shape preserving rational cubic fractal interpolation function abstract a new type of c 1 fractal interpolation function fif is developed using the iterated function system ifs which contains the rational spline the numerator of this rational spline contains cubic polynomial and the denominator of the rational spline contains quadratic polynomial we find uniform error bound between the original function which belongs to the class c 2 and the fif we described suitable conditions on scaling factors and shape parameters such that they preserve the shape properties which are inherited in the data

cubic hermite interpolation with minimal derivative oscillation in this paper a new optimal cubic hermite interpolation method is presented the method is to optimize the derivative of the interpolant the diagonally dominant property of the obtained system of normal equations and the error bound are better than some of the existing cubic interpolants for parametric curve design the vector valued interpolation method is given some numerical examples are provided to illustrate the satisfactory shape of the interpolation curves

an always successful method in univariate convex c 2 interpolation we consider convex interpolation with cubic c 2 splines on grids built by adding two knots in each subinterval of neighbouring data sites the additional knots have to be variable in order to get a chance to always retain convexity by means of the staircase algorithm we provide computable intervals for the added knots such that all knots from these intervals allow convexity preserving spline interpolation of c 2 continuity

c 1 comonotone hermite interpolation via parametric cubics abstract we present a local scheme for c 1 hermite interpolation the interpolant constructed is a parametric piecewise cubic and it is comonotone with the data the approximation order is also investigated under appropriate assumptions the method is fourth order accurate

on shape preserving quadratic spline interpolation in this paper we discuss the design of algorithms for interpolating discrete data using c 1 quadratic splines in such a way that the monotonicity and or convexity of the data is preserved the analysis culminates in an interactive algorithm which takes full advantage of the flexibility which quadratic splines permit

staircase algorithm and construction of convex spline interpolants up to the continuity c3 abstract this paper is concerned with the convex interpolation of data sets based on the staircase algorithm several methods are presented which allow the construction of convex spline interpolants up to the continuity c 3

convexity and monotonicity properties of the local integro cubic spline in this paper we consider shape preserving properties of the local integro cubic spline the sufficient conditions for convex monotone or positive approximation are given some examples are provided to illustrate shape preserving properties of the splines

fuzzy system reliability analysis by interval of confidence abstract the use of fuzzy number over interval of confidence instead of probability and possibilitic considerations for evaluating the range value and interval of the fuzzy reliability is proposed this approach of evaluating fuzzy system reliability by interval arithmetic and cuts has a considerable advantage we can calculate reliability problems easily by the use of derive and mathcad packages through theoretical analysis an illustrative example and computational results we show that the proposed approach is more general and simplified

convex interval interpolation using a three term staircase algorithm motivated by earlier considerations of interval interpolation problems as well as a particular application to the reconstruction of railway bridges we deal with the problem of univariate convexity preserving interval interpolation to allow convex interpolation the given data intervals have to be in strictly convex position this property is checked by applying an abstract three term staircase algorithm which is presented in this paper additionally the algorithm provides strictly convex ordinates belonging to the data intervals therefore the known methods in convex lagrange interpolation can be used to obtain interval interpolants in particular we refer to methods based on polynomial splines defined on grids with additional knots

a new approach for shape preserving interpolating curves in this paper we present an new approach to construct the so called shape preserving interpolation curves the basic idea is first to approximate the set of interpolated points with a class of mq quasi interpolation operators and then pass through the set with the use of multivariate interpolation by using compactly supported radial basis functions this approach possesses the advantages of certain shape preserving and good approximation behaviors the proposed algorithm is easy to implement

symmetric powers of the p 1 dimensional indecomposable module of a cyclic p group and the λ structure of its green ring let k be a field of order p and let g be a cyclic group of order pk k 2 we explicitly decompose the symmetric powers of the indecomposable kg module of dimension p 1 into indecomposable kg modules using this result for every odd prime p we give a negative answer to the conjecture posed by kouwenhoven f m kouwenhoven the structure of the green rings of cyclic p groups proc sympos pure math 47 1987 451 466

on the numerical approximation of the rotation number abstract the starting point of this paper is a polygonal approximation of an invariant curve of a map using this polygonal approximation an approximation for the circle map the restriction of the map to the invariant curve is obtained the rotation number of the circle map is then approximated by the rotation number of the approximated circle map the error in the obtained approximate rotation number is discussed and related to the error in the polygonal approximation of the invariant curve simple algorithms for the approximation of the rotation number are described a numerical example illustrates the theory

algorithms for computing shape preserving spline approximations to data we treat the problem of approximating data that are sampled with error from a function known to be convex and increasing the approximating function is a polynomial spline with knots at the data points this paper presents results analogous to those in 7 and 9 that describe some approximation properties of polynomial splines and algorithms for determining the existence of a shape preserving approximant for given data

generalized multiset theory abstract the paper gives a formal axiomatic theory of collections such that elements may occur some number of times in these collections and this number is allowed to be an arbitrary real number e g a negative one the presented theory is a generalization of multiset theories and some fuzzy set theories some important consequences of the theory are revealed among potential applications of the theory expressing of vague properties and relations is anticipated the usefulness of the theory and its consequences for applications involving vague properties is demonstrated by an example of multi criteria decision making based on vague properties of alternatives

c 2 positivity preserving rational interpolation splines in one and two dimensions abstract a class of rational quartic cubic interpolation spline with two local control parameters is presented which can be c 2 continuous without solving a linear system of consistency equations for the derivative values at the knots the effects of the local control parameters on generating interpolation curves are illustrated for c 2 interpolation the given interpolant can locally reproduce quadratic polynomials and has o h 2 or o h 3 convergence simple schemes for the c 2 interpolant to preserve the shape of 2d positive data are developed moreover based on the boolean sum of quintic interpolating operators a class of bi quintic partially blended rational quartic cubic interpolation surfaces is also constructed the given interpolation surface provides four local control parameters and can be c 2 continuous without using the second or higher mixed partial derivatives on a rectangular grid simple sufficient data dependent constraints are also derived on the local control parameters to preserve the shape of a 3d positive data set arranged over a rectangular grid

convexity preserving piecewise rational cubic interpolation a twice continuously differentiable piecewise rational cubic interpolant f to data s x j y j j 1 cdots n delta j 2 ne 0 x j 1 x j is defined in terms of turn parameters tau j j 1 cdots n 1 and end conditions f x 1 f x n 0 the necessary and sufficient condition f x j delta j 2 geqq 0 j 2 cdots n 1 is derived which ensures that the interpolant f preserves the local convexity concavity properties of s a fast simple and robust procedure for generating f is described and the results obtained for a specific data set s are compared with some standard cubic spline interpolants

shape preserving piecewise rational interpolation with higher order continuity abstract a united form of the classical hermite interpolation and shape preserving interpolation is presented in this paper the presented interpolation method provides higher order continuous shape preserving interpolation splines the given interpolants are explicit piecewise rational expressions without solving a linear or nonlinear system of consistency equations by setting parameter values the interpolation curve can be generated by choosing the classical piecewise hermite interpolation polynomials or the presented piecewise rational expressions for monotonicity preserving and convexity preserving interpolation the appropriate values of a parameter are given on each subinterval numerical examples indicate that the given method produces visually pleasing curves

comonotone shape preserving spline histopolation algorithms are presented for the construction of histopolating splines consisting of linear linear rational or quadratic polynomial pieces a unique comonotone histospline of such kind exists for any histogram with weak alternation of data in general case without weak alternation of data a modified comonotone spline histopolation strategy should be used the method is implemented via the representation with histogram heights and knot values of first derivatives of the spline numerical examples are given

convexity preserving piecewise rational quartic interpolation a family of piecewise rational quartic interpolants is given identified uniquely by the value of a tension parameter lambda i each interpolant of the family can be c 2 spline without solving a linear or nonlinear system of consistency equations for the derivative values at the knots the interpolant can preserve the local convexity concavity properties of the given data a proper choice of lambda i to guarantee shape preservation is given a convergence analysis establishes an error bound in terms of lambda i and shows that o h 3 accuracy is obtained for c 2 continuity several examples are supplied to support the practical value of the method

convexity preserving approximation by univariate cubic splines a convexity preserving approximation method is presented similar to the cubic spline interpolation the given approximation function is univariate cubic spline with c 2 continuity a very simple computing method is described the approximation order and polynomial reproducing property of this convexity preserving approximant are discussed the computing method for interpolating a small quantity of data is given some numerical examples are given to show the effectivity

a general parametric framework for functional tension schemes the paper describes a general approach that can be used to obtain local tension schemes for functional approximation and interpolation examples of one and two dimensional schemes are presented and analysed

a fuzzy set approach to fault tree and reliability analysis abstract conventional fault tree and reliability analysis does not give a picture concerning the tolerances of the probability values of hazards to overcome this shortcoming the relative frequencies of the basic envents are considered as fuzzy numbers using standard approximations for the membership functions n ary possibilistic and or and neg operators are derived allowing a straightforward evaluation of the frequencies of hazard events in complex systems simultaneously with their tolerances the paper shows the use of these operators for constructing possibilistic fault trees the use of the method for numerical calculation is demonstrated on a real like safety problem

shape preserving approximations by polynomials and splines a review of shape preserving approximation methods and algorithms for approxi mating univariate functions or discrete data is given the notion of shape refers to the geometrical behavior of a function s or approximant s graph and usually includes positivity monotonicity and or convexity but in the recent literature the broader concept of shape also includes symmetry gen eralized convexity unimodality lipschitz property possessing peaks or discontinuities etc special stress is put on shape preserving interpolation methods by polynomials and splines of course this text has no pretensions to be complete geywords shape preserving approximation approximation of univariate functions approxima tion of discrete data shape preserving interpolation polynomial spline positivity monotonicity convexity generalized convexity unimodality lipschitz property

similarity of fuzzy choice functions in this paper two new concepts are introduced the similarity and the d equality of fuzzy choice functions we investigate the way the similarity and d equality behave with respect to some fundamental concepts of fuzzy revealed preference theory

d equalities of fuzzy sets abstract a equality of two fuzzy sets means the two fuzzy sets are equal to each other to an extent of s in this paper we present basic properties of 6 equality and various results on 6 equality with respect to algebraic operations fuzzy relations extension principle and triangular norms we also present a number of reliability examples to show usability of equality in reality

shape preserving curves using quadratic trigonometric splines this paper develops a smooth trigonometric quadratic spline it consists of three parameters which provide flexibility of tangents in their magnitude and direction at the end points of each sub interval furthermore constraints have been derived on two of the parameters to demonstrate positive constrained monotone and convex shapes of the data the third parameter is left free to achieve extra control on shape if needed the error bound of developed spline has been achieved of order three

shape preserving interpolation by parametrically defined curves criteria are discussed for interpolation which preserves local convexity and local monotonicity particular schemes are then given for interpolation by parametric piecewise polynomials the interpolating curves can have any desired order of geometric continuity and they are convex in a region where the data are convex if the data arise from a function then the curve is also single valued and in this case the curve can also preserve local monotonicity if desired

λ ring structure of the green ring of a cyclic p group abstract let f p be a field of order p g a cyclic group of order p k and rs p k the green ring of g over f p this paper concerns the conjecture on the ring structure of a certain quotient ring rs p k i p k of rs p k when k 2 which was originally due to kouwenhoven to be more precise he conjectured that the ideal i p k is closed under the exterior powers and rs p k i p k is equipped with the ring structure for the induced exterior powers we show that kouwenhoven s conjecture turns out to be true when p 2 but false when p 3 for other primes except p 2 3 it will be demonstrated that rs p k i p k cannot have the ring structure for the induced exterior powers

a new c2 rational interpolation based on function values and constrained control of the interpolant curves in this paper a new method is developed to create a high order smoothness interpolation using values of the function being interpolated this is a kind of rational cubic interpolation with quadratic denominator this rational spline not only belongs to c 2 in the interpolating interval but could also be used to constrain the shape of the interpolant curve such as to force it to be in the given region all because of the selectable parameters in the rational spline itself the more important achievement mathematically of this method is that the uniqueness of the interpolating function for the given data would be replaced by the uniqueness of the interpolating curve for the given data and the selected parameters

shape preserving piecewise rational interpolant with quartic numerator and quadratic denominator an explicit representation of a piecewise rational interpolant with quartic numerator and quadratic denominator is presented for positivity data monotone data and convex data the shape preserving properties of the interpolant are given the interpolant is c 2 continuous spline with a shape parameter w i on each subinterval the values of w i to guarantee shape preservation are estimated a convergence analysis establishes an error bound in terms of w i and shows that the interpolant is o h 2 or o h 3 accurate several examples are supplied to support the practical value of the given interpolation method

quadratic and related exponential splines in shape preserving interpolation abstract for quadratic and related exponential splines necessary and sufficient conditions are given under which the properties of convexity or monotonicity carry over from the data set to the interpolants it turns out that for quadratic splines the problems of convex or monotone interpolation may be not solvable however when using the more general exponential splines the shape is preserved if the parameters occuring now are chosen appropriately furthermore since convex or monotone spline interpolants are in general not uniquely determined a strategy for selecting one of them is proposed

fuzzy reasoning as a control problem different from the dominant view of treating fuzzy reasoning as generalization of classical logical inference in this paper fuzzy reasoning is treated as a control problem a new fuzzy reasoning method is proposed that employs an explicit feedback mechanism to improve the robustness of fuzzy reasoning the fuzzy rule base given a priori serves as a controlled object and the fuzzy reasoning method serves as the corresponding controller the fuzzy rule base and the fuzzy reasoning method constitute a control system that may be open loop or closed loop depending on the underlying reasoning goals constraints the fuzzy rule base the fuzzy reasoning method and the corresponding reasoning goals constraints define the three distinct ingredients of fuzzy reasoning while various existing fuzzy reasoning methods are essentially a static mapping from the universe of single fuzzy premises to the universe of single fuzzy consequences the new fuzzy reasoning method maps sequences of fuzzy premises to sequences of fuzzy consequences and is a function of the underlying reasoning goals constraints the monte carlo simulation shows that the new fuzzy reasoning method is much more robust than the optimal fuzzy reasoning method proposed in our previous work the explicit feedback mechanism embedded in the fuzzy reasoning method does significantly improve the robustness of fuzzy reasoning which is concerned with the effects of perturbations associated with given fuzzy rule bases and or fuzzy premises on fuzzy consequences the work presented in this paper sets a new starting point for various principles of feedback control and optimization to be applied in fuzzy reasoning or logical inference and to explore new forms of reasoning including robust reasoning and adaptive reasoning it can be also expected that the new fuzzy reasoning method presented in this paper can be used for modeling and control of complex systems and for decision making under complex environments

exterior and symmetric powers of modules for cyclic 2 groups abstract we prove a recursive formula for the exterior and symmetric powers of modules for a cyclic 2 group this makes computation straightforward previously a complete description was only known for cyclic groups of prime order

on univariate function identification problems applications in the areas of data fitting forecasting and estimation naturally lead to a rich class of constrained infinite dimensional optimization problems over extended real valued semicontinuous functions we discuss a framework for dealing with such applications even in the presence of nearly arbitrary constraints on the functions we formulate computationally tractable approximating problems relying on piecewise polynomial semicontinuous approximations of the actual functions the approximations enable the study of evolving problems caused by incrementally arriving data and other information in contrast to an earlier more general treatment we focus on optimization over functions defined on a compact interval of the real line which still addresses a large number of applications the paper provides an introduction to the subject through simplified derivations and illustrative examples

shape preserving c1 hermite interpolants generated by a gori pitolli subdivision scheme in this paper we present a method for the construction of c 1 hermite interpolants obtained from a particular family of refinable spline functions introduced by gori and pitolli they constitute a one parameter subfamily of the hermite interpolants generated by the general merrien s subdivision scheme we compare this family to the other one parameter subfamily studied by merrien and sablonniere and lyche and merrien on the solution of two points hermite interpolation problems with arbitrary monotonicity or convexity constraints

positive quartic monotone quintic c 2 spline interpolation in one and two dimensions abstract this paper is concerned with shape preserving interpolation of discrete data by polynomial splines we show that positivity can be always preserved by quartic c2 splines and monotonicity by quintic c2 splines this is proved for one dimensional interpolation as well as for two dimensional interpolation on rectangular grids

on kolmogorov s representation of functions of several variables by functions of one variable abstract this paper proposes a multivariate nonparametric regression estimator motivated by kolmogorov doklady 114 1957 679 we consider functions of the form k 1 t g k k 1 k x 1 k d k x d 1 t

an energy minimization framework for monotonic cubic spline interpolation this paper describes the use of cubic splines for interpolating monotonic data sets interpolating cubic splines are popular for fitting data because they use low order polynomials and have c2 continuity a property that permits them to satisfy a desirable smoothness constraint unfortunately that same constraint often violates another desirable property monotonicity it is possible for a set of monotonically increasing or decreasing data points to yield a curve that is not monotonic i e the spline may oscillate in such cases it is necessary to sacrifice some smoothness in order to preserve monotonicity the goal of this work is to determine the smoothest possible curve that passes through its control points while simultaneously satisfying the monotonicity constraint we first describe a set of conditions that form the basis of the monotonic cubic spline interpolation algorithm presented in this paper the conditions are simplified and consolidated to yield a fast method for determining monotonicity this result is applied within an energy minimization framework to yield linear and nonlinear optimization based methods we consider various energy measures for the optimization objective functions comparisons among the different techniques are given and superior monotonic c2 cubic spline interpolation results are presented extensions to shape preserving splines and data smoothing are described

c1 sign monotonicity and convexity preserving hermite polynomial splines of variable degree abstract this paper proposes an algorithm for constructing interpolatory hermite polynomial splines of variable degree which preserve the sign the monotonicity and the convexity of the data the polynomial segments are represented as bezier curves the degree of each segment plays the role of the tension parameter of the spline we discuss extensively the monotonicity and convexity criteria detailing a strict and a weak form of monotonicity preservation as well as their implications on the hermite interpolation we also propose a global method for estimating the nodal derivatives of the spline we evaluate the results of this method in an extensive set of examples comparing them with a number of local derivative estimation methods from the pertinent literature the algorithm is numerically stable simple to implement and it is of linear complexity since the related mathematical conditions can be expressed as linear inequalities with respect to the control points of the spline

a smoothing stochastic simulated annealing method for localized shapes approximation abstract in b splines approximation setting it is known that monotony and convexity or concavity shapes can easily be controlled by the spline coefficients in this paper we deal with the general context of combinations of localized shape constraints we prove that unimodality constraint is fulfilled simply by an increasing and decreasing sequence of the spline coefficients by using the descartes sign rule then the local support property of b splines is used to locate each constraint on a given interval we formulate a smoothing spline approximation under inequality constraints in function of the spline coefficients we also give a simulated annealing algorithm to solve the optimization problem and we establish the almost sure convergence of the efficient solution

a note on the value similarity of fuzzy systems variables abstract the notion of similarity of fuzzy variables is discussed in arbitrary set in this paper it is shown that the property of similarity is preserved when the sup min compositional rule of inference is applied to similar values of fuzzy variables and fuzzy relations which extends the results of pappis 1991

scattered data interpolation subject to piecewise quadratic range restrictions abstract the constriction of range restricted univariate and bivariate c1 interpolants to scattered data is considered in particular we deal with quadratic spline interpolation on a refined univariate grid respectively on a powell sabin refinement of a triangulation of the data sites subject to piecewise quadratic lower and upper obstacles to the values of the interpolant the derived sufficient conditions for the fulfillment of the range restrictions result in a system of linear inequalities for the slopes respectively gradients as parameters which is separated with respect to the data sites this system is shown to be always solvable for important special forms of the obstacles if at all in general there exist an infinite number of spline interpolants meeting the constraints the selection of a visually pleasant one is based on the minimization of a suitable choice functional

perturbation of fuzzy sets and fuzzy reasoning based on normalized minkowski distances in this paper we extend the concept of the perturbation of fuzzy sets based on normalized minkowski distances and present some new conclusions on perturbation raised by various operations of fuzzy sets these operations are induced by triangular norms and conorms furthermore we discuss the perturbation of fuzzy reasoning

monotone and convex approximation by splines error estimates and a curve fitting algorithm we obtain jackson type estimates for the approximation of increasing or convex functions by splines with the same property these estimates are new when the knots are not uniformly spaced the method of proof motivates an algorithm for fitting a curve to data with local convexity preservation the properties of this algorithm are developed and some examples are given

admissible slopes for monotone and convex interpolation in many applications interpolation of experimental data exhibiting some geometric property such as nonnegativity monotonicity or convexity is unacceptable unless the interpolant reflects these characteristics this paper identifies admissible slopes at data points of variousc 1 interpolants which ensure a desirable shape we discuss this question in turn for the following function classes commonly used for shape preserving interpolations monotone polynomials c 1 monotone piecewise polynomials convex polynomials parametric cubic curves and rational functions

adams operations on the green ring of a cyclic group of prime power order we consider the green ring r kc for a cyclic p group c over a field k of prime characteristic p and determine the adams operations psi n in the case where n is not divisible by p this gives information on the decomposition into indecomposables of exterior powers and symmetric powers of kc modules

robustness of fuzzy reasoning and spl delta equalities of fuzzy sets fuzzy reasoning methods or approximate reasoning methods are extensively used in intelligent systems and fuzzy control in this paper the author discusses how errors in premises affect conclusions in fuzzy reasoning that is he discusses the robustness of fuzzy reasoning after reviewing his previous work 1996 he presents robustness results for various implication operators and inference rules all the robustness results are formulated in terms of spl delta equalities of fuzzy sets two fuzzy sets are said to be spl delta equal if they are equal to an extent of spl delta

nonnegativity monotonicity or convexity preserving cubic and quintic hermite interpolation the hermite polynomials are simple effective interpolants of discrete data these interpolants can preserve local positivity monotonicity and convexity of the data if we restrict their derivatives to satisfy constraints at the data points this paper de scribes the conditions that must be satisfied for cubic and quintic hermite interpolants to preserve these properties when they exist in the discrete data we construct algorithms to ensure that these constraints are satisfied and give numerical examples to illustrate the effectiveness of the algorithms on locally smooth and rough data 1 introduction piecewise polynomial interpolants especially those based on hermite polynomials polynomials determined by their values and values of one or more derivatives at both ends of an interval have a number of desirable properties they are easy to compute once the derivative values are chosen if the derivative values are chosen locally e g by finite difference methods then the interpolant at a given point will depend only on the given data at nearby mesh points if the derivatives are computed by spline methods then the interpolant will have an extra degree of continuity at the mesh points in either case the interpolant is linear in the given function values and has excellent convergence properties as the mesh spacing decreases these methods however do not necessarily preserve the shape of the given data when the data arise from a physical experiment it may be vital that the interpolant preserve nonnegativity f x 0 nonpositivity f x 0 or f x 0 or concavity f x 0 in this and other cases geometric considerations such as preventing spurious behavior near rapid changes in the data may be more important than the asymptotic accuracy of the interpolation method one can construct a shape preserving interpolant by constraining the derivatives for the hermite polynomials to meet conditions which imply the desired properties 4 5 8 11 15 20 by adding new mesh points

shape preserving interpolation by quadratic splines abstract an interpolating quadratic spline was constructed which preserves the shape of data the spline contains parameters which can be used to match it to additional information

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

monotonicity preserving interpolatory subdivision schemes a class of local nonlinear stationary subdivision schemes that interpolate equidistant data and that preserve monotonicity in the data is examined the limit function obtained after repeated application of these schemes exists and is monotone for arbitrary monotone initial data next a class of rational subdivision schemes is investigated these schemes generate limit functions that are continuously differentiable for any strictly monotone data the approximation order of the schemes is four some generalisations such as preservation of piecewise monotonicity and application to homogeneous grid refinement are briefly discussed