Manuel Fernández-Martínez

• PhD in Mathematics
• Professor (Assistant) at Centro Universitario de la Defensa, San Javier, Spain

The main goal in this paper was to provide a novel chaos indicator based on a topological model which allows to calculate the fractal dimension of any curve. A fractal structure is a topological tool whose recursiveness becomes ideal to generalize the concept of fractal dimension. In this paper, we provide an algorithm to calculate a new fractal di...

In this paper, we provide the first known overall algorithm to calculate the Hausdorff dimension of any compact Euclidean subset. This novel approach is based on both a new discrete model of fractal dimension for a fractal structure which considers finite coverings and a theoretical result that the authors contributed previously in [14]. This new p...

In this paper, we use fractal structures to study a new approach to the Hausdorff dimension from both continuous and discrete points of view. We show that it is possible to generalize the Hausdorff dimension in the context of Euclidean spaces equipped with their natural fractal structure. To do this, we provide three definitions of fractal dimensio...

The main goal of this paper is to provide a generalized definition of fractal dimension for any space equipped with a fractal structure. This novel theory generalizes the classical box-counting dimension theory on the more general context of GF-spaces. In this way, if we select the so-called natural fractal structure on any Euclidean space, then th...

This paper provides a new model to compute the fractal dimension of a subset on a generalized-fractal space. Recall that fractal structures are a perfect place where a new definition of fractal dimension can be given, so we perform a suitable discretization of the Hausdorff theory of fractal dimension. We also find some connections between our defi...

This paper contributes the first-known study involving international military high-performance aircrafts together with their weaponry systems for defense purposes. This technical study had the invaluable help from an advisory group of instructor pilots whose expertise in combat aircrafts allowed the calculation of the relative importance of qualita...

This short communication addresses that question by comparing both the SFS and the TFS versions of the TOPSIS approach to rank a set of alternatives regarding a matter of planetary defense.

Extensions of fuzzy sets to broader contexts constitute one of the leading areas of research in the context of problems in artificial intelligence. Their aim is to address decision-making problems in the real world whenever obtaining accurate and sufficient data is not a straightforward task. In this way, spherical fuzzy sets were recently introduc...

In this lecture, we shall introduce and explore the so-called level separation property (LSP), a novel separation condition for attractors that is equivalent to a Moran type theorem for both fractal dimensions III and IV (i.e., Hausdorff type dimensions introduced previously by the authors for a fractal structure) of the self-similar set. Interesti...

We conducted a longitudinal study involving 240 patients grouped according to the classification of periodontal diseases agreed in the World Workshop by the different groups of specialists gathered there. We proceed to select images of Cone Beam Computed Tomography (CBCT) that were used to perform a study of bone density through a precise algorithm...

In this paper, we rediscover in detail a series of unknown attempts that some Spanish mathematicians carried out in the 1930s to address a challenge posed by Mr. la Cierva in 1934, which consisted of mathematically justifying the stability of la Cierva's autogiro, the first practical use of the direct-lift rotary wing and one of the first helicopte...

In this contribution, the role of biphosphonate as the fundamental chemical component in newly analyzed biochemical processes is studied, using both fractal dimension analysis on the mathematical and theoretical side, and empirical evaluations of the results of some pathology problems. The main goal of this paper is to throw some empirical evidence...

One of the main characteristics of cryptocurrencies is the high volatility of their exchange rates. In a previous work, the authors found that a process with volatility clusters displays a volatility series with a high Hurst exponent. In this paper, we provide a novel methodology to calculate the probability of volatility clusters with a special em...

The Posterior Superior Alveolar Artery (PSAA) provides vascular support to molars, gingiva, and maxillary sinus. A tear of the PSAA may cause profuse hemorrhages which may lead to complications at a surgical level. As such, it becomes crucial to anatomically analyse several features regarding the PSAA as well as the area surrounding it. In this pap...

Current statistics on the flux of impactors on the Earth reveal that asteroids with a diameter larger than few hundreds of meters are unlikely to reach the Earth’s orbit, although their potential effects on the life of our planet could be quite catastrophic. The present article assesses a set consisting of four deflection approaches, namely: kineti...

In this paper, some mathematical support is provided to properly justify the validity of the so-called multifractal height cross-correlation analysis (MFHXA), first contributed by Kristoufek (2011)[1]. With this aim, we extend several concepts from univariate random functions and their increments to the bivariate case. Specifically, we introduce th...

We conducted a longitudinal study involving 240 patients grouped according to the classification of periodontal diseases agreed in the World Workshop by the different groups of specialists gathered there. We proceed to select images of Cone Beam Computed Tomography (CBCT) that were used to perform a study of bone density through a precise algorithm...

In this paper, some mathematical support is provided to properly justify the validity of the so-called multifractal height cross-correlation analysis (MFHXA), first contributed by Kristoufek (2011)[1]. With this aim, we extend several concepts from univariate random functions and their increments to the bivariate case. Specifically, we introduce th...

The self-similarity index has been consolidated as a widely applied measure to quantify long-memory in stock markets. In this article, though, we shall provide a novel methodology allowing the detection of clusters of volatility in series of asset returns. With this aim, the concept of a volatility series is introduced. We found that the existence...

In this paper, we explore the (in)efficiency of the continuum Bitcoin-USD market in the period ranging from mid 2010 to early 2019. To deal with, we dynamically analyse the evolution of the self-similarity exponent of Bitcoin-USD daily returns via accurate FD4 approach by a 512 day sliding window with overlapping data. Further, we define the memory...

This book provides a generalised approach to fractal dimension theory from the standpoint of asymmetric topology by employing the concept of a fractal structure. The fractal dimension is the main invariant of a fractal set, and provides useful information regarding the irregularities it presents when examined at a suitable level of detail. New theo...

The dynamics of asteroids’ trajectories constitute potential threats to the Earth in the hypothetical case where the orbit of such an object crosses the orbit of the Earth. For this reason, advanced monitoring systems such as NASA JPL Sentry continually scan trajectories of Near-Earth Asteroids (NEAs), in addition to other celestial bodies. A large...

In this chapter, we shall study how to generalize the Hausdorff dimension for Euclidean sets throughout three new approaches of fractal dimension for a fractal structure. Thus, while two of such fractal dimensions will consist of appropriate discretizations regarding the classical Hausdorff dimension (the so-called fractal dimensions IV and V), the...

The main goal of this chapter is to generalize the classical box dimension in the broader context of fractal structures. We state that whether the so-called natural fractal structure (which any Euclidean subset can be always endowed with) is selected, then the box dimension remains as a particular case of the generalized fractal dimension models. T...

The main purpose of this chapter is to recall some definitions, results, and notations that are useful to develop a new theory of fractal dimension for fractal structures. In this way, we will be focused on quasi-pseudometrics, fractal structures, iterated function systems, and box-counting and Hausdorff dimension topics.

In this chapter, we explore a new model to calculate the fractal dimension of a subset with respect to a fractal structure. The new definition we provide presents better analytical properties than box dimension and can be calculated with easiness. It is worth mentioning that such a fractal dimension will be formulated as a discretization of Hausdor...

In this paper, we prove the identity dimH(F)=d·dimH(α−1(F)), where dimH denotes Hausdorff dimension, F⊆Rd, and α:[0,1]→[0,1]d is a function whose constructive definition is addressed from the viewpoint of the powerful concept of a fractal structure. Such a result stands particularly from some other results stated in a more general setting. Thus, Ha...

In this paper, we prove the identity $\dim_{\textrm H}(F)=d\cdot \dim_{\textrm H}(\alpha^{-1}(F))$, where $\dim_{\textrm H}$ denotes Hausdorff dimension, $F\subseteq \mathbb{R}^d$, and $\alpha:[0,1]\to [0,1]^d$ is a function whose constructive definition is addressed from the viewpoint of the powerful concept of a fractal structure. Such a result s...

The objective of the present paper is to describe all the anatomical considerations surrounding the nasopalatine foramen by relating them to the study of bone structure density via an accurate fractal dimension analysis in that area. We consecutively selected a sample of 130 patients, all of them with cone beam computed tomography (CBCT) images per...

A combination of two Multi-Criteria Decision Making methods is employed to determine the weights of the criteria to assess hazardous NEOs and obtain a ranking.

Owing to the complexity of decision environment, not all the attributes in multiple attribute decision making are quantitative. There are also some qualitative attributes, which are related to the integration of multiple attribute decision making (MADM) and linguistic multiple attribute decision making (LMADM). The specific method for composite mul...

In this paper, we re-explore in detail the techniques employed in P. A. P. Moran’s original proof for a key result in fractal geometry allowing the calculation of the Hausdorff dimension of attractors of iterated function systems constructed by similitudes.

Along this paper, we shall update the state-of-the-art concerning the application of fractal-based techniques to test for fractal patterns in physiological time series. As such, the first half of the present work deals with some selected approaches to deal with the calculation of the self-similarity exponent of time series. They include broadly-use...

One of the milestones in Fractal Geometry is the so-called Moran’s Theorem, which allows the calculation of the similarity dimension of any strict self-similar set under the open set condition. In this paper, we contribute a generalized version of the Moran’s theorem, which does not require the \(\mathrm{OSC}\) to be satisfied by the similitudes th...

In this paper, we explore the fractal dimension of Cone Beam Computed Tomography images to analyze the trabecular bone structure of healthy subjects. That quantity, computed throughout three distinct approaches, provided us accurate values of normality concerning the radiographic density of this kind of bones and will allow us to establish comparis...

In this paper, we highlight an intelligent system to properly construct a function between a pair of generalized-fractal spaces: the d-cube [0, 1] d endowed with its natural fractal structure, and the closed unit interval endowed with a natural-like fractal structure. Such a function allows the definition of space-filling curves by levels as well a...

Fractal dimension and specifically, box-counting dimension, is the main tool applied in many fields such as odontology to detect fractal patterns applied to the study of bone quality. However, the effective computation of such invariant has not been carried out accurately in literature. In this paper, we propose a novel approach to properly calcula...

A self-similar set is described as the unique (nonempty) compact subset remaining invariant under the action of a finite collection of similitudes on a complete metric space. Among this kind of fractals, those satisfying the so-called Moran's open set condition are especially appropriate to deal with applications of Fractal Geometry since their Hau...

Moran's Theorem is one of the milestones in Fractal Geometry. It allows the calculation of the similarity dimension of any (strict) self-similar set lying under the open set condition. Throughout a new fractal dimension we provide in the context of fractal structures, we generalize such a classical result for attractors which are required to satisf...

Along this talk, we shall deal with a classical problem in Fractal Geometry consisting of the calculation of the similarity dimension of self-similar sets. Clasically, the open set condition has been understood as the right separation condition for IFS-attractors since it becomes a sufficient (though not necessary) condition allowing to easily calc...

In this paper, we shall illustrate the numerical calculation of the effective temperature in Coulomb glasses by excitation probability provided that the system has been placed in a stationary state after applying a strong electric field. The excitation probability becomes a better alternative than the occupation probability, which has been classica...

Previous works have highlighted the suitability of the concept of fractal structure, which derives from asymmetric topology, to propound generalized definitions of fractal dimension. The aim of the present article is to collect some results and approaches allowing to connect the self-similarity index and the fractal dimension of a broad spectrum of...

In this paper, we characterize a novel separation property for IFS-attractors on complete metric spaces. Such a separation property is weaker than the strong open set condition (SOSC) and becomes necessary to reach the equality between the similarity and the Hausdorff dimensions of strict self-similar sets. We also investigate the size of the overl...

In this paper, we characterize a novel separation property for IFS-attractors on complete metric spaces. Such a separation property is weaker than the strong open set condition (SOSC) and becomes necessary to reach the equality between the similarity and the Hausdorff dimensions of strict self-similar sets. We also investigate the size of the overl...

Inthispaper,theclassicalTaylor’sexpansionseriesforagivencontinuous and k-times differentiable real function is obtained as the unique solution of a cer- tain class of initial value problems. Further, through some subsequent generalizations regarding that problem in terms of certain derivative-based operators, we obtain some generalized Taylor’s typ...

In this paper, the classical Taylor’s expansion series for a given continuous and k-times differentiable real function is obtained as the unique solution of a certain class of initial value problems. Further, through some subsequent generalizations regarding that problem in terms of certain derivative-based operators, we obtain some generalized Tay...

In this paper, we explore the chaotic behavior of resistively and capacitively shunted Josephson junctions via the so-called Network Simulation Method. Such a numerical approach establishes a formal equivalence among physical transport processes and electrical networks, and hence, it can be applied to efficiently deal with a wide range of different...

Since the pioneer contributions due to Vandewalle and Ausloos, the Hurst exponent has been applied by econophysicists as a useful indicator to deal with investment strategies when such a value is above or below 0.5, the Hurst exponent of a Brownian motion. In this paper, we hypothesize that the self-similarity exponent of financial time series prov...

In this paper, we explore the chaotic behavior of resistively and capacitively shunted Josephson junctions via the so-called Network Simulation Method. Such a numerical approach establishes a formal equivalence among physical transport processes and electrical networks, and hence, it can be applied to efficiently deal with a wide range of different...

In this paper, we provide some computational evidence concerning the dependence of conductivity on the system thickness for Coulomb glasses. We also verify the Efros–Shklovskii law and deal with the calculation of its characteristic parameter as a function of the thickness. Our results strengthen the link between theoretical and experimental fields...

In this paper, we provide some computational evidence concerning the dependence of conductivity on the system thickness for Coulomb glasses. We also verify the Efros-Shklovskii law and deal with the calculation of its characteristic parameter as a function of the thickness. Our results strengthen the link between theoretical and experimental fields...

The impact of a near-Earth object (NEO) may release large amounts of energy and cause serious damage. Several NEO hazard studies conducted over the past few years provide forecasts, impact probabilities and assessment ratings, such as the Torino and Palermo scales. These high-risk NEO assessments involve several criteria, including impact energy, m...

Along the years, the foundations of Fractal Geometry have received contributions starting from mathematicians like Cantor, Peano, Hilbert, Hausdorff, Carathéodory, Sierpinski, and Besicovitch, to quote some of them. They were some of the pioneers exploring objects having self-similar patterns or showing anomalous properties with respect to standard...

A fractal structure is a countable family of coverings which displays accurate information about the irregularities that a set presents when being explored with enough level of detail. It is worth noting that fractal structures become especially appropriate to provide new definitions of fractal dimension, which constitutes a valuable measure to tes...

In this paper, we deal with the part of Fractal Theory related to finite families of (weak) contractions, called iterated function systems (IFS, herein). An attractor is a compact set which remains invariant for such a family. Thus, we consider spaces homeomorphic to attractors of either IFS or weak IFS, as well, which we will refer to as Banach an...

Fractal dimension constitutes the main tool to test for fractal patterns in Euclidean contexts. For this purpose, it is always used the box dimension, since it is easy to calculate, though the Hausdorff dimension, which is the oldest and also the most accurate fractal dimension, presents the best analytical properties. Additionally, fractal structu...

The bird strike damage on aircrafts is a widely studied matter [1] with a high economic impact on stakeholders finances. Some authors estimate it in about USD1.2 Billion for nowadays commercial worldwide activity [2], and more than USD937 million in direct and other monetary losses per year just for the United States, as an example of civil aviatio...

In this paper, we introduce a new theoretical model to calculate the fractal dimension especially appropriate for curves. This is based on the novel concept of induced fractal structure on the image set of any curve. Some theoretical properties of this new definition of fractal dimension are provided as well as a result which allows to construct sp...

In this paper, we introduce a new theoretical model to calculate the fractal dimension especially appropriate for curves. This is based on the novel concept of induced fractal structure on the image set of any curve. Some theoretical properties of this new definition of fractal dimension are provided as well as a result which allows to construct sp...

The main goal in this paper was to provide a novel chaos indicator based on a topological model which allows to calculate the fractal dimension of any curve. A fractal structure is a topological tool whose recursiveness becomes ideal to generalize the concept of fractal dimension. In this paper, we provide an algorithm to calculate a new fractal di...

In the present paper, we study regular and chaotic dynamics from planar oscillations of a dumbbell satellite under the influence of the gravity field generated by an oblate body, considering the effect of the zonal harmonic parameter J2. We theoretically show the existence of chaotic oscillations provided that the eccentricity becomes arbitrarily s...

A pattern of interpolation nodes on the disk is studied, for which the interpolation problem is theoretically unisolvent, and which renders a minimal numerical condition for the collocation matrix when the standard basis of Zernike polynomials is used. It is shown that these nodes have an excellent performance also from several alternative points o...

In this paper, three approaches to calculate the self-similarity exponent of a time series are compared in order to determine which one performs best to identify the transition from random efficient market behavior (EM) to herding behavior (HB) and hence, to find out the beginning of a market bubble. In particular, classical Detrended Fluctuation A...

Hausdorff dimension, which is the oldest and also the most accurate model for fractal dimension, constitutes the main reference for any fractal dimension definition that could be provided. In fact, its definition is quite general, and is based on a measure, which makes the Hausdorff model pretty desirable from a theoretical point of view. On the ot...

In this paper, we explain how to generate adequate pre-fractals in order to properly approximate attractors of iterated function systems on the real line within a priori known Hausdorff dimension. To deal with, we have applied the classical Moran's Theorem, so we have been focused on nonoverlapping strict self-similar sets. This involves a quite si...

A Brain Computer Interface (BCI) system is a tool not requiring any muscle action to transmit information. Acquisition, preprocessing, feature extraction (FE), and classification of electroencephalograph (EEG) signals constitute the main steps of a motor imagery BCI. Among them, FE becomes crucial for BCI, since the underlying EEG knowledge must be...

In this paper, the classical Taylor’s expansion series for a given continuous and k-times differentiable real function is obtained as the unique solution of a certain class of initial value problems. Further, through some subsequent generalizations regarding that problem in terms of certain derivative-based operators, we obtain some generalized Tay...

In the present paper, we study regular and chaotic dynamics from planar oscillations of a dumbbell satellite under the influence of the gravity field generated by an oblate body, considering the effect of the zonal harmonic parameter J2. We theoretically show the existence of chaotic oscillations provided that the eccentricity becomes arbitrarily s...

Background and objective To analyze the effects of an aquatic biodance based therapy on sleep quality, anxiety, depression, pain and quality of life in fibromyalgia patients. Patients and method Randomized controlled trial with 2 groups. Fifty-nine patients were assigned to 2 groups: experimental group (aquatic biodance) and control group (stretch...

BACKGROUND AND OBJECTIVE: To analyze the effects of an aquatic biodance based therapy on sleep quality, anxiety, depression, pain and quality of life in fibromyalgia patients. PATIENTS AND METHOD: Randomized controlled trial with 2 groups. Fifty-nine patients were assigned to 2 groups: experimental group (aquatic biodance) and control group (stretc...

A fractal structure is a tool that is used to study the fractal behavior of a space. In this paper, we show how to apply a new concept of fractal dimension for fractal structures, extending the use of the box-counting dimension to new contexts. In particular, we define a fractal structure on the domain of words and show how to use the new fractal d...

Comparación entre biodanza en medio acuático y stretching en la mejora de la calidad de vida y dolor en los pacientes con fibromialgia Resumen Objetivo: Comparar qué grado de mejoría pueden alcanzar los pacientes con fibromialgia en las variables dolor, impacto de la fibromialgia y depresión, mediante la biodanza acuática frente al stretching. Dise...

OBJECTIVE To determine the level of improvement, as regards pain, impact on fibromyalgia and depression, achieved by patients with fibromyalgia by comparing aquatic biodanza and stretching exercises. DESIGN Randomised controlled trial with two intervention groups. LOCATION Five health centres (Almeria). PATIENTS A total of 82 fibromyalgia patien...

In this paper, three new algorithms are introduced in order to explore long memory in financial time series. They are based on a new concept of fractal dimension of a curve. A mathematical support is provided for each algorithm and its accuracy is tested for different length time series by Monte Carlo simulations. In particular, in the case of shor...

Geometric method-based procedures, which we will call GM algorithms hereafter, were introduced in M.A. Sánchez-Granero, J.E. Trinidad Segovia, J. García Pérez, Some comments on Hurst exponent and the long memory processes on capital markets, Phys. A 387 (2008) 5543–5551, to calculate the Hurst exponent of a time series. The authors proved that GM a...

In this paper, we show how a new fractal dimension definition for fractal structures can be used to study curves, and then we use this fractal dimension to study financial type series. In particular, we relate this dimension with the Hurst exponent for a wide range of random processes, including fractional Brownian motions and fractional Levy stabl...