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More General fractal functions on the Sphere

MEDITERRANEAN JOURNAL OF MATHEMATICS 2019年第6期16卷 134-134页

作者： Akhtar, Md. Nasim Guru Prem Prasad, M. Navascues, M. A. Indian Inst Technol Guwahati Dept Math Gauhati 781039 Assam India Univ Zaragoza Dept Matemat Aplicada Zaragoza Spain

In this article, a family of continuous functions on the unit sphere S subset of R-3 generalizing the spherical harmonics, is considered. Using fractal methodology, the fractal version of this family of continuous functions on the sphere S is constructed. To do this, an iterated function system (IFS) and a linear bounded operator that maps classical functions to its fractal analogues is defined. Some approximation properties of fractal functions on the sphere are investigated. Restricting the scale vector involved in the IFS, a fractal Hilbert basis is established for the functions on the sphere.

关键词： fractal interpolation functions alpha-fractal interpolation functions spherical harmonics best approximations Hilbert bases

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Dependence of friction coefficient on the resolution of asperities in metallic rough surfaces under cyclic loading

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INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES 2017年 108卷 85-97页

作者： Panagouli, Olympia K. Mastrodimou, Konstantina Univ Thessaly Dept Civil Engn Lab Struct Anal & Design Volos 38334 Volos Greece

In this paper the friction mechanism that is developed between metallic rough interfaces with irregularities of different scales, submitted to cyclic loading is studied. The main purpose of the paper is to investigate how the resolution delta(n), which takes into account asperities of different scales of the rough interfaces and the intensity of the excitation, affect the friction that develops between the two surfaces. Friction is assumed to be the result of gradual plastification of the interface asperities in a metallic body with elastic-plastic behaviour. The interface of the body is described by a fractal interpolation function f With iterative regular construction, which permits the analysis of interfaces with irregularities of different scales. In the sequel, the structures resulting from the different iterations (i.e. resolutions) of the rough interface are submitted to different acceleration histories that induce forces leading to plastification of the interface asperities. The apparent dynamic friction coefficient is studied macroscopically, as the ratio between the forces that develop parallel and normal to the interface. Finally, the influence of the applied normal pressure to the friction mechanism is investigated with respect to the fractal resolution and the acceleration history of the interfaces. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Friction coefficient Roughness fractal interpolation functions Asperity resolution Normal pressure Cyclic loading

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Box dimension of α-fractal function with variable scaling factors in subintervals

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CHAOS SOLITONS & fractalS 2017年 103卷 440-449页

作者： Akhtar, Md Nasim Prasad, M. Guru Prem Navascures, M. A. IIT Guwahati Dept Math Gauhati 781039 India Univ Zaragoza Dept Matemat Aplicada Zaragoza Spain

The box dimension of the graph of non-affine alpha-fractal interpolation function f(alpha) with variable scaling factors is estimated in the interval [0, 1]. Due to the non- affinity of f(alpha), the behavior of the graph is nonuniform in the subintervals. An attempt is made to estimate the box dimension of the graph of f(alpha) in the subintervals as well and it is compared with the box dimension of f(alpha) on the whole interval. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： fractal interpolation functions alpha-fractal interpolation functions Variable scaling factors Box-counting dimensions

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fractal interpolation functions ON POST CRITICALLY FINITE SELF-SIMILAR SETS

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2010年第1期18卷 119-125页

作者： Ruan, Huo-Jun Zhejiang Univ Dept Math Hangzhou 310027 Peoples R China Cornell Univ Dept Math Ithaca NY 14853 USA

In this paper, we introduce fractal interpolation functions (FIFs) and linear FIFs on a post critically finite (p.c.f. for short) self-similar set K. We present a sufficient condition such that linear FIFs have finite energy and prove that the solution of Dirichlet problem - Delta(mu)u = f, u vertical bar(partial derivative K) = 0 is a linear FIF on K if f is a linear FIF.

关键词： fractal interpolation functions PCF Self-Similar Sets Energy Laplacian

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Convexity/Concavity and Stability Aspects of Rational Cubic fractal interpolation Surfaces

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Computational Mathematics and Modeling 2017年第3期28卷 407-430页

作者： Chand, A.K.B. Vijender, N. Navascués, M.A. Indian Institute of Technology Madras Chennai India Vellore Institute of Technology University Chennai India Centro Politécnico Superior de Ingenieros Universidad de Zaragoza Zaragoza Spain

fractal interpolation is more general than the classical piecewise interpolation due to the presence of the scaling factors that describe smooth or non-smooth shape of a fractal curve/surface. We develop the rational cubic fractal interpolation surfaces (FISs) by using the blending functions and rational cubic fractal interpolation functions (FIFs) with two shape parameters in each sub-interval along the grid lines of the interpolation domain. The properties of blending functions and C¹-smoothness of rational cubic FIFs render C¹-smoothness to our rational cubic FISs. We study the stability aspects of the rational cubic FIS with respect to its independent variables, dependent variable, and first order partial derivatives at the grids. The scaling factors and shape parameters seeded in the rational cubic FIFs are constrained so that these rational cubic FIFs are convex/concave whenever the univariate data sets along the grid lines are convex/concave. For these constrained scaling factors and shape parameters, our rational cubic FIS is convex/concave to given convex/concave surface data. © 2017, Springer Science+Business Media, LLC.

关键词： blending functions concavity convexity fractal interpolation functions fractal interpolation surfaces fractals

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PARTIALLY BLENDED CONSTRAINED RATIONAL CUBIC TRIGONOMETRIC fractal interpolation SURFACES

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2016年第3期24卷 1650027-1650027页

作者： Chand, A. K. B. Tyada, K. R. Indian Inst Technol Dept Math Madras 600036 Tamil Nadu India

fractal interpolation is an advance technique for visualization of scientific shaped data. In this paper, we present a new family of partially blended rational cubic trigonometric fractal interpolation surfaces (RCTFISs) with a combination of blending functions and univariate rational trigonometric fractal interpolation functions (FIFs) along the grid lines of the interpolation domain. The developed FIFs use rational trigonometric functions p(i, j)(theta)/q(i,j)(theta), where p(i,j)(theta) and q(i,j)(theta) are cubic trigonometric polynomials with four shape parameters. The convergence analysis of partially blended RCTFIS with the original surface data generating function is discussed. We derive sufficient data-dependent conditions on the scaling factors and shape parameters such that the fractal grid line functions lie above the grid lines of a plane Pi, and consequently the proposed partially blended RCTFIS lies above the plane Pi. Positivity preserving partially blended RCTFIS is a special case of the constrained partially blended RCTFIS. Numerical examples are provided to support the proposed theoretical results.

关键词： Iterated Function Systems fractal interpolation functions Rational Cubic Trigonometric interpolation Blending functions fractal interpolation Surfaces Positivity

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BOX DIMENSIONS OF α-fractal functions

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2016年第3期24卷 1650037-1650037页

作者： Akhtar, Md. Nasim Prasad, M. Guru Prem Navascues, M. A. Indian Inst Technol Guwahati Dept Math Gauhati 781039 Assam India Univ Zaragoza Dept Mate Aplicada E-50009 Zaragoza Spain

The box dimension of the graph of non-affine, continuous, nowhere differentiable function fa which is a fractal analogue of a continuous function f corresponding to a certain iterated function system (IFS), is investigated in the present paper. The estimates for box dimension of the graph of alpha-fractal function f(alpha) for equally spaced as well as arbitrary data sets are found.

关键词： fractal interpolation functions alpha-fractal interpolation functions Variation of Continuous functions Box-Counting Dimensions

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A NEW CLASS OF fractal interpolation SURFACES BASED ON FUNCTIONAL VALUES

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2016年第1期24卷 1650007-1650007页

作者： Chand, A. K. B. Vijender, N. Indian Inst Technol Dept Math Madras 600036 Tamil Nadu India VIT Univ Chennai Dept Math Madras 600127 Tamil Nadu India

fractal interpolation is a modern technique for fitting of smooth/non-smooth data. Based on only functional values, we develop two types of C-1-rational fractal interpolation surfaces (FISs) on a rectangular grid in the present paper that contain scaling factors in both directions and two types of positive real parameters which are referred as shape parameters. The graphs of these C-1-rational FISs are the attractors of suitable rational iterated function systems (IFSs) in R-3 which use a collection of rational IFSs in the x-direction and y-direction and hence these FISs are self-referential in nature. Using upper bounds of the interpolation error of the x-direction and y-direction fractal interpolants along the grid lines, we study the convergence results of C-1-rational FISs toward the original function. A numerical illustration is provided to explain the visual quality of our rational FISs. An extra feature of these fractal surface schemes is that it allows subsequent interactive alteration of the shape of the surfaces by changing the scaling factors and shape parameters.

关键词： fractals Iterated Function Systems fractal interpolation functions Self-Referential fractal interpolation Surfaces Blending Function Convergence

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Positive blending Hermite rational cubic spline fractal interpolation surfaces

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CALCOLO 2015年第1期52卷 1-24页

作者： Chand, A. K. B. Vijender, N. Indian Inst Technol Madras Dept Math Madras 600036 Tamil Nadu India

fractal interpolation provides an efficient way to describe data that have smooth and non-smooth structures. Based on the theory of fractal interpolation functions (FIFs), the Hermite rational cubic spline FIFs (fractal boundary curves) are constructed to approximate an original function along the grid lines of interpolation domain. Then the blending Hermite rational cubic spline fractal interpolation surface (FIS) is generated by using the blending functions with these fractal boundary curves. The convergence of the Hermite rational cubic spline FIS towards an original function is studied. The scaling factors and shape parameters involved in fractal boundary curves are constrained suitably such that these fractal boundary curves are positive whenever the given interpolation data along the grid lines are positive. Our Hermite blending rational cubic spline FIS is positive whenever the corresponding fractal boundary curves are positive. Various collections of fractal boundary curves can be adapted with suitable modifications in the associated scaling parameters or/and shape parameters, and consequently our construction allows interactive alteration in the shape of rational FIS.

关键词： fractals Iterated function systems fractal interpolation functions Blending functions fractal interpolation surfaces Positivity

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A Monotonic Rational fractal interpolation Surface and Its Analytical Properties 1

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2nd International Conference on Mathematics and Computing (ICMC)

作者： Chand, A. K. B. Vijender, N. Indian Inst Technol Madras Dept Math Madras 600036 Tamil Nadu India VIT Univ Dept Math Madras 600127 Tamil Nadu India

ISBN: (数字)9788132224525

ISBN: (纸本)9788132224525;9788132224518

A l(1)-continuous rational cubic fractal interpolation function was introduced and its monotonicity aspect was investigated in [Adv. Difference Eq. (30) 2014]. Using this univariate interpolant and a blending technique, in this article, we develop a monotonic rational fractal interpolation surface (FIS) for given monotonic surface data arranged on the rectangular grid. The analytical properties like convergence and stability of the rational cubic FIS are studied. Under some suitable hypotheses on the original function, the convergence of the rational cubic FIS is studied by calculating an upper bound for the uniform error of the surface interpolation. The stability results are studied when there is a small perturbation in the corresponding scaling factors. We also provide numerical examples to corroborate our theoretical results.

关键词： fractals fractal interpolation functions fractal interpolation surfaces Monotonicity Blending functions

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