evolution equations of lurie type in hilbert spaces on etudie le probleme de lurie pour le systeme dx d ax b d dx o c w p ou l operateur a est soit borne soit suppose engendrer un groupe co fortement continu t t t e sur un espace de hilbert reel x de produit interieur et de norme b c x p e et o e e est une fonction non lineaire lipschitzienne uniformement continue

a least squares method for finding the preisach hysteresis operator from measurements a finite element like least squares method is introduced for determining the density function in the preisach hysteresis model from overdeterined measured data it is shown that the least squares error depends on the quality of the data and the best approximations to the analytic density for consistent data criteria are given for convergence of the approximate density and preisach operator with increasing measurements

closed asymptotic couples the derivation of a hardy field induces on its value group a certain function if a hardy field extends the real field and is closed under powers then its value group is also a vector space over such ordered vector spaces with function are called h couples we define closed h couples and show that every h couple can be embedded into a closed one the key fact is that closed h couples have an elimination theory solvability of an arbitrary system of equations and inequalities built up from vector space operations the function parameters and the unknowns to be solved for is equivalent to an effective condition on the parameters of the system the h couple of a maximal hardy field is closed and this is also the case for the h couple of the field of logarithmic exponential series over we analyze in detail finitely generated extensions of a given h couple

on the computational efficiency of multiple objective metaheuristics the knapsack problem case study abstract the paper describes a computational experiment which goal is to evaluate computational efficiency of three multiple objective evolutionary metaheuristics on the multiple objective multiple constraints knapsack problem the relative efficiency of the multiple objective algorithms is evaluated with respect to a single objective evolutionary algorithm ea we use a methodology that allows consistent evaluation of the quality of approximately pareto optimal solutions generated by both multiple and single objective metaheuristics then we compare computational efforts needed to generate solutions of approximately the same quality by the two kinds of methods the results indicate that computational efficiency of multiple objective eas deteriorates with the growth of the number of objectives furthermore significant differences in the performance of the three algorithms are observed

multiplicity of fixed points and growth of e neighborhoods of orbits abstract we study the relationship between the multiplicity of a fixed point of a function g and the dependence on e of the length of e neighborhood of any orbit of g tending to the fixed point the relationship between these two notions was discovered in elezovic ubrinic and upanovic 2007 5 in the differentiable case and related to the box dimension of the orbit here we generalize these results to non differentiable cases introducing a new notion of critical minkowski order we study the space of functions having a development in a chebyshev scale and use multiplicity with respect to this space of functions with the new definition we recover the relationship between multiplicity of fixed points and the dependence on e of the length of e neighborhoods of orbits in non differentiable cases applications include in particular poincare maps near homoclinic loops and hyperbolic 2 cycles and abelian integrals this is a new approach to estimate the cyclicity by computing the length of the e neighborhood of one orbit of the poincare map for example numerically and by comparing it to the appropriate scale

solving the discrete multiple criteria problem using linear prospect theory abstract prospect theory developed by kahneman and tversky is a popular model of choice in decision problems under uncertainty prospect theory has recently been extended to multiple criteria choice problems in this paper an interactive method for solving discrete multiple criteria decision problems based on prospect theory type value functions has been developed piecewise linear marginal value functions are assumed to approximate the s shaped value functions of prospect theory therefore the proposed procedure is valid only for convex preferences

alien limit cycles in lienard equations this paper aims at providing an example of a family of polynomial lienard equations exhibiting an alien limit cycle this limit cycle is perturbed from a 2 saddle cycle in the boundary of an annulus of periodic orbits given by a hamiltonian vector field the hamiltonian represents a truncated pendulum of degree 4 in comparison to a former polynomial example not only the equations are simpler but a lot of tedious calculations can be avoided making the example also interesting with respect to simplicity in treatment

on the number of limit cycles near a homoclinic loop with a nilpotent singular point abstract in this article we study the expansion of the first melnikov function appearing by perturbing an integrable and reversible system with a homoclinic loop passing through a nilpotent singular point and obtain formulas for computing the first coefficients of the expansion based on these coefficients we obtain a lower bound for the maximal number of limit cycles near the homoclinic loop moreover as an application of our main results we consider a type of integrable and reversible polynomial systems obtaining at least 3 4 or 5 limit cycles respectively

a chebyshev criterion for abelian integrals we present a criterion that provides an easy sufficient condition in order for a collection of abelian integrals to have the chebyshev property this condition involves the functions in the integrand of the abelian integrals and can be checked in many cases in a purely algebraic way by using this criterion several known results are obtained in a shorter way and some new results which could not be tackled by the known standard methods can also be deduced

on the finite cyclicity of open period annuli let be an open relatively compact period annulus of real analytic vector fieldx0 on an analytic surface we prove that the maximal number of limit cycles which bifurcate from under a given multiparameter analytic deformation x of x0 is finite provided that x0 is either a hamiltonian or generic darbouxian vector field 1 statement of the result let s be a real analytic surface without border compact or not and let x0 be a real analytic vector field on it an open period annulus of x0 is a union of period orbits of x0 which is bianalytic to the standard annulus s1 0 1 the image of each circle s1 u being a periodic orbit of x0 let x r 0 be an analytic family of analytic vector fields on s and let be an open period annulus of x0 the cyclicity cycl x of with respect to the deformation x is the maximal number of limit cycles of x which tend to as tends to zero see definition 2 clearly the vector field x0 has an analytic first integral f in the period annulus which has no critical points in what follows we suppose that the open period annulus is relatively compact i e its closure s is compact definition 1 we say that x0 is a hamiltonian vector field provided that it has a first integral with isolated critical points in a complex neighborhood of we say that x0 is a generic darbouxian vector field provided that all singular points of x0 in a neighborhood of are orbitally analytically equivalent to linear saddles x y with 0 remark 1 note that if x0 is a plane vector field with a first integral h as above then x0 h1 hy x hx y duke mathematical journal vol 152 no 1 c 2010 doi 10 1215 00127094 2010 005 received 3 july 2008 revision received 1 may 2009 2000 mathematics subject classification primary 34c07 secondary 34c10

bounding the number of zeros of certain abelian integrals abstract in this paper we prove a criterion that provides an easy sufficient condition in order for any nontrivial linear combination of n abelian integrals to have at most n k 1 zeros counted with multiplicities this condition involves the functions in the integrand of the abelian integrals and it can be checked in many cases in a purely algebraic way

an interactive evolutionary multi objective optimization algorithm with a limited number of decision maker calls this paper presents a preference based method to handle optimization problems with multiple objectives with an increase in the number of objectives the computational cost in solving a multi objective optimization problem rises exponentially and it becomes increasingly difficult for evolutionary multi objective techniques to produce the entire pareto optimal front in this paper an evolutionary multi objective procedure is combined with preference information from the decision maker during the intermediate stages of the algorithm leading to the most preferred point the proposed approach is different from the existing approaches as it tries to find the most preferred point with a limited budget of decision maker calls in this paper we incorporate the idea into a progressively interactive technique based on polyhedral cones the idea is also tested on another progressively interactive approach based on value functions results are provided on two to five objective unconstrained as well as constrained test problems

on the number of zeros of abelian integrals a constructive solution of the infinitesimal hilbert sixteenth problem we prove that the number of limit cycles generated from nonsingular energy level ovals periodic trajectories in a small non conservative perturbation of a hamiltonian polynomial vector field on the plane is bounded by a double exponential of the degree of the fields this solves the long standing infinitesimal hilbert 16th problem

interquad an interactive quad tree based procedure for solving the discrete alternative multiple criteria problem abstract in this paper an interactive procedure based upon a data structure called a quad tree is developed for solving the discrete alternative multiple criteria problem called interquad the procedure has been designed with large discrete alternative problems in mind interquad takes advantage of the ability of a quad tree to identify store and retrieve nondominated criterion vectors then the user interacts with the nondominated criterion vectors stored in the quad tree in a fashion similar to that of the combined tchebycheff aspiration criterion vector procedure of steuer silverman and whisman

abelian integrals and limit cycles abstract the paper deals with generic perturbations from a hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles in this paper we show that near a 2 saddle cycle the number of limit cycles produced in unfoldings with one unbroken connection can exceed the number of zeros of the related abelian integral even if the latter represents a stable elementary catastrophe we however also show that in general finite codimension of the abelian integral leads to a finite upper bound on the local cyclicity in the treatment we introduce the notion of simple asymptotic scale deformation

the field of reals with multisummable series and the exponential function we show that the field of real numbers with multisummable real power series is model complete o minimal and polynomially bounded further expansion by the exponential function yields again a model complete and o minimal structure which is exponentially bounded and in which the gamma function on the positive real line is definable 2000 mathematics subject classification primary 03c10 32b05 32b20 secondary 26e05

identification of hysteresis loops abstract a classical model representing several hysteresis phenomena as they appear in ferromagnetism and in porous media filtration e g is considered here it is demonstrated that the parameters of this model can be determined with any desired precision from suitable experiments we use the mathematical model of hysteresis as introduced by preisach who explained the magnetization curve and saturation loop of the material by a superposition of elementary rectangular loops a certain properly defined measure is responsible for the shape of the loop we describe a new algorithm which determines this measure completely our results are related to previous papers of biorci and pescetti extensive numerical computations conclude our investigations

a multiple criteria method for choosing among discrete alternatives abstract an interactive method is presented for solving a discrete alternative multiple criteria problem under certainty the method is based on the approach of zionts and wallenius a computer program embodying the method has been written and the results to date have been favorable extensions to the case in which some of the criteria are not readily quantifiable are also considered

limit cycle bifurcations near a 2 polycycle or double 2 polycycle of planar systems abstract in this paper we study the bifurcation problem of limit cycles for a near hamiltonian system near a 2 polycycle or double 2 polycycle we obtain new results on the number of limit cycles by using the method of stability changing of a homoclinic loop this method also provides a new way to find alien limit cycles especially we found some alien limit cycles of new type

can a linear value function explain choices an experimental study abstract we investigate in a simple bi criteria experimental study whether subjects are consistent with a linear value function while making binary choices many inconsistencies appeared in our experiment however the impact of inconsistencies on the linearity vs non linearity of the value function was minor moreover a linear value function seems to predict choices for bi criteria problems quite well this ability to predict is independent of whether the value function is diagnosed linear or not inconsistencies in responses did not necessarily change the original diagnosis of the form of the value function our findings have implications for the design and development of decision support tools for multiple criteria decision making problems

hardy type derivations on generalised series fields abstract we consider the valued field k r of generalised series with real coefficients and monomials in a totally ordered multiplicative group we investigate how to endow k with a series derivation that is a derivation that satisfies some natural properties such as commuting with infinite sums strong linearity and an infinite version of leibniz rule we characterise when such a derivation is of hardy type that is when it behaves like differentiation of germs of real valued functions in a hardy field we provide a necessary and sufficient condition for a series derivation of hardy type to be surjective

differential galois theory of linear difference equations we present a galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations with this we are able to reprove holder s theorem that the gamma function satisfies no polynomial differential equation and are able to give general results that imply for example that no differential relationship holds among solutions of certain classes of q hypergeometric equations

nonlinear scheme with high accuracy for nonlinear coupled parabolic hyperbolic system a nonlinear finite difference scheme with high accuracy is studied for a class of two dimensional nonlinear coupled parabolic hyperbolic system rigorous theoretical analysis is made for the stability and convergence properties of the scheme which shows it is unconditionally stable and convergent with second order rate for both spatial and temporal variables in the argument of theoretical results difficulties arising from the nonlinearity and coupling between parabolic and hyperbolic equations are overcome by an ingenious use of the method of energy estimation and inductive hypothesis reasoning the reasoning method here differs from those used for linear implicit schemes and can be widely applied to the studies of stability and convergence for a variety of nonlinear schemes for nonlinear pde problems numerical tests verify the results of the theoretical analysis particularly it is shown that the scheme is more accurate and faster than a previous two level nonlinear scheme with first order temporal accuracy

an approach for preference ranking of alternatives abstract an approach for determining the ranking of a set of alternatives is developed it is based on concepts of conditional probability the approach evaluates the set of alternatives and arrives at a preference ranking of either the entire set or a subset of available alternatives results of preliminary tests involving generated problems are favorable

ilyashenko algebras based on transserial asymptotic expansions abstract we construct a hardy field that contains ilyashenko s class of germs at of almost regular functions found in 12 as well as all log exp analytic germs this implies non oscillatory behaviour of almost regular germs with respect to all log exp analytic germs in addition each germ in this hardy field is uniquely characterized by an asymptotic expansion that is an le series as defined by van den dries et al 7 as these series generally have support of order type larger than the notion of asymptotic expansion itself needs to be generalized

zero one programming with multiple criteria abstract the study of multiple criteria problems is not new but during the last seventeen years more and more study has been done in this area the study of 0 1 multiple criteria problems started much later than the study of continuous multiple criteria problems some of the earliest works on 0 1 problems with multiple criteria was first presented in 1973 quite a few solution methods have been developed since then especially during the last seven years in this paper most of these methods and some applications will be treated

algebraic limit cycles in polynomial systems of differential equations using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4 5 and 6 we also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3 moreover we show that there are polynomial systems of differential equations of arbitrary degree that have algebraic limit cycles of degree 3 as well as give an example of a cubic polynomial system of differential equations with two algebraic limit cycles of degree 4

preference ranking of discrete multiattribute instances abstract this paper presents a procedure for preference ranking of discrete multiattribute instances based on the observations that 1 a difficult question could be answered in terms of answers to a sequence of easier ones 2 in general the decision maker s preferences can easily be expressed only relative to pairs of instances that differ over one attribute and 3 the decision maker is able and willing to express his uncertainty regarding a quantity via a subjective probability distribution the procedure is illustrated through a hypothetical situation

an interactive branch and bound algorithm for multiple criteria optimization we present a new interactive algorithm for multiple criteria optimization the algorithm is of the branch and bound type and differs from previous interactive algorithms in several ways first the field of application is wider because it applies to two important classes of multiple criteria decision problems those for which the feasible set is convex and those for which the feasible set is discrete secondly the algorithm does not require a great deal from the decision maker he is merely required to indicate his preference between two vectors whenever the algorithm so demands

evaluation of nondominated solution sets for k objective optimization problems an exact method and approximations abstract integrated preference functional ipf is a set functional that given a discrete set of points for a multiple objective optimization problem assigns a numerical value to that point set this value provides a quantitative measure for comparing different sets of points generated by solution procedures for difficult multiple objective optimization problems we introduced the ipf for bi criteria optimization problems in carlyle w m fowler j w gel e kim b 2003 quantitative comparison of approximate solution sets for bi criteria optimization problems decision sciences 34 1 63 82 as indicated in that paper the computational effort to obtain ipf is negligible for bi criteria problems for three or more objective function cases however the exact calculation of ipf is computationally demanding since this requires k 3 dimensional integration in this paper we suggest a theoretical framework for obtaining ipf for k 3 objectives the exact method includes solving two main sub problems 1 finding the optimality region of weights for all potentially optimal points and 2 computing volumes of k dimensional convex polytopes several different algorithms for both sub problems can be found in the literature we use existing methods from computational geometry i e triangulation and convex hull algorithms to develop a reasonable exact method for obtaining ipf we have also experimented with a monte carlo approximation method and compared the results to those with the exact ipf method

solving the discrete multiple criteria problem using convex cones an interactive method employing pairwise comparisons of attainable solutions is developed for solving the discrete deterministic multiple criteria problem assuming a single decision maker who has an implicit quasi concave increasing utility or value function the method chooses an arbitrary set of positive multipliers to generate a proxy composite linear objective function which is then maximized over the set of solutions the maximizing solution is compared with several solutions using pairwise judgments asked of the decision maker responses are used to eliminate alternatives using convex cones based on expressed preferences and then a new set of weights is found that satisfies the indicated preferences the requisite theory and proofs as well as a detailed numerical example are included in addition the results of some computational experiments to test the effectiveness of the method are described

the use of domains in multicriteria decision making abstract this paper describes a domain criterion for a multicriteria problem given decision maker it then outlines possibilities for aggregating individual preference structures in the case of multiple decision makers

a nonlinear iteration method for solving a two dimensional nonlinear coupled system of parabolic and hyperbolic equations a nonlinear iteration method for solving a class of two dimensional nonlinear coupled systems of parabolic and hyperbolic equations is studied a simple iterative finite difference scheme is designed the calculation complexity is reduced by decoupling the nonlinear system and the precision is assured by timely evaluation updating a strict theoretical analysis is carried out as regards the convergence and approximation properties of the iterative scheme and the related stability and approximation properties of the nonlinear fully implicit finite difference fifd scheme the iterative algorithm has a linear constringent ratio its solution gives a second order spatial approximation and first order temporal approximation to the real solution the corresponding nonlinear fifd scheme is stable and gives the same order of approximation numerical tests verify the results of the theoretical analysis the discrete functional analysis and inductive hypothesis reasoning techniques used in this paper are helpful for overcoming difficulties arising from the nonlinearity and coupling and lead to a related theoretical analysis for nonlinear fi schemes

pruning of decision alternatives in multiple criteria decision making based on the uta method for estimating utilities abstract a new interactive method for the progressive elimination of elements from a finite set of decision alternatives is proposed a sequence of alternatives is presented to the decision maker who places each new alternative presented in rank order relative to the earlier alternatives evaluated this ranking of elements in a subset of the decision space is used to eliminate other alternatives from further consideration the approach is broadly based on the uta method for utility assessment the emphasis is not however on assessing a utility function as such instead the class of utility functions is used to eliminate elements of the decision space when the optimality of such elements is inconsistent with the presumed properties of the utility function and the rank orderings given in this way the decision maker need only evaluate a relatively small subset of the decision space before all remaining alternatives are eliminated the new procedure is illustrated by two numerical examples

hopf takens bifurcations and centres the paper deals with hopf takens bifurcations both in generic families and in families containing centres attention goes to the relation between normal forms lyapunov coefficients and the bautin ideal in case the bautin ideal has only one generator we pay attention to the first non zero melnikov function

aspiration level approach in stochastic mcdm problems the paper considers a discrete stochastic multiple criteria decision making problem this problem is defined by a finite set of actions a a set of attributes x and a set of evaluations of actions with respect to attributes e in stochastic case the evaluation of each action with respect to each attribute takes form of a probability distribution thus the comparison of two actions leads to the comparison of two vectors of probability distributions in the paper a new procedure for solving this problem is proposed it is based on three concepts stochastic dominance interactive approach and preference threshold the idea of the procedure comes from the interactive multiple objective goal programming approach the set of actions is progressively reduced as the decision maker specifies additional requirements at the beginning the decision maker is asked to define preference threshold for each attribute next at each iteration the decision maker is confronted with the set of considered actions if the decision maker is able to make a final choice then the procedure ends otherwise he she is asked to specify aspiration level a didactical example is presented to illustrate the proposed technique

fdm for multi dimensional nonlinear coupled system of parabolic and hyperbolic equations a finite difference method fdm for multi dimensional nonlinear coupled system of parabolic and hyperbolic equations is studied nonlinear coupled property is fully considered variant techniques of discrete functional analysis such as summation by parts and deductive reasoning are used to make prior estimates the fd scheme is uniquely solvable and unconditionally stable with the approximation property of the scheme the convergence is obtained with a rate of o h 2 dt 2 order

an interactive approach for placing alternatives in preference classes abstract in this paper we consider the multiple criteria decision making problem of partitioning alternatives into preference classes we develop an interactive procedure based on the assumption that the decision maker dm has a linear utility function the approach requires the dm to place an alternative in a preference class from time to time based on the preference information derived from the dm s placement as well as from dominance and linearity we try to place other alternatives we present results on the performance of the algorithm applied to four different sets of alternatives

dual cone approach to convex cone dominance in multiple criteria decision making abstract in a paper published in management science in 1984 korhonen wallenius and zionts presented the idea and method based on convex cone dominance in the discrete multiple criteria decision making framework in our current paper we revisit the old idea from a new standpoint and provide the mathematical theory leading to a dual cone based approach to solving such problems our paper makes the old results computationally more tractable the results provided in the present paper also help extend the theory

bayesian decision process for cost efficient dynamic ranking via crowdsourcing rank aggregation based on pairwise comparisons over a set of items has a wide range of applications although considerable research has been devoted to the development of rank aggregation algorithms one basic question is how to efficiently collect a large amount of high quality pairwise comparisons for the ranking purpose because of the advent of many crowdsourcing services a crowd of workers are often hired to conduct pairwise comparisons with a small monetary reward for each pair they compare since different workers have different levels of reliability and different pairs have different levels of ambiguity it is desirable to wisely allocate the limited budget for comparisons among the pairs of items and workers so that the global ranking can be accurately inferred from the comparison results to this end we model the active sampling problem in crowdsourced ranking as a bayesian markov decision process which dynamically selects item pairs and workers to improve the ranking accuracy under a budget constraint we further develop a computationally efficient sampling policy based on knowledge gradient as well as a moment matching technique for posterior approximation experimental evaluations on both synthetic and real data show that the proposed policy achieves high ranking accuracy with a lower labeling cost

a hardy field extension of szemeredi s theorem abstract in 1975 szemeredi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions bergelson and leibman showed in 1996 that the common difference of the arithmetic progression can be a square a cube or more generally of the form p n where p n is any integer polynomial with zero constant term we produce a variety of new results of this type related to sequences that are not polynomial we show that the common difference of the progression in szemeredi s theorem can be of the form n where is any positive real number and x denotes the integer part of x more generally the common difference can be of the form a n where a x is any function that is a member of a hardy field and satisfies a x x k and a x x k 1 0 for some non negative integer k the proof combines a new structural result for hardy sequences techniques from ergodic theory and some recent equidistribution results of sequences on nilmanifolds

matrix estimation by universal singular value thresholding consider the problem of estimating the entries of a large matrix when the observed entries are noisy versions of a small random fraction of the original entries this problem has received widespread attention in recent times especially after the pioneering works of emmanuel candes and collaborators this paper introduces a simple estimation procedure called universal singular value thresholding usvt that works for any matrix that has a little bit of structure surprisingly this simple estimator achieves the minimax error rate up to a constant factor the method is applied to solve problems related to low rank matrix estimation blockmodels distance matrix completion latent space models positive definite matrix completion graphon estimation and generalized bradley terry models for pairwise comparison

feeling the bern adaptive estimators for bernoulli probabilities of pairwise comparisons we study methods for aggregating pairwise comparison data among a collection of n items with the goal of estimating the outcome probabilities for future comparisons working within a flexible model that only imposes a form of strong stochastic transitivity we introduce an adaptivity index which compares the risk of our estimator to that of an oracle over appropriate sub models where the oracle knows the specific sub model in the ground truth in addition to measuring the usual worst case risk of an estimator this adaptivity index also captures the extent to which the estimator adapts to instance specific difficulty relative to an oracle estimator first we propose a three step estimator termed count randomize least squares and show that it has adaptivity index upper bounded by sqrt n up to logarithmic factors we then show that conditional on the planted clique hypothesis no computationally efficient estimator can achieve an adaptivity index smaller than sqrt n second we show that a regularized least squares estimator can achieve a poly logarithmic adaptivity index thereby demonstrating a sqrt n gap between optimal and computationally achievable adaptivity finally we prove that the standard least squares estimator which is known to be optimally adaptive in several closely related problems fails to adapt in the context of estimating pairwise probabilities

monte carlo simulation techniques for group decision making with incomplete information abstract in this paper we deal with group decision making problems where several decision makers elicit their own preferences separately the decision makers preferences are quantified using a decision support system which admits incomplete information concerning the decision makers responses to the questions they are asked consequently each decision maker proposes classes of utility functions and attribute weight intervals for the different attributes we introduce an approach based on monte carlo simulation techniques for aggregating decision maker preferences that could be the starting point for a negotiation process if necessary the negotiation process would basically involve the decision maker tightening the imprecise component utilities and weights to output more meaningful results and achieve a consensus alternative we focus on how attribute weights and the component utilities associated with a consequence are randomly generated in the aggregation process taking into account the decision makers preferences i e their respective attribute weight intervals and classes of utility functions finally an application to the evaluation of intervention strategies for restoring a radionuclide contaminated lake illustrates the usefulness and flexibility of this iterative process

properties of the preisach model for hysteresis as shown by krasnosel skii the classical preisach model allows to construct a hysteresis operator wbetween spaces of real functions of time this construction via the definition of a measure mu in the so called preisach plane is recalled characterizations in terms of mu are given for several mapping and continuity properties of w in various function spaces for the invertibility of w and for the corresponding mapping and continuity properties of the inverse

growth properties of functions in hardy fields this paper continues the author s earlier work on the notion of rank in a hardy field further results are given on functions in hardy fields of finite rank including extensions of hardy s results on the rates of growth of his logarithmicoexponential functions 1 comparability classes nonzero elements a b of an ordered abelian group are called comparable if there are positive integers m n such that mlal iilt and n1 i1 lal nonzero elements f g of a hardy field k are accordingly called comparable if both lim f x and limxx g x are 0 or no and their rates of approach to 0 or xo or xo as x no are comparable that is if the nonzero elements v f v g of the value group v k of k are comparable as in 5 3 we assume some familiarity with 5 whose notation we employ more generally germs f g of continuous real valued functions on positive half lines in r which are nowhere zero on some half line and are such that lim x x f x and lim x x g x are either 0 or no or o will be called comparable if on some half line each of if l lgl is bounded above and below by suitable integral powers of the other comparability is an equivalence relation among such germs in particular comparability is an equivalence relation among all nonzero elements f of hardy fields with the property that lim5 f x is one of 0 oo or oo that is among all nonzero elements f such that v f 0 for f a germ of a nowhere zero continuous real valued function on a positive half line that approaches 0 oo or oo as x no denote the comparability class of f by cl f noting that f f 1 f and 1 f are comparable and that precisely one of these is infinitely increasing that is approaches no we define cl f cl v if and only if v u u v v v which holds if and only if v log u v log v received by the editors september 20 1985 1980 mathematics subject classification 1985 revision primary 34e05 41a60 secondary 12h05 13n05 26a12 research supported by national science foundation grant number dms 8303286 31987 american mathematical society 0002 9947 87 1 00 25 per page

estimation from pairwise comparisons sharp minimax bounds with topology dependence data in the form of pairwise comparisons arises in many domains including preference elicitation sporting competitions and peer grading among others we consider parametric ordinal models for such pairwise comparison data involving a latent vector w e rd that represents the qualities of the d items being compared this class of models includes the two most widely used parametric models the bradley terry luce btl and the thurstone models working within a standard minimax framework we provide tight upper and lower bounds on the optimal error in estimating the quality score vector w under this class of models the bounds depend on the topology of the comparison graph induced by the subset of pairs being compared via the spectrum of the laplacian of the comparison graph thus in settings where the subset of pairs may be chosen our results provide principled guidelines for making this choice finally we compare these error rates to those under cardinal measurement models and show that the error rates in the ordinal and cardinal settings have identical scalings apart from constant pre factors

interactive approaches for discrete alternative multiple criteria decision making with monotone utility functions in this paper we develop interactive approaches for the discrete alternative multiple criteria decision making problem we develop an algorithm that finds the most preferred alternative of a decision maker dm assuming only that the dm has a monotonic utility function the algorithm divides the criteria space into a number of smaller subspaces and then uses the ideal points of these subspaces to eliminate alternatives we also develop a more efficient version of the algorithm for the more restrictive case of a monotonic quasiconcave utility function we present favorable computational results in terms of the required number of pairwise comparisons for both versions of the algorithm we then develop a general algorithm that first identifies the type of the dm s utility function and then employs the approach that is compatible with the identified utility function type we also present computational results for the general algorithm

rank centrality ranking from pairwise comparisons the question of aggregating pairwise comparisons to obtain a global ranking over a collection of objects has been of interest for a very long time be it ranking of online gamers e g msr s trueskill system and chess players aggregating social opinions or deciding which product to sell based on transactions in most settings in addition to obtaining a ranking finding scores for each object e g player s rating is of interest for understanding the intensity of the preferences in this paper we propose rank centrality an iterative rank aggregation algorithm for discovering scores for objects or items from pairwise comparisons the algorithm has a natural random walk interpretation over the graph of objects with an edge present between a pair of objects if they are compared the score which we call rank centrality of an object turns out to be its stationary probability under this random walk to study the efficacy of the algorithm we consider the popular bradley terry luce btl model equ

testing the efficiency of pairwise comparison methods in discrete multiple criteria problems we test the efficiency of two pairwise comparison methods the methods studied are salminen s piecewise linear prospect theory plp method and the convex cone method by korhonen wallenius and zionts kwz the plp method is based on a piecewise linear difference function the kwz method assumes a quasi concave utility or value function these methods are tested using randomly generated data sets with 50 100 150 and 200 alternatives and from 2 up to 6 criteria the dm s preference statements are simulated using linear quadratic and chebycheff value functions and piecewise linear difference functions both methods lose their efficiency very rapidly with increasing number of criteria our results indicate that pairwise comparison methods are not as efficient as reported in literature

decision making with incomplete information abstract decision situations with incomplete information are characterized by a decision maker without a precisely defined stable preference structure by probability distributions not known completely or by an inexact evaluation of consequences within the paper a general framework for decision making with incomplete information is presented which shows how to solve problems from descriptive as well as prescriptive decision theory based on this framework an overview of existing methods which are particularly suitable for handling incomplete information is given

unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops abstract in this paper we present a complete study of quadratic 3 parameter unfoldings of some integrable system belonging to the classqr3 and having two centers and two unbounded heteroclinic loops we restrict to unfoldings that are transverse toqr3 obtain a versal bifurcation diagram and all global phase portraits including the precise number and configuration of the limit cycles it is proved that 3 is the maximal number of limit cycles surrounding a single focus and only the 1 1 configuration can occur in case of simultaneous nests of limit cycles essentially the proof relies on a careful analysis of a related non conservative abelian integral

cyclicity of several planar graphics and ensembles through three singular points without generic conditions abstract this paper investigates the number and distribution of the limit cycles bifurcated from several graphics and ensembles through a saddle node p 0 and two hyperbolic saddles p 1 and p 2 for the non generic cases of r 1 0 1 r 2 0 1 and r 1 0 1 r 2 0 1 where r 1 0 and r 2 0 are the hyperbolicity ratio of the saddles p 1 and p 2 respectively for the case of r 1 0 1 r 2 0 1 we suppose that the connection from p 0 to p 2 and the connection from p 0 to p 1 keep unbroken we prove that these graphics and ensembles are of finite cyclicity respectively moreover the cyclicity is linearly dependent on the order of the neutral saddle p 1 if p 2 is contractive and r 2 0 q we also show that the nearer r 2 0 is close to 1 the more the limit cycles are bifurcated for the case of r 1 0 1 r 2 0 1 we obtain that these graphics and ensembles are of finite cyclicity respectively if p 1 is of finite order and the hp connection from p 0 to p 2 keeps unbroken

transseries as germs of surreal functions we show that ecalle s transseries and their variants le and el series can be interpreted as functions from positive infinite surreal numbers to surreal numbers the same holds for a much larger class of formal series here called omega series omega series are the smallest subfield of the surreal numbers containing the reals the ordinal omega and closed under the exp and log functions and all possible infinite sums they form a proper class can be composed and differentiated and are surreal analytic the surreal numbers themselves can be interpreted as a large field of transseries containing the omega series but unlike omega series they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper

interactive algorithms for a broad underlying family of preference functions abstract in multi criteria decision making approaches it is typical to consider an underlying preference function that is assumed to represent the decision maker s preferences in this paper we introduce a broad family of preference functions that can represent a wide variety of preference structures we develop the necessary theory and interactive algorithms for both the general family of the preference functions and for its special cases the algorithms guarantee to find the most preferred solution point of the decision maker under the assumed conditions the convergence of the algorithms are achieved by progressively reducing the solution space based on the preference information obtained from the decision maker and the properties of the assumed underlying preference functions we first demonstrate the algorithms on a simple bi criteria problem with a given set of available points we also test the performances of the algorithms on three criteria knapsack problems and show that they work well

saddle quantities and applications abstract in this paper we make the connection between the theoretical study of the generalized homoclinic loop bifurcation ghb and the practical computational aspects for this purpose we first compare the dulac normal form with the joyal normal form these forms were both used to prove the ghb theorem but the second one is far more practical from the algorithmic point of view we then show that the information carried by these normal forms can be computed in a much simpler way using what we shall call dual lyapunov constants the coefficients of a normal form or the dual lyapunov quantities are particular cases of what we shall call saddle quantities we calculate the saddle quantities for quadratic systems and we show that no more than three limit cycles can appear in a homoclinic loop bifurcation we also study the homoclinic loop bifurcation of order 5 appearing in a 6 parameter family close to a hamiltonian system to our knowledge this is the first time that one can find a complete description of a ghb of such high order finally we calculate the saddle quantities for a symmetric cubic vector field and we deduce a bound for the number of limit cycles that appear in a ghb

limit cycle bifurcations by perturbing a class of integrable systems with a polycycle abstract in this paper we deal with the problem of limit cycle bifurcation near a 2 polycycle or 3 polycycle for a class of integrable systems by using the first order melnikov function we first get the formal expansion of the melnikov function corresponding to the heteroclinic loop and then give some computational formulas for the first coefficients of the expansion based on the coefficients we obtain a lower bound for the maximal number of limit cycles near the polycycle as an application of our main results we consider quadratic integrable polynomial systems obtaining at least two limit cycles

computer assisted techniques for the verification of the chebyshev property of abelian integrals abstract we develop techniques for the verification of the chebyshev property of abelian integrals these techniques are a combination of theoretical results analysis of asymptotic behavior of wronskians and rigorous computations based on interval arithmetic we apply this approach to tackle a conjecture formulated by dumortier and roussarie in f dumortier r roussarie birth of canard cycles discrete contin dyn syst 2 2009 723 781 which we are able to prove for q 2

h fields and their liouville extensions we introduce h fields as ordered differential fields of a certain kind hardy fields extending mathbb r as well as the field of logarithmic exponential series over mathbb r are h fields we study liouville extensions in the category of h fields as a step towards a model theory of h fields the main result is that an h field has at most two liouville closures

hardy type derivations on fields of exponential logarithmic series abstract we consider the valued field k r of formal series with real coefficients and monomials in a totally ordered multiplicative group we investigate how to endow k with a logarithm l which satisfies some natural properties such as commuting with infinite products of monomials we studied derivations on k kuhlmann and matusinski in press km10 here we investigate compatibility conditions between the logarithm and the derivation i e when the logarithmic derivative is the derivative of the logarithm we analyze sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives in kuhlmann 2000 kuh00 the first author described the exponential closure k el of k l here we show how to extend such a log compatible derivation on k to k el

an efficient nonlinear iteration scheme for nonlinear parabolic hyperbolic system abstract a nonlinear iteration method named the picard newton iteration is studied for a two dimensional nonlinear coupled parabolic hyperbolic system it serves as an efficient method to solve a nonlinear discrete scheme with second spatial and temporal accuracy the nonlinear iteration scheme is constructed with a linearization discretization approach through discretizing the linearized systems of the original nonlinear partial differential equations it can be viewed as an improved picard iteration and can accelerate convergence over the standard picard iteration moreover the discretization with second order accuracy in both spatial and temporal variants is introduced to get the picard newton iteration scheme by using the energy estimate and inductive hypothesis reasoning the difficulties arising from the nonlinearity and the coupling of different equation types are overcome it follows that the rigorous theoretical analysis on the approximation of the solution of the picard newton iteration scheme to the solution of the original continuous problem is obtained which is different from the traditional error estimate that usually estimates the error between the solution of the nonlinear discrete scheme and the solution of the original problem moreover such approximation is independent of the iteration number numerical experiments verify the theoretical result and show that the picard newton iteration scheme with second order spatial and temporal accuracy is more accurate and efficient than that of first order temporal accuracy

alien limit cycles in abel equations abstract the aim of this paper is to study the existence of limit cycles for a family of generalized abel equations x a t x m b t x n m n 2 under certain assumptions it is proved that there exists a non trivial limit cycle this limit cycle has the characteristic that it arises from neither a hopf bifurcation nor a perturbation of periodic orbits in a period annulus around the centre at the origin