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

on monotone and convex spline interpolation this paper is concerned with the problem of existence of monotone and or convex splines having degree n and order of continuity k which interpolate to a set of data at the knots the interpolating splines are obtained by using bernstein polynomials of suitable continuous piecewise linear functions they satisfy the inequality k n k the theorems presented here are useful in developing algorithms for the construction of shape preserving splines interpolating arbitrary sets of data points earlier results of mcallister passow and roulier can be deduced from those given in this paper

a fractal procedure for monotonicity preserving interpolation interpolation subject to strip condition on first derivative is considered with a fif shape parameters involved in the fif controls shape of the curve near left and right end points some traditional rational splines emerge as special cases of the fif derivative of the monotonic rational fif may be nondifferentiable in a finite or dense subset of interval this paper is concerned with interpolation subject to a strip condition on the first order derivative using a class of rational cubic fractal interpolation functions fifs this facilitates the fif to generate monotonic curves for a given set of monotonic data the proposed monotonicity preserving rational fif subsumes and supplements a classical monotonic rational cubic spline in models leading to the monotonicity preserving interpolation problem wherein the first order derivative of the constructed interpolant is to be nondifferentiable in a finite or dense subset of the interpolation interval the developed fractal scheme is well suited in contrast to its classical nonrecursive counterpart it is shown that the present fractal interpolation scheme has o h 4 accuracy provided the original function belongs to c 4 i and the parameters involved in the fif are appropriately chosen

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

moment preserving spline approximation on finite intervals continuing previous wotk we discuss the problem of approximating a functionf on the interval 0 1 by a spline function of degreem withn variable knots matching as many of the initial moments off as possible additional constraints on the derivatives of the approximation at one endpoint of 0 1 may also be imposed we show that if the approximations exist they can be represented in terms of generalized gauss lobatto and gauss radau quadrature rules relative to appropriate moment functionals or measures depending off pointwise convergence asn for fixedm 0 is shown for functionsf that are completely monotonic on 0 1 among others numerical examples conclude the paper

free vibration analysis of cylindrical helical springs by the pseudospectral method abstract the pseudospectral method is applied to the free vibration analysis of cylindrical helical springs the displacements and the rotations are approximated by the series expansions of chebyshev polynomials and the governing equations are collocated the number of collocation points is chosen to be less than the number of the expansion terms to handle the boundary condition numerical examples are provided for fixed fixed free free fixed free and hinged hinged boundary conditions the results show good agreement with those of the transfer matrix method and the dynamic stiffness method the formulation of the pseudospectral method is straightforward and shows an exponential rate of convergence with mesh refinement

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

the grothendieck duality theorem via bousfield s techniques and brown representability grothendieck proved that if f x y is a proper morphism of nice schemes then rf has a right adjoint which is given as tensor product with the relative canonical bundle the original proof was by patching local data deligne proved the existence of the adjoint by a global argument and verdier showed that this global adjoint may be computed locally in this article we show that the existence of the adjoint is an immediate consequence of brown s representability theorem lit follows almost as imme diately by smashing arguments that the adjoint is given by tensor product with a dualising complex verdier s base change theorem is an easy conse quence department of mathematics university of virginia charlottesville virginia 22903 e mail address an3rqvirginia edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 06 26 19 utc all use subject to http about jstor org terms

an innovative eigenvalue problem solver for free vibration of uniform timoshenko beams by using the adomian modified decomposition method this paper deals with free vibration problems of uniform timoshenko beams under various supporting boundary conditions the technique we have used is based on applying the adomian modified decomposition method amdm to our vibration problems doing some simple mathematical operations on the method we can obtain ith natural frequencies and mode shapes one at a time the computed results agree well with those analytical and numerical results given in the literature these results indicate that the present analysis is accurate and provides a unified and systematic procedure which is simpler and more straightforward than the other modal analysis

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

an algorithm for computing shape preserving interpolating splines of arbitrary degree abstract the aim of this paper is to describe an algorithm for computing co monotone and or co convex splines of degree m and deficiency m k at the knots 0 k m k which are interpolant or osculatory to a given set of data the method is based upon some existence properties recently developed graphical examples and a listing of the fortran code spisp1 are given

restricted range approximation by splines and variational inequalities error estimates are obtained for the approximation of nonnegative functions from below by nonnegative splines the maximum norm results are of the same order as the jackson type estimates for unconstrained approximation by splines the l 2 results are used to estimate the rate of convergence of the ritz method for certain variational inequalities the more general problem of approximating functions f in the set w g in c a b l x leqq g x leqq u x x in a b by splines in w is also discussed

parametric representation of fuzzy numbers and application to fuzzy calculus we present several models to obtain simple parametric representations of the fuzzy numbers or intervals based on the use of piecewise monotonic functions of different forms the representations have the advantage of allowing flexible and easy to control shapes of the fuzzy numbers we use the standard cuts setting but also the membership functions are obtained immediately and can be used directly to obtain error controlled approximations of the fuzzy calculus in terms of a finite set of parameters the general setting is the hermite type interpolation where the values and the slopes of the monotonic interpolators are given by appropriate parameters and the overall errors of the fuzzy computations can be controlled within a prefixed tolerance by eventually augmenting the total number of pieces and of the parameters by which the results are obtained the representations are designed to model the lower and the upper extremal values of each cut compact intervals of the fuzzy numbers and are able to produce almost any possible configuration differentiable continuous or with a finite number of discontinuity points by using parametric monotonic functions of different types we show applications in the standard fuzzy calculus and we stress the generality and the applicability of the proposed representation to a large class of problems including the numerical solution of fuzzy differential equations the fuzzy linear regression and the stochastic extensions of the fuzzy mathematics the proposed model is called the lower upper representation and we denote the associated fuzzy numbers or intervals as lu fuzzy

the theoretical analysis and algorithm on a class of optimal curve fitting problems in this paper the theory and algorithm on a class of optimal curve fitting problems which can be extensively applied to the engineering are established and completed with regard to the curve fitting problems the available theories and methods are largely concerned with the spline interpolation and linear or nonlinear least squares methods and there are lots of ripe results to my best knowledge however in view of the complexity the third method that is to resort to solve functional extremal problems remains completed both in theories and algorithms regarding the class of this problems that is here called as the optimal curve fitting problems under relaxation constraints the analytic expression of solution and solving algorithms were discussed in p cheng k c zhang on a class of constrained functional minimization problems and their numerical solution journal of optimization theory and applications 78 2 1993 p267 p287 p cheng kc zhang unified method for a kind of constrained functional minimization computing technology and automation 10 2 1991 13 25 k c zhang a numerical method for solving a class of variation problems with constraint and its application to engineering numerical mathematics a journal of chinese universities 11 2 1989 1 9 k c zhang the algorithm and analysis of numerical computation beijing science press 2003 pp 172 184 but the existence of optimal solution has not been solved so far in the paper with help of variational principle and convexity analysis we give the proof of existence in details and a different proof about the analytic expression of optimal solution moreover we prove that for every smoothing parameter m this class of optimal curve fitting problems is equivalent to solve a quadratic programming with a spherical constraint which usually appears in the trust region and sqp subproblems and its algorithms have been quite ripe therefore a novel algorithm different from zhang 2003 for this class problems is established finally we give an example of m 2 to validate our theory and method numerical test shows that the method is effective and practical

post processing piecewise cubics for monotonicity monotone interpolation schemes of the fit and modify type are considered algorithms with optimal order error properties are developed numerical examples are given comparing these methods with others the theoretical underpinnings of the algorithms include sufficient conditions for the existence of monotone splines with specified error and smoothness properties and a result concerning approximate projections onto convex sets

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

a nonlinear algorithm for monotone piecewise bicubic interpolation we present an algorithm for monotone interpolation on a rectangular mesh we use the sufficient conditions for monotonicity of carlton and fritsch we use nonlinear techniques to approximate the partial derivatives at the grid points we develop piecewise bicubic hermite interpolants with these approximations we present some numerical examples where we compare different results in this paper we present an algorithm for monotone interpolation of monotone data on a rectangular mesh by piecewise bicubic functions carlton and fritsch 1985 develop conditions on the hermite derivatives that are sufficient for such a function to be monotone here we extend our results of arandiga 2013 to obtain nonlinear approximations to the first partial and first mixed partial derivatives at the mesh points that allow us to construct a monotone piecewise bicubic interpolants we analyze its order of approximation and present some numerical experiments

vibration analysis of circular mindlin plates using the differential quadrature method abstract axisymmetric free vibrations of moderately thick circular plates described by the linear shear deformation mindlin theory are analyzed by the differential quadrature dq method the first fifteen natural frequencies of vibration are calculated for uniform circular plates with free simply supported and clamped edges through these computations the capability and simplicity of the differential quadrature method for moderately thick plate eigenvalue analysis is demonstrated and convergence and accuracy are thoughtfully examined the case of rigid point support at the plate centre is also considered in the present paper for which special attention is paid to the capability and convergence of the current method

axisymmetric vibrations of polar orthotropic mindlin annular plates of variable thickness abstract analysis and numerical results are presented for the axisymmetric vibrations of polar orthotropic annular plates with linear variation in thickness according to mindlin s shear theory of plates a chebyshev collocation technique has been employed to obtain the frequency equations for the transverse motion of such plates for three different boundary conditions frequencies mode shapes and moments for the first three modes of vibration have been computed for different plate parameters a comparison of frequencies with the corresponding values obtained by classical plate theory leads to some interesting conclusions

monotonicity preserving c2 rational cubic spline for monotone data abstract designers in industries need to generate splines which can interpolate the data points in such a way that they preserve the inherited shape characteristics positivity monotonicity convexity of data among the properties that the spline for curves and surfaces need to satisfy smoothness and shape preservation of given data are mostly needed by all the designers in this paper a rational cubic function with three shape parameters has been developed data dependent sufficient constraints are derived for one of these shape parameters to preserve the inherited shape feature like monotonicity of data remaining two shape parameters are left free for designer to refine the shape of the monotone curve as desired numerical examples and interpolation error analysis show that the interpolant is not only c 2 local computationally economical and visually pleasant but also smooth the error of rational cubic function is also calculated when the arbitrary function being interpolated is c 3 in an interpolating interval the order of approximation of interpolant is o h i 3

a method for computing the tension parameters in convexity preserving spline in tension interpolation this paper is concerned with the problem of convexity preservng orc preserving interpolation by using exponential splines in tension or est s for this purpose the notion of ac preserving interpolant which is usually employed in spline in tension interpolation is refined and the existence ofc preserving est s is established for the so calledc admissible data sets the problem of constructing ac preserving and visually pleasing est is then treated by combining a generalized newton raphson method due to ben israel with a step length technique which serves the need for visual pleasantness the numerical performance of the so formed iterative scheme is discussed for several examples

eigenvalue analysis of timoshenko beams and axisymmetric mindlin plates by the pseudospectral method a study of the free vibration of timoshenko beams and axisymmetric mindlin plates is presented the analysis is based on the chebyshev pseudospectral method which has been widely used in the solution of fluid mechanics problems clamped simply supported free and sliding boundary conditions of timoshenko beams are treated and numerical results are presented for different thickness to length ratios eigenvalues of the axisymmetric vibration of mindlin plates with clamped simply supported and free boundary conditions are presented for various thickness to radius ratios

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

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

variational derivation of governing differential equations for truncated version of bresse timoshenko beams abstract in this paper we provide a variational derivation of modified bresse timoshenko equations this process leads to the additional term to so called truncated set of bresse timoshenko equations this new term is associated with the modified slope inertia the two sets of truncated versions of the bresse timoshenko equations are contrasted with each other as well as with the original bresse timoshenko equations on the example of the a beam that is simply supported at its both ends and b the cantilever beam with end mass it is concluded that application of either truncated sets of equations is advantageous over the original bresse timoshenko equations as far as two truncated sets of equations are concerned the variationally derived set appears to be preferable

discrete approximations to spherically symmetric distributions we consider the approximation of spherically symmetric distributions in d by linear combinations of heaviside step functions or dirac delta functions the approximations are required to faithfully reproduce as many moments as possible we discuss stable methods of computing such approximations taking advantage of the close connection with gauss christoffel quadrature numerical results are presented for the distributions of maxwell bose einstein and fermi dirac

vibration of mindlin plates using boundary characteristic orthogonal polynomials abstract the vibration analysis of shear deformable plates formulated on the basis of first order mindlin theory is presented the displacement and rotational functions of the plates are approximated by sets of boundary characteristic orthogonal polynomials the ease of generation and manipulation of these polynomial functions greatly enhances the computational efficiency of the numerical method the energy functional of the shear deformable plates derived from the mindlin plate theory is minimized in the ritz procedure to arrive at the governing eigenvalue equation corresponding natural frequencies and mode shapes can be obtained by solving the eigenvalue equation computed frequency results for various boundary conditions aspect ratios a b and thickness ratios t b are presented to demonstrate the effects of each factor on the vibration frequencies of moderately thick plates vibration mode shapes in the form of contour plots are presented for a thickness ratio of t b 0 1 these plots are believed to be the first known in the open literature

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

piecewise cubic curve fitting algorithm we give a piecewise cubic curve fitting algorithm which preserves monotonicity of the data the algorithm has a higher order of convergence than the fritsch carlson algorithm and is simpler than the eisenstat jackson lewis algorithm

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

the order of monotone piecewise cubic interpolation fritsch and carlson siam j numer anal 17 1980 pp 238 246 developed an algorithm which produces a monotone c 1 piecewise cubic interpolant to a monotone function we show that their algorithm yields a third order approximation while a modification is fourth order accurate

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

a bivariate rational cubic interpolating spline with biquadratic denominator a bivariate rational bicubic interpolating spline bris with biquadratic denominator and six shape parameters is constructed in a rectangle domain the c1 continuous condition of bris discussed bris is proved to be bounded and its error is estimated in the case of the equally spaced knots the matrix expression and symmetry of bris are presented some properties of the basis of bris are given in the end a numerical example is given to illustrat the effect of the shape parameters on the shape of bris surface

interpolation of lipschitz functions this paper describes a new computational approach to multivariate scattered data interpolation it is assumed that the data is generated by a lipschitz continuous function f the proposed approach uses the central interpolation scheme which produces an optimal interpolant in the worst case scenario it provides best uniform error bounds on f and thus translates into reliable learning of f this paper develops a computationally efficient algorithm for evaluating the interpolant in the multivariate case we compare the proposed method with the radial basis functions and natural neighbor interpolation provide the details of the algorithm and illustrate it on numerical experiments the efficiency of this method surpasses alternative interpolation methods for scattered data

computation of natural frequencies of shear deformable beams and plates by an rbf pseudospectral method a study of free vibration of timoshenko beams and mindlin plates is presented the analysis is based on a new numerical scheme where collocation by radial basis functions and pseudospectral methods are combined to produce highly accurate results numerical results are presented and discussed for various thickness to length ratios

auslander reiten sequences and t structures on the homotopy category of an abelian category we use t structures on the homotopy category kb r mod for an artin algebra r and watts representability theorem to give an existence proof for auslander reiten sequences of r modules this framework naturally leads to a notion of generalized or higher auslander reiten sequences

higher auslander reiten sequences and t structures abstract let r be an artin algebra and c an additive subcategory of mod r we construct a t structure on the homotopy category k c and argue that its heart h c is a natural domain for higher auslander reiten ar theory in the paper 5 we showed that k mod r is a natural domain for classical ar theory here we show that the abelian categories h mod r and h c interact via various functors if c is functorially finite then h c is a quotient category of h mod r we illustrate our theory with two examples when c is a maximal n orthogonal subcategory iyama developed a higher ar theory see 10 in this case we show that the simple objects of h c correspond to iyama s higher ar sequences and derive his higher ar duality from the existence of a serre functor on the derived category d b h c the category o of a complex semi simple lie algebra fits into higher ar theory in the situation when r is the coinvariant algebra of the weyl group

monotone approximation of decision problems many decision problems exhibit structural properties in the sense that the objective function is a composition of different component functions that can be identified using empirical data we consider the approximation of such objective functions subject to general monotonicity constraints on the component functions using a constrained b spline approximation we provide a data driven robust optimization method for environments that can be sample sparse the method which simultaneously identifies and solves the decision problem is illustrated for the problem of optimal debt settlement in the credit card industry

a spectral tchebychev technique for solving linear and nonlinear beam equations abstract this paper presents a spectral tchebychev technique for solving linear and nonlinear beam problems the technique uses tchebychev polynomials as spatial basis functions and applies galerkin s method to obtain the spatially discretized equations of motion unlike alternative techniques that require different admissible functions for each different set of boundary conditions the spectral tchebychev technique incorporates the boundary conditions into the derivation and thereby enables the utilization of the solution for any linear boundary conditions without re derivation furthermore the proposed technique produces symmetric system matrices for self adjoint problems in this work the spectral tchebychev solutions for euler bernoulli and timoshenko beams are derived the convergence and accuracy characteristics of the spectral tchebychev technique is studied by solving eigenvalue problems with different boundary conditions it is found that the convergence is exponential and a small number of polynomials is sufficient to obtain machine precision accuracy the application of the technique is demonstrated by solving 1 eigenvalue problems for tapered timoshenko beams with different boundary conditions taper ratios and beam lengths 2 an euler bernoulli beam problem with spatially and temporally varying forcing elastic boundary and damping 3 large deflection nonlinear euler bernoulli beam problems with different boundary conditions and 4 a micro beam problem with nonlinear electrostatic excitation the results obtained from the spectral tchebychev solutions are seen to be in excellent agreement with those presented in the literature

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

direction consistent tangent vectors for generating interpolation curves abstract in this paper monotonicity preserving interpolation is generalized to direction consistent interpolation the conditions for constructing direction consistent tangent vectors are given the conditions on the tangent vectors are also obtained such that the piecewise cubic hermite interpolation curves are tangent direction consistent with the direction of data polygon based on geometric insights the balanced tangent vectors are presented and proved to be direction consistent tangent vectors with the balanced tangent vectors the generated cubic hermite interpolation curves are tangent direction consistent and the generated quintic hermite interpolation curves are also tangent direction consistent provided that we use the accumulative chord length parametrization some graphic examples are given to show that the generated interpolation curves preserve satisfactorily the shape of the given data control polygon

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

approximate fuzzy arithmetic operations using monotonic interpolations abstract we suggest the use of piecewise monotonic interpolations to approximate and represent a fuzzy number or interval and to derive a procedure to control the absolute error associated to the arithmetic operations between fuzzy numbers in order to reduce the distance between the true result of the operation and its approximation the monotonic functions are then used to define a parametric representation of a large class of fuzzy numbers having general shapes of the membership function and a simple and accurate procedure is introduced for the fuzzy arithmetic several computational experiments are given to show the good performance of the proposed procedure

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

explicit error estimates for quintic and biquintic spline interpolation ii abstract the purpose of this paper is to provide explicit error estimates between a given function x t pc n a b 2 n 6 and its quintic spline interpolate s x t the results obtained are sharp and improve on or supplement those established by hall 1 and schultz 2 these results are then used to acquire precise error bounds for the approximated and biquintic spline interpolates sufficient numerical illustration which dwells upon the importance of the obtained results is also included

accurate c 2 rational interpolants in tension a family of rational interpolants is defined for a set of real data points x i f i 1 leqq i leqq n with x 1 x 2 cdots x n each interpolant of the family is a c 2 spline with a cubic numerator and a quadratic denominator which is identified uniquely by the value of a positive tension parameter c 0 the solution constructed is nonparametric a value of c 0 close to zero will produce piecewise linear behaviour while a large value of c 0 will give approximately the cubic spline solution the method used arises from a study of other rational functional forms for effecting monotonic or convex interpolation a convergence analysis establishes an error bound in terms of c 0 and shows that o h 4 accuracy is obtained when exact boundary derivatives are prescribed several examples are supplied to support the practical value of the method

higher dimensional auslander reiten theory on maximal orthogonal subcategories abstract we introduce the concept of maximal n 1 orthogonal subcategories over artin algebras and orders and develop n 1 dimensional auslander reiten theory on them we give the n auslander reiten translation and the n auslander reiten duality then show the existence of n almost split sequences and n fundamental sequences we give some examples

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

hyperbolic splines and nonlinear distortion using mobius transformations we consider how to interpolate a set of monotone increasing data points x sub k y sub k so that the interpolant is a smooth invertible function and that the interpolant and its inverse have the same simple functional form we use our method to cancel jammer induced distortion in a nonlinear channel

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

abelian quotients of categories of short exact sequences abstract we mainly investigate abelian quotients of categories of short exact sequences the natural framework to consider the question is via identifying quotients of morphism categories as module categories these ideas not only can be used to recover the abelian quotients produced by cluster tilting subcategories of both exact categories and triangulated categories but also can be used to reach our goal let c e be an exact category we denote by e c the category of bounded complexes whose objects are given by short exact sequences in e and by s e c the full subcategory formed by split short exact sequences in general e c is just an exact category but the quotient e c s e c turns out to be abelian in particular if c e is frobenius we present three equivalent abelian quotients of e c and point out that the equivalences are actually given by left and right rotations the abelian quotient e c s e c admits some nice properties we explicitly describe the abelian structure projective objects injective objects and simple objects which provide a new viewpoint to understanding hilton rees theorem and auslander reiten theory furthermore we present some analogous results for triangulated versions

an approach to the interpolation of nonuniformly spaced data in this paper we consider the spline in tension as an interpolatory tool that helps us avoid large projector norm tension parameters are computed in advance in such a way that the prescribed exponential decay of fundamental functions is assured further an interpolation error bound for a smooth function is produced this bound depends on the choice of tension parameters and is of order 4 for small tension values but reduces to 2 as parameters grow to infinity

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