improved derivative bounds of the rational quadratic bezier curves a new derivative bound of the rational quadratic bezier curves is obtained it is an improvement over the bound which was regarded as a sharp bound in a recent paper also the new bound is superior to the existing bounds given by some other authors

pseudo integral and generalized choquet integral abstract due to many applications the choquet integral as a powerful tool for modeling non deterministic problems needs to be further extended therefore the paper is devoted to a generalization of the choquet integral as a basis the pseudo integral for bounded integrand is extended to the case for nonnegative integrands at first and then the generalized choquet integral is defined as special cases pseudo choquet stieltjes integrals pseudo fuzzy stieltjes integrals g choquet integrals pseudo n fuzzy integrals and pseudo s fuzzy integrals are obtained and various kinds of properties and convergence theorems are shown meanwhile markov jensen minkowski and holder inequalities are proved in the end the generalized discrete choquet integral is discussed the obtained results for the generalized choquet integral cover some previous results on different types of nonadditive integrals

a curse of dimensionality free numerical method for solution of certain hjb pdes in previous works of the author and others max plus methods have been explored for the solution of first order nonlinear hamilton jacobi bellman partial differential equations hjb pdes and corresponding nonlinear control problems these methods exploit the max plus linearity of the associated semigroups in particular although the problems are nonlinear the semigroups are linear in the max plus sense these methods have been used successfully to compute solutions although they provide certain computational speed advantages they still generally suffer from the curse of dimensionality here we consider hjb pdes in which the hamiltonian takes the form of a pointwise maximum of linear quadratic forms the approach to the solution will be rather general but in order to ground the work we consider only constituent hamiltonians corresponding to long run average cost per unit time optimal control problems for the development we obtain a numerical method not subject to the curse of dimensionality the method is based on construction of the dual space semigroup corresponding to the hjb pde this dual space semigroup is constructed from the dual space semigroups corresponding to the constituent linear quadratic hamiltonians the dual space semigroup is particularly useful due to its form as a max plus integral operator with a kernel obtained from the originating semigroup one considers repeated application of the dual space semigroup to obtain the solution

lebesgue theorems in non additive measure theory in this paper the sufficient and necessary conditions for different kinds of lebesgue theorem in non additive measure theory are presented respectively the equivalence among the lebesgue theorem the monotone convergence theorems of fuzzy and of choquet integral are shown as direct results of the lebesgue theorems the necessary conditions for egoroff s theorem are given

on the equality of integrals considering a finite space x several necessary conditions and one rather general sufficient condition describing when the choquet integral coincides with the pan integral with respect to the standard arithmetic operations are shown these conditions are characterized by using the minimal atoms of monotone measure under the introduced constraints the calculation of these coinciding two integrals is also given

several inequalities for the pan integral several inequalities for the pan integral are investigated it is shown that the chebyshev inequality holds for an arbitrary subadditive measure if and only if the integrands f g are comonotone thus we provide a new characterization for nonnegative comonotone functions it is also shown that the holder and minkowski inequalities for the pan integral hold if the monotone measure is subadditive since the pan integral coincides with the concave integral when is subadditive our results can also be applied to the concave integral

autocontinuity and convergence theorems for the choquet integral we provide convergence theorems for the choquet integral for various notions of convergence monotone almost uniform convergence from above below is associated to monotone autocontinuity from above below the dominated almost uniform convergence is associated to monotone autocontinuity autocontinuity can be associated to the dominated convergence in measure in mean or in strict measure

a unified approach to the monotone convergence theorem for nonlinear integrals abstract we present a unified approach to the monotone convergence theorem for nonlinear integrals such as the choquet the sipos the sugeno and the shilkret integral a nonlinear integral may be viewed as a nonlinear functional defined on a set of pairs of a nonadditive measure and a measurable function we thus formulate our general type of monotone convergence theorem for such a functional the key tool is a perturbation of functional that manages not only the monotonicity of the functional but also the small change of the functional value arising as a result of adding small amounts to a measure and a function in the domain of the functional our approach is also applicable to the lebesgue integral when a nonadditive measure is additive

a sufficient condition of equivalence of the choquet and the pan integral abstract in this note we continue to investigate the relationship between the choquet integral and the pan integral on infinite space we will show that the m property of monotone measures is a sufficient condition that the choquet integral coincides with the pan integral in this discussion the spaces are not restricted to be finite thus the previous results obtained in finite space are further generalized and developed a characteristic of the m property of monotone measures is also presented

max plus fundamental solution semigroups for a class of difference riccati equations recently a max plus dual space fundamental solution semigroup for a class of difference riccati equations dres has been developed this fundamental solution semigroup is represented in terms of the kernel of a specific max plus linear operator that plays the role of the dynamic programming evolution operator in a max plus dual space in order to fully understand connections between this dual space fundamental solution semigroup and evolution of the value function of the underlying optimal control problem a new max plus primal space fundamental solution semigroup for the same class of difference riccati equations is presented connections and commutation results between this new primal space fundamental solution semigroup and the recently developed dual space fundamental solution semigroup are established

a max plus based fundamental solution for a class of discrete time linear regulator problems abstract efficient riccati equation based techniques for the approximate solution of discrete time linear regulator problems are restricted in their application to problems with quadratic terminal payoffs where non quadratic terminal payoffs are required these techniques fail due to the attendant non quadratic value functions involved in order to compute these non quadratic value functions it is often necessary to appeal directly to dynamic programming in the form of grid or element based iterations for the value function these iterations suffer from poor scalability with respect to problem dimension and time horizon in this paper a new max plus based method is developed for the approximate solution of discrete time linear regulator problems with non quadratic payoffs this new method is underpinned by the development of new fundamental solutions to such linear regulator problems via max plus duality in comparison with a typical grid based approach a substantial reduction in computational effort is observed in applying this new max plus method a number of simple examples are presented that illustrate this and other observations

a note on the space of fuzzy measurable functions for a monotone measure abstract in this paper we give a necessary and sufficient condition to guarantee that the space of all finite measurable functions for a monotonic measure is a topological vector space with a countable local base and in this space convergence with respect to this topology is equivalent to the convergence in measure

on the determination of strength of belief for decision support under uncertainty part ii fusing strengths of belief the problem of decision making under uncertainty is described and the difficulty of comparing actions with uncertain outcomes is noted we discuss the role of valuation functions which convert an uncertain outcome into a single representative value and hence aid in the comparison of uncertain alternatives the need for information about the strength of belief that a given outcome will occur is pointed out we then look at various methods for obtaining this information we consider in particular detail the situation in which the strength of belief is dependent on the outcome payoff

sandwich games the extension of set functions or capacities in a concave fashion namely a concavification is an important issue in decision theory and combinatorics it turns out that some set functions cannot be properly extended if the domain is restricted to be bounded this paper examines the structure of those capacities that can be extended over a bounded domain in a concave manner we present a property termed the sandwich property that is necessary and sufficient for a capacity to be concavifiable over a bounded domain we show that when a capacity is interpreted as the product of any sub group of workers per a unit of time the sandwich property provides a linkage between optimality of time allocations and efficiency

max plus eigenvector methods for nonlinear h _ infty problems error analysis the h infty problem for a nonlinear system is considered the corresponding dynamic programming equation is a fully nonlinear first order steady state partial differential equation pde possessing a term which is quadratic in the gradient the solutions are typically nonsmooth and further there is nonuniqueness among the class of viscosity solutions in the case where one tests a feedback control to see if it yields an h infty controller or where either the controller or disturbance sufficiently dominates the pde is a hamilton jacobi bellman equation the computation of the solution of a nonlinear steady state first order pde is typically quite difficult in a companion paper we developed an entirely new class of methods for obtaining the correct solution of such pdes these methods are based on the linearity of the associated semigroup over the max plus or in some cases min plus algebra in particular solution of the pde is reduced to solution of a max plus or min plus eigenvector problem for known unique eigenvalue 0 the max plus multiplicative identity it was demonstrated that the eigenvector is unique and that the power method converges to it in the companion paper the basic method was laid out without discussion of errors and convergence in this paper we both approach the error analysis for such an algorithm and demonstrate convergence the errors are due to both the truncation of the basis expansion and computation of the matrix whose eigenvector one computes

on the continuity of the concave integral whenever a functional is concave it is natural to ask whether its sendograph is a closed convex set if so the hahn banach theory implies that the functional can be represented as the infimum of all continuous linear functionals greater than or equal to it we refer to such representation as a dual representation dominated convergence of the concave integral for capacities is characterized in terms of dual representation whenever sequences of functions converge pointwise outside a set of zero capacity

a note on the coincidence of decomposition integrals and superdecomposition integrals abstract it is known that the decomposition integral i h and the superdecomposition integral i h related to the system h of all finite chains from a where a is a algebra of the subsets of a nonvoid set x coincide with each other for each monotone measure and for each nonnegative measurable function and they are equal to the choquet integral in this note we show that the converse is also true that is for a complete system h of finite set systems from a if i h i h holds for each monotone measure and for each nonnegative measurable function then the system h and the system of all finite chains from a are equivalent in some sense and the integrals involved are the choquet integral

improved bounds on the magnitude of the derivative of rational bezier curves abstract in this paper we derive some new derivative bounds of rational bezier curves according to some existing identities and inequalities the comparison of the new bounds with some existing ones is also presented

convergence theorems for monotone measures in classical measure theory there are a number of convergence theorems such as the egorov the riesz and the lusin theorem among others we consider monotone measures i e monotone set functions vanishing in the empty set and defined on a measurable space and discuss how and to which extent classical convergence theorems can be carried over to this more general case

on the equivalence of the choquet pan and concave integrals on finite spaces abstract in this paper we introduce the concept of maximal cluster of minimal atoms on monotone measure spaces and by means of this new concept we continue to investigate the relation between the choquet integral and the pan integral on finite spaces it is proved that the m property of a monotone measure is a sufficient condition that the choquet integral coincides with the pan integral based on the usual addition and multiplication thus combining our recent results we provide a necessary and sufficient condition that the choquet integral is equivalent to the pan integral on finite spaces meanwhile we also use the characteristics of minimal atoms of monotone measure to present another necessary and sufficient condition that these two kinds of integrals are equivalent on finite spaces the relationships among the choquet integral the pan integral and the concave integral are summarized

the dual theory of choice under risk in this essay a new theory of choice under risk is being proposed it is a theory which in a sense that will become clear is dual to expected utility theory hence the title dual theory risky prospects are evaluated in this theory by a cardinal numerical scale which resembles an expected utility except that the roles of payments and probabilities are reversed this theme the reversal of the roles of probabilities and payments will recur throughout the paper i should emphasize that playing games with probabilities masquerading as payments and payments masquerading as probabilities is not my object rather i hope to convince the reader that the dual theory has intrinsic economic significance and that in some areas its predictions are superior to those of expected utility theory while in other areas the reverse will be the case two reasons have prompted me to look for an alternative to expected utility theory the first reason is methodological in expected utility theory the agent s attitude towards risk and the agent s attitude towards wealth are forever bonded together at the level of fundamental principles risk aversion and diminishing marginal utility of wealth which are synonymous under expected utility theory are horses of different colors the former expresses an attitute towards risk increased uncertainty hurts while the latter expresses an attitude towards wealth the loss of a sheep hurts more when the agent is poor than when the agent is rich a question arises therefore as to whether these two notions can be kept separate from each other in a full fledged theory of cardinal utility the dual theory will have this property the second reason that leads me to look for an alternative to expected utility theory is empirical behavior patterns which are systematic yet inconsistent with expected utility theory have often been observed two prominent references among many others are allais 1953 and kahneman tversky 1979 so deeply

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

control and separating points of modular functions let l be a complemented or sectionally complemented lattice and let x be a locally convex linear space we prove the existence of an x valued control for uniformly exhaustive sequences of x valued modular functions on l moreover if l is cr complete we prove that every countable family of cr o c nonatomic group valued modular functions on l admits separating points

order continuous of monotone set function and convergence of measurable functions sequence in this paper we introduce the concept of strongly order continuity of a set function and use it to investigate the convergence of measurable functions sequence and of integral sequence on fuzzy measure space several equivalent conditions of strongly order continuity are presented it is shown that a necessary and sufficient condition that lebesgue s theorem remains valid for a monotone set function is that the set function possesses strongly order continuity

some remarks on the double asymptotic null additivity of monotonic measures and related applications in this paper we discuss the inheriting of convergence of the monotonic measures under the following operations addition multiplication max and min and the uniqueness of convergence of a monotonic measure moreover we also point out that autocontinuity from above cannot imply double asymptotic null additivity for monotonic measures using a counterexample contrary to the case of fuzzy measures proved by ha et al fundamental convergence of sequences of measurable functions on fuzzy measure space fuzzy sets and systems 95 1998 77 81 finally we show that for monotonic measure convergence double asymptotic null additivity is better than autocontinuity from above

sandwich theorems for set functions abstract in the present paper the following problems are treated when do there exist a a precontent and b a precharge which lies between two given set functions the obtained results yield among others several well known theorems such as bondareva s core theorem kelley s intersection number theorem and sandwich theorems of kranz gol dberg and frank as immediate consequences

conditions for egoroff s theorem in non additive measure theory abstract this paper gives necessary and or sufficient conditions for egoroff s theorem in non additive measure theory a necessary and sufficient condition described without measurable functions two sufficient conditions and a necessary condition one of the two sufficient conditions is strong order total continuity continuity at measurable sets of measure zero with respect to net convergence and the other is strong order continuity sequential continuity at measurable sets of measure zero together with property s the necessary condition is strong order continuity in addition the paper shows the following continuity from above and below which is a known sufficient condition and the above mentioned two sufficient conditions are independent of each other the disjunction of these three sufficient conditions is not a necessary condition if the underlying set is at most countable then strong order continuity is necessary and sufficient and generally strong order continuity is not sufficient

a max plus dual space fundamental solution for a class of operator differential riccati equations a new fundamental solution semigroup for operator differential riccati equations is developed this fundamental solution semigroup is constructed via an auxiliary finite horizon optimal control problem whose value functional growth with respect to time horizon is determined by a particular solution of the operator differential riccati equation of interest by exploiting semiconvexity of this value functional and the attendant max plus linearity and semigroup properties of the associated dynamic programming evolution operator a semigroup of max plus integral operators is constructed in a dual space defined via the legendre fenchel transform it is demonstrated that this semigroup of max plus integral operators can be used to propagate all solutions of the operator differential riccati equation that are initialized from a specified class of initial conditions as this semigroup of max plus integral operators can be identified with a semigroup of quadratic kernels an explicit recipe for the aforementione

generalized convergence theorems for monotone measures abstract in this paper we propose three types of absolute continuity for monotone measures and present some of their basic properties by means of these three types of absolute continuity we establish generalized egoroff s theorem generalized riesz s theorem and generalized lebesgue s theorem in the framework involving the ordered pair of monotone measures the egoroff theorem the riesz theorem and the lebesgue theorem in the traditional sense concerning a unique monotone measure are extended to the general case these three generalized convergence theorems include as special cases several previous versions of egoroff like theorem riesz like theorem and lebesgue like theorem for monotone measures respectively

the vitali convergence in measure theorem of nonlinear integrals abstract the vitali convergence in measure theorem for the abstract lebesgue integral is fundamental in lebesgue integration theory and yields the bounded convergence theorem and the dominated convergence theorem as its applications in this paper in a unified way using the perturbation method the vitali convergence in measure theorem is established for nonlinear integrals such as the choquet sipos sugeno and shilkret integrals and their symmetric and asymmetric extensions it is derived from the fatou and the reverse fatou type lemmas for perturbative nonlinear integral functionals

the max plus finite element method for solving deterministic optimal control problems basic properties and convergence analysis we introduce a max plus analogue of the petrov galerkin finite element method to solve finite horizon deterministic optimal control problems the method relies on a max plus variational formulation we show that the error in the sup norm can be bounded from the difference between the value function and its projections on max plus and min plus semimodules when the max plus analogue of the stiffness matrix is exactly known in general the stiffness matrix must be approximated this requires approximating the operation of the lax oleinik semigroup on finite elements we consider two approximations relying on the hamiltonian we derive a convergence result in arbitrary dimension showing that for a class of problems the error estimate is of order delta delta x delta 1 or sqrt delta delta x delta 1 depending on the choice of the approximation where delta and delta x are respectively the time and space discretization steps we compare our method with another max plus based discretization method previously introduced by fleming and mceneaney we give numerical examples in dimensions 1 and 2

a probabilistic representation of exact games on σ algebras the purpose of this paper is to establish the intrinsic relations between the cores of exact games on s algebras and the extensions of exact games to function spaces given a probability space exact functionals are defined on l as an extension of exact games to derive a probabilistic representation for exact functionals we endow them with two probabilistic conditions law invariance and the fatou property the representation theorem for exact functionals lays a probabilistic foundation for nonatomic scalar measure games based on the notion of p convexity we also investigate the equivalent conditions for the representation of anonymous convex games

relationship between two types of superdecomposition integrals on finite spaces abstract this paper investigates the relationship between two types of superdecomposition integrals namely the convex integral and the pan integral from above on finite spaces to this end we introduce two new concepts related to monotone measures superadditivity with respect to singletons and minimal strictly subadditive set and discuss some of their properties in the case that the monotone measure is superadditive with respect to singletons we show that these two types of integrals are equivalent in other cases by means of the characteristics of minimal strictly subadditive sets we provide a set of necessary and sufficient conditions for which these two types of integrals coincide with each other

a max plus based algorithm for a hamilton jacobi bellman equation of nonlinear filtering the hamilton jacobi bellman hjb equation associated with the robust hinfty filter as well as the mortensen filter is considered these filters employ a model where the disturbances have finite power the hjb equation for the filter information state is a first order equation with a term that is quadratic in the gradient yet the solution operator is linear in the max plus algebra this property is exploited by the development of a numerical algorithm where the effect of the solution operator on a set of basis functions is computed off line the precomputed solutions are stored as vectors of coefficients of the basis functions these coefficients are then used directly in the real time computations

pseudo optimal measures based on a pseudo addition on 0 pseudo optimal measures related to non negative set functions vanishing at the empty set non negative games are introduced for a fixed pseudo addition an equivalence on non negative set functions is studied a special stress is given to pure pseudo optimal measures admitting no other equivalent set function

superdecomposition integrals this study introduces and discusses a new class of integrals based on superdecompositions of integrated functions including an analysis of their relationship with decomposition integrals which were introduced recently by even and lehrer the proposed superdecomposition integrals have several properties that are similar or dual with respect to decomposition integrals but they also have some significant differences the convex integral is obtained by considering all possible superdecompositions with no constraints on the applied sets which can be treated as the greatest convex homogeneous functional that is bounded from above by the measure we consider the relationship with the universal integral of klement et al is also discussed finally some possible generalizations are outlined

sandwich theorems for set functions an application of the lehrer teper integral abstract we show that a new integral recently introduced by lehrer and teper 2008 11 provides all the tools to determine the necessary and sufficient conditions for the existence of an additive set function m sandwiched between two arbitrary set functions and m

monotonicity and stabilizability properties of solutions of the riccati difference equation propositions lemmas theorems fallacious conjectures and counterexamples abstract the problem considered is that of selecting an initial covariance matrix for the kalman filter to ensure that the closed loop filter at every subsequent time instant is exponentially asymptotically stable as a time invariant filter sufficient conditions are derived based on monotonicity properties of the solution of the riccati difference equation the results have application in observer design and the cases of filtering for nonstabilizable systems and systems with singular system matrices are included

on policy capturing with fuzzy measures policy capturing methods generally apply linear regression analysis to model human judgment in this paper we examine the application of fuzzy set and fuzzy measure theories to obtain subjective descriptions of cue importance for policy capturing at the heart of the approach is a method of learning fuzzy measures the shapley values associated with the fuzzy measures provide a basis for comparison with the results of linear regression however the fuzzy measure theoretical approach provides additional insight into interaction effects corresponding to the nonlinear noncompensatory nature of the underlying decision model to illustrate the methodology we estimated the importance of factors and the interactions among them that influence decisions related to strategic investments in telecommunications infrastructure and compared the results from the fuzzy approach to those obtained from traditional statistical methods

integrals based on monotone set functions an overview of various integrals is given which can be defined on arbitrary monotone set functions vanishing in the empty set called here monotone measures our survey includes not only the choquet integral 1954 10 the shilkret integral 1971 66 and the sugeno integral 1974 71 and some of their properties but also some more general and more recent concepts as universal integrals klement et al 2010 27 and decomposition integrals even 2014 13 together with some of their properties such as integral inequalities and convergence theorems

symmetric upper probabilities as a first step toward developing statistical models based on upper and lower probabilities we study upper probabilities and upper expectations on the unit interval that are symmetric by which we mean invariant with respect to equimeasurability these upper probabilities are generalizations of uniform probability measures we give some characterizations of these upper probabilities specifically we show that symmetry of the upper expectation functional is equivalent to the underlying set of densities being closed under majorization we also show that a function is the upper distribution for a symmetric upper probability if and only if its lower graph is star shaped with respect to the origin and to the point 1 1 we derive inner and outer approximations to symmetric classes of probabilities based on the upper probability the class of symmetric upper expectations that are completely determined by their values on the indicator functions is characterized we provide a geometric characterization of a hierarchy of upper probabilities including fine s generalized upper probabilities and 2 alternating choquet capacities in particular we establish a 1 1 correspondence between symmetric 2 alternating capacities and nonincreasing density functions we prove that undominated generalized upper probabilities do not exist in the symmetric case examples from robust statistics are considered an example is given that shows that symmetry of upper probabilities does not imply symmetry of upper expectations a corollary is that symmetry of the choquet integral does not imply symmetry of the upper expectation functional

on the bounds of the derivative of rational bezier curves according to some identities and inequalities about the intermediate weights and control points of the de casteljau algorithm for rational bezier curve we derive a new bound of the derivative of rational bezier curves the comparison of the new bounds with some existing ones is also presented based on some previous bounds the bound presented in this paper and many numerical examples we propose a conjecture of the bound for the derivative of rational bezier curves

a note on the trigonometric approximation of lip ω t p class recently some results on the trigonometric approximation of lip t p class have been given by srivastava and singh 2014 in this note we point out that one of the assumption conditions in their theorems is invalid in some cases thus all conclusions in the paper based on the condition lose the foundation moreover we obtain the similar results for a class of more extensive functions after deleting the mentioned condition and weakening another condition in their theorems

a set valued egoroff type theorem in this paper an egoroff type theorem for a fuzzy multimeasure defined on a s algebra of subsets of an abstract space and taking values in the family of all nonvoid closed subsets of a real normed space is established some necessary and or sufficient conditions under which egoroff s theorem is valid are also presented

totally balanced games and games of flow a class of characteristic function games arising from maximum flow problems is introduced and is shown to coincide with the class of totally balanced games the proof relies on the max flow min cut theorem of ford and fulkerson and on the observation that the class of totally balanced games is the span of the additive games with the minimum operation

a sandwich theorem for precharges abstract in the present paper the following question is treated given two arbitrary set functions on an arbitrary set system when can one find a precharge that lies in between a number of necessary and sufficient conditions for the existence of such a precharge is derived and it is shown how these conditions can be simplified in special cases

decompositions of supermodular functions and s decomposable measures abstract we consider supermodular functions and decomposable measures defined on complete lattice and with values in complete lattice ordered semigroup for such supermodular functions and decomposable measures on lattice with relative complement hewitt yosida decomposition type theorem and lebesgue decomposition theorem are proved

the equivalence of convergences of sequences of fuzzy numbers and its applications to the characterization of compact sets in this paper with the help of the convergence of sequences of fuzzy numbers with respect to the lebesgue measure we study the relationship between convergences of sequences of fuzzy numbers with respect to the endograph metric the sendograph metric and the l p metric we prove that these convergences are equivalent under proper conditions in addition by applying our result we give a new characterization of compact sets in fuzzy number space with the sendograph metric

a discrete choquet integral for ordered systems a model for a choquet integral for arbitrary finite set systems is presented the model includes in particular the classical model on the system of all subsets of a finite set the general model associates canonical non negative and positively homogeneous superadditive functionals with generalized belief functions relative to an ordered system which are then extended to arbitrary valuations on the set system it is shown that the general choquet integral can be computed by a simple monge type algorithm for so called intersection systems which include as a special case weakly union closed families generalizing lovasz classical characterization we give a characterization of the superadditivity of the choquet integral relative to a capacity on a union closed system in terms of an appropriate model of supermodularity of such capacities

relationship between the concave integrals and the pan integrals on finite spaces a r t i c l e i n f o a b s t r a c t this study discusses the relationship between the concave integrals and the pan integrals on finite spaces the minimal atom of a monotone measure is introduced and some properties are investigated by means of two important structure characteristics related to minimal atoms minimal atoms disjointness property and subadditivity for minimal atoms a necessary and sufficient condition is given that the concave integral coincides with the pan integral with respect to the standard arithmetic operations a nd on finite spaces following this result we have shown that these two integrals coincide if the underlying monotone measure is sub additive

max plus eigenvector representations for solution of nonlinear h sub spl infin problems basic concepts the h sub spl infin problem for a nonlinear system is considered the corresponding dynamic programming equation is a fully nonlinear first order steady state partial differential equation pde possessing a term which is quadratic in the gradient the solutions are typically nonsmooth and further there is nonuniqueness among the class of viscosity solutions in the case where one tests a feedback control to see if it yields an h sub spl infin controller the pde is a hamilton jacobi bellman equation in the case where the optimal feedback control is being determined as well the problem takes the form of a differential game and the pde is in general an isaacs equation the computation of the solution of a nonlinear steady state first order pde is typically quite difficult in this paper we develop an entirely new class of methods for obtaining the correct solution of such pdes these methods are based on the linearity of the associated semigroup over the max plus or in some cases min plus algebra in particular solution of the pde is reduced to solution of a max plus or min plus eigenvector problem for known unique eigenvalue 0 the max plus multiplicative identity it is demonstrated that the eigenvector is unique and that the power method converges to it an example is included

a new fundamental solution for differential riccati equations arising in control the matrix differential riccati equation dre is ubiquitous in control and systems theory the presence of the quadratic term implies that a simple linear systems fundamental solution does not exist of course it is well known that the bernoulli substitution may be applied to obtain a linear system of doubled size here however tools from max plus analysis and semiconvex duality are brought to bear on the dre we consider the dre as a finite dimensional solution to a deterministic linear quadratic control problem taking the semiconvex dual of the associated semigroup one obtains the solution operator as a max plus integral operator with quadratic kernel the kernel is equivalently represented as a matrix using the semigroup property of the dual operator one obtains a matrix operation whereby the kernel matrix propagates as a semigroup the propagation forward is through some simple matrix operations this time indexed family of matrices forms a new fundamental solution for the dre solution for any initial condition is obtained by a few matrix operations on the fundamental solution and the initial condition in analogy with standard algebra linear systems the fundamental solution can be viewed as an exponential form over a certain idempotent semiring this fundamental solution has a particularly nice control interpretation and might lead to improved dre solution speeds

new minimal bounds for the derivatives of rational bezier paths and rational rectangular bezier surfaces new minimal bounds are derived for the magnitudes of the derivatives of the rational bezier paths and the rational rectangular bezier surface patches of arbitrary degree which improve previous work of this type in many cases moreover our new bounds are explicitly given by simple and closed form expressions an important advantage of the closed form expressions is that they allow us to prove that our bounds are sharp under certain well defined conditions some numerical examples highlighting the potential of the new bounds in providing improved estimates are given in an appendix

when fuzzy measures are upper envelopes of probability measures abstract fuzzy measures which are upper envelopes of probability measures may play an important role to develop a general theory of bayesian statistics especially from a technical point of view a widely accepted generalized bayes rule would be applicable for those kind of fuzzy measures we give sufficient general conditions to ensure that fuzzy measures are upper envelopes of probability measures they are applied to some special classes of important types of fuzzy measures namely sugeno fuzzy plausibility and possibility measures the proof of the main result is based on recently systemized inner extension procedures within abstract measure theory

time continuity and nonadditive expected utility information consisting of probabilities of some but possibly not all events induces an integral with respect to a probability specified on a subalgebra a decision maker evaluates the alternatives using only the available information and completely ignoring unavailable information assume now that the decision maker assesses the worth of a different lottery at each point in a discrete time assume also that each such lottery is preferred to some fixed alternative lottery now consider the situation where the sequence of lotteries converges in some sense if the limiting lottery is preferred to the fixed alternative then the preference order is referred to as time continuous this paper studies time continuity for two preference functionals the choquet integral and the integral with respect to a probability specified on a subalgebra the integral with respect to probability specified on a subalgebra is determined by the structure of the available information by relating it to the choquet integral we characterize the structure of available information that would yield time continuity

ambiguity in the small and in the large this paper considers local and global multiple prior representations of ambiguity for preferences that are i monotonic ii bernoullian that is admit an affine utility representation when restricted to constant acts and iii locally lipschitz continuous we do not require either certainty independence or uncertainty aversion we show that the set of priors identified by ghirardato maccheroni and marinacci s 2004 unambiguous preference relation can be characterized as a union of clarke differentials we then introduce a behavioral notion of locally better deviation at an act and show that it characterizes the clarke differential of the preference representation at that act these results suggest that the priors identified by these preference statements are directly related to local optimizing behavior

the concave integral over large spaces this paper investigates the concave integral for capacities defined over large spaces we characterize when the integral with respect to capacity v can be represented as the infimum over all integrals with respect to additive measures that are greater than or equal to v we introduce the notion of loose extendability and study its relation to the concave integral a non additive version for the levi theorem and the fatou lemma are proven finally we provide several convergence theorems for capacities with large cores

on the equivalence of the choquet integral and the pan integrals from above abstract a monotone measure is said to have dual m property if for any a b there exists c with a c b such that c a and b c b c by using this concept we study the relationship of the choquet integral and the pan integral from above we prove that the choquet integral coincides with the pan integral from above if has dual m property when the underlying space is finite we prove that the dual m property is also necessary for the coincidence of these two integrals thus we provide a necessary and sufficient condition for the equivalence of the choquet integral and the pan integral from above on a finite space

on the m property of monotone measures and integrals on atoms abstract we further investigate m property an important structure characteristic of monotone measures some necessary and or sufficient conditions of m property are shown and some characteristics of m property are described it is shown that the restriction of a monotone measure to an atom of its own possesses m property on this atom by means of the m property we characterize the equivalence among the choquet integral the pan integral the concave integral and the shilkret integral on atoms of monotone measures and obtain some special results we also show a necessary and sufficient condition for the concave integral and the pan integral to coincide on atoms of monotone measures

a theory of fuzzy measures representations the choquet integral and null sets abstract some theorems of t murofushi and m sugeno fuzzy sets and systems 29 1989 201 227 concerning representation of fuzzy measures and the choquet integral are generalized it is shown that if there is a certain relation between two measurable functions then the choquet integral is additive for these two functions in addition this article discusses null sets with respect to fuzzy measures and also fuzzy measures defined on a class which is not a algebra