goal programming for decision making an overview of the current state of the art there have been significant advances in the theory of goal programming gp in recent years particularly in the area of intelligent modelling and solution analysis the intention of this paper is to provide an overview of these developments to detail and assess the current state of the art in the subject and to highlight areas which seem promising for future research modelling techniques such as detection and restoration of pareto efficiency normalisation redundancy checking and non standard utility function modelling are overviewed the connection between gp and other multi objective programming techniques as well as a utility interpretation of gp are examined the rationality of ranking multi criteria decision making techniques and of placing gp in such a ranking is discussed

a parametric model for determining consensus priority vectors from fuzzy comparison matrices abstract we consider a group decision making problem where a set of alternatives have to be ranked according to fuzzy preference judgments given by multiple experts we assume that expert assessments are expressed in the form of fuzzy multiplicative preference relations or fuzzy comparison matrices in this paper we propose a general model to generate crisp priority weights of the alternatives from possibly inconsistent and conflicting fuzzy preference relations we express our model in terms of matrix approximations to address the consistency problem and we use weighted metrics to simulate the group dynamics matrix approximation techniques for deriving a common crisp consistent matrix are extended to work with fuzzy matrices by using the concept of cuts in the aggregation process the importance of each expert is taken into consideration according to the agreement of the group with the expert this results in a parametric optimization problem for which a computational formulation is given

max algebra and pairwise comparison matrices abstract this paper is a continuation of our 2004 paper max algebra and pairwise comparison matrices in which the max eigenvector of a symmetrically reciprocal matrix was used to approximate such a matrix by a transitive matrix this approximation was based on minimizing the maximal relative error in a later paper by dahl a different error measure was used and led to a slightly different approximating transitive matrix here some geometric properties of this approximation problem are discussed these lead among other results to a new characterization of a max eigenvector of an irreducible nonnegative matrix the case of toeplitz matrices is discussed in detail and an application to music theory that uses toeplitz symmetrically reciprocal matrices is given

deriving weights from pairwise comparison ratio matrices an axiomatic approach abstract this paper examines the problem of extracting object or attribute weights from a pairwise comparison ratio matrix this problem is approached from the point of view of a distance measure on the space of all such matrices a set of axioms is presented which such a distance measure should satisfy and the uniqueness of the measure is proven the problem of weight derivation is then shown to be equivalent to that of finding a totally transitive matrix which is a minimum distance from the given matrix this problem reduces to a goal programming model finally it is shown that the problem of weight derivation is related to that of ranking players in a round robin tournament the space of all binary tournament matrices is proven to be isometric to a subset of the space of ratio matrices

equidistribution towards the bifurcation current i multipliers and degree d polynomials in the moduli space mathcal p d of degree d polynomials the set text per n w of classes f for which f admits a cycle of exact period n and multiplier multiplier w is known to be an algebraic hypersurface we prove that given w in mathbb c these hypersurfaces equidistribute towards the bifurcation current as n tends to infinity

thermodynamics dimension and the weil petersson metric in this paper we study branched coverings of metrized simplicial trees f t t which arise from polynomial maps f c c with disconnected julia sets we show that the collection of all such trees up to scale forms a contractible space ptd compactifying the moduli space of polynomials of degree d that f records the asymptotic behavior of the multipliers of f and that any meromorphic family of polynomials over can be completed by a unique tree at its central fiber in the cubic case we give a combinatorial enumeration of the trees that arise and show that pt3 is itself a tree resume dans ce travail nous etudions des revetements ramifies d arbres metriques simpliciaux f t t qui sont obtenus a partir d applications polynomiales f c c possedant un ensemble de julia non connexe nous montrons que la collection de tous ces arbres a un facteur d echelle pres forme un espace contractile ptd qui compactifier l espace des modules des polynomes de degre d nous montrons aussi que f enrigistre le comportement asymptotique des multiplicateurs de f et que tout famille meromorphe de polynomes definis sur peut etre completee par un unique arbre comme sa fibre centrale dans le cas cubique nous donnons une enumeration combinatoire des arbres ainsi obtenus et montrons que pt3 est lui meme un arbre

the interpolation theorem for narrow quadrilateral isoparametric finite elements the interpolation theorem for convex quadrilateral isoparametric finite elements is proved in the case when the condition varrho k h k ge varrho 0 0 is not satisfied where h k is the diameter of the element k and varrho k is the radius of an inscribed circle in k the interpolation error is o h k 2 in the l 2 k norm and o h k in the h 1 k norm provided that the interpolated function belongs to h 2 k in the case when the long sides of the quadrilateral k are parallel the constants appearing in the estimates are evaluated

fiedler ptak scaling in max algebra abstract this is essentially the text of my talk on fiedler ptak scaling in max algebra delivered in the invited minisymposium in honor of miroslav fiedler at the 17th ilas conference in braunschweig germany

a parametric gp model dealing with incomplete information for group decision making abstract we consider a group decision making problem where preferences given by the experts are articulated into the form of pairwise comparison matrices in many cases experts are not able to efficiently provide their preferences on some aspects of the problem because of a large number of alternatives limited expertise related to some problem domain unavailable data etc resulting in incomplete pairwise comparison matrices our goal is to develop a computational method to retrieve a group priority vector of the considered alternatives dealing with incomplete information for that purpose we have established an optimization problem in which a similarity function and a parametric compromise function are defined associated to this problem a logarithmic goal programming formulation is considered to provide an effective procedure to compute the solution moreover the parameters involved in the method have a clear meaning in the context of group problems

critical heights on the moduli space of polynomials abstract let m d be the moduli space of one dimensional degree d 2 complex polynomial dynamical systems the escape rates of the critical points determine a critical heights map g m d r d 1 for generic values of g we show that each connected component of a fiber of g is the deformation space for twist deformations on the basin of infinity we analyze the quotient space t d obtained by collapsing each connected component of a fiber of g to a point the space t d is a parameter space analog of the polynomial tree t f associated to a polynomial f c c studied in demarco and mcmullen 2008 6 and there is a natural projection from t d to the space of trees t d we show that the projectivization p t d is compact and contractible further the shift locus in p t d has a canonical locally finite simplicial structure the top dimensional simplices are in one to one correspondence with topological conjugacy classes of structurally stable polynomials in the shift locus

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

a method for approximating symmetrically reciprocal matrices by transitive matrices the problem of approximating symmetrically reciprocal matrices by transitive matrices has received some attention recently this problem has applications in multicriteria decision theory several approximation approaches have been suggested and analyzed we here suggest another approach called the multiplicative approach we show that the optimal approximation in this sense may be found efficiently by transforming the problem into a known combinatorial optimization problem the minimum cycle mean problem for which efficient and simple combinatorial algorithms exist

the combinatorial rigidity conjecture is false for cubic polynomials we show that there exist two cubic polynomials with connected julia sets which are combinatorially equivalent but not topologically conjugate on their julia sets this disproves a conjecture by mcmullen from 1995

tropical implementation of the analytical hierarchy process decision method abstract we apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the analytic hierarchy process in multi criteria problems of rating alternatives from pairwise comparison data in this framework we first consider the minimax chebyshev approximation of pairwise comparison matrices by consistent matrices in the logarithmic scale recasting this approximation problem as a problem of tropical pseudo quadratic programming we then write out a closed form solution to it this solution might be either a unique score vector up to a positive factor or a set of different score vectors to handle the problem when the solution is not unique we develop tropical optimization techniques of maximizing and minimizing the hilbert seminorm to find those vectors from the solution set that are the most and least differentiating between the alternatives with the highest and lowest scores and thus are well representative of the entire solution set

an anisotropic superconvergent nonconforming plate finite element the classical finite element convergence analysis relies on the following regularity condition there exists a constant c independent of the element k and the mesh such that h k r k

modifying the power method in max algebra in the max algebra system the eigenequation for an n n irreducible nonnegative matrix a aij is a x a x here a x i maxjaijxj and a is the maximum circuit geometric mean the complexity of the power method given in l elsner p van den driessche linear algebra appl to appear to compute a and x is considered under some assumptions on the critical matrix it is shown that the algorithm may have time complexity o n4 a modified power method based on karp s formula is presented for this new algorithm with no assumptions on the critical matrix a and x can be computed in o n3 time furthermore this algorithm can be used to compute all linearly independent eigenvectors corresponding to a

normality and repelling periodic points let k 3 2 be an integer and f be a family of functions meromorphic in a domain d c c all of whose poles have multiplicity at least 2 at least 3 if in d each f f has neither repelling fixed points nor repelling periodic points of period k then f is a normal family in d examples are given to show that the conditions on poles are necessary and sharp

the analytic hierarchy process max algebra and multi objective optimisation the analytic hierarchy process ahp is widely used for decision making involving multiple criteria elsner and van den driessche 2004 2010 10 11 introduced a max algebraic approach to the single criterion ahp we extend this to the multi criteria ahp by considering multi objective generalisations of the single objective optimisation problem solved in these earlier papers we relate the existence of globally optimal solutions to the commutativity properties of the associated matrices we relate min max optimal solutions to the generalised spectral radius and we prove that pareto optimal solutions are guaranteed to exist

polynomials without repelling periodic point of given period abstract bergweiler proved that for any given integer k 2 every polynomial p of degree d 2 has at least one repelling periodic cycle of period k unless k d 2 2 2 3 2 4 3 2 here we classified these exceptional polynomials we also showed that the julia sets of these exceptional polynomials are connected

double set parameter finite element method for two sided displacement obstacle problem of clamped plate abstract double set parameter nonconforming finite element method fem for the two sided displacement obstacle problem of clamped plate is considered firstly a double set parameter nonconforming triangular element named veubeke zienkiewicz element is introduced which has the same global dimension as zienkiewicz element and can pass the generalized patch test for any mesh secondly we skillfully construct an enriching operator for connecting the veubeke zienkiewicz and the c 1 conforming bell element spaces with the help of this operator and an auxiliary obstacle problem built as a bridge between the continuous and discrete obstacle problems we present the optimal order error estimates in the broken energy norm and l norm respectively finally we establish a general framework of double set parameter nonconforming elements approximating for the two sided displacement obstacle problem of clamped plate

on the optimal consistent approximation to pairwise comparison matrices abstract consistency retrieval from a biased relative preference table is an imperative task in decision theory this paper considers the least squares approximation of a pairwise comparison matrix by consistent matrices it is observed that the highly nonlinear manifold of consistent matrices can be changed into a linear subspace by the componentwise logarithmic transformation a first order optimality condition therefore can be described in terms of coordinates in the linear subspace this approach facilitates the otherwise much more complicated optimality condition if working with the variables in the original manifold fast nonlinear equation solvers can be employed to solve the problem efficiently

a new superconvergence property of wilson nonconforming finite element in this paper the wilson nonconforming finite element method is considered to solve a class of two dimensional second order elliptic boundary value problems a new superconvergence property at the vertices and the midpoints of four edges of rectangular meshes is obtained

a new error analysis of nonconforming bergan s energy orthogonal element for the extended fisher kolmogorov equation abstract a nonconforming finite element method fem for short is proposed and analyzed for the fourth order extended fisher kolmogorov efk for short equation by employing the bergan s energy orthogonal plate element because the shape function and its first derivatives of this element are discontinuous at the element s vertices which is quite different from the conventional finite elements used in the existing literature a series of novel approaches including some a priori bounds and interpolation function splitting are developed to present a new error analysis for deriving optimal estimates of order o h in energy norm for both the semi discrete and backward euler fully discrete schemes at last numerical experiments are also provided to verify the theoretical analysis

continuity of the generalized spectral radius in max algebra abstract let be an induced matrix norm associated with a monotone norm on r n and be the collection of all nonempty closed and bounded subsets of n n nonnegative matrices under this matrix norm for the hausdorff metric h between and is given by h max sup a inf b a b sup b inf a a b the max algebra system consists of the set of nonnegative numbers with sum a b max a b and the standard product ab for a b 0 for n n nonnegative matrices a b their product is denoted by a b where a b ij max 1 k n a ik b kj for each the max algebra version of the generalized spectral radius of is limsup m sup a m a 1 m where m a 1 a 2 a m a i here a is the maximum circuit geometric mean in this paper we prove that the max algebra version of the generalized spectral radius is continuous on the hausdorff metric space h the notion of the max algebra version of simultaneous nilpotence of matrices is also proposed necessary and sufficient conditions for the max algebra version of simultaneous nilpotence of matrices are presented as well

combinatorics and topology of straightening maps i compactness and bijectivity abstract we study the parameter space structure of degree d 3 one complex variable polynomials as dynamical systems acting on c we introduce and study straightening maps these maps are a natural higher degree generalization of the ones introduced by douady and hubbard to prove the existence of small copies of the mandelbrot set inside itself we establish that straightening maps are always injective and that their image contains all the corresponding hyperbolic systems also we characterize straightening maps with compact domains moreover we give two classes of bijective straightening maps the first produces an infinite collection of embedded copies of the d 1 fold product of the mandelbrot set in the connectedness locus of degree d 3 the second produces an infinite collection of full families of quadratic connected filled julia sets in the cubic connectedness locus such that each filled julia set is quasiconformally embedded

spectral properties of matrix polynomials in the max algebra we consider the spectral properties of matrix polynomials over the max algebra in particular we show how the perron frobenius theorem for the max algebra extends to such polynomials and illustrate the relevance of this for multistep difference equations in the max algebra we also present a number of inequalities for the largest max eigenvalue of a matrix polynomial

on the cycles of components of disconnected julia sets for any integers d ge 3 and n ge 1 we construct a hyperbolic rational map of degree d such that it has n cycles of the connected components of its julia set except single points and jordan curves

multilevel schwarz methods we consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations a class of multilevel algorithms are studied using the additive schwarz framework we establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels this is an improvement on dryja and widlund s result on a multilevel additive schwarz algorithm as well as bramble pasciak and xu s result on the bpx algorithm some multiplicative variants of the multilevel methods are also considered we establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one which is independent of the number of levels for a proper ordering the iteration operators correspond to the error propagation operators of certain v cycle multigrid methods using gauss seidel and damped jacobi methods as smoothers respectively

on the continuity of the generalized spectral radius in max algebra abstract given a bounded set of n n non negative matrices let and denote the generalized spectral radius of and its max version respectively we show that sup t 0 n 1 t 1 t where t denotes the hadamard power of we apply this result to give a new short proof of a known fact that is continuous on the hausdorff metric space h of all nonempty compact collections of n n non negative matrices

transitive matrices and their applications abstract transitive matrices and symmetrically reciprocal sr matrices are defined and spectral properties of certain sr perturbations of transitive matrices are studied the results are applied to a multicriteria decision making method the analytic hierarchy process ahp which uses an sr matrix with positive entries a proof is given that rank reversal is inherent in this method if its input matrix is perturbed

a matrix based ranking method with application to tennis abstract we introduce and investigate a ranking methodology which may be of interest in sports like tennis the approach may also be of interest in decision making situations based on pairwise comparisons the method is based on linear algebra and one computes a score for each player by solving a certain linear system of equations from these scores one finds the ranking the input is a set of matches and weights representing the importance of the matches this is represented by a weighted directed graph we prove a number of properties of the method including uniqueness of scores connection to m matrices and combinatorial interpretations a case study from ranking in professional tennis is discussed in detail

the logarithmic least squares and the generalized pseudoinverse in estimating ratios abstract this paper concerns the pairwise comparison method used in analytic hierarchy process ahp the logarithmic least square method is one of the methods used to rank a finite number of stimuli based on their pairwise comparison in the case of one decision maker the problem can be solved using the geometric mean method it is then assumed that the solution is geometrically normalized in the case of multiple decision makers a set of linear equations is obtained and if we have a different number of judgments for each pair of the compared objects the geometric normalization assumption can not be used directly the aim of this paper is to show that applying the generalized pseudoinverse we obtain the solution that is geometrically normalized and consistent with the case of one decision maker to define the pseudoinverse the spectral decomposition is used the structure of the general solution is presented and the existence of the general solution is discussed

backpropagation multi layer perceptron for incomplete pairwise comparison matrices in analytic hierarchy process abstract analytic hierarchy process ahp is a widely used decision making method in many areas such as management sciences the performance ratings of multiple criteria and alternatives can be elicited from pairwise comparisons obtained by expressing the decision maker s perceive however it may be difficult for the decision maker to prudently assign values to comparisons for large number of criteria or alternatives since there exist distinct relationships between any two elements in the real world the relationship between the missing comparison and the assigned comparisons should be taken into account the aim of this paper is to propose a novel method using a well known regression tool namely the backpropagation multi layer perceptron i e mlp to realize the above implicit relationship so as to estimate a missing pairwise judgment from the other assigned entries a computer simulation is employed to demonstrate that the proposed method can effectively find a missing entry of an incomplete pairwise matrix such that its consistency index is minimized

nonconforming finite element approximation of crystalline microstructure we consider a class of nonconforming nite element approxima tions of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can un dergo either an orthorhombic to monoclinic double well or a cubic to tetrag onal triple well transformation we rst establish a series of error bounds in terms of elastic energies for the l 2 approximation of derivatives of the de formation in the direction tangential to parallel layers of the laminate for the l 2 approximation of the deformation for the weak approximation of the de formation gradient for the approximation of volume fractions of deformation gradients and for the approximation of nonlinear integrals of the deformation gradient we then use these bounds to give corresponding convergence rates for quasi optimal nite element approximations

superconvergence analysis and error expansion for the wilson nonconforming finite element in this paper the wilson nonconforming finite element is considered for solving a class of two dimensional second order elliptic boundary value problems superconvergence estimates and error expansions are obtained for both uniform and non uniform rectangular meshes a new lower bound of the error shows that the usual error estimates are optimal finally a discussion on the error behaviour in negative norms shows that there is generally no improvement in the order by going to weaker norms

on the spectrum of pairwise comparison matrices abstract this paper deals with the spectrum of symmetrically reciprocal sr perturbations of transitive matrices of nonzero complex entries by introducing the so called pseudo circulant matrices the spectral decomposition of specific sr perturbations of n n transitive matrices and that of arbitrary 3 3 sr matrices are provided closed forms are given for all eigenvalues and their associated eigenvectors of these sr matrices also known as pairwise comparison matrices pcms to improve the evaluations at a given pcm as well as the priority ranking of the alternatives these results are applied to the analytic hierarchy process ahp using positive sr matrices explicit intervals for the ranges of possible rank reversals between the alternatives are established the spectrum of certain sr perturbations of transitive matrices with nonzero complex entries is also derived which can be applied in vehicle system dynamics three application examples are included as well

extremal properties of tropical eigenvalues and solutions to tropical optimization problems abstract an unconstrained optimization problem is formulated in terms of tropical mathematics to minimize a functional that is defined on a vector set by a matrix and calculated through multiplicative conjugate transposition for some particular cases the minimum in the problem is known to be equal to the tropical spectral radius of the matrix we examine the problem in the common setting of a general idempotent semifield a complete direct solution in a compact vector form is obtained to this problem under fairly general conditions the result is extended to solve new tropical optimization problems with more general objective functions and inequality constraints applications to real world problems that arise in project scheduling are presented to illustrate the results obtained numerical examples are also provided

a max version of the generalized spectral radius theorem abstract let be a bounded set of n n nonnegative matrices in max algebra in this paper we propose the notions of the max algebra version of the generalized spectral radius of and the max algebra version of the joint spectral radius of the max algebra version of the generalized spectral radius theorem is established we propose the relationship between the generalized spectral radius of in the sense of daubechies and lagarias and its max algebra version moreover a generalization of elsner and van den driessche s lemma is presented as well

pairwise comparisons simplified this study examines the notion of generators of a pairwise comparisons matrix such approach decreases the number of pairwise comparisons from n n 1 to n 1 an algorithm of reconstructing of the pc matrix from its set of generators is presented

antipode preserving cubic maps the fjord theorem this note will study a family of cubic rational maps which carry antipodal points of the riemann sphere to antipodal points we focus particularly on the fjords which are part of the central hyperbolic component but stretch out to infinity these serve to decompose the parameter plane into subsets each of which is characterized by a corresponding rotation number

pairwise ranking choice of method can produce arbitrarily different rank order abstract we examine three methods for ranking by pairwise comparison perronrank principal eigenvector hodgerank and tropicalrank we show that the choice of method can produce arbitrarily different rank order to be precise for any two of the three methods and for any pair of rankings of at least four items there exists a comparison matrix for the items such that the rankings found by the two methods are the prescribed ones we discuss the implications of this result in practice study the geometry of the methods and state some open problems

puzzle geometry and rigidity the fibonacci cycle is hyperbolic we describe a new and robust method to prove rigidity results in complex dynamics the new ingredient is the geometry of the critical puzzle pieces under control of geometry and complex bounds two generalized polynomial like maps which admit a topological conjugacy quasiconformal outside the filled in julia set are indeed quasiconformally conjugate the proof uses a new abstract removability type result for quasiconformal maps following ideas of heinonen and koskela and of kallunki and koskela optimized for applications in complex dynamics we prove as the first application of this new method that for even criticalities distinct from two the period two cycle of the fibonacci renormalization operator is hyperbolic with dimensional unstable manifold

a new robust c0 type nonconforming triangular element for singular perturbation problems abstract in this paper a new robust c 0 triangular element is proposed for the fourth order elliptic singular perturbation problem with double set parameter method and bubble function technique and a general convergence theorem for c 0 nonconforming elements is presented the convergence of the new element is proved in the energy norm uniformly with respect to the perturbation parameter numerical experiments are also carried out to demonstrate the efficiency of the new element

a method for dealing with inconsistencies in pairwise comparisons abstract the pairwise comparison method is a powerful inference tool for assessing the relative importance of a set of items formally its objective is to make compatible decision maker s assignments paired comparison with properties needed for obtaining an overall rank in this paper we propose a distance based framework for analysing this kind of compatibility in this context goal programming is proposed as an attractive and flexible tool

a monte carlo investigation of incomplete pairwise comparison matrices in ahp the analytic hierarchy process ahp is a decision analysis technique used to evaluate complex multiattribute alternatives among one or more decision makers it imposes a hierarchical structure on any complex multicriterion problem however a major drawback of the ahp is that a large number of pairwise comparisons is needed to calibrate the hierarchy when there are a few levels and sublevels the ahp can be applied in a straightforward manner to derive the weights relative preference for each alternative as the size of the hierarchy increases the number of pairwise comparisons increases rapidly it is well established in the marketing and consumer behavior literature that in a very long interview even under the best circumstances the respondent is likely to suffer from information overload recognition of this problem was the motivation which led to the investigation of a modification of ahp which required less data collection i e a reduction in the threat of information overload the first question to be answered is the effect on ahp weights due to different patterns of missing data likely to result from reallife data collection in this study a monte carlo simulation was conducted which uses the incomplete pairwise comparisons ipc algorithm 14 to investigate the effect of reduced sets of pairwise comparisons in the ahp data for the study were generated with known structure and comparisons made between complete and incomplete matrices the results of the simulation suggest that incomplete sets of pairwise comparison matrices can capture the attribute level weights without significant loss of accuracy and independent of decision model form and amount of error considered

the boundary of chaos for interval mappings a goal in the study of dynamics on the interval is to understand the transition to positive topological entropy there is a conjecture from the 1980s that the only route to positive topological entropy is through a cascade of period doubling bifurcations we prove this conjecture in natural families of smooth interval maps and use it to study the structure of the boundary of mappings with positive entropy in particular we show that in families of mappings with a fixed number of critical points the boundary is locally connected and for analytic mappings that it is a cellular set

preference elicitation from inconsistent judgments using multi objective optimization abstract several decision making techniques involve pairwise comparisons to elicit the preferences of a decision maker dm this paper proposes a new approach for prioritization from pairwise comparisons using the concept of indirect judgments no method exists that simultaneously minimizes deviations from both direct and indirect judgments in order to estimate preferences it is sensible to consider both the acquired judgments and the other judgments latent in the dm s mind hence a technique is developed here to minimize the deviations from both types of judgments estimated preferences are generally evaluated based on two criteria their deviation from the provided judgments and the number of judgments that have been ordinally violated here it is proposed to optimize three objectives simultaneously the deviations from both direct and indirect judgments and the number of judgments violated a prototype application has been developed to generate all non dominated solutions using a multiple objective evolutionary algorithm the new approach is shown to offer users greater flexibility than all other tested methods

the iteration of cubic polynomials part i the global topology of parameter space fonctions univalentes en dynamique analytique complexe torsion de la structure complexe topologie globale de l espace parametre

on the asymptotic stability of nonnegative matrices in max algebra abstract in the max algebra system for an n n nonnegative matrix a a ij the eigenequation for max eigenvalue and corresponding max eigenvector x is a x x where a x i max 1 j n a ij x j and a is the maximum circuit geometric mean it is shown that the following conditions are mutually equivalent i a r n ii a 1 iii a lim k a k 0 where a max x 1 x 0 a x and a limsup k a k 1 k

on the max version of the generalized spectral radius theorem abstract let be a bounded set of n n non negative matrices recently the max algebra version of the generalized spectral radius of was introduced we show that lim t i t 1 t where denotes the generalized spectral radius and t the hadamard power of this provides a description of that uses no max terminology as an application we give a short proof of the max version of the generalized spectral radius theorem

approximation of positive matrices by transitive matrices a newton kantorovich algorithm used in the approximation of symmetrically reciprocalpositive matrices by transitive matrices is found to be more generally useful in the approximation of merely positive matrices the occurrence of multiple stationary values when the approximated matrix has certain symmetries is investigated

a max version of the perron frobenius theorem abstract if a an n n nonnegative irreducible matrix then there exists a 0 and a positive vector x such that maxjaijxj a xi i 1 2 n furthermore a is the maximum geometric mean of a circuit in the weighted directed graph corresponding to a this theorem which we refer to as the max version of the perron frobenius theorem is well known in the context of matrices over the max algebra and also in the context of matrix scalings in the present work which is partly expository we bring out the intimate connection between this result and the perron frobenius theory we present several proofs of the result some of which use the perron frobenius theorem structure of max eigenvalues and max eigenvectors is described possible ways to unify the perron frobenius theorem and its max version are indicated some inequalities for a are proved

harmonic measure and polynomial julia sets there is a natural conjecture that the universal bounds for the dimension spectrum of harmonic measure are the same for simply connected and for non simply connected domains in the plane because of the close relation to conformal mapping theory the simply connected case is much better understood and proving the above statement would give new results concerning the properties of harmonic measure in the general case we establish the conjecture in the category of domains bounded by polynomial julia sets the idea is to consider the coefficients of the dynamical zeta function as subharmonic functions on a slice of teichmuller s space of the polynomial and then to apply the maximum principle 1 dimension spectrum of harmonic measure in this paper we discuss some properties of harmonic measure in the complex plane for a domain and a point a let a denote the harmonic measure of evaluated at a the measure a can be defined for instance as the hitting distribution of a brownian motion started at a if e then a e is the probability that a random brownian path first hits the boundary at a point of e much work has been devoted to describing dimensional properties of when the domain is as general as possible in particular jones and wolff 7 proved that no matter what the domain is harmonic measure is concentrated on a borel set of hausdorff dimension at most one in other words dim 1 for all plane domains 1 1 we are interested in finding similar but stronger universal estimates involving the dimension spectrum of 1 1 universal spectrum for every positive we denote f dim z where z is the lower pointwise dimension of z lim inf 0 log b z log b z is a general notation for the disc with center z and radius the universal dimension spectrum is the function sup f 1 2 where the supremum is taken over harmonic measures of all planar domains the first author is supported by n s f grant dms 9970283 the second author is supported by n s f grant dms 9800714

hodgerank is the limit of perron rank in the context of pairwise comparison ranking we show that hodgerank row geometric mean is the limit of perron rank ranking with principal eigenvector as a certain parameter k goes to 0 this result provides a novel mathematical link between two important pairwise ranking methods it complements the known result that as k approaches infinity perron rank converges to tropical rank thus these three pairwise ranking methods belong to the same parametrized family our proof technique is useful for mathematical comparison of these methods as a sample application we show that for ranking models with i i d noise hodgerank is a linear approximation of perron rank in this particular setup for large numbers of items with sufficiently large score differences the two methods yield identical ordinal rankings

combining minsum and minmax a goal programming approach a number of methods for multiple objective optimization problems mop give as solution to mop the set of optimal solutions for some single objective optimization problems associated with it well known examples of these single objective optimization problems are the minsum and the minmax in this note we propose a new parametric single objective optimization problem associated with mop by means of goal programming ideas we show that the minsum and minmax are particular instances so we are somehow combining minsum and minmax by means of a parameter moreover such parameter has a clear meaning in the value space applications of this parametric problem to classical models in locational analysis are discussed

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

on the power method in max algebra abstract the eigenvalue problem for an irreducible nonnegative matrix a a ij in the max algebra system is a otimes x lambda x where a otimes x i mathop max j a ij x j and lambda turns out to be the maximum circuit geometric mean mu a a power method algorithm is given to compute mu a and eigenvector x the algorithm is developed by using results on the convergence of max powers of a which are proved using nonnegative matrix theory in contrast to an algorithm developed in 4 this new method works for any irreducible nonnegative a and calculates eigenvectors in a simpler and more efficient way some asymptotic formulas relating mu a the spectral radius and norms are also given

efficient weight vectors from pairwise comparison matrices abstract pairwise comparison matrices are frequently applied in multi criteria decision making a weight vector is called efficient if no other weight vector is at least as good in approximating the elements of the pairwise comparison matrix and strictly better in at least one position a weight vector is weakly efficient if the pairwise ratios cannot be improved in all non diagonal positions we show that the principal eigenvector is always weakly efficient but numerical examples show that it can be inefficient the linear programs proposed test whether a given weight vector is weakly efficient and in case of strong inefficiency an efficient strongly dominating weight vector is calculated the proposed algorithms are implemented in pairwise comparison matrix calculator available at pcmc online