Cristina Serpa

• Phd
• Professor (Assistant) at Polytechnical Institute of Lisbon

In this research, the doping of SrTiO3 with Mn⁴⁺ was performed in order to evaluate the potential application as a photocatalyst for the degradation of organic dye pollutants. Since photocatalytic activity depends on grain microstructure, fractal analysis was used to estimate the Hausdorff dimension to provide a more thorough investigation of Mn@Sr...

The Living Planet Index (LPI) is a global index which measures the state of the world’s biodiversity. Analyzing the LPI solely by statistical trends provides, however, limited insight. Fractal Regression Analysis is a recently developed tool that has been successfully applied in multidisciplinary scientific contexts for time series analysis. This m...

In this paper, we have compared the fractal calculus on fractal sets and fractal curves. The analogues of the Riemann-Liouville fractional integrals and derivatives and Caputo fractional derivatives are defined on the fractal curves which are non-local derivatives. The analogues fractional Laplace is defined to solve fractal non-local differential...

The microstructure–property relationship in poly(methyl methacrylate) PMMA composites is very important for understanding interface phenomena and the future prediction of properties that further help in designing improved materials. In this research, field emission scanning electron microscopy (FESEM) images of denture PMMA composites with SrTiO3,...

In this paper, random and stochastic processes are defined on fractal curves. Fractal calculus is used to define cumulative distribution function, probability density function, moments, variance and correlation function of stochastic process on fractal curve. A new framework which is a generalization of mean square calculus is formulated. Sequence...

The paper covers the foundations of fractal calculus on fractal curves, defines different function classes, establishes vector spaces for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}...

Fractal analysis was used to estimate the fractal dimensions and further define the structure of lanthanum magnesium titanium oxide. The microstructures and space groups of the materials were determined. This is relevant to the electrical properties at microwave frequencies, which are important for miniaturising electronic components. Fractal analy...

The concept of Laplace transform has been extended to fractal curves, enabling the solution of fractal differential equations with constant coefficients. This extension, known as the fractal Laplace transform, is particularly useful for handling inhomogeneous differential equations that involve delta Dirac functions and step functions within the re...

In this paper, new results on the [Formula: see text]-fractal function with variable parameters are presented. The Weyl–Marchaud variable order fractional derivative of an [Formula: see text]-fractal function with variable parameters is examined by imposing certain conditions on the scaling factors. Following the investigation of fractional derivat...

In this paper, we summarize fractal calculus on fractal curves and nonstandard analysis. Using nonstandard analysis which includes hyperreal and hyperinteger numbers, we define left and right limits and derivatives on fractal curves. Fractal integral and differential forms are defined using nonstandard analysis. Some examples are solved to show det...

In this paper, the integral of classical fractal interpolation function (FIF) and A-fractal function is explored for both the cases of constant and variable scaling factors. The definite integral for the classical FIF in the closed interval of [Formula: see text] is estimated. The novel notion of affine-quadratic FIF is introduced and integrated fo...

In this manuscript, fractal and fuzzy calculus are summarized. Fuzzy calculus in terms of fractal limit, continuity, its derivative, and integral are formulated. The fractal fuzzy calculus is a new framework that includes fractal fuzzy derivatives and fractal fuzzy integral. In this framework, fuzzy number-valued functions with fractal support are...

The challenges in productivity of satellite mobile devices are growing rapidly to overcome the question of miniaturization. The intention is to supply the electrical and microwave properties of materials by discovering their outstanding electronic properties. Neodymium Zinc Titanate (NZT) can be a promising ferroelectric material due to its stable...

Polymers and polymer matrix composites are commonly used materials with applications extending from packaging materials to delicate electronic devices. Epoxy resins and fiber-reinforced epoxy-based composites have been used as adhesives and construction parts. Fractal analysis has been recognized in materials science as a valuable tool for the micr...

A recently proposed fractal analysis approach (Serpa, 2022) is applied to estimate the fractal dimension (FD) of segmented brain white matter (WM) MRI image in healthy and traumatic brain injury (TBI) subjects to investigate the differences among the two groups. The WM volume is extracted from the structural MRI data with an image processing algori...

This chapter explores the Katugampola fractional integral of a multivariate vector-valued function defined on \(\mathbb {R}^n\). Alongside, it is shown that the prescribed fractional operator preserves some analytical properties of the original function like continuity and boundedness. Further, this chapter discusses applying one of the fractional...

In this paper, we summarize fractal and fuzzy calculus. We formulate fuzzy calculus in terms of fractal limit, continuity, derivative, and integral. The fractal fuzzy calculus is a new framework that includes fractal fuzzy derivatives and fractal fuzzy integral. In this framework, fuzzy number-valued functions with fractal support are the solutions...

This paper develops a method to find fractal curves to fit real data. With a formulation for fractal functions through a type of affine systems of iterative functional equations, we apply the procedure of minimizing the sum of square residuals that is used in the classical linear regression. We develop formulas for approximation and exact fractal f...

This contributed volume will be a selection of chapters by diverse academics, scholars, researchers, educators, and experts from a wide range of disciplines, who address different aspects of the Fractal concept through a wide array of expertise, including but not limited to mathematics, physics, engineering, astrology, meteorology, oceanology, geol...

In this paper, fractal calculus, which is called Fα-calculus, is reviewed. Fractal calculus is implemented on fractal interpolation functions and Weierstrass functions, which may be non-differentiable and non-integrable in the sense of ordinary calculus. Graphical representations of fractal calculus of fractal interpolation functions and Weierstras...

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 st...

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 an...

Proposition of new methods for the study of complex systems and chaos is working on the frontiers of knowledge and hence it calls for philosophical contemplation besides well-set empirical researches. Complex system studies still suffer methodological paucity. Hence, the research goal is proposing a new generic methodology in the domain of chaotic...

This presentation shows step by step how to find a fractal function on the Sahara Desert and estimate its fractal dimension.

This is a second file explaining how to do a Fractal Analysis of Real Data.

Find the program online: https://cristinaserpa.selz.com/item/fractal-real-finder

We consider systems of non-affine iterative functional equations. From the constructive form of the solutions, recently established by the authors, representations of these systems in terms of symbolic spaces as well as associated fractal structures are constructed. These results are then used to derive upper bounds both for the appropriate fractal...

We formulate a general theoretical framework for systems of iterative functional equations between general spaces X and Y. We find general necessary conditions for the existence of solutions. When X and Y are topological spaces, we characterize continuity of solutions; when X, Y are metric spaces, we find sufficient conditions for existence and uni...

Conjugacy equations arise from the problem of identifying dynamical systems from the topological point of view. It is well known that when conjugacies exist they cannot, in general, be expected to be smooth. We show that even in the simplest cases, e.g. piecewise affine maps, solutions of functional equations arising from conjugacy problems may hav...

We construct an explicit formula for the fractal interpolation function associated to an IFS with variable parameters. The solution is given in terms of the base p representation of numbers. This construction is a consequence of the formulation of the problem in a general functional equation setting. We introduce compatibility conditions as essenti...

Topological invariants of interval maps are preserved by conjugacy. We investigate some features of the conjugacy equations associated to piecewise expanding maps. For special cases, it is possible to construct explicitly a conjugacy function in terms of the a-base expansion of numbers through a solution of the corresponding functional equations.

We establish combinatorial properties of the dynamics of piecewise increasing, continuous, expanding maps of the interval such as description of periodic and pre-periodic points, primitiveness of truncated itineraries and length of pre-periodic itineraries. We include a relation between the dynamics of a family of circle maps and the properties of...

Neste artigo mostra-se como a congruência associada ao Pequeno teorema de Fermat admite versões muito mais gerais, que se podem demonstrar por métodos de sistemas dinâmicos e de combinatória.ilustram-se também as contribuições mais importantes para a obtenção destes resultados.

There are many possible proofs of Fermat’s little theorem. Among them we exemplify those using necklaces and dynamical systems. Both methods lead to a generalization. It is a congruence theorem which is already known from Gauss and Gegenbauer. A natural result from these proofs is a bijection between aperiodic necklaces and circle maps.

We introduce concepts of formal languages, such as necklaces and Lyndon words. We give examples, some classical and others visually more attractive and playful. In parallel, in terms of dynamical systems, we define a particular circle map. Associating all the listed concepts we construct simultaneously aperiodic necklaces and periodic orbits of the...