a family of non stationary subdivision schemes reproducing exponential polynomials we present a family of non stationary subdivision schemes taking the exponential quasi spline as the basic limit function reaching up to interpolatory refinable functions the exponential quasi spline generalizes the exponential b spline the associated non stationary subdivision scheme can reproduce the desired number of exponential polynomials due to the built in parameter permitting a wide range of tension control our scheme can avoid the undesirable artifact generated by usual non stationary interpolatory schemes on the irregularly spaced region of control points under a natural condition on the rate of convergence of the shape parameters all the subdivision schemes in the family have the same smoothness and approximation order as those of the asymptotically equivalent stationary schemes we present numerical examples to verify that our subdivision schemes have high flexibility and versatility in design

construction of binary four and five point non stationary subdivision schemes from hyperbolic b splines binary non stationary four and five point subdivision schemes have been developed using hyperbolic b spline basis functions the proposed subdivision schemes generate pleasing curves near the control polygon as compared to the existing non stationary subdivision schemes developed using trigonometric b splines basis functions the main speciality of the hyperbolic subdivision schemes is that they can regenerate hyperbolas and parabolas quite efficiently

reproduction of exponential polynomials by multivariate non stationary subdivision schemes with a general dilation matrix we study scalar multivariate non stationary subdivision schemes with a general integer dilation matrix we characterize the capability of such schemes to reproduce exponential polynomials in terms of simple algebraic conditions on their symbols these algebraic conditions provide a useful theoretical tool for checking the reproduction properties of existing schemes and for constructing new schemes with desired reproduction capabilities and other enhanced properties we illustrate our results with several examples

generation of fractal curves and surfaces using ternary 4 point interpolatory subdivision scheme in this paper the generation of fractal curves and surfaces along with their properties using ternary 4 point interpolatory subdivision scheme with one parameter are analyzed the relationship between the tension parameter and the fractal behavior of the limiting curve is demonstrated through different examples the specific range of the tension parameter has also been depicted which provides a clear way to generate fractal curves since the fractal scheme introduces in the paper have more number of control points therefore it gives more degree of freedom to control the shape of the fractal curve

a family of non uniform subdivision schemes with variable parameters for curve design in this paper we present non uniform subdivision schemes with variable parameter sequences a locally different tension parameter is set at each edge of the initial control polygon to control locally the shape of the resulting curve such that the scheme becomes non uniform due to the variable parameters the scheme can reproduce locally different analytic curves such as conics lissajous trigonometric and catenary curves hence blending curves including such analytic components can be successfully generated we discuss the convergence and smoothness of the proposed non uniform schemes and present some numerical results to demonstrate their advantages in geometric modeling furthermore as an application we propose a chamfering algorithm which can be used in designing automobile and mechanical products

fractal behavior of ternary 4 point interpolatory subdivision scheme with tension parameter in this paper the fractal curves and surfaces of ternary 4 point interpolatory subdivision scheme are developed and analyzed the fractal range of the tension parameter has also been determined the behavior of fractal curves and surfaces corresponding to different values of parameter has been depicted through six examples which offers direct means for an effective generation of subdivision fractals the real life applications of the proposed fractal subdivision scheme has also been demonstrated

shape controlled interpolatory ternary subdivision ternary subdivision schemes compare favorably with their binary analogues because they are able to generate limit functions with the same or higher smoothness but smaller support in this work we consider the two issues of local tension control and conics reproduction in univariate interpolating ternary refinements we show that both these features can be included in a unique interpolating 4 point subdivision method by means of non stationary insertion rules that do not affect the improved smoothness and locality of ternary schemes this is realized by exploiting local shape parameters associated with the initial polyline edges

convergence of nonstationary cascade algorithms a nonstationary multiresolution of l 2 mathbb r s is generated by a sequence of scaling functions phi k in l 2 mathbb r s k in mathbb z we consider phi k that is the solution of the nonstationary refinement equations phi k m sum j h k 1 j phi k 1 m cdot j k in mathbb z where h k is finitely supported for each k and m is a dilation matrix we study various forms of convergence in l 2 mathbb r s of the corresponding nonstationary cascade algorithm phi k n m sum j h k 1 j phi k 1 n 1 m cdot j as k or n tends to infty it is assumed that there is a stationary refinement equation at infty with filter sequence h and that sum k h k j h j infty the results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h

hyperbolic forms of ternary non stationary subdivision schemes originated from hyperbolic b splines in this work ternary non stationary subdivision schemes based on hyperbolic b spline basis functions have been presented the proposed hyperbolic ternary three point and four point subdivision schemes give pleasing as well as consistent curves with the control polygons as compared to the existing non stationary subdivision schemes obtained from trigonometric b splines the main speciality of the hyperbolic schemes is that they can reproduce hyperbolas and parabolas quite efficiently which has been demonstrated with the help of curvature plots

a comparison of compatible finite and inductive graph properties abstract in the theory of hyperedge replacement grammars and languages one encounters three types of graph properties that play an important role in proving decidability and structural results the three types are called compatible finite and inductive graph properties all three of them cover graph properties that are well behaved with respect to certain operations on hypergraphs in this paper we show that the three notions are essentially equivalent consequently three lines of investigation in the theory of hyperedge replacement so far separated merge into one

a non stationary binary three point approximating subdivision scheme in this paper we present a non stationary binary three point approximating subdivision scheme which can generate a wide variety of c3 continuous limit curves with different initial control parameter 0 the continuity of the scheme is proved by using the conclusion in dyn and levin 1995 5 the advantage of the scheme is illustrated by a number of examples

from approximating subdivision schemes for exponential splines to high performance interpolating algorithms in this work we construct three novel families of approximating subdivision schemes that generate piecewise exponential polynomials and we show how to convert these into interpolating schemes of great interest in curve design for their ability to reproduce important analytical shapes and to provide highly smooth limit curves with a controllable tension in particular throughout this paper we will focus on the derivation of 6 point interpolating schemes that turn out to be unique in combining vital ingredients like c 2 continuity simplicity of definition ease of implementation user independency tension control and ability to reproduce salient trigonometric and transcendental curves

non uniform interpolatory subdivision surface abstract this paper presents an interpolatory subdivision scheme with non uniform parametrization for arbitrary polygon meshes with arbitrary manifold topology this is the first attempt to generalize the non uniform four point interpolatory curve subdivision to surface with extraordinary points the scheme is constructed from the inspiration of the relation between the non uniform four point interpolatory subdivision scheme and the non uniform b spline refinement rule numerical examples and comparisons with the uniform interpolatory subdivision schemes indicate that the quality of the limit surface can be improved by using non uniform parameter values for non uniform initial data

on four point penalized lagrange subdivision schemes this paper is devoted to the definition and analysis of new subdivision schemes called penalized lagrange their construction is based on an original reformulation for the construction of the coefficients of the mask associated to the classical 4 points lagrange interpolatory subdivision scheme these coefficients can be formally interpreted as the solution of a linear system similar to the one resulting from the constrained minimization problem in kriging theory which is commonly used for reconstruction in geostatistical studies in such a framework the introduction in the formulation of a so called error variance can be viewed as a penalization of the oscillations of the coefficients following this idea we propose to penalize the 4 points lagrange system this penalization transforms the interpolatory schemes into approximating ones with specific properties suitable for the subdivision of locally noisy or strongly oscillating data according to a so called penalization vector a family of schemes can be generated a full theoretical study is first performed to analyze this new type of non stationary subdivision schemes then in the framework of position dependant penalization vector several numerical tests are provided to point out the efficiency of these schemes compared to standard approaches

equivariant constrained willmore tori in the 3 sphere in this paper we study equivariant constrained willmore tori in the 3 sphere these tori admit a 1 parameter group of mobius symmetries and are critical points of the willmore energy under conformal variations we show that the spectral curve associated to an equivariant torus is given by a double covering of mathbb c and classify equivariant constrained willmore tori by the genus g of their spectral curve in this case the spectral genus satisfies g le 3

a family of ternary subdivision schemes for curves a family of ternary 3 point subdivision schemes is introduced and has a degree of smoothness c1 continuity a family of ternary 4 point subdivision schemes is derived and has degrees of smoothness c1 and c2 continuities also a ternary 4 point approximating subdivision scheme is proposed that generates the limiting curve of c3 continuity the eigenvalue analysis and generating function formalism are used to analyze the continuity properties of these subdivision schemes also these subdivision schemes are compared with the established subdivision schemes

binary 3 point and 4 point non stationary subdivision schemes using hyperbolic function in this paper binary 3 point and 4 point non stationary subdivision schemes are presented using hyperbolic function as basis function asymptotic equivalence method is used to investigate the continuity of the proposed schemes comparison between the proposed schemes their stationary counterparts and some existing non stationary schemes has been depicted through examples it can be observed that the proposed schemes have the capability to reproduce conics particularly parabola

a non stationary combined subdivision scheme generating exponential polynomials abstract in this paper a non stationary combined subdivision scheme is presented which can unify several existing non stationary approximating and interpolatory subdivision schemes this scheme is obtained by generalizing the connection between the approximating and interpolatory schemes in the stationary case first formalized by maillot stam using a push back operator to the non stationary case for such a combined scheme we investigate its c l convergence and exponential polynomial generation reproduction property and get that it can reach c 4 degree of smoothness and generate reproduce certain exponential polynomials with suitable choices of the parameters besides we give a more generalized combined scheme for the purpose of generating and reproducing more general exponential polynomials the performance of our new schemes is illustrated by several numerical examples

improved binary four point subdivision scheme and new corner cutting scheme a binary 4 point approximating subdivision scheme presented by siddiqi and ahmad 2006 9 has been improved by introducing a global tension parameter an improved form of the scheme generates a family of c n n 1 2 3 4 limiting curves for certain range of global tension parameter and a c 5 limiting curve for a fixed value also a new corner cutting subdivision scheme has been proposed which generates a limiting curve of c 1 continuity the laurent polynomial method has been used to investigate the order of the derivative continuity of both subdivision schemes the performance of the subdivision schemes has been exposed by considering examples and compared with the existing schemes

a combined approximating and interpolating ternary 4 point subdivision scheme abstract in this paper a new combined approximating and interpolating ternary 4 point subdivision scheme with multiple parameters is proposed a set of nice properties such as support continuity and polynomial generation are briefly discussed the new combined scheme not only contains a lot of classical ternary schemes as special cases but also generates brand new ternary schemes compared to other approximating subdivision schemes limit curves generated by the new scheme are more consistent with the corresponding control polygons and keep detail features better examples are given to show the effectiveness of the scheme furthermore fractal property is analyzed and fractal curves are also given

a unique extremal metric for the least eigenvalue of the laplacian on the klein bottle we prove the following conjecture recently formulated by jakobson nadirashvili and polterovich cite jnp on the klein bottle mathbb k the metric of revolution g 0 9 1 8 cos 2v 2 over 1 8 cos 2v left du 2 dv 2 over 1 8 cos 2v right 0 le u frac pi 2 0 le v pi is the emph unique extremal metric of the first eigenvalue of the laplacian viewed as a functional on the space of all riemannian metrics of given area the proof leads us to study a hamiltonian dynamical system which turns out to be completely integrable by quadratures

a new non stationary binary 6 point subdivision scheme in this paper the hyperbolic form of binary 6 point interpolating non stationary subdivision scheme has been constructed using the hyperbolic function some of the important properties of the proposed scheme has been discussed the comparison of the proposed scheme with the trigonometric form of binary 6 point interpolatory non stationary scheme is depicted through examples which indicate that the proposed scheme not only accommodates and is more consistent with the control polygon but also generates pleasing curves corresponding to the larger parametric values in and outside the parametric interval as compared to the trigonometric form of binary 6 point scheme developed by daniel et al comparison with some other interpolating non stationary subdivision schemes has also been demonstrated

analysis of asymptotically equivalent binary subdivision schemes non stationary uniform subdivision schemes are presented including some important examples as the up function and exponential splines the multivariate analysis of subdivision schemes is extended to a class of non stationary schemes which are asymptotically equivalent to converging stationary or non stationary schemes

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

an interpolating 6 point c2 non stationary subdivision scheme in this paper we consider a 6 point interpolating c 2 binary non stationary subdivision scheme and present some of its important properties we highlight some advantages of the scheme and demonstrate its performance by some examples

modified form of binary and ternary 3 point subdivision schemes binary 3 point scheme developed by hormann and sabin hormann k and sabin malcolm a 2008 a family of subdivision schemes with cubic precision computer aided geometric design 25 41 52 has been modified by introducing a tension parameter which generates a family of c 1 limiting curves for certain range of tension parameter ternary 3 point scheme introduced by siddiqi and rehan siddiqi shahid s and rehan k 2009 a ternary three point scheme for curve designing international journal of computer mathematics in press doi 10 1080 00207160802428220 has also been modified by introducing a tension parameter which generates family of c 1 and c 2 limiting curves for certain range of tension parameter laurent polynomial method is used to investigate the continuity of the subdivision schemes the performance of modified schemes has been demonstrated by considering different examples along with its comparison with the established subdivision schemes

limits of level and parameter dependent subdivision schemes in this paper we present a new matrix approach for the analysis of subdivision schemes whose non stationarity is due to linear dependency on parameters whose values vary in a compact set indeed we show how to check the convergence in c r s and determine the holder regularity of such level and parameter dependent schemes efficiently via the joint spectral radius approach the efficiency of this method and the important role of the parameter dependency are demonstrated on several examples of subdivision schemes whose properties improve the properties of the corresponding stationary schemes

monadic second order evaluations on tree decomposable graphs abstract every graph generated by a hyperedge replacement graph grammar can be represented by a tree namely the derivation tree of the derivation sequence that produced it certain functions on graphs can be computed recursively on the derivation trees of these graphs by using monadic second order logic and semiring homomorphisms we describe in a single formalism a large class of such functions polynomial and even linear algorithms can be constructed for some of these functions we unify similar results obtained by takamizawa 1982 bern 1987 arnborg 1991 and habel 1989

analysis of non stationary hermite subdivision schemes reproducing exponential polynomials abstract the aim of this paper is to study the convergence and smoothness of non stationary hermite subdivision schemes of order 2 in conti et al 2017 provided sufficient conditions for the convergence of a non stationary hermite subdivision scheme that reproduces a set of functions including exponential polynomials the analysis has been focused on the non stationary hermite scheme with the order 3 but the case of 2 which is practically most useful is yet to be investigated in this regard the first goal of this paper is to fill the gap we analyze the convergence of non stationary hermite subdivision schemes of order 2 next we provide a tool which allows us to estimate the smoothness of a non stationary hermite scheme by developing a novel factorization framework of non stationary vector subdivision operators using the proposed non stationary factorization framework we estimate the smoothness of the non stationary hermite subdivision schemes the non stationary interpolatory hermite scheme proposed by conti et al 2015 and a new class of non stationary dual hermite subdivision schemes of de rham type

a ternary 4 point approximating subdivision scheme in the implementation of subdivision scheme three of the most important issues are smoothness size of support and approximation order our objective is to introduce an improved ternary 4 point approximating subdivision scheme derived from cubic polynomial interpolation which has smaller support and higher smoothness comparing to binary 4 point and 6 point schemes ternary 3 point and 4 point schemes see table 2 the method is easily generalized to ternary 2n 2 point approximating subdivision schemes we choose a ternary scheme because a way to get smaller support is to raise arity and we use polynomial reproduction to get higher approximation order easily 2007 elsevier inc all rights reserved

on a stable family of four point nonlinear subdivision schemes eliminating the gibbs phenomenon abstract the chaikin s scheme presents the interesting property of not producing gibbs phenomenon the problem arises when we need a scheme that provides a higher order of approximation in this case linear schemes are not a good option as they produce gibbs phenomenon close to discontinuities in this article a new four point nonlinear family of subdivision schemes that eliminate the gibbs phenomenon is presented it is based on the linear family of four point subdivision schemes depending on a tension parameter introduced in dyn et al 2005 a simple algebraic transformation leads to an easy way of introducing nonlinearity in the original family of schemes the non interpolatory characteristic of the nonlinear scheme can be modulated just varying the value of the tension parameter results about the stability convergence and the elimination of the gibbs phenomenon are presented some numerical comparisons of the results obtained in the generation of curves are also shown leading to the conclusion that the high order nonlinear schemes are more suitable for this purpose

regularity of non stationary subdivision a matrix approach in this paper we study scalar multivariate non stationary subdivision schemes with integer dilation matrix m and present a unifying general approach for checking their convergence and for determining their holder regularity latter in the case m mi m ge 2 m mi m 2 the combination of the concepts of asymptotic similarity and approximate sum rules allows us to link stationary and non stationary settings and to employ recent advances in methods for exact computation of the joint spectral radius as an application we prove a recent conjecture by dyn et al on the holder regularity of the generalized daubechies wavelets we illustrate our results with several examples

nonuniform cascade algorithms abstract for a sequence of bounded linear operators hk k 0 1 on a banach space s the algorithm k n hk k 1 n 1 generates a family of sequences k n n 0 k 0 1 from an initial family of vectors k 0 s k 0 1 we study the convergence of k n as n and give an application on the convergence of cascade algorithms for nonuniform splines when s is the space of all sequences i i z with norm sup i z i and i i z belong to the banach space x l 2 r

on a nonlinear 4 point ternary and non interpolatory subdivision scheme eliminating the gibbs phenomenon abstract a nonlinear ternary 4 point non interpolatory subdivision scheme is presented it is based on a nonlinear perturbation of the 4 point subdivision scheme studied in 16 the convergence of the scheme and the regularity of the limit function are analyzed it is shown that the gibbs phenomenon that is classical in linear schemes is eliminated we also establish the stability of the subdivision scheme that is not a consequence of its convergence due to its non linearity to the best of our knowledge this is the first ternary non interpolatory subdivision scheme that can be found in the literature

fractal generation using ternary 5 point interpolatory subdivision scheme this paper is devoted to explore the generation of fractal curves and surfaces using ternary 5 point interpolatory subdivision scheme the fractal behavior of the limiting curves and surfaces is analyzed with the help of tension parameter it may be noted that the proposed fractal scheme determines two intervals of range of shape control parameter the behavior of the shape control parameter for the generation of fractals has been visualized through five examples which provides faster rate of generation of fractals

interpolation through an iterative scheme abstract a new method of interpolation is generated through an iterative scheme on dyadic rationals the process is linear local and produces almost twice differentiable functions a fundamental interpolating function is analyzed error formulas are derived the method is applied for curve fitting