sl 3 r as the group of symmetry transformations for all one dimensional linear systems ii realizations of the lie algebra the two dimensional space time realizations of the lie algebra of sl 3 r are obtained when the group acts as the maximal point symmetry group of any given one dimensional newtonian linear system it is shown that these realizations are isomorphic with the realization of the lie algebra of the projective group in the plane next an active point of view is introduced and sl 3 r is interpreted as a group of mappings that transform one admissible world line of the system into another thus a new mechanical realization of the sl 3 r algebra comes to the fore some miscellaneous examples are included

an axiomatic approach to the definition of the entropy of a discrete choquet capacity to extend the classical shannon entropy to nonadditive measures marichal recently introduced the concept of generalized entropy for discrete choquet capacities we provide a first axiomatization of this new concept on the basis of three axioms the symmetry property a boundary condition for which the entropy reduces to the shannon entropy and a generalized version of the well known recursivity property we also show that this generalized entropy fulfills several properties considered as requisites for defining an entropy like measure lastly we provide an interpretation of it in the framework of aggregation by the discrete choquet integral

infinitesimal symmetry transformations of some one dimensional linear systems the converse problem of similarity analysis is solved in general for the infinitesimal symmetry transformations of any given inhomogeneous ordinary differential equation of the second order x f2 t x f1 t x f0 t the completely general associated lie algebra is obtained for equations of this kind which structure constants depend only on the chosen set of initial values f0 0 f1 0 f2 0 and f2 0 the infinitesimal elements of the dynamical group of a newtonian one dimensional linear system are also discussed and some miscellaneous examples are given

determination of weights of interacting criteria from a reference set abstract in this paper we present a model allowing to determine the weights related to interacting criteria this is done on the basis of the knowledge of a partial ranking over a reference set of alternatives prototypes a partial ranking over the set of criteria and a partial ranking over the set of interactions between pairs of criteria

a group theoretical identification of integrable cases of the lienard type equation x f x x g x 0 i equations having nonmaximal number of lie point symmetries we carry out a detailed lie point symmetry group classification of the lienard type equation x f x x g x 0 where f x and g x are arbitrary smooth functions of x we divide our analysis into two parts in the present first part we isolate equations that admit lesser parameter lie point symmetries namely one two and three parameter symmetries and in the second part we identify equations that admit maximal eight parameter lie point symmetries in the former case the invariant equations form a family of integrable equations and in the latter case they form a class of linearizable equations under point transformations further we prove the integrability of all the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent hamiltonians several of these equations are being identified for the first time from the group theoretical analysis

minimum classification error training for choquet integrals with applications to landmine detection a novel algorithm for discriminative training of choquet integral based fusion operators is described fusion is performed by choquet integration of classifier outputs with respect to fuzzy measures the fusion operators are determined by the parameters of fuzzy measures these parameters are found by minimizing a minimum classification error mce objective function the minimization is performed with respect to a special class of measures the sugeno lambda measures an analytic expression is derived for the gradient of the choquet integral with respect to the sugeno lambda measure the new algorithm is applied to a landmine detection problem and compared to previous techniques

smooth attractors have zero thickness abstract a finite dimensional global attractor a can be embedded using some linear map l into a euclidean space r k of sufficiently high dimension the holder exponent of l 1 depends upon k and upon a the thickness exponent of a we show that global attractors which are uniformly bounded in the sobolev spaces h s for all s 0 have a 0 it follows using a result of b r hunt and v y kaloshin that the holder constant of the inverse of a typical linear embedding into r k or rank k orthogonal projection can be chosen arbitrarily close to 1 if k is large enough

extension of the fuzzy integral for general fuzzy set valued information the fuzzy integral fi is an extremely flexible aggregation operator it is used in numerous applications such as image processing multicriteria decision making skeletal age at death estimation and multisource e g feature algorithm sensor and confidence fusion to date a few works have appeared on the topic of generalizing sugeno s original real valued integrand and fuzzy measure fm for the case of higher order uncertain information both integrand and measure for the most part these extensions are motivated by and are consistent with zadeh s extension principle ep namely existing extensions focus on fuzzy number fn i e convex and normal fuzzy set fs valued integrands herein we put forth a new definition called the generalized fi gfi and efficient algorithm for calculation for fs valued integrands in addition we compare the gfi numerically and theoretically with our non ep based fi extension called the nondirect fi ndfi examples are investigated in the areas of skeletal age at death estimation in forensic anthropology and multisource fusion these applications help demonstrate the need and benefit of the proposed work in particular we show there is not one supreme technique instead multiple extensions are of benefit in different contexts and applications

on the extension of pseudo boolean functions for the aggregation of interacting criteria abstract the paper presents an analysis on the use of integrals defined for non additive measures or capacities as the choquet and the sipos integral and the multilinear model all seen as extensions of pseudo boolean functions and used as a means to model interaction between criteria in a multicriteria decision making problem the emphasis is put on the use besides classical comparative information of information about difference of attractiveness between acts and on the existence for each point of view of a neutral level allowing to introduce the absolute notion of attractive or repulsive act it is shown that in this case the sipos integral is a suitable solution although not unique properties of the sipos integral as a new way of aggregating criteria are shown with emphasis on the interaction among criteria

on the behavior of the lorenz equation backward in time abstract the sets of solutions to the lorenz equations that exist backward in time and are bounded at an exponential rate determined by the eigenvalues of the linear part of the equation are examined the set associated with the middle eigenvalue is shown to project surjectively onto a plane thereby providing a lower estimate for its dimension specific bounds are also found for a cone containing this set

on a leray α model of turbulence in this paper we introduce and study a new model for three dimensional turbulence the leray model this model is inspired by the lagrangian averaged navier stokes model of turbulence also known navier stokes model or the viscous camassa holm equations as in the case of the lagrangian averaged navier stokes model the leray model compares successfully with empirical data from turbulent channel and pipe flows for a wide range of reynolds numbers we establish here an upper bound for the dimension of the global attractor the number of degrees of freedom of the leray model of the order of l l d 12 7 where l is the size of the domain and l d is the dissipation length scale this upper bound is much smaller than what one would expect for three dimensional models i e l l d 3 this remarkable result suggests that the leray model has a great potential to become a good sub grid scale large eddy simulation model of turbulence we support this observation by studying analytically and computationally the energy spectrum and show that in addition to the usual k 5 3 kolmogorov power law the inertial range has a steeper power law spectrum for wavenumbers larger than 1 finally we propose a prandtl like boundary layer model induced by the leray model and show a very good agreement of this model with empirical data for turbulent boundary layers

on the dual cascade in two dimensional turbulence we study the dual cascade scenario for two dimensional turbulence driven by a spectrally localized forcing applied over a finite wavenumber range kmin kmax with kmin 0 such that the respective energy and enstrophy injection rates and satisfy k 2 k 2 the classical kraichnan leith batchelor paradigm based on the simultaneous conservation of energy and enstrophy and the scaleselectivity of the molecular viscosity requires that the domain be unbounded in both directions for two dimensional turbulence either in a doubly periodic domain or in an unbounded channel with a periodic boundary condition in the acrosschannel direction a direct enstrophy cascade is not possible in the usual case where the forcing wavenumber is no greater than the geometric mean of the integral and dissipation wavenumbers constant spectral slopes must satisfy 5 and 8 where is the asymptotic slope of the range of wavenumbers lower higher than the forcing wavenumber the influence of a large scale dissipation on the realizability of a dual cascade is analyzed we discuss the consequences for numerical simulations attempting to mimic the classical unbounded picture in a bounded domain

universal bounds on the attractor of the navier stokes equation in the energy enstrophy plane we continue our examination of the projection of the global attractor of the two dimensional navier stokes equations in a normalized dimensionless energy enstrophy plane in this plane the attractor must lie above the line of slope 1 through the origin an immediate consequence of the poincare inequality we showed in dascaliuc et al j dyn differ equ 17 629 649 2005 that the attractor must also lie below a parabola the square root curve here we optimize this parabola to obtain a sharper bound these bounds are universal they are independent of viscosity domain size and strength of the force the effectiveness of the bounds is demonstrated on galerkin approximations of modest size up to 1372 modes

on the complete symmetry group of the classical kepler system a rather strong concept of symmetry is introduced in classical mechanics in the sense that some mechanical systems can be completely characterized by the symmetry laws they obey accordingly a complete symmetry group realization in mechanics must be endowed with the following two features 1 the group acts freely and transitively on the manifold of all allowed motions of the system 2 the given equations of motion are the only ordinary differential equations that remain invariant under the specified action of the group this program is applied successfully to the classical kepler problem since the complete symmetry group for this particular system is here obtained the importance of this result for the quantum kinematic theory of the kepler system is emphasized

stochastic multiobjective acceptability analysis for the choquet integral preference model and the scale construction problem abstract the choquet integral preference model is adopted in multiple criteria decision aiding mcda to deal with interactions between criteria while the stochastic multiobjective acceptability analysis smaa is an mcda methodology considered to take into account uncertainty or imprecision on the considered data and preference parameters in this paper we propose to combine the choquet integral preference model with the smaa methodology in order to get robust recommendations taking into account all parameters compatible with the preference information provided by the decision maker dm in case the criteria are on a common scale one has to elicit only a set of non additive weights technically a capacity compatible with the dm s preference information instead if the criteria are on different scales besides the capacity one has to elicit also a common scale compatible with the preferences given by the dm our approach permits to explore the whole space of capacities and common scales compatible with the dm s preference information

random generation of k interactive capacities abstract the theory of capacities provides powerful formal methodology to account for criteria dependencies in multiple criteria decision problems the discrete choquet and sugeno integrals aggregate criteria valuations accounting for criteria synergies and redundancies we address an important problem of randomly generating capacities of special classes for simulation studies and for capacity learning through evolutionary algorithms we discuss two efficient methods suitable for k interactive capacities the results are supported by the extensive numerical evidence and provide a useful tool for large scale simulations

minimum variance capacity identification abstract in the framework of multi criteria decision making whose aggregation process is based on the choquet integral we present a maximum entropy like method enabling to determine if it exists the least specific capacity compatible with the initial preferences of the decision maker the proposed approach consists in solving a strictly convex quadratic program whose objective function is equivalently either the opposite of a generalized entropy measure or the variance of the capacity the application of the proposed approach is illustrated on two examples

the application of fuzzy integrals in multicriteria decision making abstract this paper presents a synthesis on the application of fuzzy integrals as an innovative tool for criteria aggregation in decision problems the main point is that fuzzy integrals are able to model interaction between criteria in a flexible way the methodology has been elaborated mainly in japan and has been applied there successfully in various fields such as design reliability evaluation of goods etc it seems however that this technique is still very little known in europe it is one of the aims of this review to disseminate this emerging technology in many industrial fields

tolerant or intolerant character of interacting criteria in aggregation by the choquet integral abstract in many multi criteria decision making problems the decision criteria present some interaction whose nature may vary from one situation to another for example some criteria may be statistically correlated thus making them somewhat redundant or opposed some others may be somewhat substitutive or complementary depending on the behavior of the decision maker some others may be decisive in the sense that the global score of any alternative obtained by aggregation is bounded by the partial score along one of them in this paper we analyze this latter form of interaction when a criterion bounds the global score from above it is called a blocker or a veto due to its rather intolerant character when it bounds the global score from below it is then called a pusher or a favor we thus investigate the tolerance of criteria or equivalently the tolerance of the weighted aggregation operator here the choquet integral that is used to aggregate criteria more specifically we propose axiomatically indices to appraise the extent to which each criterion behaves like a veto or a favor in the aggregation by the choquet integral previous to this we also propose global tolerance degrees measuring the extent to which the choquet integral is conjunctive or disjunctive

sl 3 r as the group of symmetry transformations for all one dimensional linear systems the converse problem of similarity analysis is solved in general for the finite symmetry transformations of any inhomogeneous ordinary linear differential equation of the second order x f2 t x f1 t x f0 t the eight parameter realizations of the symmetry group are obtained in the form f 1p2 f where f stands for transformations of t x that depend exclusively on the fundamental solutions of the equation and where p2 is an arbitrary projective transformation in the plane thus it is shown that the full point symmetry group corresponds to sl 3 r indeed without recourse to the lie algebra also a technique is obtained for calculating the finite point symmetry realization of sl 3 r for any given one dimensional linear system some miscellaneous examples are given

parametric conditions for a monotone tsk fuzzy inference system to be an n ary aggregation function despite the popularity and practical importance of the fuzzy inference system fis the use of an fis model as an n ary aggregation function which is characterized by both the monotonicity and boundary properties is yet to be established this is because research on ensuring that fis models satisfy the monotonicity property i e monotone fis is relatively new not to mention the additional requirement of satisfying the boundary property the aim of this paper therefore is to establish the parametric conditions for the takagi sugeno kang tsk fis model to operate as an n ary aggregation function hereafter denoted as n tsk fis via the specifications of fuzzy membership functions fmfs and fuzzy rules an absorption property with fuzzy rules interpretation is outlined and the use of n tsk fis as a uni norm is explained exploiting the established parametric conditions a framework for which an n tsk fis model can be constructed from data samples is formulated and analyzed along with a number of remarks synthetic data sets and a benchmark example on education assessment are presented and discussed to be best of the authors knowledge this study serves as the first use of the tsk fis model as an n ary aggregation function

the midweight method to measure attitudes toward risk and ambiguity this paper introduces a parameter free method for measuring the weighting functions of prospect theory and rank dependent utility these weighting functions capture risk attitudes subjective beliefs and ambiguity attitudes our method called the midweight method is based on a convenient way to obtain midpoints in the weighting function scale it can be used both for risk known probabilities and for uncertainty unknown probabilities the resulting integrated treatment of risk and uncertainty is particularly useful for measuring ambiguity i e the difference between uncertainty and risk compared to existing methods to measure weighting functions and attitudes toward uncertainty and ambiguity our method is more efficient and can accommodate violations of expected utility under risk an experiment demonstrates the tractability of our method yielding plausible results such as ambiguity aversion for moderate and high likelihoods but ambiguity seeking for low likelihoods as predicted by ellsberg this paper was accepted by george wu decision analysis

preference learning using the choquet integral the case of multipartite ranking we propose a novel method for preference learning or more specifically learning to rank where the task is to learn a ranking model that takes a subset of alternatives as input and produces a ranking of these alternatives as output just like in the case of conventional classifier learning training information is provided in the form of a set of labeled instances with labels or say preference degrees taken from an ordered categorical scale this setting is known as multipartite ranking in the literature our approach is based on the idea of using the discrete choquet integral as an underlying model for representing ranking functions being an established aggregation function in fields such as multiple criteria decision making and information fusion the choquet integral offers a number of interesting properties that make it attractive from a machine learning perspective too the learning problem itself comes down to properly specifying the fuzzy measure on which the choquet integral is defined this problem is formalized as a margin maximization problem and solved by means of a cutting plane algorithm the performance of our method is tested on a number of benchmark datasets

inverse energy cascade in two dimensional turbulence deviations from gaussian behavior high resolution numerical simulations of stationary inverse energy cascade in two dimensional turbulence are presented deviations from gaussian behavior of velocity differences statistics are quantitatively investigated the level of statistical convergence is pushed enough to permit reliable measurement of the asymmetries in the probability distribution functions of longitudinal increments and odd order moments which bring the signature of the inverse energy flux no measurable intermittency corrections could be found in their scaling laws the seventh order skewness increases by almost two orders of magnitude with respect to the third thus becoming of order unity

the problem of collective identity in a fuzzy environment we define the problem of group identification in a fuzzy environment we concentrate on the case where the society is required to self determine the belongingness of each member to a specific group characterized by a single attribute in general terms this case consists of a collective identity issue that can be regarded as an aggregation problem of individual assessments within a group here we introduce the possibility that both the original assessments and the resulting output attach partial memberships to the collectivity for each potential member we propose relevant classes of rules and some are axiomatically characterized our new approach provides a way to circumvent classical impossibility results like kasher and rubinstein s

choquet fuzzy integral based hierarchical networks for decision analysis a choquet fuzzy integral based approach to hierarchical network implementation is investigated in this approach we generalized the fuzzy integral as an excellent component for decision analysis the generalization involves replacing the max or min operator in information aggregation with a fuzzy integral based neuron resulting in increased flexibility the characteristics of the choquet fuzzy integral are studied and a network based decision analysis framework is proposed the trainable hierarchical network can be implemented utilizing the fuzzy integral based neurons and connectives the training algorithms are derived and several examples given to illustrate the behaviors of the networks also we present a decision making experiment using the proposed network to learn appropriate functional relationships in the defective numeric fields detection domain

estimates on enstrophy palinstrophy and invariant measures for 2 d turbulence we construct semi integral curves which bound the projection of the global attractor of the 2 d navier stokes equations in the plane spanned by enstrophy and palinstrophy of particular interest are certain regions of the plane where palinstrophy dominates enstrophy previous work shows that if solutions on the global attractor spend a significant amount of time in such a region then there is a cascade of enstrophy to smaller length scales one of the main features of 2 d turbulence theory the semi integral curves divide the plane into regions having limited ranges for the direction of the flow this allows us to estimate the average time it would take for an intermittent solution to burst into a region of large palinstrophy we also derive a sharp universal upper bound on the average palinstrophy and show that it is achieved only for forces that admit statistical steady states where the nonlinear term is zero

some remarks on the lie group of point transformations for the harmonic oscillator some aspects of the symmetry group sl 3 r of the simple harmonic oscillator are revisited a realisation of the general element of the group in newtonian spacetime is obtained with the eight parameters included the group is shown to be sl 3 r in fact without recourse to the lie algebra finally in this way the group law of binary composition of the parameters is calculated easily in the group manifold

on the definition of penalty functions in data aggregation abstract in this paper we point out several problems in the different definitions and related results of penalty functions found in the literature then we propose a new standard definition of penalty functions that overcomes such problems some results related to averaging aggregation functions in terms of penalty functions are presented as the characterization of averaging aggregation functions based on penalty functions some examples are shown as the penalty functions based on spread measures which happen to be continuous we also discuss the definition of quasi penalty functions in order to deal with non monotonic or weakly directionally monotonic averaging functions

a review of methods for capacity identification in choquet integral based multi attribute utility theory applications of the kappalab r package the application of multi attribute utility theory whose aggregation process is based on the choquet integral requires the prior identification of a capacity the main approaches to capacity identification proposed in the literature are reviewed and their advantages and inconveniences are discussed all the reviewed methods have been implemented within the kappalab r package their application is illustrated on a detailed example

k order additive discrete fuzzy measures and their representation abstract in order to face with the complexity of discrete fuzzy measures we propose the concept of k orderadditive fuzzy measure including usual additive measures and fuzzy measures every discrete fuzzy measure is a k order additive fuzzy measure for a unique k a related topic of the paper is to introduce an alternative representation of fuzzy measures called the interaction representation which sets and extends in a common framework the shapley value and the interaction index proposed by murofushi and soneda

learning choquet integral based metrics for semisupervised clustering we consider an application of fuzzy measures to the problem of metric learning in semisupervised clustering we investigate the necessary and sufficient conditions on the underlying fuzzy measure that make the discrete choquet integral suitable for defining a metric as a byproduct we can obtain the analogous conditions for the ordered weighted averaging owa operators which constitute a special case we then generalize these results for power based choquet and owa operators we show that this metric learning problem can be formulated as a linear programming problem and specify the required sets of linear constraints we present the results of numerical experiments on artificial and real world datasets which illustrate the potential usefulness and limitations of this construction

contact symmetry algebras of scalar second order ordinary differential equations there is a natural link between contact symmetries symmetries that leave invariant the first order contact form and first integrals of a given second order ordinary differential equation it will be shown that the contact symmetry algebra of a general second order ordinary differential equation is infinite dimensional and it is generated by the functionally independent first integrals of the equation moreover the contact symmetry algebras of second order equations admitting one two three and eight point symmetries are obtained

non additive robust ordinal regression a multiple criteria decision model based on the choquet integral abstract within the multicriteria aggregation disaggregation framework ordinal regression aims at inducing the parameters of a decision model for example those of a utility function which have to represent some holistic preference comparisons of a decision maker dm usually among the many utility functions representing the dm s preference information only one utility function is selected since such a choice is arbitrary to some extent recently robust ordinal regression has been proposed with the purpose of taking into account all the sets of parameters compatible with the dm s preference information until now robust ordinal regression has been implemented to additive utility functions under the assumption of criteria independence in this paper we propose a non additive robust ordinal regression on a set of alternatives a whose utility is evaluated in terms of the choquet integral which permits to represent the interaction among criteria modelled by the fuzzy measures parameterizing our approach in our methodology besides holistic pairwise preference comparisons of alternatives from a subset of reference alternatives a a the dm is also requested to express the intensity of preference on pairs of alternatives from a and to supply pairwise comparisons on the importance of criteria and the sign and intensity of interaction among pairs of criteria the output of our approach defines a set of fuzzy measures capacities such that the corresponding choquet integral is compatible with the dm s preference information moreover using linear programming our decision model establishes two preference relations for any a b a the necessary weak preference relation if for all compatible fuzzy measures the utility of a is not smaller than the utility of b and the possible weak preference relation if for at least one compatible fuzzy measure the utility of a is not smaller than the utility of b

an axiomatic approach of the discrete choquet integral as a tool to aggregate interacting criteria the most often used operator to aggregate criteria in decision making problems is the classical weighted arithmetic mean however in many problems the criteria considered interact and a substitute to the weighted arithmetic mean has to be adopted we show that under rather natural conditions the discrete choquet integral is an adequate aggregation operator that extends the weighted arithmetic mean by taking into consideration of the interaction among criteria the axiomatic that supports the choquet integral is presented and an intuitive approach is proposed as well

on the global attractor of 2d incompressible turbulence with random forcing abstract this study revisits bounds on the projection of the global attractor in the energy enstrophy plane for 2d incompressible turbulence dascaliuc foias and jolly 2005 2010 in addition to providing more elegant proofs of some of the required nonlinear identities the treatment is extended from the case of constant forcing to the more realistic case of random forcing numerical simulations in particular often use a stochastic white noise forcing to achieve a prescribed mean energy injection rate the analytical bounds are demonstrated numerically for the case of white noise forcing

efficient multiple kernel classification using feature and decision level fusion kernel methods for classification is a well studied area in which data are implicitly mapped from a lower dimensional space to a higher dimensional space to improve classification accuracy however for most kernel methods one must still choose a kernel to use for the problem since there is in general no way of knowing which kernel is the best multiple kernel learning mkl is a technique used to learn the aggregation of a set of valid kernels into a single ideally superior kernel the aggregation can be done using weighted sums of the precomputed kernels but determining the summation weights is not a trivial task furthermore mkl does not work well with large datasets because of limited storage space and prediction speed in this paper we address all three of these multiple kernel challenges first we introduce a new linear feature level fusion technique and learning algorithm gamklp second we put forth three new algorithms defimkl degamkl and delsmkl for nonlinear fusion of kernels at the decision level to address mkl s storage and speed drawbacks we apply the nystrom approximation to the kernel matrices we compare our methods to a successful and state of the art technique called mkl group lasso mklgl and experiments on several benchmark datasets show that some of our proposed algorithms outperform mklgl when applied to support vector machine svm based classification however to no surprise there does not seem to be a global winner but instead different strategies that a user can employ experiments with our kernel approximation method show that we can routinely discard most of the training data and at least double prediction speed without sacrificing classification accuracy these results suggest that mkl based classification techniques can be applied to big data efficiently which is confirmed by an experiment using a large dataset

the choquet integral for the aggregation of interval scales in multicriteria decision making this paper addresses the question of which models fit with information concerning the preferences of the decision maker over each attribute and his preferences about aggregation of criteria interacting criteria we show that the conditions induced by these information plus some intuitive conditions lead to a unique possible aggregation operator the choquet integral

robust optimization of the choquet integral we study the problem of the choquet integral maximization for the case when preferences of the decision maker do not define a unique capacity but rather some convex set we introduce a robust version of the problem based on the minimax regret criterion and construct a global optimization algorithm it is shown that the decision rule based on regret minimization over a set of capacities is equivalent to maximization of the choquet integral with respect to a certain capacity belonging to this set we compare our method to various capacity identification approaches developed in the literature and provide a semantic interpretation in terms of information value

classification of heterogeneous fuzzy data by choquet integral with fuzzy valued integrand as a fuzzification of the choquet integral the defuzzified choquet integral with fuzzy valued integrand dcifi takes a fuzzy valued integrand and gives a crisp valued integration result in this paper the dcifi acts as a projection to project high dimensional heterogeneous fuzzy data to one dimensional crisp data to handle the classification problems involving different data forms such as crisp data interval values fuzzy numbers and linguistic variables simultaneously the nonadditivity of the signed fuzzy measure applied in the dcifi can represent the interaction among the measurements of features towards the discrimination of classes values of the signed fuzzy measure in the dcifi are considered to be unknown parameters which should be learned before the classifier is used to classify new data we have implemented a genetic algorithm ga based adaptive classifier learning algorithm to optimally learn the signed fuzzy measure values and the classified boundaries simultaneously the performance of our algorithm has been tested both on synthetic and real data the experimental results are satisfactory and outperform those of existing methods such as the fuzzy decision trees and the fuzzy neuro networks

data driven compression and efficient learning of the choquet integral the choquet integral chi is a parametric nonlinear aggregation function defined with respect to the fuzzy measure fm to date application of the chi has sadly been restricted to problems with relatively few numbers of inputs primarily as the fm has 2 n variables for n inputs and n 2 n 1 1 monotonicity constraints in return the community has turned to density based imputation e g sugeno lambda fm or the number of interactions fm variables are restricted e g k additivity herein we propose a new scalable data driven way to represent and learn the chi making learning computationally manageable for larger n first data supported variables are identified and used in optimization identification of these variables also allows us recognize future ill posed fusion scenarios chis involving variable subsets not supported by data second we outline an imputation function framework to address data unsupported variables third we present a lossless way to compress redundant variables and associated monotonicity constraints finally we outline a lossy approximation method to further compress the chi if when desired computational complexity analysis and experiments conducted on synthetic datasets with known fms demonstrate the effectiveness and efficiency of the proposed theory

statistical estimates for the navier stokes equations and the kraichnan theory of 2 d fully developed turbulence a mathematical formulation of the kraichnan theory for 2 d fully developed turbulence is given in terms of ensemble averages of solutions to the navier stokes equations a simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber up to a logarithmic correction this is followed by partial rigorous support for kraichnan s eddy breakup mechanism a rigorous estimate for the total energy is found to be consistent with kraichnan s theory finally it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case

quantum kinematics of the harmonic oscillator the formalism of non abelian quantum kinematics is applied to the newtonian symmetry group of the harmonic oscillator within the regular ray representation of the group the schrodinger operator as well as two other new invariant operators are obtained as casimir operators of the extended kinematic algebra superselection rules are then introduced which permit the identification and the explicit calculation of the physical states of the system next a complementary ray representation attached to the space time realization of the group casts the schrodinger operator into the familiar time dependent space time differential operator of the harmonic oscillator and thus by means of the superselection rules one obtains the time dependent schrodinger equation of the sytem finally the evaluation of a hurwitz invariant integral over the group manifold affords the well known feynman space time propagator t x t x of the simple harmonic oscillator everything comes out from the assumed symmetries of the system the whole approach is group theoretic and relativistic no classical analog is used in this quantization scheme

on the landau lifschitz degrees of freedom in 2 d turbulence we show that if the kraichnan theory of fully developed turbulence holds then the landau lifschitz degrees of freedom is bounded up to a logarithmic term by g1 2 where g is the grashof number

robustness of the inverse cascade in two dimensional turbulence we study quasisteady inverse cascades in unbounded and bounded two dimensional turbulence driven by time independent injection and dissipated by molecular viscosity it is shown that an inverse cascade that carries only a fraction r of the energy input to the largest scales requires the enstrophy range energy spectrum to be steeper than k ensuremath 5 ruling out a direct cascade unless 1 ensuremath r ensuremath ll 1 a direct cascade requires the presence of an inverse cascade that carries virtually all energy to the largest scales 1 ensuremath r ensuremath ll 1 these facts underlie the robustness of the kolmogorov kraichnan k ensuremath 5 3 inverse cascade which is readily observable in numerical simulations without an accompanying direct enstrophy cascade we numerically demonstrate an instance where the k ensuremath 5 3 inverse cascading range is realizable with 79 of the energy injection dissipated within the energy range and virtually all of the enstrophy dissipated in the vicinity of the forcing region as equilibrium is approached the respective logarithmic slopes ensuremath alpha and ensuremath beta of the ranges of wave numbers lower and higher than the forcing wave number satisfy ensuremath alpha ensuremath beta ensuremath approx 8 these results are consistent with recent theoretical predictions

inconsistency and non additive capacities the analytic hierarchy process in the framework of choquet integration we examine the ahp in the framework of choquet integration and we propose an extension of the standard ahp aggregation scheme on the basis of the shapley values associated with the criteria in our model a measure of dominance inconsistency between criteria is defined in terms of the totally inconsistent matrix associated with the main pairwise comparison matrix of the ahp the measure of dominance inconsistency is then used to construct a non additive capacity whose associated shapley values reduce to the standard ahp priority weights in the consistency case in the general inconsistency case however the extended aggregation scheme based on the shapley weighted mean tends to attenuate resp emphasize the priority weights of the criteria with higher resp lower average dominance inconsistency with respect to the other criteria

asymptotic analysis of the navier stokes equations abstract new bounds are established on the number of modes which determine the solutions of the navier stokes equations in two dimensions the best bound available at present is nearly proportional to the generalized grashof number defined in the paper and less than logarithmically dependent on the spatial structure or the shape of the force driving the flow to the extent than for the case of 2 dimensional rayleigh benard convection the generalized grashof number may be identified with the usual grashof number the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

sl 3 r as the group of symmetry transformations for all one dimensional linear systems iii equivalent lagrangian formalisms the sl 3 r theory of projective transformations of the plane is applied to the lagrangians of all one dimensional newtonian linear systems noether and non noether equivalent lagrangians as well as the associated noether and non noether constants of motion are thus obtained in a completely general and systematic way complete unification is achieved by this group theoretic approach to lagrangians of one dimensional linear systems

determining the fuzzy measures in multiple criteria decision aiding from the tolerance perspective abstract we consider multiple criteria decision aiding mcda in the case of interactions between criteria in dealing with interactions between criteria fuzzy measures and integrals have demonstrated great advantages nevertheless the determination of fuzzy measures has proven difficult because the capacities of not only single criterion but also all subsets of criteria need to be identified due to the value judgment essence of mcda the attitudes of the decision maker dm are typically modeled to identify fuzzy measures in this paper the tolerance attitudes of the dm which implies a direct requirement instead of partial preference are modeled with regard to the determination of fuzzy measures for the first time with two scales developed in this paper the dm can directly express the tolerance attitudes to certain criteria other than providing partial preference through pairwise comparison as a result it requires less prior knowledge and is more efficient to some extent further the inherent interacting mechanism of criteria under different tolerance attitudes is explored at last the tolerance attitudes are applied to the process of multiple criteria analysis using a choquet integral a classic student evaluation problem is given as an example the evaluation results are compared with additive models this paper not only provides a new inspiration to the determination of fuzzy measures but also improves the descriptive capacity of fuzzy measures to the real world