bernstein operators for extended chebyshev systems abstract let u n c n a b be an extended chebyshev space of dimension n 1 suppose that f 0 u n is strictly positive and f 1 u n has the property that f 1 f 0 is strictly increasing we search for conditions ensuring the existence of points t 0 t n a b and positive coefficients 0 n such that for all f c a b the operator b n c a b u n defined by b n f k 0 n f t k k p n k satisfies b n f 0 f 0 and b n f 1 f 1 here it is assumed that p n k k 0 n is a bernstein basis defined by the property that each p n k has a zero of order k at a and a zero of order n k at b

fuzzy markovian decision process in this paper a markovian decision process with fuzzy states is considered it is shown that if the markov chain associated with the process of non fuzzy states is irreducible then the corresponding markov chain associated with the fuzzy states is also irreducible the importance of this result lies in the fact that the traditional methods for solving the markovian decision process can be used in the case with fuzzy states also

a note on the optimal stability of bases of univariate functions this note is concerned with the characterizations and uniqueness of bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation with respect to bases whose elements have no sign changes

a note on measure expansive diffeomorphisms abstract in this note we prove that a homeomorphism is countably expansive if and only if it is measure expansive this result is applied for showing that the c 1 interior of the sets of expansive measure expansive and continuum wise expansive c 1 diffeomorphisms coincide

error analysis of efficient evaluation algorithms for tensor product surfaces backward stability of the casteljau algorithm and two more efficient algorithms for polynomial tensor product surfaces with interest in cagd is shown the conditioning of the corresponding bases are compared these algorithms are also compared with the corresponding horner algorithm and their higher accuracy is shown a running error analysis of the algorithms is also carried out providing algorithms which calculate a posteriori sharp error bounds simultaneously to the evaluation of the surface without increasing significantly the computational cost

a continuous time dynamic fuzzy system i a limit theorem abstract a continuous time dynamic fuzzy system with possibility transition is constructed and a differential of fuzzy states is given by paths a limit theorem for fuzzy states is discussed under a contractive condition on fuzzy relations

evaluation of the derivative of a polynomial in bernstein form in geometric modelling and computer aided design a polynomial is usually represented in bernstein form forward error analysis for the computation of the derivative of a polynomial in bernstein form is performed the corresponding running error is also analyzed we have compared our results with those corresponding to a polynomial represented in power basis and using legendre and chebyshev forms since the conversions between the bernstein and the power bases and between bernstein and chebyshev polynomials were already studied in the literature for the sake of completeness the conversion between bernstein and chebyshev polynomials of the first kind is analyzed in an appendix

various expansive measures for flows abstract we discuss a characterization of countably expansive flows in measure theoretical terms as in the discrete case 2 more precisely we define the countably expansive flows and prove that a homeomorphism of a compact metric space is countable expansive just when its suspension flow is moreover we exhibit a measure expansive flow in the sense of 4 which is not countably expansive next we define the weak expansive measures for flows and prove that a flow of a compact metric space is countable expansive if and only if it is weak measure expansive i e every orbit vanishing measure is weak expansive furthermore unlike the measure expansive ones the weak measure expansive flows may exist on closed surfaces finally it is shown that the integrated flow of a c 1 vector field on a compact smooth manifold is c 1 stably expansive if and only if it is c 1 stably weak measure expansive

decision theoretic foundations of qualitative possibility theory abstract this paper presents a justification of two qualitative counterparts of the expected utility criterion for decision under uncertainty which only require bounded linearly ordered valuation sets for expressing uncertainty and preferences this is carried out in the style of savage starting with a set of acts equipped with a complete preordering relation conditions on acts are given that imply a possibilistic representation of the decision maker uncertainty in this framework pessimistic i e uncertainty averse as well as optimistic attitudes can be explicitly captured the approach thus proposes an operationally testable description of possibility theory

optimal bases for a class of mixed spaces and their associated spline spaces we consider spaces of the form span where the functions u i i 1 4 are algebraic polynomials or trigonometric or hyperbolic functions we find intervals 0 a where we can guarantee that the spaces possess normalized totally positive bases and so shape preserving representations we construct their normalized b bases with optimal shape preserving properties we also present a unified approach to deal with the associated spline spaces a recursive procedure to obtain bases with optimal shape preserving and stability properties is presented

a continuous time dynamic fuzzy system ii fuzzy potentials abstract this paper is a sequel of yoshida 9 fuzzy potentials are introduced in a continuous time dynamic fuzzy system and a fuzzy relational differential equation is given to characterize the fuzzy potentials

descriptive set theory for expansive systems abstract kato 5 and artigue 3 merged the theory of expansive systems 10 and foliations with the continuum theory 14 here we merge the expansive systems but with the descriptive set theory 6 instead more precisely we define meagre expansivity for both homeomorphisms and measures by requiring the interior of the dynamical balls up to some prefixed radio to be either empty or with zero measure respectively we first prove that every cw expansive homeomorphism of a locally connected metric space without isolated points is meagre expansive but not conversely second that a homeomorphism of a metric space is meagre expansive if and only if every borel probability measure is meagre expansive next that the space of meagre expansive measures of a homeomorphism of a compact metric space x is an f subset of the space of borel probability measures equipped with the weak topology in the sequel we prove that every homeomorphism with a meagre expansive measure of a compact metric space has an invariant meagre expansive measure also that the set of periodic points of every meagre expansive homeomorphism of a compact metric space has empty interior in the circle or the interval we prove that there are no meagre expansive homeomorphisms of the circle or the interval moreover the meagre expansive measures of an interval homeomorphism or a circle homeomorphism with rational rotation number are precisely the finite convex combinations of dirac measures supported on isolated periodic points a circle homeomorphism with irrational rotation number has a meagre expansive measure if and only if it is a denjoy map in such a case the meagre expansive measures are precisely those measures supported on the unique minimal set of the map to obtain some of our results we will consider a measurable version of the classical baire category

expansive measures for flows abstract we extend the concept of expansive measure 2 from homeomorphism to flows we prove for continuous flows on compact spaces that every expansive measure has no singularities in the support is aperiodic is expansive with respect to time t maps but not conversely remains expansive under topological equivalence vanishes along the orbits and is natural under suspensions we apply these properties to prove that there are no expansive flows in the sense of 26 of any closed surface

running relative error for the evaluation of polynomials running relative error for evaluating polynomials is analyzed and applied to different evaluation algorithms horner vs de casteljau and clenshaw algorithms numerical experiments show that the derived running relative error bounds are sharp and that the de casteljau algorithm presents better stability properties than the remaining algorithms

measure expansive diffeomorphisms abstract in this paper measure expansive diffeomorphisms are considered and the characterizations of the c 1 interiors of the set of measure expansive diffeomorphisms are obtained

measure theoretic pressure for subadditive potentials abstract by using the definition of topological pressure for subadditive potentials we introduce a new definition of measure theoretic pressure for ergodic measures of subadditive potentials on a compact metric space under some assumption we show that this definition has a similar formalism to the definition of measure theoretic pressure for ergodic measures of additive potentials on a compact metric space as an application assume that the subadditive potential is holder continuous then for an expansive homeomorphism with the specification property we study the relationship between the measure theoretic pressure and the periodic points

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

multifractal analysis of local entropies for expansive homeomorphisms with specification in the present paper we study the multifractal spectrum of local entropies we obtain results similar to those of the multifractal analysis of pointwise dimensions but under much weaker assumptions on the dynamical systems we assume our dynamical system to be defined by an expansive homeomorphism with the specification property we establish the variational relation between the multifractal spectrum and other thermodynamical characteristics of the dynamical system including the spectrum of correlation entropies

topological stability and pseudo orbit tracing property for expansive measures abstract we extend the concepts of topological stability and pseudo orbit tracing properties from homeomorphisms to borel measures we prove that every expansive measure with the pseudo orbit tracing property is topologically stable this represents a measurable version of the classical result by walters 28 afterwards we derive some properties of the topologically stable measures and measures with the pseudo orbit tracing property on compact metric spaces

on the optimal stability of the bernstein basis we show that the bernstein polynomial basis on a given interval is optimally stable in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval this result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations we further show by means of some low degree examples that the bernstein form is not uniquely optimal in this respect however it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval these ideas are illustrated by comparing the stability properties of the power bernstein and generalized ball bases

weak measure expansive flows abstract a notion of measure expansivity for flows was introduced by carrasco olivera and morales in 3 as a generalization of expansivity and they proved that there were no measure expansive flows on closed surfaces in this paper we introduce a concept of weak measure expansivity for flows which is really weaker than that of measure expansivity and show that there is a weak measure expansive flow on a closed surface moreover we show that any c 1 stably weak measure expansive flow on a c closed manifold m is stable and any c 1 stably measure expansive flow on m satisfies both axiom a and the quasi transversality condition

least supported bases and local linear independence we introduce the concept of least supported basis which is very useful for numerical purposes we prove that this concept is equivalent to the local linear independence of the basis for any given locally linearly independent basis we characterize all the bases of the space sharing the same property several examples for spline spaces are given

expansiveness hyperbolicity and hausdorff dimension we show that there exists a simple upper bound on the dimension of a hyperbolic compact set of a dynamical system in terms of topological entropy and a uniform contraction rate on the stable and unstable manifolds this allows us to give proofs of several apparently unrelated theorems

shape preserving properties of generalized bernstein operators on extended chebyshev spaces we study the existence and shape preserving properties of a generalized bernstein operator b n fixing a strictly positive function f 0 and a second function f 1 such that f 1 f 0 is strictly increasing within the framework of extended chebyshev spaces u n the first main result gives an inductive criterion for existence suppose there exists a bernstein operator b n c a b u n with strictly increasing nodes fixing f 0 f 1 in u n if u n subset u n 1 and u n 1 has a non negative bernstein basis then there exists a bernstein operator b n 1 c a b u n 1 with strictly increasing nodes fixing f 0 and f 1 in particular if f 0 f 1 f n is a basis of u n such that the linear span of f 0 f k is an extended chebyshev space over a b for each k 0 n then there exists a bernstein operator b n with increasing nodes fixing f 0 and f 1 the second main result says that under the above assumptions the following inequalities hold b n f geq b n 1 f geq f furthermore b nf is f0 f1 convex for all f0 f1 convex functions f in c a b

fuzzy dynamic programming main developments and applications abstract we survey major developments and applications of fuzzy dynamic programming which is here advocated as a promising attempt at making dynamic programming models more realistic by a relaxation of often artificial assumptions of precision as to the constraints goals states and their transitions termination time etc issues related to numerical efficiency are considered applications in r d planning health care and medicine socioeconomic regional development energy systems water and environmental systems chemical engineering etc are surveyed

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

beyond topological hyperbolicity the l shadowing property abstract in this paper we further explore the l shadowing property defined in 20 for dynamical systems on compact spaces we prove that structurally stable diffeomorphisms and some pseudo anosov diffeomorphisms of the two dimensional sphere satisfy this property homeomorphisms satisfying the l shadowing property have a spectral decomposition where the basic sets are either expansive or contain arbitrarily small topological semi horseshoes periodic sets where the restriction is semiconjugate to a shift to this end we characterize the l shadowing property using local stable and unstable sets and the classical shadowing property we exhibit homeomorphisms with the l shadowing property and arbitrarily small topological semi horseshoes without periodic points at the end we show that positive finite expansivity jointly with the shadowing property imply that the space is finite

on the optimal stability of bases of univariate functions this paper is concerned with bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation the only bases considered are those whose elements have no sign changes among these an optimally stable basis is characterized under the assumption that the set of points where each basis function is nonzero is an interval a uniqueness result and many examples of such optimally stable bases are also provided

a corner cutting algorithm for evaluating rational bezier surfaces and the optimal stability of the basis the usual method for evaluating rational bezier surfaces uses the projection operator and the representation provided by the bernstein basis we prove the optimal stability of the basis used in this representation we also propose an alternative method for evaluating rational surfaces through that representation we show the stability properties of this last method and prove that it has better properties than other known methods from the point of view of avoiding underflow and overflow

dynamic programming for fuzzy systems with fuzzy environment abstract a dynamic programming functional equation for a multi stage decision problem with fuzzy dynamics and environment is formulated and solved by a process of fuzzy interpolation this is an extension of the bellman zadeh model

measure n expansive systems abstract the n expansive systems have been recently studied in the literature 6 7 9 14 here we characterize them as those homeomorphisms for which every borel probability measure is n expansive in particular the strongly measure expansive homeomorphisms in the sense of 8 are precisely the homeomorphisms for which every invariant measure is 1 expansive we also characterize the 1 expansive measures for equicontinuous homeomorphisms as the convex sum of finitely many dirac measures supported on isolated points in particular such measures do not exist on metric spaces without isolated points furthermore we consider n expansive measure for flows and prove that a flow is n expansive in the sense of 9 if and only if every borel probability measure is n expansive finally we obtain a lower bound of the topological entropy of the n expansive flows as the exponential growth rate of the number of periodic orbits

b splines and optimal stability it is proved that among all nonnegative bases of its space the b spline basis is optimally stable for evaluating spline functions

running error for the evaluation of rational bezier surfaces through a robust algorithm running error analysis of the corner cutting algorithm for rational bezier surfaces is carried out and the sharpness of the corresponding error bounds is shown

a general class of bernstein like bases this paper presents a unified approach to deal with spaces containing simultaneously algebraic and trigonometric or hyperbolic polynomials bases with optimal shape preserving and stability properties are constructed evaluation and subdivision algorithms for them are provided bases for the corresponding and mixed spline spaces are also constructed some nice properties of these bases and the generated curves are shown

continuous time fuzzy decision processes with discounted rewards a continuous time decision process with fuzzy transitions which is termed fuzzy decision process is discussed from a viewpoint of a dynamic fuzzy system by the authors the discounted total reward is given by a fuzzy number and is discussed by the fuzzy max order this paper gives the optimality equation for the fuzzy decision model by a fuzzy differential equation the optimization problem of the discounted total rewards is also discussed by a fuzzy expectation generated from a fuzzy goal and an optimal policy and the permissible range of the decision maker s optimal expected payoffs are given for this problem also an illustrated example is given to explain the idea of the paper

pointwise topological stability and persistence abstract we prove that if a homeomorphism of a compact metric space is equicontinuous and pointwise topologically stable then it is persistent in the sense of lewowicz 12 the proof relies on the notion of persistent measure which has its own interest we compute the borel hierarchy of these measures and prove that they are not necessarily topologically stable in the sense of 9

markov type fuzzy decision processes with a discounted reward on a closed interval abstract we formulate a new multi stage decision process with markov type fuzzy transition which is termed markov type fuzzy decision process in the general framework of the decision process both of state and action are assumed to be fuzzy itself the transition of states is defined using the fuzzy relation with markov property and the discounted total reward is described as a fuzzy number on a closed bounded interval to discuss the optimization problem a partial order of convex fuzzy numbers is introduced in this paper the discounted total reward associated with an admissible stationary policy is characterized by a unique fixed point of the contractive mapping moreover the optimality equation for the fuzzy decision model is derived under some continuity conditions also an illustrated example is given to explain the theoretical results and the computation in the paper

running error for the evaluation of rational bezier surfaces error analysis of the usual method to evaluate rational bezier surfaces is performed the corresponding running error analysis is also carried out and the sharpness of our running error bounds is shown we also modify the evaluation algorithm to include such error bounds without increasing significantly its computational cost

topological stability for fuzzy expansive maps abstract we introduce the definitions of expansivity and topological stability for homeomorphisms on fuzzy metric spaces we show some basic properties of fuzzy expansive homeomorphisms moreover we prove walters s theorem in the context of fuzzy metric spaces i e a fuzzy expansive system with the fuzzy shadowing property is fuzzy topologically stable