisfying overline r 1313 overline r 2323 ge 2 kappa 2 with non empty boundary satisfying lambda 1 lambda 2 2 kappa is diffeomorphic to a 1 handlebody the main technical advance is the pointwise curvature derivative estimate the proof of this estimate requires a new argument as the existing techniques for mean curvature flow due to huisken sinestrari haslhofer kleiner and brian white cannot be generalized to the fully nonlinear setting to establish this estimate we employ an induction on scales argument this relies on a combination of several ingredients including the almost convexity estimate the inscribed radius estimate as well as a regularity result for radial graphs we expect that this technique will be useful in other situations as well

no mass drop for mean curvature flow of mean convex hypersurfaces a possible evolution of a compact hypersurface in rn 1 by mean curvature past singularities is defined via the level set flow in the case where the initial hypersurface has positive mean curvature we show that the brakke flow associated to the level set flow is actually a brakke flow with equality as a consequence we obtain the fact that no mass drop can occur along such a flow a further application of the techniques used above is to give a new variational formulation for mean curvature flow of mean convex hypersurfaces

self similar solutions and translating solitons for lagrangian mean curvature flow we construct many self similar and translating solitons for lagrangian mean curvature flow including self expanders and translating solitons with arbitrarily small oscillation on the lagrangian angle our translating solitons play the same role as cigar solitons in ricci flow and are important in studying the regularity of lagrangian mean curvature flow given two transverse lagrangian planes wn in cn with sum of characteristic angles less than a we show there exists a lagrangian self expander asymptotic to this pair of planes the maslov class of these self expanders is zero thus they can serve as local models for surgeries on lagrangian mean curvature flow families of self shrinkers and self expanders with different topologies are also constructed this paper generalizes the work of anciaux i joyce 12 lawlor 15 and lee and wang 17 18

generalized kato decomposition single valued extension property and approximate point spectrum in this paper we define the generalized kato spectrum of an operator and obtain that the generalized kato spectrum differs from the semi regular spectrum on at most countably many points we study the localized version of the single valued extension property at the points which are not limit points of the approximate point spectrum as well as of the surjectivity spectrum in particular we shall characterize the single valued extension property at a point 0 c in the case that 0i t admits a generalized kato decomposition from this characterization we shall deduce several results on cluster points of some distinguished parts of the spectrum

volume growth and bernstein theorems for translating solitons abstract in this paper we study properties of the potential function of a translating soliton m in r n 1 and the volume growth of the intersection of euclidean balls with m we show that if m is a convex translating soliton then the infimum of the mean curvature function on m is zero i e inf m h 0 we give a condition to obtain the bernstein theorem for the translating solitons we also give an outline of a simple proof of the bernstein theorem due to bombieri de giorgi miranda

convexity estimates for surfaces moving by curvature functions we consider the evolution of compact surfaces by fully nonlinear parabolic curvature ows for which the normal speed is given by a smooth degree one homogeneous function of the principal curvatures of the evolving surface under no further restrictions on the speed function we prove that initial surfaces on which the speed is positive become weakly convex at a singularity of the flow this generalises the corresponding result 26 of huisken and sinestrari for the mean curvature ow to the largest possible class of degree one homogeneous surface flows

mean curvature flow of mean convex hypersurfaces in the last 15 years white and huisken sinestrari developed a far reaching structure theory for the mean curvature flow of mean convex hypersurfaces their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow in the present paper we give a new treatment of the theory of mean convex and k convex flows this includes 1 an estimate for derivatives of curvatures 2 a convexity estimate 3 a cylindrical estimate 4 a global convergence theorem 5 a structure theorem for ancient solutions and 6 a partial regularity theorem our new proofs are both more elementary and substantially shorter than the original arguments our estimates are local and universal a key ingredient in our new approach is the new noncollapsing result of andrews 2 some parts are also inspired by the work of perelman 32 33 in a forthcoming paper 17 we will give a new construction of mean curvature flow with surgery based on the methods established in the present paper note added in may 2015 since the first version of this paper was posted on arxiv in april 2013 the estimates have been used to construct mean convex flow with surgery in 3 by brendle and huisken 5 in september 2013 and in another paper by the authors in april 2014 2016 wiley periodicals inc

a monotonicity formula for mean curvature flow with surgery abstract we prove a monotonicity formula for mean curvature flow with surgery this formula differs from huisken s monotonicity formula by an extra term involving the mean curvature as a consequence we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is in fact free of surgeries this result is used in the analysis of mean curvature flow with surgery in riemannian three manifolds cf 5

consistency analysis and priority derivation of triangular fuzzy preference relations based on modal value and geometric mean existing consistency notions of triangular fuzzy preference relations tfprs are illustrated to be non robust new consistency and acceptable consistency definitions are put forward for tfprs we propose a notion of normalized triangular fuzzy multiplicative weights ntfmws transformation formulae are provided to convert ntfmws into consistent tfprs a logarithmic least square model is developed to derive a ntfmw vector from a tfpr triangular fuzzy preference relation tfpr is an effective framework to model pairwise estimations with imprecision and vagueness in order to obtain a reliable and rational decision result it is important to investigate consistency and priority derivation of tfprs the paper analyzes existing definitions and properties of consistent tfprs and illustrates that they have no invariance with respect to permutations of decision alternatives a new triangular fuzzy arithmetic based transitivity equation is introduced to define consistent tfprs the new transitivity equation reflects multiplicative consistency of modal values and multiplicative consistency of geometric means of triangular fuzzy estimations some properties are presented for consistent tfprs and a notion of acceptable consistency is put forward for tfprs geometric mean and uncertainty ratio based transformation formulae are devised to convert normalized triangular fuzzy multiplicative weights into consistent tfprs a logarithmic least square model is further established for deriving a normalized triangular fuzzy multiplicative weight vector from a tfpr with acceptable consistency a geometric mean based method is developed to compare and rank triangular fuzzy multiplicative weights three numerical examples including a group decision making problem are examined to demonstrate validity and advantages of the proposed models

on training rbf neural networks using input output fuzzy clustering and particle swarm optimization this paper elaborates on the use of particle swarm optimization in training gaussian type radial basis function neural networks under the umbrella of input output fuzzy clustering the problem being investigated concerns the selection of basis function centers that contribute most in network s performance given that the clustering process in the input space is guided by the clustering in the output space to accomplish this task we quantify the effect of the input space fuzzy partition upon network s square error in terms of an objective function that describes the ability of the partition to accurately reconstruct the input training samples we then theoretically prove that the minimization of the above function acts to minimize an upper bound of the network s square error therefore the resulting solution corresponds to a minimal square error while at the same time it maintains the structure of the input data due to the peculiarity of the aforementioned objective function we treat it as the fitness function used by the particle swarm optimizer the proposed methodology encompasses three design steps the first step implements an independent fuzzy clustering in the output space to obtain a set of cluster centers in the second step unlike other approaches the above centers are directly involved in the estimation of the membership degrees in the input output space in the third step these membership degrees are used by the particle swarm optimizer in order to obtain optimal values for the centers to summarize the novelty of our contribution lies in a the way we handle the information flow from output to input space and b the way we handle the effect of the input space partition upon network s performance the experiments indicate that the fitness function decreases as the number of hidden node increases finally a comparison between the proposed method and other sophisticated approaches shows its statistically significant superiority

motion of level sets by mean curvature ii we give a new proof of short time existence for the classical motion by mean curvature of a smooth hypersurface our method consists in studying a fully nonlinear uniformly parabolic equation satisfied by the signed distance function to the surface

a technical note on two inconsistency indices for preference relations a case of functional relation abstract different representations of valued preferences have been studied and used over the years however one common factor has been the prominence of the concept of consistency inconsistency indices have been introduced to estimate the deviation of preferences from a fully consistent form in this note we shall recall two types of preference relations reciprocal relations and multiplicative preference relations and show that two inconsistency indices introduced in these two different frameworks are functionally related besides this main result some reflections on the consequences of being functionally related are presented

some issues on consistency of fuzzy preference relations in decision making in order to avoid misleading solutions the study of consistency when the decision makers express their opinions by means of preference relations becomes a very important aspect in order to avoid misleading solutions in decision making problems based on fuzzy preference relations the study of consistency is associated with the study of the transitivity property in this paper a new characterization of the consistency property defined by the additive transitivity property of the fuzzy preference relations is presented using this new characterization a method for constructing consistent fuzzy preference relations from a set of n 1 preference data is proposed applying this method it is possible to assure better consistency of the fuzzy preference relations provided by the decision makers and in such a way to avoid the inconsistent solutions in the decision making processes additionally a similar study of consistency is developed for the case of multiplicative preference relations 2002 elsevier b v all rights reserved

transitivity frameworks for reciprocal relations cycle transitivity versus fg transitivity for a reciprocal relation q on a set of alternatives a two transitivity frameworks which generalize both t transitivity and stochastic transitivity are compared the framework of cycle transitivity introduced by the present authors soc choice welf to appear and which is based upon the ordering of the numbers q a b q b c and q c a for all a b c a 3 and the framework of fg transitivity introduced by switalski fuzzy sets and systems 137 2003 85 as an immediate generalization of stochastic transitivity the rules that enable to express fg transitivity in the form of cycle transitivity and cycle transitivity in the form of fg transitivity illustrate that for reciprocal relations the concept of cycle transitivity provides a framework that can cover more types of transitivity than does the concept of fg transitivity

mean curvature flow with surgery we give a new proof for the existence of mean curvature flow with surgery of 2 convex hypersurfaces in r as announced in hk13a our proof works for all n 3 including mean convex surfaces in r we also derive a priori estimates for a more general class of flows in a local and flexible setting

a sharp bound for the inscribed radius under mean curvature flow we consider a family of embedded mean convex hypersurfaces which evolve by the mean curvature flow it follows from general results of white that the inscribed radius at each point on the hypersurface is at least frac c h where c is a positive constant that depends only on the initial data andrews recently gave a new proof of that fact using the maximum principle in this paper we show that the inscribed radius is at least frac 1 1 delta h at each point where the curvature is sufficiently large

consistency analysis and group decision making based on triangular fuzzy additive reciprocal preference relations a positive triangular fuzzy number is expressed by a cross ratio expressed triplet operational laws are established for cross ratio expressed triangular fuzzy numbers cretfns multiplicative consistency is analyzed for triangular fuzzy additive reciprocal preference relations a triangular fuzzy weighted geometric operator is developed to aggregate cretfnsa novel method is put forward for comparing two cretfns triangular fuzzy numbers are effective in modeling imprecise and uncertain information and have been widely applied in decision making this paper uses a cross ratio expressed triplet to characterize a positive triangular fuzzy number and introduces notions of cross ratio expressed triangular fuzzy numbers cretfns and triangular fuzzy additive reciprocal preference relations tfarprs we present transformation methods between tfarprs and triangular fuzzy multiplicative reciprocal preference relations and develop operational laws of cretfns such as complement addition multiplication and power a cross ratio expressed triangular fuzzy multiplication based transitivity equation is established to define multiplicative consistency of tfarprs the new consistency captures tanino s multiplicative consistency among the cross ratio expressed modal values and geometric consistency of the interval fuzzy preference relation constructed from lower and upper support values of cross ratio expressed triangular fuzzy judgments some desirable properties are furnished for multiplicatively consistent tfarprs we propose a cross ratio expressed triangular fuzzy weighted geometric operator to aggregate cretfns and extend it to fuse tfarprs score and uncertainty index functions are defined and employed to devise a novel comparison method for cretfns a detailed procedure is put forward to solve group decision making problems with tfarprs six numerical examples are provided to illustrate the validity and applicability of the proposed models