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A novel approach to zipper fractal interpolation theory for functions of several variables is presented. Multivariate zipper fractal functions are constructed and then perturbed through free choices of base functions, scaling functions, and a binary matrix called signature to obtain their zipper α-fractal versions. In particular, we propose a multivariate Bernstein zipper fractal function and study its coordinate-wise monotonicity which depends on the values of signature. We derive bounds for the graph of a multivariate zipper fractal function by imposing conditions on the scaling factors and the Hölder exponent of the associated germ function and base function. The box dimension result for multivariate Bernstein zipper fractal function is derived. Finally, we study some constrained approximation properties for multivariate zipper Bernstein fractal functions. [ABSTRACT FROM AUTHOR]

The main concern of this paper is to develop a new class of A-stable fourth-order numerical scheme for solving initial value problems. The idea is an evolutionary and heuristic approach; using the Grasshopper optimization, along with the Hermite interpolation for stages, we obtain a class of A-stable methods. Four types of weighting rules are introduced for the current formulation. The fundamental weighting rule (FWR) is the most important rule, which emphasizes on the symmetric and central structure of the method. A systematic strategy is proposed to obtain the FWR based on swarm intelligence and regression. The new techniques are called α I -(e + i)P, where e and i are the number of terminal and internal points, respectively. The numerical experiments demonstrate the reasonable behaviour of the algorithms on several test problems from different applications. Finally, we find that the new formulas are well suited for long time behaviour of the time-history analysis in earthquake engineering. [ABSTRACT FROM AUTHOR]

Fractal approximation is a well studied concept, but the convergence of all the existing fractal approximants towards the original function follows usually if the magnitude of the corresponding scaling factors approaches zero. In this article, for a given function f ∈ C (I) , by exploiting fractal approximation theory and considering the classical q-Bernstein polynomials as base functions, we construct a sequence { f n (q , α) (x) } n = 1 ∞ of (q , α) -fractal functions that converges uniformly to f even if the norm/magnitude of the scaling functions/scaling factors does not tend to zero. The convergence of the sequence { f n (q , α) (x) } n = 1 ∞ of (q , α) -fractal functions towards f follows from the convergence of the sequence of q-Bernstein polynomials of f towards f. If we consider a sequence { f m (x) } m = 1 ∞ of positive functions on a compact real interval that converges uniformly to a function f, we develop a double sequence { { f m , n (q , α) (x) } n = 1 ∞ } m = 1 ∞ of (q , α) -fractal functions that converges uniformly to f. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction and analysis of some new non-linear subdivision and multiresolution schemes using the Lehmer means. Our main objective is to attain adaption close to discontinuities. We present theoretical, numerical results and applications for different schemes. The main theoretical result is related to the four-point interpolatory scheme, that we write as a perturbation of a linear scheme. Our aim is to establish a one-step contraction property that allows to prove the stability of the new scheme. Indeed with a one-step contraction property for the scheme of differences, it is possible to prove the stability of the 2D algorithm constructed using a tensor product approach. In this article, we also consider the associated three points cell-average scheme, that we will use to present some results for image compression, and a non-interpolatory scheme, that we will use to introduce an application to subdivision curves in 2D. These applications show that the use of the Lehmer mean is suitable for the design of subdivision schemes for the generation of curves and for image processing. [ABSTRACT FROM AUTHOR]

This paper presents a computational technique based on cubic trigonometric B-splines for the time fractional Burgers' equation. The nonlinear advection term is approximated by a new linearization technique which is very efficient and significantly reduces the computational cost. The usual finite difference formulation is used to approximate the Caputo time fractional derivative while the derivative in space is discretized using cubic trigonometric B-spline functions. The proposed technique is proved to be globally unconditionally stable. A convergence analysis is discussed to measure the accuracy of the solution. Computational experiments are performed to further confirm the accuracy and stability of the method. Numerical results are compared with those obtained by a scheme based on parametric spline functions. The comparison reveals that the proposed scheme is quite accurate and effective. [ABSTRACT FROM AUTHOR]

We study some procedures for the approximation of three-dimensional data on a grid with a hypothesis of periodicity. The first part proposes a generalization of a discrete periodic approximation defined by Dunham Jackson. The functions used have the advantage of owning an analytical explicit expression in terms of the samples (specific values) of the original function or data. In the second part, we describe a continuous approximation function for the same problem, defined through an integral. Some results of the rate of convergence and bounds of the approximation error are presented, with the single hypothesis of Hölder continuity or continuity of the original function. [ABSTRACT FROM AUTHOR]

In this paper, we study how to construct a planar cubic G1 Hermite interpolation curve with minimal length and curvature variation energy. This problem is solved by a bi-objective optimization model. We prove that there is a unique solution to this problem and give the expression of the solution. Some numerical examples show that the proposed method makes the curvature variation energy of the curve as small as possible when the length of the curve as short as possible. [ABSTRACT FROM AUTHOR]

In this work, we introduce a modification into the technique, presented in A. Cordero, J.L. Hueso, E. Martínez, and J.R. Torregrosa [Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett. 25 (2012), pp. 2369–2374], that increases by two units the convergence order of an iterative method. The main idea is to compose a given iterative method of order p with a modification of Newton's method that introduces just one evaluation of the function, obtaining a new method of order p+2, avoiding the need to compute more than one derivative, so we improve the efficiency index in the scalar case. This procedure can be repeated n times, with the same approximation to the derivative, obtaining new iterative methods of order p+2n. We perform different numerical tests that confirm the theoretical results. By applying this procedure to Newton's method one obtains the well known fourth order Ostrowski's method. We finally analyse its dynamical behaviour on second and third degree real polynomials. [ABSTRACT FROM AUTHOR]

As this paper is concerned with a class of multistep numerical difference techniques to solve one-dimensional parabolic inverse problems with source control parameter , we apply the linear multistep method combining with Lagrange interpolation to develop three different numerical difference schemes. The problem of numerical differentiation with noisy scattered data is mildly ill-posed, the smoothing spline model based on Tikhonov regularization method is developed to compute numerical derivative contaminated by noise error. Simultaneously, the truncation error estimations and the convergence conclusions are proposed for the above difference methods respectively. The results of numerical tests with different noise levels are given to show that the presented algorithms are accurate and effective. [ABSTRACT FROM AUTHOR]

One of the open problems in hypercomplex analysis is the interpolation of monogenic functions by monogenic polynomials. We consider the case of monogenic functions defined in a domain ofwith values in the algebra of quaternions. The idea is to interpolate these functions by a special system of monogenic polynomials, the so-called pseudo complex polynomials. Quaternionic analysis shows a lot of analogies to complex analysis and the monogenic functions play the role of holomorphic functions. Due to the non-commutative multiplication a fundamental theorem of algebra is not available, and standard tools from linear algebra cannot be applied in the usual way. So, the main goal of this paper is to show that the interpolation problem is solvable for arbitrarily given interpolation nodes and general interpolation data. [ABSTRACT FROM AUTHOR]

The unsteady two-dimensional Navier–Stokes system of equations, for viscous incompressible fluids are solved using a global method of approximated particular solutions (MAPS) in terms of a Stokes formulation, where the velocity and pressure fields are approximated from a linear superposition of particular solutions of a non-homogeneous Stokes system of equations, with a multiquadric (MQ) radial basis function (RBF) as non-homogeneous term. Steady-state solution of the flow problems considered in this work can be unstable at high Reynolds numbers (Re), corresponding to bifurcation of solutions that result in the appearance of new stable steady-state or periodic solutions. The main objective of this work is to present a global meshless numerical scheme able to predict these bifurcation points and concurrent new stable or periodic solutions. This is well known to be a very difficult task for any numerical scheme. An implicit first-order time-stepping scheme is used to approximate the transient term and the obtained nonlinear system of algebraic equations is solved by a Newton–Raphson method with variable step. Two steady-state and two transient problems are considered to validate the numerical scheme: the lid-driven cavity and backward-facing step (BFS) flows (steady-state problems) and the decaying Taylor–Green vortex and two-sided lid-driven cavity flows (transient problems). The first two problems are solved up toRe=10,000 and 2300, respectively. Results obtained are compared with corresponding benchmark numerical solutions, showing excellent agreement. Obtained numerical solutions for the decaying vortices atRe=100 shown excellent agreement with the corresponding analytical results. The transient problem of a rectangular two-sided lid-driven cavity flow is solved atRe=700. The influence of the cavity length,l, in determining the different structures of the flow pattern is studied for values of, showing that the scheme is able to reproduce the previously reported change in the flow pattern whenl=2. Finally, the global Stokes MAPS are used to carry out nonlinear stability analyses of three steady-state problems: the sudden expansion, lid-driven cavity and BFS flows. Stable and unstable steady-state solutions atRevalues greater than critical are predicted with the proposed numerical scheme. Our numerical results are consistent with previously stability analysis reported in the literature. [ABSTRACT FROM AUTHOR]

The theory of symmetric multivariate Lagrange interpolation is a beautiful but rather unknown tool that has many applications. Here we derive from it an Exchange Lemma that allows to explain in a simple and natural way the full description of the double sum expressions introduced by Sylvester in 1853 in terms of subresultants and their Bézout coefficients. [ABSTRACT FROM AUTHOR]

High-order finite difference discontinuity detectors are essential for the location of discontinuities on discretized functions, especially in the application of high-order numerical methods for high-speed compressible flows for shock detection. The detectors are used mainly for switching between numerical schemes in regions of discontinuity to include artificial dissipation and avoid spurious oscillations. In this work a discontinuity detector is analysed by the construction of a piecewise polynomial function that incorporates jump discontinuities present on the function or its derivatives (up to third order) and the discussion on the selection of a cut-off value required by the detector. The detector function is also compared with other discontinuity detectors through numerical examples. [ABSTRACT FROM PUBLISHER]

This paper investigates the use of a surrogate model based on Proper Orthogonal Decomposition (POD) and Radial Basis Functions (RBF) for calibrating the nanoindentation-based loading of an elastic–plastic material. Using Taguchi design of experiments and Analysis of Variance (ANOVA), the total number of finite element-based training points is reduced for input parameters that exhibited lower significance. It is found that ANOVA-based sensitivity information can be used to reduce the number of training points without significantly affecting model accuracy. It is also observed that RBFs capable of conforming nonlinearly perform better when the spatial distance between training points increases. Furthermore, for some RBFs the performance is further tuned by choice of the shape parameters. Finally, it is demonstrated that the surrogate model's performance remains stable under the effect of random noise. Thereby, this study provides a general framework for solving a nanoindentation-based material modelling inverse problem using the POD–RBF technique. [ABSTRACT FROM AUTHOR]

We introduce the notion of a confluent Vandermonde matrix with quaternion entries and discuss its connection with Lagrange–Hermite interpolation over quaternions. The formula for the rank of a confluent Vandermonde matrix is obtained as well as the representation formula for divided differences of quaternion polynomials. Extensions of these results to the power series setting include the formula for the rank of a confluent Cauchy matrix and norm-constrained Lagrange–Hermite interpolation by square summable power series over quaternions. [ABSTRACT FROM PUBLISHER]

In this paper, we present a symbolic algorithm for a mixed interpolation of the form a cos kx + b sin kx + ... cixi, s ≥2, where k > 0 is a given parameter and the coefficients a, b, and c0,..., cs-2 are determined by a given set of independent integral conditions at arbitrary nodes. Implementation of the proposed algorithm in Maple is described and sample computations are provided. This algorithm will help to implement the manual calculations in commercial packages such as Mathematica, Matlab, Singular, Scilab, etc. [ABSTRACT FROM AUTHOR]

In this paper, a recursion numerical technique is considered to solve the inverse heat conduction problems, with an unknown time-dependent heat source and the Neumann boundary conditions. The numerical solutions of the heat diffusion equations are constructed using the Gaussian radial basis functions. The details of algorithms in the one-dimensional and two-dimensional cases, involving the global or partial initial conditions, are proposed, respectively. The Tikhonov regularization method, with the generalized cross-validation criterion, is used to obtain more stable numerical results, since the linear systems are badly ill-conditioned. Moreover, we propose some results of the condition number estimates to a class of positive define matrices constructed by the Gaussian radial basis functions. Some numerical experiments are given to show that the presented schemes are favourably accurate and effective. [ABSTRACT FROM AUTHOR]

A family of smooth rational spline fractal interpolation function (SRFIF) is presented with the help of classical rational cubic spline. Each SRFIF of the family is identified uniquely by the values of vertical scaling factorand shape parametersand, and the shape of the fractal interpolating curves can be constrained and modified by selecting suitable parameters and vertical scaling factor. In order to meet the needs of practical design, a new method is developed to visualize constrained data in the view of constrained curves. In the special case, the methods of linear constraint and quadratic constraint are discussed. Also, convergence analysis shows that SRFIF gives a good approximation to the original function. [ABSTRACT FROM AUTHOR]

An algorithm for computing an analytic function of a matrixis described. The algorithm is intended for the case wherehas some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This algorithm is a modification of some well known and widely used algorithms. A novel feature is an approximate calculation of divided differences for the Newton interpolating polynomial in a special way. This modification does not require to reorder the Schur triangular form and to solve Sylvester equations. [ABSTRACT FROM AUTHOR]

Constructing shape-preserving interpolation spline has been a hot topic in industrial design and scientific data visualization during the past 30 years. In the existing shape preservinginterpolation spline methods, however, some methods can be only used to preserve the monotonic data set, while others can be only used to preserve the convex data set, and often forcontinuity, it is requested to solve a linear system of consistency equations for the derivative values at the knots. In this paper, a new explicit representation of arational quartic interpolation spline with two local tension parameters is developed. A convergence analysis establishes an error bound and shows that the order of approximation isaccuracy. Sufficient conditions for the proposed interpolation spline to preserve the shape of positive, monotonic, and convex set of data are derived. [ABSTRACT FROM AUTHOR]

Decomposing multivariate functions in terms of less variate components such as univariate or bivariate structures is an efficient way to reduce the mathematical and computational complexity of the related problem in computer-based applications. The enhanced multivariance product representation (EMPR) method is an extension to high-dimensional model representation (HDMR) which has a divide-and-conquer philosophy. The EMPR method has some additional structures named as support functions in its expansion when compared with HDMR and we have an important flexibility in selecting these support function structures. This selection process makes the method more successful than HDMR in most cases. In this sense, this work aims to apply the EMPR method to the multivariate data modelling problems having orthogonal geometries. The numerical results also show that the idea of this work is successfully applied to the considered problems and we obtain good representations in data modelling. [ABSTRACT FROM PUBLISHER]

In dynamical systems saddle points partition the domain into basins of attractions of the remaining locally stable equilibria. This situation is rather common especially in population dynamics models, like prey–predator or competition systems. Focusing on squirrels population models with niche, in this paper we design algorithms for the detection and the refinement of points lying on the separatrix manifold partitioning the phase space. We consider both the two populations and the three populations cases. To reconstruct the separatrix curve and surface, we apply the Partition of Unity method, which makes use of Wendland's functions as local approximants. [ABSTRACT FROM PUBLISHER]

In this paper, we construct a numerical method based on cubic splines in tension for solving regularized long wave equation. The truncation error is analysed and the method shows that by choosing suitably parameters we can obtain various accuracy schemes. Numerical stability of the method has been studied by using a linearized stability analysis. Test problems are dealt with. The numerical simulations can validate and demonstrate the advantages of the method. [ABSTRACT FROM AUTHOR]

In many mathematical models with complex parameter functions, it is critical to suggest a strategy for estimating those functions. This paper considers the problem of fitting parameter functions to given data by using a finite number of piecewise functions. Here, the primary aim is to propose several necessary and sufficient conditions that can guarantee the existence of parameter functions. With these criteria, the paper provides an algorithm for solving the parameter estimation problem efficiently. Numerical examples verify that the proposed algorithm is effective in estimating parameter functions in differential equations. [ABSTRACT FROM AUTHOR]

This article presents a new model for pitch-and-rhythm interrelationships and demonstrates the importance of these interrelationships as features for music analysis and classification. We show that the model is capable of identifying certain musical patterns and categorising pieces created by Mozart's dice game by identifying their quality. We discuss the cognitive foundation of this model and present future directions and applications. We employ the subdivision-based music analysis methods introduced earlier by the present authors to model certain pitch-and-rhythm interdependencies and symmetry patterns in selected pieces, shedding more light on the analogy proposed between subdivision schemes and musical structures. Our approach to pitch-and-rhythm interactions points out an interesting phenomenon, according to which the relations between neighbour pitches may be related to their timing through a relatively simple model. We propose hierarchical reductions and indicate that their levels are interrelated through simple subdivision schemes using new criteria, yielding features for classification. [ABSTRACT FROM PUBLISHER]

Conic sections have many applications in industrial design, however, they cannot be exactly represented in polynomial form. Hence approximating conic sections with polynomials is a challenging problem. In this paper, we use the monomial form of Delgado and Peña (DP) curves and present a matrix representation for them. Using the matrix form and the least squares method, we propose a simple and efficient algorithm for approximating conic sections by DP curves of arbitrary degree with endpoint interpolation. Finally, we test and compare the proposed algorithm on some numerical examples which validates and confirms efficiency of it. [ABSTRACT FROM AUTHOR]

In this paper an efficient numerical method for solving a class of multipoint boundary value problems with special boundary conditions of Birkhoff-type is presented. After a quick reference to Birkhoff-type interpolation polynomial which satisfies the particular conditions, and a result on the existence and uniqueness of solution of the given problem, an algorithm is introduced to find a polynomial that approximates the solution. It is a general collocation method. Then an a priori estimation of the error of this approximation is given. Finally, to show the efficiency and the applicability of the method, numerical results are presented. These numerical experiments provide favourable comparisons with the NDSolve command of Mathematica. [ABSTRACT FROM AUTHOR]

We construct norming meshes with cardinality 𝒪(ns),s = 3, for polynomials of total degree at mostnon the closure of bounded planar Lipschitz domains. Such cardinality is intermediate between optimality (s = 2), recently obtained by Kroó on multidimensionalC2star-like domains, and that arising from a general construction on Markov compact sets due to Calvi and Levenberg (s = 4). [ABSTRACT FROM AUTHOR]

The classical composite mid-point rectangle rule for the computation of Cauchy principal value integrals is discussed. When the singular point coincides with some priori known point, the convergence rate of the mid-point rectangle rule is higher than the globally one the same as the Riemann integral which is called as superconvergence phenomenon. The superconvergence phenomenon of the composite mid-rectangle rule occurs at certain local coordinate of each subinterval and the corresponding superconvergence error estimate is obtained. At last,some numerical examples are provided to validate the theoretical analysis. [ABSTRACT FROM PUBLISHER]

Leveraged (inverse) exchange-traded funds (LETFs) seek to deliver multiples (opposite) of the performance of the index or benchmark they track. LETFs typically are designed to achieve their stated performance objectives on a daily basis. Many real-life and hypothetical examples have been given to show that the performance of these ETFs over a period longer than one day can differ from their stated daily performance objectives. Formulae have been found using both continuous method and discrete method. A discrete method was used to find a formula linking the return of a leveraged fund with the corresponding multiple of the return of the unleveraged fund and its realized variance but the method needs to use some assumptions and statistical properties to create the volatility term. A CME report finds a very simple way to include volatility in their formula but fails to link to the return of the corresponding unleveraged product. In this paper, we find a natural way to link a leveraged fund with its corresponding unleveraged product and its realized variance in a discrete manner. Our derivation process is similar to that in the CME report, so we do not need to use assumptions and statistical properties to create the volatility term. Unlike the CME method, we use geometric return as opposed to arithmetic return. So, we are able to connect with the return of the corresponding unleveraged product. [ABSTRACT FROM AUTHOR]

The objective of the present paper is the study of affine transformations of the plane, which provide self-affine curves as attractors. The properties of these curves depend decisively of the coefficients of the system of affnities involved. The corresponding functions are continuous on a compact interval. If the scale factors are properly chosen one can define Schauder bases ofC[a,b] composed by affine fractal functions close to polygonals. They can be chosen bounded. The basis constants and the biorthogonal sequence of coefficient functionals are studied. [ABSTRACT FROM AUTHOR]

Subdivision schemes are multi-resolution methods used in computer-aided geometric design to generate smooth curves or surfaces. We propose two new models for data analysis and compression based on subdivision schemes:(a) The ‘subdivision regression’ model, which can be viewed as a special multi-resolution decomposition.(b) The ‘tree regression’ model, which allows the identification of certain patterns within the data. The paper focuses on analysis and mentions compression as a byproduct. We suggest applying certain criteria on the output of these models as features for data analysis. Differently from existing multi-resolution analysis methods, these new models and criteria provide data features related to the schemes (the filters) themselves, based on a decomposition of the data into different resolution levels, and they also allow analysing data of non-smooth functions and working with varying-resolution subdivision rules. Finally, applications of these methods for music analysis and other potential usages are mentioned. [ABSTRACT FROM PUBLISHER]

We consider the problem of interpolating a multivariable function defined on a bounded domain using its traces on parametric hypersurfaces. Our approach is based on the theory of operator polynomial interpolation. We construct the corresponding operator interpolation polynomial for a given function and analyse in detail particular two- and three-dimensional cases. Numerical examples presented in the paper show the flexibility of the proposed approach. [ABSTRACT FROM PUBLISHER]

Several methods based on smoothing or statistical criteria have been used for deriving disaggregated values compatible with observed annual totals. The present method is based on the artificial neural networks. This article evaluates the use of artificial neural networks (ANNs) for the disaggregation of annual US GDP data to quarterly time increments. A feed-forward neural network with back-propagation algorithm for learning was used. An ANN model is introduced and evaluated in this paper. The proposed method is considered as a temporal disaggregation method without related series. A comparison with previous temporal disaggregation methods without related series has been done. The disaggregated quarterly GDP data compared well with observed quarterly data. In addition, they preserved all the basic statistics such as summing to the annual data value, cross correlation structure among quarterly flows, etc. [ABSTRACT FROM AUTHOR]

An alternative approach for the numerical approximation of ODEs is presented in this article. It is based on a variational framework recently introduced in S. Amat and P. Pedregal [A variational approach to implicit ODEs and differential inclusions, ESAIM: COCV 15 (2009), 149–172] where the solution is sought as the minimizer of an error functional tailored after the ODE in a rather straightforward way. A suitable discretization of this error functional is pursued, and it is performed using Hermite's interpolation and quadrature formulae. Notice that only Hermite's interpolation is necessary when polynomial systems of ODEs are considered (many models in practice use these types of equations). A comparison with implicit Runge–Kutta methods is analysed. With this variational strategy not only some classical collocation methods, but alsonewschemes that seem to havebetternumerical behaviour can be recovered. Although the driving idea is very simple, the strategy turns out to be very general and flexible. At the same time, it can be implemented efficiently. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate the numerical integration problem of a real valued function generally known only on multivariate scattered points using Lobachevsky splines, a pioneering version of cardinal B-splines. Starting from their interpolation properties, we focus on the construction of new integration formulas, which are quite flexible requiring no special distribution of nodes. Numerical results using Lobachevsky splines turn out to be interesting and promising for both accuracy and simplicity in computation. Finally, a comparison with integration by radial basis functions confirms the validity of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, a construction of a cubic Bézier spline surface that interpolates prescribed spatial points and the corresponding normal directions of tangent planes is proposed. Boundary curves of each triangular patch minimize the approximated strain energy. A comparison of optimal boundary curves is given. The interpolant minimizes Willmore energy functional. Some numerical examples and applications of the interpolation scheme are presented: surface approximation, hole filling and condensation of parameters. [ABSTRACT FROM AUTHOR]

A family of discrete delay advection–reaction operators is introduced along with an infinite matrix formulation in order to investigate the asymptotic behaviour of the orbits of their iterates. The infinite matrices obtained are triangular matrices with only one non-zero subdiagonal. We show that the elements of powers of these matrices can be written as distinctive products of two factors, one of them involving derivatives of the Lagrange polynomials of basic functions with the diagonal elements as nodes. The other factor consists of products of the subdiagonal elements. Consequently the convergence of the iterates of the operators depends on their eigenvalues and the products of their subdiagonal elements. [ABSTRACT FROM PUBLISHER]

This paper describes a method to smooth an arc spline. Arc splines areG1continuous segments made of circular arcs and straight lines. We have proposed a smooth version of the arc spline by replacing its parts with C-, S-, and J-shaped spiral transitions, stitched withG2continuity, by using asinglesegment of Pythagorean hodograph quintic function. Use of asinglepolynomial function rather than two has the benefit that designers have fewer entities to deal with. Spiral transitions are important in manufacturing industries because of their use in the cutting paths for numerically controlled cutting machinery, highway or railway designing, non-holonomic robot path planning and spur gear designing. [ABSTRACT FROM AUTHOR]

We show that the property of being a (weakly) admissible mesh for multivariate polynomials is preserved by small perturbations on real and complex Markov compacts. Applications are given to smooth transformations of polynomial meshes and to polynomial interpolation. [ABSTRACT FROM PUBLISHER]

Based on the construction of bivariate fractal interpolation functions (FIFs), a class of FIFs with vertical scaling factor functions are presented and the analytical properties of smoothness and stability are proved. [ABSTRACT FROM PUBLISHER]

A class of cardinal basis functions is proposed in order to achieve a generalization to Banach spaces of Hermite-Birkhoff interpolation on arbitrarily distributed data. First, a constructive characterization of the class of cardinal basis functions is given. Then, the interpolation problem is solved by using a suitable combination of such functions and Taylor-Fréchet expansions. The performance of the obtained interpolants is improved by applying a localizing scheme, and the corresponding approximation error is estimated. A noteworthy case in Hilbert spaces and a numerical test comparing the Hermite-Birkhoff and Lagrange interpolants complete the presentation. [ABSTRACT FROM AUTHOR]

In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations (SPDEs). Using an implicit time-stepping scheme, we transform stochastic parabolic equations into stochastic elliptic equations. Our main attention is concentrated on the numerical solution of the elliptic equations at each time step. The estimator of the solution of the elliptic equations is given as a linear combination of reproducing kernels derived from the differential and boundary operators of the SPDE centred at collocation points to be chosen by the user. The random expansion coefficients are computed by solving a random system of linear equations. Numerical experiments demonstrate the feasibility of the method. [ABSTRACT FROM AUTHOR]

This work is a contribution towards the graphical display of 2D data when they are convex, monotone and positive. A piecewise rational function in a cubic/cubic form is proposed, which, in each interval, involves four free parameters in its construction. These four free parameters have a direct geometric interpretation, making their use straightforward. Illustrations of their effect on the shape of the rational function are given. Two of these free parameters are constrained to preserve the shape of convex, monotone and positive data, while the other two parameters are utilized for the modification of positive, monotone and convex curves to obtain a visually pleasing curve. The problem of shape preservation of data lying above a line is also discussed. The method that is presented applies equally well to data or data with derivatives. The developed scheme is computationally economical and pleasing. The error of rational interpolating function is also derived when the arbitrary function being interpolated is C 3 in an interpolating interval. The order of approximation is . [ABSTRACT FROM PUBLISHER]

Subdivision schemes are multi-resolution methods used in computer-aided geometric design to generate smooth curves or surfaces. In this paper, we are interested in both smooth and non-smooth subdivision schemes. We propose two models that generalize the subdivision operation and can yield both smooth and non-smooth schemes in a controllable way: (1) The ‘varying-resolution’ model allows a structured access to the various resolutions of the refined data, yielding certain patterns. This model generalizes the standard subdivision iterative operation and has interesting interpretations in the geometrical space and also in creativity-oriented domains, such as music. As an infrastructure for this model, we propose representing a subdivision scheme by two dual rules trees. The dual tree is a permuted rules tree that gives a new operator-oriented view on the subdivision process, from which we derive an ‘adjoint scheme’. (2) The ‘generalized perturbed schemes’ model can be viewed as a special multi-resolution representation that allows a more flexible control on adding the details. For this model, we define the terms ‘template mask’ and ‘tension vector parameter’. The non-smooth schemes are created by the permutations of the ‘varying-resolution’ model or by certain choices of the ‘generalized perturbed schemes’ model. We then present procedures that integrate and demonstrate these models and some enhancements that bear a special meaning in creative contexts, such as music, imaging and texture. We describe two new applications for our models: (a) data and music analysis and synthesis, which also manifests the usefulness of the non-smooth schemes and the approximations proposed, and (b) the acceleration of convergence and smoothness analysis, using the ‘dual rules tree’. [ABSTRACT FROM AUTHOR]

In this paper, a spline collocation method is applied to solve a system of fourth-order boundary-value problems associated with obstacle, unilateral and contact problems. The presented method is dependent on four collocation points to be satisfied by four parameters θ j ∈(0, 1], j=1(1) 4 in each subinterval. It turns out that the proposed method when applied to the concerned system is a fourth-order convergent method and gives numerical results which are better than those produced by other spline methods [E.A. Al-Said and M.A. Noor, Finite difference method for solving fourth-order obstacle problems, Int. J. Comput. Math. 81(6) (2004), pp. 741–748; F. Geng and Y. Lin, Numerical solution of a system of fourth order boundary value problems using variational iteration method, Appl. Math. Comput. 200 (2008), pp. 231–241; J. Rashidinia, R. Mohammadi, R. Jalilian, and M. Ghasemi, Convergence of cubic-spline approach to the solution of a system of boundary-value problems, Appl. Math. Comput. 192 (2007), pp. 319–331; S.S. Siddiqi and G. Akram, Solution of the system of fourth order boundary value problems using non polynomial spline technique, Appl. Math. Comput. 185 (2007), pp. 128–135; S.S. Siddiqi and G. Akram, Numerical solution of a system of fourth order boundary value problems using cubic non-polynomial spline method, Appl. Math. Comput. 190(1) (2007), pp. 652–661; S.S. Siddiqi and G. Akram, Solution of the system of fourth order boundary value problems using cubic spline, Appl. Math. Comput. 187(2) (2007), pp. 1219–1227; Siraj-ul-Islam, I.A. Tirmizi, F. Haq, and S.K. Taseer, Family of numerical methods based on non-polynomial splines for solution of contact problems, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), pp. 1448–1460]. Moreover, the absolute stability properties appear that the method is A-stable. Two numerical examples (one for each case of boundary conditions) are given to illustrate practical usefulness of the method developed. [ABSTRACT FROM PUBLISHER]

This paper develops a smoothing domain-based energy (SDE) error indicator and an efficient adaptive procedure using edge-based smoothed point interpolation methods (ES-PIM), in which the strain field is constructed via the generalized smoothing operation over smoothing domains associated with edges of three-node triangular background cells. Because the ES-PIM can produce a close-to-exact stiffness and achieve 'super-convergence' and 'ultra-accurate' solutions, it is an ideal candidate for adaptive analysis. A SDE error indicator is first devised to make use of the features of the ES-PIM. A local refinement technique based on the Delaunay algorithm is then implemented to achieve high efficiency. The refinement of nodal neighbourhood is accomplished simply by adjusting a scaling factor assigned to control local nodal density. Intensive numerical studies, including the problems with stress concentration and solution singularity, demonstrate that the proposed adaptive procedure is effective and efficient in producing solutions with desired accuracy. [ABSTRACT FROM AUTHOR]

In this paper, two sextic-spline collocation methods are developed and analysed for approximating solutions of nonlinear fifth-order boundary-value problems. The first method uses a spline interpolant and the second one is based on a spline quasi-interpolant, which are constructed from sextic splines. They are both proved to be second-order convergent. Numerical results confirm the order of convergence predicted by the analysis. It has been observed that the methods developed in this paper are better than the others given in the literature. [ABSTRACT FROM AUTHOR]

The Lagrange interpolation problem in Banach spaces is approached by cardinal basis interpolation. Some error estimates are given and the results of several numerical tests are reported in order to show the approximation performances of the proposed interpolants. A comparison between some examples of interpolants is presented in the noteworthy case of Hilbert spaces, with some considerations about the possible localization of the formulas. Finally, some remarks about the cardinal basis interpolation framework are made from the application point of view. [ABSTRACT FROM AUTHOR]

The radial basis functions (RBFs) method is employed to handle a class of multi-dimensional parabolic inverse problems. Because they are not modelled by classical parabolic initial-boundary value problems, theoretical behaviour and numerical approximation of these problems have been active areas of research. Based on the idea of RBF approximation, a fast and highly accurate meshless method is developed for solving an inverse problem with a control parameter. Moreover, with the meshless property, it can be used to handle multi-dimensional parabolic inverse problems defined on very complicated geometries. [ABSTRACT FROM AUTHOR]