choquet integral with stochastic entries abstract we analyze the probabilistic features of the choquet integral with respect to a capacity where the inputs are random variables only a few papers deal with this issue we give two different formulas for the density function of the output when the input variables are independent and identically distributed random variables we also calculate the first moment of the choquet integral and we compare our results with the ones obtained in the literature which mainly concerns the cases where the common distribution of the input variables is either uniform or exponential

n point deformation of algebraic k3 surfaces we construct a set of noncommutative geometries by performing n point deformation of algebraic k3 surfaces first we consider two point deformation of algebraic k3 surfaces by performing algebraic deformation of a pair of commutative algebraic k3 surfaces in this case the moduli space of the noncommutative deformations is of dimension 19 the same as the moduli dimension of the complex deformations of commutative algebraic k3 surfaces then we extend this method to the n point case in the n point case the dimension of deformation moduli space becomes 19n n 1 2

spectra of wess zumino witten models with arbitrary simple groups we consider the wess zumino witten two dimensional sigma models with fields taking values in an arbitrary connected but not necessarily simply connected simple lie groupg the quantum states of the theory are realized geometrically as sections of a line bundle over the loop grouplg the action of the current algebra of the theory is decomposed into highest weight representations by explicit construction of the highest weight states this solves for the spectrum of the models as a by product we obtain modular invariant partition functions of the theory on tori the present paper extends the results of 7 where the casesg su 2 andso 3 were treated

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

quantum isometry groups examples and computations in this follow up of 4 where the quantum isometry group of a noncommutative manifold has been defined we explicitly compute such quantum groups for a number of classical as well as noncommutative manifolds including the spheres and the tori it is also proved that the quantum isometry group of an isospectral deformation of a classical or noncommutative manifold is a suitable deformation of the quantum isometry group of the original undeformed manifold

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

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

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

a lie algebraic approach to the local index theorem on compact homogeneous spaces abstract using a k theory point of view bott related the atiyah singer index theorem for elliptic operators on compact homogeneous spaces to the weyl character formula this article explains how to prove the local index theorem for compact homogenous spaces using lie algebra methods the method follows in outline the proof of the local index theorem due to berline and vergne but the use of kostant s cubic dirac operator in place of the riemannian dirac operator leads to substantial simplifications an important role is also played by the quantum weil algebra of alekseev and meinrenken

a theoretical study on six classifier fusion strategies we look at a single point in feature space two classes and l classifiers estimating the posterior probability for class spl omega sub 1 assuming that the estimates are independent and identically distributed normal or uniform we give formulas for the classification error for the following fusion methods average minimum maximum median majority vote and oracle

variable step size control in the numerical solution of stochastic differential equations we introduce a variable step size method for the numerical approximation of pathwise solutions to stochastic differential equations sdes the method which is dependent on a representation of bro

numerical methods for stochastic differential equations stochastic differential equations sde s play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability a general strategy for developing accurate and efficient schemes for solving stochastic equations is outlined here high order numerical methods are developed for the integration of stochastic differential equations with strong solutions we demonstrate the accuracy of the resulting integration schemes by computing the errors in approximate solutions for sde s which have known exact solutions

comment on numerical methods for stochastic differential equations wilkie phys rev e 70 017701 2004 used a heuristic approach to derive runge kutta based numerical methods for stochastic differential equations based on methods used for solving ordinary differential equations the aim was to follow solution paths with high order we point out that this approach is invalid in the general case and does not lead to high order methods we warn readers against the inappropriate use of deterministic calculus in a stochastic setting

the action functional in non commutative geometry we establish the equality between the restriction of the adler manin wodzicki residue or non commutative residue to pseudodifferential operators of order n on ann dimensional compact manifoldm with the trace which j dixmier constructed on the macaev ideal we then use the latter trace to recover the yang mills interaction in the context of non commutative differential geometry

hermitian symmetric superspaces of type iv a construction of the hermitian symmetric superspaces which are the supermanifold analogs of the cartan domains of type iv is presented herein natural generalizations of the jordan triple product and bergman operator for the superdomains are defined and their properties are studied

geometry on the quantum heisenberg manifold abstract a class of c algebras called quantum heisenberg manifolds were introduced by rieffel in comm math phys 122 1989 531 as strict deformation quantization of heisenberg manifolds using the ergodic action of heisenberg group we construct a family of spectral triples it is shown that associated kasparov modules are homotopic we also show that they induce cohomologous elements in entire cyclic cohomology the space of connes derham forms have been explicitly calculated then we characterize torsionless unitary connections and show that there does not exist a connection that is simultaneously torsionless and unitary explicit examples of connections are produced with negative scalar curvature this part illustrates computations involving some of the concepts introduced in frohlich et al comm math phys 203 1999 119 for which to the best of our knowledge no infinite dimensional example is known other that the noncommutative torus

fuzzy versus nonfuzzy in combining classifiers designed by boosting boosting is recognized as one of the most successful techniques for generating classifier ensembles typically the classifier outputs are combined by the weighted majority vote the purpose of this study is to demonstrate the advantages of some fuzzy combination methods for ensembles of classifiers designed by boosting we ran two fold cross validation experiments on six benchmark data sets to compare the fuzzy and nonfuzzy combination methods on the fuzzy side we used the fuzzy integral and the decision templates with different similarity measures on the nonfuzzy side we tried the weighted majority vote as well as simple combiners such as the majority vote minimum maximum average product and the naive bayes combination in our experiments the fuzzy combination methods performed consistently better than the nonfuzzy methods the weighted majority vote showed a stable performance though slightly inferior to the performance of the fuzzy combiners

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

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

quantum group of orientation preserving riemannian isometries we formulate a quantum group analogue of the group of orientation preserving riemannian isometries of a compact riemannian spin manifold more generally of a possibly r twisted and of compact type spectral triple the main advantage of this formulation which is directly in terms of the dirac operator is that it does not need the existence of any good laplacian as in our previous works on quantum isometry groups several interesting examples including those coming from rieffel type deformation as well as the equivariant spectral triples on su 2 and s c2 are discussed

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

quantum riemann surfaces i the unit disc we construct a non commutative algebra c mu bar u which is a quantum deformation of the algebra of continuous functions on the closed unit disc bar u c mu bar u is generated by the toeplitz operators on a suitable hilbert space of holomorphic functions onu

exponential integrators for stochastic schrodinger equations we present a class of exponential integrators to compute solutions of the stochastic schrodinger equations arising from the modeling of open quantum systems to be able to implement the methods within the same framework as the deterministic counterpart we express the solution using kunita s representation with appropriate truncations the solution operator can be written as matrix exponentials which can be efficiently implemented by the krylov subspace projection the accuracy is examined in terms of the strong convergence by comparing trajectories and in terms of the weak convergence by comparing the density matrix operators we show that the local accuracy can be further improved by introducing third order commutators in the exponential the effectiveness of the proposed methods is tested using the example from di ventra et al j phys condens matter 16 8025 2004 jcomel0953 898410 1088 0953 8984 16 45 024

graded differential geometry of graded matrix algebras we study the graded derivation based noncommutative differential geometry of the z2 graded algebra m n m of complex n m n m matrices with the usual block matrix grading for n m beside the infinite dimensional algebra of graded forms the graded cartan calculus graded symplectic structure graded vector bundles graded connections and curvature are introduced and investigated in particular we prove the universality of the graded derivation based first order differential calculus and show that m n m is a noncommutative graded manifold in a stricter sense there is a natural body map and the cohomologies of m n m and its body coincide as in the case of ordinary graded manifolds

on the moments and distribution of discrete choquet integrals from continuous distributions we study the moments and the distribution of the discrete choquet integral when regarded as a real function of a random sample drawn from a continuous distribution since the discrete choquet integral includes weighted arithmetic means ordered weighted averaging functions and lattice polynomial functions as particular cases our results encompass the corresponding results for these aggregation functions after detailing the results obtained in j l marichal i kojadinovic distribution functions of linear combinations of lattice polynomials from the uniform distribution statistics probability letters 78 2008 985 991 in the uniform case we present results for the standard exponential case show how approximations of the moments can be obtained for other continuous distributions such as the standard normal and elaborate on the asymptotic distribution of the choquet integral the results presented in this work can be used to improve the interpretation of discrete choquet integrals when employed as aggregation functions

supersymmetric quantum theory and non commutative geometry classical differential geometry can be encoded in spectral data such as connes spectral triples involving supersymmetry algebras in this paper we formulate non commutative geometry in terms of supersymmetric spectral data this leads to generalizations of connes non commutative spin geometry encompassing non commutative riemannian symplectic complex hermitian and hyper kahler geometry a general framework for non commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples in particular the non commutative torus and the non commutative 3 sphere are studied in some detail

fuzzy integral in multicriteria decision making abstract this paper presents a synthesis on the use of fuzzy integral as an aggregation operator in multicriteria decision making definitions essential properties are given and compared to those of usual aggregation operators a quick survey on applications is given

order conditions of stochastic runge kutta methods by b series in this paper general order conditions and a global convergence proof are given for stochastic runge kutta methods applied to stochastic ordinary differential equations sodes of stratonovich type this work generalizes the ideas of b series as applied to deterministic ordinary differential equations odes to the stochastic case and allows a completely general formalism for constructing high order stochastic methods either explicit or implicit some numerical results will be given to illustrate this theory

the non commutative weil algebra for any compact lie group g together with an invariant inner product on its lie algebra g we define the non commutative well algebra w g as a tensor product of the universal enveloping algebra u g and the clifford algebra cl 8 just like the usual we

high strong order methods for non commutative stochastic ordinary differential equation systems and the magnus formula in recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations sodes as sodes are often more appropriate than their deterministic counterparts in many modelling situations however unlike the deterministic case numerical methods for sodes are considerably less sophisticated due to the difficulty in representing the possibly large number of random variable approximations to the stochastic integrals although burrage and burrage high strong order explicit runge kutta methods for stochastic ordinary differential equations applied numerical mathematics 22 1996 81 101 were able to construct strong local order 1 5 stochastic runge kutta methods for certain cases it is known that all extant stochastic runge kutta methods suffer an order reduction down to strong order 0 5 if there is non commutativity between the functions associated with the multiple wiener processes this order reduction down to that of the euler maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques an additional difficulty in solving sodes arises even in the linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple wiener processes thus in this present paper first the work of magnus on the exponential solution of differential equations for a linear operator communications on pure and applied mathematics 7 1954 649 673 applied to deterministic non commutative linear problems will be applied to non commutative linear sodes and methods of strong order 1 5 for arbitrary linear non commutative sode systems will be constructed hence giving an accurate approximation to the general linear problem secondly for general nonlinear non commutative systems with an arbitrary number d of wiener processes it is shown that strong local order i runge kutta methods with d 1 stages can be constructed by evaluated a set of lie brackets as well as the standard function evaluations a method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of wiener processes finally some numerical results are presented which illustrate the efficacy of these approaches c 1999 elsevier science b v all rights reserved

gravity in non commutative geometry we study general relativity in the framework of non commutative differential geometry as a prerequisite we develop the basic notions of non commutative riemannian geometry including analogues of riemannian metric curvature and scalar curvature this enables us to introduce a generalized einstein hilbert action for non commutative riemannian spaces as an example we study a space time which is the product of a four dimensional manifold by a two point space using the tools of non commutative riemannian geometry and derive its generalized einstein hilbert action in the simplest situation where the riemannian metric is taken to be the same on the two copies of the manifold one obtains a model of a scalar field coupled to einstein gravity this field is geometrically interpreted as describing the distance between the two points in the internal space

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

error bounds in divided difference expansions a probabilistic perspective we give error bounds for the remainder term as well as sharp upper and lower bounds for the leading term in divided difference expansions a main feature of this paper is the use of a probabilistic approach in the spirit of ignatov and kaishev z g ignatov v k kaishev b splines and linear combinations of uniform order statistics math res cent tsr 2817 univ of wisconsin madison 1985 z g ignatov v k kaishev a probabilistic interpretation of multivariate b splines and some applications serdica 15 1989 91 99 and karlin et al s karlin c a micchelli y rinott multivariate splines a probabilistic perspective j multivariate anal 20 1986 69 90 based on the representation of divided differences as mathematical expectations involving linear combinations of uniform order statistics

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

numerical treatment of stochastic differential equations we define general runge kutta approximations for the solution of stochastic differential equations sde these approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift the explicit form of the correction term is given concerning the order of convergence we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than o h 3 this order is attained e g by the stochastic analogue of heun s method in the n dimensional case the highest order of convergence is in general only o h 2 attained by euler s method the order o h 3 can only be reached if nabla x sigma r sigma s nabla x sigma s sigma r for the diffusion matrix

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

noncommutative cohomological field theory and gms soliton we show that it is possible to construct a quantum field theory that is invariant under the translation of the noncommutative parameter this is realized in a noncommutative cohomological field theory as an example a noncommutative cohomological scalar field theory is constructed and its partition function is calculated the partition function is the euler number of gopakumar minwalla and strominger gms soliton space

noncommutative hyper kahler structure for k3 surfaces we apply the method of algebraic deformation to n tuple of algebraic k3 surfaces when n 3 we show that the deformed triplet of algebraic k3 surfaces exhibits a deformed hyper kahler structure the deformation moduli space of this family of noncommutative k3 surfaces turns out to be of dimension 57 which is three times that of complex deformations of algebraic k3 surfaces

ward identities for non commutative geometry we interpret the cocycle condition in quantum field theory as a set of integrated ward identities for non commutative geometry

quantum group of isometries in classical and noncommutative geometry we formulate a quantum generalization of the notion of the group of riemannian isometries for a compact riemannian manifold by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples and then proving the existence of a universal object called the quantum isometry group in the category of compact quantum groups acting smoothly and isometrically on a given possibly noncommutative manifold satisfying certain regularity assumptions the idea of quantum families due to woronowicz and soltan are relevant to our construction a number of explicit examples are given and possible applications of our results to the problem of constructing quantum group equivariant spectral triples are discussed

quantum k theory i the chern character we construct a cocycle on an infinite dimensional generalization of ap summable fredholm module our framework is related to connes cyclic cohomology and is motivated by our work on index theory on infinite dimensional manifolds thep summability condition is characteristic of dimensiono p we replace this assumption by the requirement that there exists an underlying heat kernel which is trace class then we use the heat kernel to regularize states in dimension independent fashion our cocycle may be interpreted as an infinite dimensional chern character

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

the commutative limit of a matrix geometry the noncommutative geometry of complex n n matrices is considered as an approximation to the geometry of the two sphere s2 and the differential forms defined over the matrices are mapped into the ordinary differential forms on su2 considered as a fiber bundle over s2 it is found that the extra gauge invariant vacuum studied in previous publications has as image a constant multiple of the connection on su2 that defines the fibration the dirac monopole with unit magnetic charge

spectral triples and associated connes de rham complex for the quantum su 2 and the quantum sphere in this article we construct spectral triples for the c algebra of continuous functions on the quantum su 2 group and the quantum sphere there have been various approaches towards building a calculus on quantum spaces but there seem to be very few instances of computations outlined in chapter 6 5 we give detailed computations of the associated connes de rham complex and the space of l2 forms

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

quantum isometry groups of the podles spheres abstract for 0 1 c 0 we identify the quantum group so 3 as the universal object in the category of compact quantum groups acting by orientation and volume preserving isometries in the sense of bhowmick and goswami 2009 4 on the natural spectral triple on the podles sphere s c 2 constructed by dabrowski d andrea landi and wagner 2007 in 9