binary behavior of fuzzy programming with piecewise linear membership functions the nature of vagueness imprecision and uncertainly is fuzzy rather than crisp and or random especially for a multiple objectives decision making problem a key component of fuzzy programming is the membership function that represents a mathematical expression of level function for the decision maker s preference in fact a decision making problem involves the achievement of fuzzy goals some of which are met while others are not because these fuzzy goals are subject to real world constraints to represent this situation the binary piecewise linear membership function is then employed in order to solve the problem we propose a new idea of how to formulate the binary piecewise linear membership function the formulated problem can be easily solved using common integer programming packages in addition an illustrative example is included for demonstrating the usefulness of the proposed model finally the analytical superiority of the proposed method in terms of the execution time can be seen through a computation experiment conducted on a set of generated test examples

the set of vertices with positive curvature in a planar graph with nonnegative curvature abstract in this paper we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature based on this we show that the automorphism group of a planar possibly infinite graph with nonnegative combinatorial curvature and positive total curvature is a finite group and give an upper bound estimate for the order of the group

a new proof of gromov s theorem on groups of polynomial growth we give a new proof of gromov s theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup the proof does not rely on the montgomeryzippin yamabe structure theory of locally compact groups

ricci curvature of finite markov chains via convexity of the entropy we study a new notion of ricci curvature that applies to markov chains on discrete spaces this notion relies on geodesic convexity of the entropy and is analogous to the one introduced by lott sturm and villani for geodesic measure spaces in order to apply to the discrete setting the role of the wasserstein metric is taken over by a different metric having the property that continuous time markov chains are gradient flows of the entropy using this notion of ricci curvature we prove discrete analogues of fundamental results by bakry emery and otto villani further we show that ricci curvature bounds are preserved under tensorisation as a special case we obtain the sharp ricci curvature lower bound for the discrete hypercube

on the lp spectrum of laplacians on graphs abstract we study the p independence of spectra of laplace operators on graphs arising from regular dirichlet forms on discrete spaces here a sufficient criterion is given solely by a uniform subexponential growth condition moreover under a mild assumption on the measure we show a one sided spectral inclusion without any further assumptions we study applications to normalized laplacians including symmetries of the spectrum and a characterization for positivity of the cheeger constant furthermore we consider laplacians on planar tessellations for which we relate the spectral p independence to assumptions on the curvature

comments on fuzzy programming with nonlinear membership functions abstract yang et al in their paper fuzzy programming with nonlinear membership functions published in fuzzy sets and systems 41 1991 declared that their model can solve a fuzzy program with an s shaped membership function by adding only one 0 1 variable this paper indicates that their declaration is correct only for a specific type of s shape membership functions we propose another model to treat the fuzzy programs which cannot be solved effectively by yang et al

fuzzy potential solutions in multi criteria and multi constraint level linear programming problems abstract given a multi criteria and multi constraint level mc 2 linear programming problem we can use an mc 2 simplex method to find a set of potential solutions each potential solution optimizes the mc 2 problem under a certain range of decision parameters namely the criteria weight vector and the constraint level weight vector in this paper we extend the potential solution to a fuzzy potential solution for a given fuzzy mc 2 problem with a certain range of the decision parameters a membership function of the mc 2 problem is first constructed then for such a parameter range a fuzzy potential solution is determined in terms of the membership function and a primal potential basis based on the theoretical analysis we propose a heuristic algorithm to systematically and effectively locate all fuzzy potential solutions for all possible changes of the decision parameters a numerical example is used to illustrate the algorithm

polynomial growth harmonic functions on groups of polynomial volume growth we consider harmonic functions of polynomial growth of some order d on cayley graphs of discrete groups of polynomial volume growth of order d w r t the word metric and prove the optimal estimate for the dimension of the space of such harmonic functions more precisely the dimension of this space of harmonic functions is at most of order d d 1 as in the already known riemannian case this estimate is polynomial in the growth degree more generally our techniques also apply to graphs roughly isometric to cayley graphs of groups of polynomial volume growth

lipschitz property of harmonic function on graphs this paper proves the local lipschitz property for harmonic or positive subharmonic functions on graphs an example is also obtained to show that the global lipschitz property of harmonic function on graphs does not hold

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

generalization of yang et al s method for fuzzy programming with piecewise linear membership functions li and yu fuzzy sets and systems 101 1999 109 argued that the model of yang et al for fuzzy programming fuzzy sets and systems 41 1991 39 is correct only for a specific type of piecewise linear membership function and proposed their model for other type of membership functions this study generalizes the model of yang et al a numerical example indicates that the model of li and yu is inapplicable to a fuzzy programming problem that involves more than one membership function whereas the proposed model gives the optimal solution

post optimality analysis on the membership functions of a fuzzy linear programming problem models of linear programming problems with fuzzy constraints are very well known in the current literature in almost all cases to solve these problems linear membership functions are used because they have very good properties and are very easy to manipulate in some cases however because of the knowledge that the decision maker has such membership functions could be modeled as nonlinear although the complexity of the problem could increase this paper considers the use of nonlinear membership functions in fuzzy linear programming problems to show that the corresponding solution to be obtained can be derived from a parallel linear model moreover it is easier to solve than the nonlinear model making use of a similar procedure to that of post optimal analysis in classical linear programming the case in which these membership functions are defined by means of piecewise linear approximations is also considered and analyzed

sectional curvature of polygonal complexes with planar substructures in this paper we introduce a class of polygonal complexes for which we consider a notion of sectional combinatorial curvature these complexes can be viewed as generalizations of 2 dimensional euclidean and hyperbolic buildings we focus on the case of non positive and negative combinatorial curvature as geometric results we obtain a hadamard cartan type theorem thinness of bigons gromov hyperbolicity and estimates for the cheeger constant we employ the latter to get spectral estimates show discreteness of the spectrum in the sense of a donnelly li type theorem and present corresponding eigenvalue asymptotics moreover we prove a unique continuation theorem for eigenfunctions and the solvability of the dirichlet problem at infinity

binary fuzzy goal programming goal programming is an important technique for solving many decision management problems fuzzy goal programming involves applying the fuzzy set theory to goal programming thus allowing the model to take into account the vague aspirations of a decision maker using preference based membership functions we can define the fuzzy problem through natural language terms or vague phenomena in fact decision making involves the achievement of fuzzy goals some of them are met and some not because these goals are subject to the function of environment resource constraints thus binary fuzzy goal programming is employed where the problem cannot be solved by conventional goal programming approaches this paper proposes a new idea of how to program the binary fuzzy goal programming model the binary fuzzy goal programming model can then be solved using the integer programming method finally an illustrative example is included to demonstrate the correctness and usefulness of the proposed model

nonparametric estimation and adaptive control of functional autoregressive models this paper deals with nonparametric estimation and adaptive control of nonlinear systems of the form x n 1 f x n u n xi n 1 n in bkn where the state x n is observed f is an unknown function and the control un is chosen in order to track a given reference trajectory we estimate the function f using a nonparametric estimator and study two adaptive control laws built from this nonparametric estimator and derived from the self tuning control the first one can be used for open loop stable systems and requires an additional exciting noise the second one needs some a priori knowledge on function f but allows us to control open loop unstable systems we establish some general results on the nonparametric estimator of f like the uniform almost sure convergence over dilating sets and then prove that both adaptive control laws are asymptotically optimal in quadratic mean in addition we give a strongly consistent estimator of the covariance matrix of the unobservable white noise xi n

fuzzy goal programming a parametric approach narasimhan incorporated fuzzy set theory within goal formulation in 1980 since then much research has been performed in this field and various models for solving fuzzy goal programming have been proposed one of the well known models was proposed by tiwari et al in 1987 19 where an additive model was proposed this paper is an extension to the tiwari et al model that deals with the sum of weighted negative deviations between the desirable achievement degree and the common target here properties of the model are proposed a numerical example is also given to illustrate the approach

bector chandra type duality in fuzzy linear programming with exponential membership functions in this paper we study a pair of fuzzy primal dual linear programming problems and calculate duality results using an aspiration level approach we use an exponential membership function which is in contrast to the earlier works that relied on a linear membership function as the fuzzy environment causes a duality gap we investigate how choosing the exponential membership function impacts this gap this issue is particularly important for fuzzy linear programming where in general the primal and dual objective values may not be bounded

a fuzzy goal programming approach to multi objective optimization problem with priorities abstract in goal programming problem the general equilibrium and optimization are often two conflicting factors this paper proposes a generalized varying domain optimization method for fuzzy goal programming fgp incorporating multiple priorities according to the three possible styles of the objective function the varying domain optimization method and its generalization are proposed this method can generate the results consistent with the decision maker dm s expectation that the goal with higher priority may have higher level of satisfaction using this new method it is a simple process to balance between the equilibrium and optimization and the result is the consequence of a synthetic decision between them in contrast to the previous method the proposed method can make that the higher priority achieving the higher satisfactory degree to get the global solution of the nonlinear nonconvex programming problem resulting from the original problem and the varying domain optimization method the co evolutionary genetic algorithms gas called genocopiii is used instead of the sqp method in this way the dm can get the optimum of the optimization problem we demonstrate the power of this proposed method by illustrative examples

fuzzy multicriteria integer programming via fuzzy generalized networks recent advances in network based mathematical programming methods have given rise to practical economical and computationally efficient solution methodologies for truly large scale problems today a wide variety of real world problems have been solved in such a manner even though on the surface they do not have an apparent network structure one type of network the generalized network has been successfully applied to solve large scale zero one or mixed integer mathematical programming models however one drawback to such approaches is their typical focus on but a single objective the purpose of this paper then is to document the results of an ongoing research effort which combines the generalized network concept with the technique of fuzzy programming the resultant hybrid approach has been found to provide a computationally efficient approach to multi objective zero one or integer programming problems

sobolev spaces laplacian and heat kernel on alexandrov spaces we prove the compactness of the imbedding of the sobolev space w 1 2 0 omega into l 2 omega for any relatively compact open subset omega of an alexandrov space as a corollary the generator induced from the dirichlet energy form has discrete spectrum the generator can be approximated by the laplacian induced from the dc structure on the alexandrov space we also prove the existence of the locally holder continuous heat kernel

fuzzy goal programming with different importance and priorities abstract this paper formulates fuzzy goal programming fgp incorporating different importance and preemptive priorities by using an additive model to maximize the sum of achievement degrees of all fuzzy goals in contrast to previous works the proposed approach allows the decision maker to determine a desirable achievement degree for each fuzzy goal to reflect explicitly the relative importance of these goals this approach can generate a set of achievement degrees consistent with the decision maker s expectations even though the relative importance of the goals may change furthermore in this paper we incorporate the decision maker s preemptive priority structure into a single formulation the resulting solutions satisfy both the preemptive priority structure and have the maximum achievement degrees in sum the proposed approaches effectiveness and computational superiority over the existing approaches are demonstrated and compared with examples from the literature

multi choice goal programming with utility functions abstract goal programming gp has been and still is the most widely used technique for solving multiple criteria decision problems and multiple objective decision problems by finding a set of satisfying solutions however the major limitation of goal programming is that can only use aspiration levels with scalar value for solving multiple objective problems in order to solve this problem multi choice goal programming mcgp was proposed by chang 2007a following the idea of mcgp this study proposes a new concept of level achieving in the utility functions to replace the aspiration level with scalar value in classical gp and mcgp for multiple objective problems according to this idea it is possible to use the skill of mcgp with utility functions to solve multi objective problems the major contribution of using the utility functions of mcgp is that they can be used as measuring instruments to help decision makers make the best appropriate policy corresponding to their goals with the highest level of utility achieved in addition the above properties can improve the practical utility of mcgp in solving more real world decision management problems

fuzzy goal programming with multiple priorities via generalized varying domain optimization method this paper proposes a generalized varying domain optimization method for fuzzy goal programming incorporating multiple priorities according to the three possible styles of the objective function the varying domain optimization method and its generalization are corresponding proposed in contrast to the previous method the proposed method can make that the higher priority achieving the higher satisfaction degree in this way the decision maker can get the optimal solution as well as guarantee the priorities of the multiple objective optimization problem we demonstrate the power of this proposed method by three illustrative examples and a practice application

nearest piecewise linear approximation of fuzzy numbers abstract the problem of the nearest approximation of fuzzy numbers by piecewise linear 1 knot fuzzy numbers is discussed by using 1 knot fuzzy numbers one may obtain approximations which are simple enough and flexible to reconstruct the input fuzzy concepts under study they might be also perceived as a generalization of the trapezoidal approximations moreover these approximations possess some desirable properties apart from theoretical considerations approximation algorithms that can be applied in practice are also given

an approximation approach for representing s shaped membership functions in general to formulate a fuzzy linear programming problem with n s shaped utility membership functions traditional methods require n or more extra binary variables because s shaped curves are neither convex nor concave in all places adding binary variables does not improve the bound of the linear programming relaxation on the contrary added binary variables increase the computational burden in the solution process if problems get large therefore a formulation without binary variables should be more efficient accordingly this study proposes a piecewise linear approach to formulate an s shaped membership function mf without adding any extra binary variables which improves the efficiency of fuzzy linear programming in solving decision management problems with s shaped mfs finally a computational experiment is provided to demonstrate the superiority of the proposed models an illustrative example is also provided to show the usefulness of the proposed method

geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature we apply alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature we obtain the metric classification of these graphs and construct the graphs embedded in the projective plane minus one point moreover we show the volume doubling property and the poincare inequality on such graphs the quadratic volume growth of these graphs implies the parabolicity finally we prove the polynomial growth harmonic function theorem analogous to the case of riemannian manifolds

a fuzzy multiobjective program with quasiconcave membership functions and fuzzy coefficients abstract a method is proposed for solving a fuzzy multi objective linear programming problem fmp with quasiconcave membership functions and fuzzy coefficients the proposed method first expresses a piecewise function as the summation of absolute terms then we search for the interval where the optimal solution is allocated by finding the corresponding points with same value of membership functions after that the problem is solved by goal programming techniques comparing with other fmp methods the proposed method does not need to add extra zero one variables to divide the original problem into several sub problems or transforming all original quasiconcave functions into concave functions in addition the proposed method could solve a fmp problem with fuzzy coefficients to obtain a solution closing to a global optimum

a goal programming procedure for solving problems with multiple fuzzy goals using dynamic programming abstract this paper describes how the preemptive priority based goal programming gp can be used to solve a class of fuzzy programming fp problems with the characteristics of dynamic programming dp in the proposed approach the membership functions of the objective goals of a problem with fuzzy aspiration levels are defined first then under the framework of preemptive priority based gp a multi stage dp model of the problem for achievement of the highest degree unity of each of the membership functions is developed in the decision process the goal satisficing philosophy of gp is used recursively to arrive at the most satisfactory solution two numerical examples are provided to illustrate the approach

a weighted max min model for fuzzy goal programming abstract after narasimhan s pioneering study of applying fuzzy set theory to goal programming in 1980 many achievements in the field have been recorded most of them followed the max min approach however when objectives have different levels of importance only the weighted additive model of tiwari et al seems to be applicable however the shortcoming of the additive model is that the summation of quasiconcave functions may not be quasiconcave this study proposes a novel weighted max min model for fuzzy goal programming fgp and for fuzzy multiple objective decision making the proposed model adapts well to even the most complicated membership functions numerical examples demonstrate that the proposed model can be effectively incorporated with other approaches to fgp and is superior to the weighted additive approach

an analogue of the descartes euler formula for infinite graphs and higuchi s conjecture let r be a connected 2 manifold without boundary obtained from a possibly infinite collection of polygons by identifying them along edges of equal length let v be the set of vertices and for every v v let k v denote the gaussian curvature of v 2 minus the sum of incident polygon angles descartes showed that v k 4 whenever r may be realized as the surface of a convex polytope in r 3 more generally if r is made of finitely many polygons euler s formula is equivalent to the equation v 1 r where r is the euler characteristic of r our main theorem shows that whenever v 0 for every vertex v we apply our main theorem to deduce that r is made of finitely many polygons and is homeomorphic to either the 2 sphere or to the projective plane further we show that unless is a prism antiprism or the projective planar analogue of one of these that v 3444 this resolves a recent conjecture of higuchi

an interactive satisfying method based on alternative tolerance for multiple objective optimization with fuzzy parameters an interactive satisfying method based on alternative tolerance is presented for the multiple objective optimization problem with fuzzy parameters using the alpha level sets of the fuzzy numbers all the objectives are modeled as the fuzzy goals and the tolerances of the objectives are iteratively changed according to a decision maker for a satisfying solution via a specific attainable point programming model the membership functions can be modified and then a lexicographic two phase programming procedure is constructed correspondingly to find the final solution in a special case the objective constraint is added instead of changing the membership functions therefore the dissatisfying objectives for the decision maker can be improved step by step the presented method not only acquires the alpha pareto optimal or weak alpha pareto optimal solution of the fuzzy multiple objective optimization but also satisfies the progressive preference of the decision maker a numerical example shows its power

a method for solving fuzzy goal programming problems based on minmax approach abstract narasimhan incorporated fuzzy set theory within goal programming formulation in 1980 since then numerous research has been carried out in this field one of the well known models for solving fuzzy goal programming problems was proposed by hannan in 1981 in this paper the conventional minmax approach in goal programming is applied to solve fuzzy goal programming problems it is proved that the proposed model is an extension to hannan model that deals with unbalanced triangular linear membership functions in addition it is shown that the new model is equivalent to a model proposed in 1991 by yang et al moreover a weighted model of the new approach is introduced and is compared with kim and whang s model presented in 1998 a numerical example is given to demonstrate the validity and strengths of the new models

two step interactive satisfactory method for fuzzy multiple objective optimization with preemptive priorities this paper presents a two step interactive satisfactory optimization method for fuzzy multiple objective optimization with preemptive priorities in contrast to previous works this proposed approach guarantees the order of satisfactory degrees consistent with priorities the decision maker not only acquires satisfactory solution of all the objectives but also realizes the preemptive priority requirement among them the originally complex optimization problem is simplified and divided into two subproblems that are solved in sequence numerical examples and actual application show the proposed method s effectiveness flexibility and efficiency in comparison with results from the literature

kernel density estimation and goodness of fit test in adaptive tracking we investigate the asymptotic properties of a recursive kernel density estimator of the driven noise of multivariate armax models in adaptive tracking we provide an almost sure pointwise and uniform strong law of large numbers as well as a pointwise and multivariate central limit theorem we also carry out a goodness of fit test together with some simulation experiments

fuzzy programming with nonlinear membership functions piecewise linear approximation one promising approach that has been proposed for dealing with multicriteria i e multiple objective programming models has been fuzzy linear programming flp in essence the flp approach involves the replacement of the multiple objectives with goals by means of assigning an aspiration level to each objective fuzzy membership functions are then introduced to represent the measure of the achievement of the actual solution for each goal relative to its aspired level it has been shown that if the flp problem is linear and if linear membership functions are employed to reflect goal achievement then the flp problem may be transformed into a conventional i e single objective linear programming model as such one may use any conventional linear programming algorithm to solve the transformed model however in the more general case of nonlinear membership functions the transformation process becomes considerably more involved and has in the past typically led to less desirable formulations in this paper we present what we believe to be a straightforward and computationally efficient procedure for dealing with the flp problem with any general class of nonlinear membership functions conversion of such a flp will result in either a regular linear programming model or a linear integer programming model depending upon the specific characteristics of the membership function in either case existing commercial software is readily available to solve such models

method for solving quasi concave and non concave fuzzy multi objective programming problems abstract this paper proposes a method based on linear programming techniques to treat quasi concave and non concave fuzzy multi objective programming fmop problems the proposed method initially presents a piecewise linear expression to interpreting a quasi concave membership function then we find the convex type break points and transform all quasi concave membership functions into concave functions after that the converted program is solved by linear programming techniques to obtain a global optimum in addition to not containing any of the zero one variables the proposed method does not require dividing the quasi concave fmop problem into large sub problems as in conventional methods the extension of the proposed method can treat general non concave fmop problems by merely adding less number of zero one variables

linear programming with multiple fuzzy goals abstract this paper describes the use of fuzzy set theory in goal programming gp problems in particular it is demonstrated how fuzzy or imprecise aspirations of the decision maker dm can be quantified through the use of piecewise linear and continuous functions models are then presented for the use of fuzzy goal programming with preemptive priorities with archimedean weights and with the maximization of the membership function corresponding to the minimum goal examples are also provided