Fractal and Hausdorff dimensions for systems of iterative functional equations

Buescu, Jorge; Serpa, Cristina

doi:10.1016/j.jmaa.2019.123429

Cited by 10 publications

(7 citation statements)

References 22 publications

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“…The general theoretical framework for the study of systems of iterative functional equations was established by the authors in [11]. In [12] we show that the solutions constructed in [11] have, in general, fractal structure. A crucial role with respect to basic properties of the solutions (existence, uniqueness and continuity) is played by the so-called compatibility conditions.…”

Section: Introductionmentioning

confidence: 88%

Compatibility Conditions for Systems of Iterative Functional Equations with Non-trivial Contact Sets

Buescu

Serpa

2021

Results Math

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Systems of iterative functional equations with a non-trivial set of contact points are not necessarily solvable, as the resulting intersections may lead to an overdetermination of the system. To obtain existence and uniqueness results additional conditions must be imposed on the system. These are the compatibility conditions, which we define and study in a general setting. An application to the affine and doubly affine cases allows us to solve an open problem in the theory of functional equations. In the last section we consider a special problem in a different perspective, showing that a complex compatibility condition may result in an elegant and simple property of the solution.

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Section: Introductionmentioning

confidence: 88%

Compatibility Conditions for Systems of Iterative Functional Equations with Non-trivial Contact Sets

Buescu

Serpa

2021

Results Math

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“…with this assumption and from (7), it is evident that in order to show d ≤ f ≤ D , it suffices to choose the scaling functions i (x) such that the following inequalities are satisfied:…”

Section: Fractal Perturbation Preserving the Range Of Germmentioning

confidence: 99%

“…One can get smooth or non-smooth fractal functions depending on the scaling factors choice [9,26]. In the developments of the theory of FIFs, many researchers have investigated the notion of FIFs in different ways (see, for instance, the work by Luor, Massopust, Secelean, Serpa, Simon and Tetenov in [7,22,23,31,34] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Approximation by shape preserving fractal functions with variable scalings

Jha

Chand

Navascués

2021

Calcolo

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The fractal interpolation functions with appropriate iterated function systems provide a method to perturb and approximate a continuous function on a compact interval I. This method produces a class of functions f ∈ C(I) , where is a vector with functional components. The presence of scaling function in these fractal functions helps to get a wide variety of mappings for approximation problems. The current article explores the shape-preserving properties of the -fractal functions with variable scalings, where the optimal ranges of the scaling functions are derived for fundamental shapes of the germ f. We provide several examples to illustrate the shape preserving results and apply our fractal methodologies in approximation problems. Also, it is shown that the order of convergence of the -fractal polynomial to the original shaped function matches with that of polynomial approximation. Further, based on the shape preserving properties of the -fractal functions, we provide the fractal analogue of the Chebyshev alternation theorem. To the end, we deduce the fractal version of the classical full Müntz theorem in C[0, 1].

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“…The fractal analysis of real images was performed by using the technique based on a new affine fractal regression model. This process exploits certain mathematical formulations [2][3][4] designated for obtaining the coefficients of the equations system that best fit the data. The modeled system is:…”

Section: Mathematical Fractal Analysis Techniquementioning

confidence: 99%

“…The solution of this system is a function ϕ : [0, 1) → R and is called a fractal function [2]. In fact, it is proved [4] that such functions have a mathematical fractal structure, meaning that the plot of their graph is a fractal curve. Theoretical mathematical properties and explicit solutions are provided [2,3].…”

Section: Mathematical Fractal Analysis Techniquementioning

confidence: 99%

Fractal Nature of Advanced Ni-Based Superalloys Solidified on Board the International Space Station

et al. 2021

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Materials science is highly significant in space program investigation, energy production and others. Therefore, designing, improving and predicting advanced material properties is a crucial necessity. The high temperature creep and corrosion resistance of Ni-based superalloys makes them important materials for turbine blades in aircraft engines and land-based power plants. The investment casting process of turbine blades is costly and time consuming, which makes process simulations a necessity. These simulations require fundamental models for the microstructure formation. In this paper, we present advanced analytical techniques in describing the microstructures obtained experimentally and analyzed on different sample’s cross-sectional images. The samples have been processed on board the International Space Station using the MSL-EML device based on electromagnetic levitation principles. We applied several aspects of fractal analysis and obtained important results regarding fractals and Hausdorff dimensions related to the surface and structural characteristics of CMSX-10 samples. Using scanning electron microscopy (SEM), Zeiss LEO 1550, we analyzed the microstructure of samples solidified in space and successfully performed the fractal reconstruction of the sample’s morphology. We extended the fractal analysis on the microscopic images based on samples solidified on earth and established new frontiers on the advanced structures prediction.

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Fractal and Hausdorff dimensions for systems of iterative functional equations

Cited by 10 publications

References 22 publications

Compatibility Conditions for Systems of Iterative Functional Equations with Non-trivial Contact Sets

Compatibility Conditions for Systems of Iterative Functional Equations with Non-trivial Contact Sets

Approximation by shape preserving fractal functions with variable scalings

Fractal Nature of Advanced Ni-Based Superalloys Solidified on Board the International Space Station

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