total positivity and its applications preface i spline functions total positivity of the spline kernel and its applications b bojanov total positivity properties of lb splines t lyche l l schumaker total positivity and splines k morken totally positive systems and interpolation by subspaces of generalized splines m sommer h strauss ii matrix theory preservation of total nonnegativity under the hadamard product and related topics j garloff d g wagner vertex implications for totally nonnegative under the hadamard product and related topics j garloff on factorizations of totally positive matrices j garloff on factorizations of totally positive matrices m gasca j m pena iii geometric modeling total positivity and optimal bases j m carnicer j m pena total positivity and the shape of curves t n t goodman tchebycheff curves m l mazure h pottmann iv probability and mathematical biology analytical and probabilistic methods for totally positive matrices g s goodman some inequalities of total positivity in pure and applied mathematics s karlin z ziegler new directions and problems of multivariate total positivity motivated by genetic recombination probabilities s karlin v approximation theory tchebycheff spaces and total positivity j m carnicer j m pena cebysev and weak cebysev systems r a zalik vi complex analysis zero sets of entire fourier transforms of multiply positive density functions o m katkova vii statistics total nonnegativity of moment matrices and its application to e optimal designs in polynomial regression b heiligers analysis of regression dependence in 2 xn bivariate distributions and some applications in contingency tables s kasala m b rao viii real analysis total positivity and nonlinear analysis r q jia solutions of certain functional equations s karlin alternance and snakes in the nonlinear finite moment problem k tatalian ix combinatorics the applications of total positivity to combinatorics and conversely f brenti x integral equations spectral properties of totally positive kernels and matrices a pinkus index

a measurement theory view on the granularity of partitions measurement of granularity is one of the foundational issues in granular computing this paper investigates a class of measures of granularity of partitions the granularity of a set is defined by a strictly monotonic increasing transformation of the cardinality of the set the granularity of a partition is defined as the expected granularity of all blocks of the partition with respect to the probability distribution defined by the partition many existing measures of granularity are instances of the proposed class new measures of granularity of partitions are also introduced

corner cutting algorithms for the bezier representation of free form curves abstract we give a general treatment of the problem of converting a curve expressed in a b spline expansion into its bezier form we show that corner cutting of the b spline control polygon leads to the bezier points exactly when the bezier matrix is totally positive also for spline spaces used to model space curves with curvature continuity we give an algorithm for determining the bezier representation

generalized bezier curves and surfaces based on lupas q analogue of bernstein operator in this paper a new generalization of bezier curves with one shape parameter is constructed it is based on the lupas q analogue of bernstein operator which is the first generalized bernstein operator based on the q calculus the new curves have some properties similar to classical bezier curves moreover we establish degree evaluation and de casteljau algorithms for the generalization furthermore we construct the corresponding tensor product surfaces over the rectangular domain and study the properties of the surfaces as well as the degree evaluation and de casteljau algorithms compared with q bezier curves and surfaces based on phillips q bernstein polynomials our generalizations show more flexibility in choosing the value of q and superiority in shape control of curves and surfaces the shape parameters provide more convenience for the curve and surface modeling

fuzzy probabilistic approximation spaces and their information measures rough set theory has proven to be an efficient tool for modeling and reasoning with uncertainty information by introducing probability into fuzzy approximation space a theory about fuzzy probabilistic approximation spaces is proposed in this paper which combines three types of uncertainty probability fuzziness and roughness into a rough set model we introduce shannon s entropy to measure information quantity implied in a pawlak s approximation space and then present a novel representation of shannon s entropy with a relation matrix based on the modified formulas some generalizations of the entropy are proposed to calculate the information in a fuzzy approximation space and a fuzzy probabilistic approximation space respectively as a result uniform representations of approximation spaces and their information measures are formed with this work

on the topological properties of fuzzy rough sets this paper is devoted to the discussion of the relationship between fuzzy rough set models and fuzzy topologies on a finite universe the tc axiom for fuzzy topology is proposed it is proved that the set of all lower approximation sets based on a reflexive and transitive fuzzy relation consists of a fuzzy topology which satisfies tc axiom and conversely a fuzzy topology which obey tc axiom is just the set of all lower approximation sets under a reflexive and transitive fuzzy relation that is to say there exists a one to one correspondence between the set of all reflexive and transitive fuzzy relations and the set of all fuzzy topologies which satisfy tc axiom

quantum bernstein bases and quantum bezier curves the purpose of this paper is to investigate the most general quantum bernstein bases and quantum bezier curves classical bernstein bases satisfy a two term formula for their classical derivatives quantum bernstein bases satisfy a two term formula for their quantum derivatives to study the properties of these general quantum polynomial schemes a new variant of the blossom the quantum blossom is introduced by altering the diagonal property of the classical blossom the significance of the quantum blossom is that the quantum blossom provides the dual functionals for quantum bezier curves over arbitrary intervals using the quantum blossom several fundamental identities involving the quantum bernstein bases are developed including a quantum variant of the marsden identity and the partition of unity property based on these properties of quantum bernstein bases quantum bezier curves are shown to be affine invariant and under certain conditions lie in the convex hull of their control points in addition for each quantum bezier curve of degree n a collection of n affine invariant recursive evaluation algorithms are derived using two of these recursive evaluation algorithms a recursive subdivision procedure for quantum bezier curves is constructed this subdivision procedure generates a sequence of control polygons that converges rapidly to the original quantum bezier curve

an approximation algorithm for the general max min resource sharing problem we propose an approximation algorithm for the general max min resource sharing problem with m nonnegative concave constraints on a convex set b the algorithm is based on a lagrangian decomposition method and it uses a c approximation algorithm called approximate block solver for a simpler maximization problem over the convex set b we show that our algorithm achieves within iterations or calls to the approximate block solver a solution for the general max min resource sharing problem with approximation ratio the algorithm is faster and simpler than the previous known approximation algorithms for the problem 12 13

uncertainly measures of rough set prediction abstract the main statistics used in rough set data analysis the approximation quality is of limited value when there is a choice of competing models for predicting a decision variable in keeping within the rough set philosophy of non invasive data analysis we present three model selection criteria using information theoretic entropy in the spirit of the minimum description length principle our main procedure is based on the principle of indifference combined with the maximum entropy principle thus keeping external model assumptions to a minimum the applicability of the proposed method is demonstrated by a comparison of its error rates with results of c4 5 using 14 published data sets

symmetric functions and the vandermonde matrix this work deduces the lower and the upper triangular factors of the inverse of the vandermonde matrix using symmetric functions and combinatorial identities the l and u matrices are in turn factored as bidiagonal matrices the elements of the upper triangular matrices in both the vandermonde matrix and its inverse are obtained recursively the particular value x i 1 q qi 1 in the indeterminates of the vandermonde matrix is investigated and it leads to q binomial and q stirling matrices it is also shown that q stirling matrices may be obtained from the pascal matrix

relationships between fuzzy approximation spaces and their uncertainty measures abstract fuzzy approximation spaces fa spaces can be seen as approximation spaces a spaces in a fuzzy environment this article studies relationships between fa spaces and their uncertainty measures dependence partial dependence and independence between fa spaces as characterized by the inclusion degree are first proposed then the distance between fa spaces is considered in terms of the angle of difference next algebraic map and lattice features of fa spaces are obtained finally the uncertainty of fa spaces is measured by using the dependence between them an application is given and to assess the performance of the presented measures two examples are used to perform effectiveness analysis by means of elementary statistical methods

the approximation of a totally positive band matrix by a strictly banded totally positive one every nonsingular totally positive m banded matrix is shown to be the product of m totally positive one banded matrices and therefore the limit of strictly m banded totally positive matrices this result is then extended to bi infinite m banded totally positive matrices with linearly independent rows and columns in the process such matrices are shown to possess at least one diagonal whose principal sections are all nonzero as a consequence such matrices are seen to be approximable by strictly m banded totally positive ones

a novel extension to the polynomial basis functions describing bezier curves and surfaces of degree n with multiple shape parameters the construction of bezier curves using shape control parameters is one of the most popular areas of research in computer aided geometric design cagd a class of new polynomial basis functions with n 1 local shape control parameters is presented here to allow the construction of bezier curves with n local shape control parameters which is an extension to the classical bernstein basis functions of degree n the properties of the proposed basis functions and the corresponding piecewise polynomial curve with n 1 local shape control parameters are analyzed this analysis shows that the new class of polynomial functions meets the conditions required for both c 0 c 1 and c 2 continuity as well as g 0 g 1 and g 2 continuity some curve design applications are then discussed and an extended application for surface design is also presented the modeling examples illustrate that the new extension provides not only a better approximation and mathematical description of bezier curves but allows the shape parameters to be altered making it a valuable method for the design of curves and surfaces

local path fitting a new approach to variational integrators a new approach for constructing variational integrators is presented in the general case the estimation of the action integral in a time interval t k t k 1 is used to construct a symplectic map q k q k 1 q k 1 q k 2 the basic idea is that only the partial derivatives of the estimated action integral of the lagrangian are needed in the general theory the analytic calculation of these derivatives gives rise to a new integral that depends on the euler lagrange vector itself which in the continuous and exact case vanishes and not on the lagrangian since this new integral can only be computed through a numerical method based on some internal grid points we can locally fit the exact curve by demanding the euler lagrange vector to vanish at these grid points thus the integral vanishes and the process dramatically simplifies the calculation of high order approximations the new technique is tested in high order solutions of the two body problem with high eccentricity up to 0 99 and of the outer planets of the solar system

a novel algorithm for finding reducts with fuzzy rough sets attribute reduction is one of the most meaningful research topics in the existing fuzzy rough sets and the approach of discernibility matrix is the mathematical foundation of computing reducts when computing reducts with discernibility matrix we find that only the minimal elements in a discernibility matrix are sufficient and necessary this fact motivates our idea in this paper to develop a novel algorithm to find reducts that are based on the minimal elements in the discernibility matrix relative discernibility relations of conditional attributes are defined and minimal elements in the fuzzy discernibility matrix are characterized by the relative discernibility relations then the algorithms to compute minimal elements and reducts are developed in the framework of fuzzy rough sets experimental comparison shows that the proposed algorithms are effective

information theoretic measures of uncertainty for rough sets and rough relational databases rough set theory has become well established as a mechanism for uncertainty management in a wide variety of applications including databases this paper addresses the measurement of uncertainty in rough sets and rough relational databases by introducing a measurement based on information theory this rough entropy is discussed as it applies to rough sets in general and in particular to aspects of the rough relational database model

q bernstein polynomials and bezier curves we define q bernstein polynomials which generalize the classical bernstein polynomials and show that the difference of two consecutive q bernstein polynomials of a function f can be expressed in terms of second order divided differences of f it is also shown that the approximation to a convex function by its q bernstein polynomials is one sided a parametric curve is represented using a generalized bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve we study the nature of degree elevation and degree reduction for this basis and show that degree elevation is variation diminishing as for the classical bernstein basis

weighted lupas q bezier curves this paper is concerned with a new generalization of rational bernstein bezier curves involving q integers as shape parameters a one parameter family of rational bernstein bezier curves weighted lupas q bezier curves is constructed based on a set of lupas q analogue of bernstein functions which is proved to be a normalized totally positive basis the generalized rational bezier curve is investigated from a geometric point of view the investigation provides the geometric meaning of the weights and the representation for conic sections we also obtain degree evaluation and de casteljau algorithms by means of homogeneous coordinates numerical examples show that weighted lupas q bezier curves have more modeling flexibility than classical rational bernstein bezier curves and lupas q bezier curves and meanwhile they provide better approximations to the control polygon than rational phillips q bezier curves

bezier curves based on lupas p q analogue of bernstein functions in cagd abstract this paper deals with the extension of rational lupas bernstein functions lupas bezier curves and surfaces involving p q integers as shape parameters for all p 0 and q 0 two different techniques such as de casteljau s algorithm and korovkin s type approximation based on p q integers are used a two parameter family for lupas p q bernstein functions is constructed and their degree elevation and reduction properties have been studied for lupas p q bezier curves some of their basic properties as well as degree elevation and de casteljau algorithm have been discussed the new curves have some properties similar to lupas q bezier curves similarly the corresponding tensor product for lupas bezier surfaces over the rectangular domain u v 0 1 0 1 depending on four parameters is constructed the de casteljau algorithm and degree evaluation properties of the surfaces for these generalizations and some fundamental properties are discussed we get lupas q bezier surfaces for u v 0 1 0 1 when we set the parameter p 1 p 2 1 with the help of rational lupas bernstein functions p q lupas bernstein operators are constructed based on korovkin s type approximation it has been shown that the sequence of p q analogue of lupas bernstein operators l p n q n n f x converges uniformly to f x c 0 1 if and only if 0 q n p n 1 such that lim n q n 1 lim n p n 1 and lim n p n n 1 lim n q n n 1 on the other hand for any p 0 fixed and p 1 the sequence l p q n f x converges uniformly to f x c 0 1 if and only if f x a x b for some a b r furthermore in comparison to q bezier curves and surfaces based on lupas q bernstein rational functions our generalization gives us more flexibility in controlling the shapes of curves and surfaces

entropy measures and granularity measures for set valued information systems set valued information systems are generalized models of single valued information systems in this paper we propose two new relations for set valued information systems based on these two relations the concepts of knowledge information entropy knowledge rough entropy knowledge granulation and knowledge granularity measure are defined in set valued information systems and some properties are investigated moreover relationship between knowledge information entropy and knowledge granulation knowledge rough entropy and knowledge granularity measure are studied it is also shown that knowledge information entropy and knowledge granularity measure can be used to evaluate the certainty degree of knowledge in set valued information systems and knowledge rough entropy and knowledge granulation can be used to evaluate the uncertainty degree of knowledge in set valued information systems these results may supply a further understanding the essence of uncertainty and granularity in set valued information systems

q geometric and q binomial distributions of order k abstract in this paper we generalize geometric and binomial distributions of order k to q geometric and q binomial distributions of order k using bernoulli trials with a geometrically varying success probability in particular we derive expressions for the probability mass functions of these distributions for q 1 these distributions reduce to geometric and binomial distributions of order k which have been extensively studied in the literature

communication between information systems using fuzzy rough sets communication between information systems is a basic problem in granular computing and the concept of homomorphism is a useful mathematical tool to study this problem in this paper some properties of communication between information systems based on fuzzy rough sets are investigated the concepts of fuzzy relation mappings between universes are first proposed in order to construct a fuzzy relation of one universe according to the given fuzzy relation on the other universe the main properties of the mappings are studied the notions of homomorphism of information systems based on fuzzy rough sets are then proposed and it is proved that properties of relation operations in the original information system and structural features of the system such as approximations of arbitrary fuzzy sets and attribute reductions are guaranteed in its image system under the condition of homomorphism

a novel generalization of bezier curve and surface a new formulation for the representation and designing of curves and surfaces is presented it is a novel generalization of bezier curves and surfaces firstly a class of polynomial basis functions with n adjustable shape parameters is present it is a natural extension to classical bernstein basis functions the corresponding bezier curves and surfaces the so called quasi bezier i e q bezier for short curves and surfaces are also constructed and their properties studied it has been shown that the main advantage compared to the ordinary bezier curves and surfaces is that after inputting a set of control points and values of newly introduced n shape parameters the desired curve or surface can be flexibly chosen from a set of curves or surfaces which differ either locally or globally by suitably modifying the values of the shape parameters when the control polygon is maintained the q bezier curve and surface inherit the most properties of bezier curve and surface and can be more approximated to the control polygon it is visible that the properties of end points on q bezier curve and surface can be locally controlled by these shape parameters some examples are given by figures

direct and converse results for multivariate generalized bernstein polynomials in this paper the following generalization of multivariate bernstein polynomials has been studied b n f x 1 x 2 x m email protected k 1 k 2 k m 0nfk 1b n k 2b n k mb email protected j 1mnk jx j k j 1 x j n k j where b n is a sequence of positive numbers such that lim n n b n 1 and direct and inverse theorems for this polynomials have been presented