Papers by Manuel Fernández-Martínez
In this paper, we explore the chaotic behavior of resistively and capacitively shunted Josephson ... more 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 differential systems. The generality underlying that electrical equivalence allows to apply the circuit theory to several scientific and technological problems. In this work, the Fast Fourier Transform has been applied for chaos detection purposes and the calculations have been carried out in PSpice, an electrical circuit software. Overall, it holds that such a numerical approach leads to quickly computationally solve Josephson differential models. An empirical application regarding the study of the Josephson model completes the paper.
Along the years, the foundations of Fractal Geometry have received contributions starting from ma... more Along the years, the foundations of Fractal Geometry have received contributions starting from mathematicians like Cantor, Peano, Hilbert, Hausdorff, Carathéodory, Sierpi´nski, 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 analytic attributes. Among the new tools developed to deal with this kind of objects, fractal dimension has become one of the most applied since it constitutes a single quantity which throws useful information concerning fractal patterns on sets. Several years later, fractal structures were introduced from Asymmetric Topology to characterize self-similar symbolic spaces. Our aim in this survey is to collect several results involving distinct definitions of fractal dimension we proved jointly with Prof.M.A. Sánchez-Granero in the context of fractal structures.
In this paper, the classical Taylor's expansion series for a given continuous and k-times differe... more 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 Taylor's type polynomial expansions, including the Taylor–Aleph series, which remains as particular cases. In addition to that, some analytical properties about these involved operators are also provided.
A fractal structure is a countable family of coverings which displays accurate information about ... more 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 test for chaos in dynamical systems. In this paper, we explore several approaches to calculate the fractal dimension of a subset with respect to a fractal structure. These models generalize the classical box dimension in the context of Euclidean subspaces from a discrete viewpoint. To illustrate the flexibility of the new models, we calculate the fractal dimension of a family of self-affine sets associated with certain discrete dynamical systems.
In this paper, we introduce a new theoretical model to calculate the fractal dimension especially... more 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 space-filling curves. We explore and analyze the behavior of this new fractal dimension compared to classical models for fractal dimension, namely, both the Hausdorff dimension and the box-counting dimension. This analytical study is illustrated through some examples of space-filling curves, including the classical Hilbert's curve. Finally, we contribute some results linking this fractal dimension approach with the self-similarity exponent for random processes.
Fractal dimension constitutes the main tool to test for fractal patterns in Euclidean contexts. F... more 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 structures provide an appropriate topological context where new models of fractal dimension for a fractal structure could be developed in order to generalize the classical models of fractal dimension. In this survey, we gather different definitions and counterexamples regarding these new models of fractal dimension in order to show the reader how they behave mathematically with respect to the classical models, and also to point out which features of such models can be exploited to powerful effect in applications.
In this paper, we deal with the part of Fractal Theory related to finite families of (weak) contr... more 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 at-tractors of either IFS or weak IFS, as well, which we will refer to as Banach and topological fractals, respectively. We present a collection of counterexamples in order to show that all the presented definitions are essential, though they are not equivalent in general.
Since the pioneer contributions due to Vandewalle and Ausloos, the Hurst exponent has been applie... more 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 provides a reliable indicator for
In this paper, we provide some computational evidence concerning the dependence of conductivity o... more 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 dam... more 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, mass, and absolute magnitude. The main objective of this paper is to provide the first Multi-Criteria Decision Making (MCDM) approach to classify hazardous NEOs. Our approach applies a combination of two methods from a widely utilized decision making theory. Specifically, the Analytic Hierarchy Process (AHP) methodology is employed to determine the criteria weights, which influence the decision making, and the Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) is used to obtain a ranking of alternatives (potentially hazardous NEOs). In addition, NEO datasets provided by the NASA Near-Earth Object Program are utilized. This approach allows the classification of NEOs by descending order of their TOPSIS ratio, a single quantity that contains all of the relevant information for each object. Asteroids, described as small rocky bodies with sizes consisting of a few metres to a few hundred kilometres in diameter, constitute a potential threat. While most of them might impact the Earth in the next million years with a probability close to 0.5% 1 , there is a chance of approximately 1% that an impact > 1000 MT (equivalent to 100 Tunguskas) might happen once each century 2. Hence, several studies have explored the implications of large asteroid impacts on early Earth ecosystems 3,4. It turns out that the determination of the statistical frequency of an asteroid impact is less relevant than stating whether an asteroid may impact the Earth 5. Even small objects (with diameters ranging from 50 to 100 m) may cause great damage. In fact, these types of small objects may lead to so-called Tunguska-class events whose impact energy is equivalent to 10 MT of TNT 6. Near-Earth objects (NEOs) are small asteroids that orbit close to the Earth's orbit. NEO research provides information concerning the evolution of the early solar system 7. Several measures to rate hazardous NEOs have been contributed in the scientific literature 8. Among them are the Torino scale 9 , which evaluates the a priori risk due to a potentially hazardous asteroid, and the Palermo scale 10 , which takes into account the energy at impact and the estimated probability of the event (that might happen in a time period spanning from present time to predicted impact) with respect to a record of events with a comparable or greater level 11. Several factors, including impact energy, impact velocity, estimated diameter, number of potential impacts, absolute magnitude, and impact probability, are also used to quantify the risk of NEO impacts 10. Furthermore, other non-physical factors such as Purgatorio Ratio (PR) have been used to manage the communication of impact threats; in this case, the PR is expressed by the ratio of time between the first and last observation to the time between the present and next possible impact date 12. Therefore, it is clear that an assessment of hazardous NEOs involves a wide list of varied nature criteria. The impact hazard assessment requires the development of ad hoc techniques beyond what is routinely conducted by automatic impact monitoring systems 13. In this paper, we contribute the first known Multi-Criteria Decision Making (MCDM) approach for hazardous NEO assessment. More specifically, we have applied the Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) to classify hazardous NEOs. Next, we will motivate the application of MCDM techniques to hazardous NEO assessment. A MCDM problem consists of a set of alternatives to be evaluated with respect to a list of criteria. All that information is contained in a decision matrix. The main goal is to find the best option among all the alternatives once they have
A Brain Computer Interface (BCI) system is a tool not requiring any muscle action to transmit inf... more 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 properly extracted
into a feature vector. Linear approaches have been widely applied to FE in BCI, whereas nonlinear tools are not so common in literature. Thus, the main goal of this paper is to check whether
some Hurst exponent and fractal dimension based estimators become valid indicators to FE in motor imagery BCI. The final results obtained were not optimal as expected, which may be due
to the fact that the nature of the analyzed EEG signals in these motor imagery tasks were not self-similar enough.
Background and objective: To analyze the effects of an aquatic biodance based therapy on sleep qu... more 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 (stretching). The outcome measures were quality of sleep (Pittsburgh questionnaire), anxiety (State Anxiety Inventory), depression (Center for Epidemiologic Studies Depression Scale), pain (visual analogue scale, pressure algometry and McGill) and quality of life (Fibromyalgia Impact Questionnaire) before and after a 12-week therapy.
Results: After treatment, we observed significant differences in the experimental group (P < .05) on sleep quality (49.7%), anxiety (14.1%), impact of fibromyalgia (18.3%), pain (27.9%), McGill (23.7%) and tender points (34.4%).
Conclusions: Aquatic biodance contributed to improvements in sleep quality, anxiety, pain and other fibromyalgia symptoms.
In this paper, we explain how to generate adequate pre-fractals
in order to properly approximate ... more 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 significant hypothesis:
the so-called open set condition. The main theoretical result contributed in this paper becomes quite interesting from a computational point of view, since in such a context, there is always a maximum level (of the natural fractal structure we apply in this work) that may be achieved.
Hausdorff dimension, which is the oldest and also the most accurate model for fractal dimension, ... more 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 other hand, it turns out that
fractal structures provide a perfect context where a new definition of fractal dimension could be proposed. Further, it has been already shown that both Hausdorff and box dimensions can be generalized by some definitions of fractal dimension formulated in terms of fractal structures. Given this, and being mirrored in some of the properties satisfied by Hausdorff dimension, in this paper we explore which ones are satisfied by the fractal dimension definitions for a fractal structure, that are explored along this work.
The main goal in this paper was to provide a novel chaos indicator based on a topological model w... more 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 dimension specially designed for a parametrization of a curve or a random process, whose definition is made by means of fractal structures. As an application, we explore the use of this new concept of fractal dimension as a chaos indicator for dynamical systems, in a similar way to the classical maximal Lyapunov exponent. To illustrate it, we apply
the new fractal dimension as an indicator to model the
chaotic behavior of a satellite which is moving around a planet whose gravity field is approximated by the field of a point mass.
The main goal of this paper is to provide a generalized definition of fractal dimension for any s... more 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 the box-counting dimension becomes just a particular case. This idea allows to consider a wide range of fractal structures to calculate the effective fractal dimension for any subset of this space. Unlike it happens with the classical theory of fractal dimension, the new definitions we provide may be calculated in contexts where the box-counting one can have no sense or cannot be calculated. Nevertheless, the new models can be computed for any space admitting a fractal structure, just
as easy as the box-counting dimension in empirical applications.
In the present paper, we study regular and chaotic dynamics from planar oscillations of a dumbbel... more 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 small, and the parameter J2 is of the same order of magnitude as the eccentricity. This is carried out by applying the so-called Melnikov method. Finally, for arbitrarily chosen values for the parameters involved in such a problem, we
study the transition from regular to chaotic oscillations for a dumbbell satellite via the analysis of chaotic maps and Poincaré surfaces of section, respectively.
In this paper, we introduce a new approach which generalizes the GM2 algorithm (introduced in Sán... more In this paper, we introduce a new approach which generalizes the GM2 algorithm (introduced in Sánchez Granero et al. (2008) [52]) as well as fractal dimension algorithms (FD1, FD2 and FD3) (first appeared in Sánchez-Granero et al. (2012) [51]), providing an accurate algorithm to calculate the Hurst exponent of self-similar processes. We prove that this algorithm performs properly in the case of short time series when fractional Brownian motions and Lévy stable motions are considered.
We conclude the paper with a dynamic study of the Hurst exponent evolution in the S&P500 index stocks.
In this paper, three new algorithms are introduced in order to explore long memory in financial
t... more 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 short length series, the introduced algorithms perform much better
than the classical methods. Finally, an empirical application for some stock market indexes as well as some
individual stocks is presented.
Background and objective: To analyze the effects of an aquatic biodance based therapy on sleep qu... more 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 (stretching). The outcome measures were quality of sleep (Pittsburgh questionnaire), anxiety (State Anxiety Inventory), depression (Center for Epidemiologic Studies Depression Scale), pain (visual analogue scale, pressure algometry and McGill) and quality of life (Fibromyalgia Impact Questionnaire) before and after a 12-week therapy.
Results: After treatment, we observed significant differences in the experimental group (P < .05) on sleep quality (49.7%), anxiety (14.1%), impact of fibromyalgia (18.3%), pain (27.9%), McGill (23.7%) and tender points (34.4%).
Conclusions: Aquatic biodance contributed to improvements in sleep quality, anxiety, pain and other fibromyalgia symptoms.
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Papers by Manuel Fernández-Martínez
into a feature vector. Linear approaches have been widely applied to FE in BCI, whereas nonlinear tools are not so common in literature. Thus, the main goal of this paper is to check whether
some Hurst exponent and fractal dimension based estimators become valid indicators to FE in motor imagery BCI. The final results obtained were not optimal as expected, which may be due
to the fact that the nature of the analyzed EEG signals in these motor imagery tasks were not self-similar enough.
Patients and method: Randomized controlled trial with 2 groups. Fifty-nine patients were assigned to 2 groups: experimental group (aquatic biodance) and control group (stretching). The outcome measures were quality of sleep (Pittsburgh questionnaire), anxiety (State Anxiety Inventory), depression (Center for Epidemiologic Studies Depression Scale), pain (visual analogue scale, pressure algometry and McGill) and quality of life (Fibromyalgia Impact Questionnaire) before and after a 12-week therapy.
Results: After treatment, we observed significant differences in the experimental group (P < .05) on sleep quality (49.7%), anxiety (14.1%), impact of fibromyalgia (18.3%), pain (27.9%), McGill (23.7%) and tender points (34.4%).
Conclusions: Aquatic biodance contributed to improvements in sleep quality, anxiety, pain and other fibromyalgia symptoms.
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 significant hypothesis:
the so-called open set condition. The main theoretical result contributed in this paper becomes quite interesting from a computational point of view, since in such a context, there is always a maximum level (of the natural fractal structure we apply in this work) that may be achieved.
fractal structures provide a perfect context where a new definition of fractal dimension could be proposed. Further, it has been already shown that both Hausdorff and box dimensions can be generalized by some definitions of fractal dimension formulated in terms of fractal structures. Given this, and being mirrored in some of the properties satisfied by Hausdorff dimension, in this paper we explore which ones are satisfied by the fractal dimension definitions for a fractal structure, that are explored along this work.
of fractal dimension. In this paper, we provide an algorithm
to calculate a new fractal dimension specially designed for a parametrization of a curve or a random process, whose definition is made by means of fractal structures. As an application, we explore the use of this new concept of fractal dimension as a chaos indicator for dynamical systems, in a similar way to the classical maximal Lyapunov exponent. To illustrate it, we apply
the new fractal dimension as an indicator to model the
chaotic behavior of a satellite which is moving around a planet whose gravity field is approximated by the field of a point mass.
In this way, if we select the so-called natural fractal structure on any Euclidean space, then the box-counting dimension becomes just a particular case. This idea allows to consider a wide range of fractal structures to calculate the effective fractal dimension for any subset of this space. Unlike it happens with the classical theory of fractal dimension, the new definitions we provide may be calculated in contexts where the box-counting one can have no sense or cannot be calculated. Nevertheless, the new models can be computed for any space admitting a fractal structure, just
as easy as the box-counting dimension in empirical applications.
the existence of chaotic oscillations provided that the eccentricity becomes arbitrarily small, and the parameter J2 is of the same order of magnitude as the eccentricity. This is carried out by applying the so-called Melnikov method. Finally, for arbitrarily chosen values for the parameters involved in such a problem, we
study the transition from regular to chaotic oscillations for a dumbbell satellite via the analysis of chaotic maps and Poincaré surfaces of section, respectively.
We conclude the paper with a dynamic study of the Hurst exponent evolution in the S&P500 index stocks.
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 short length series, the introduced algorithms perform much better
than the classical methods. Finally, an empirical application for some stock market indexes as well as some
individual stocks is presented.
Patients and method: Randomized controlled trial with 2 groups. Fifty-nine patients were assigned to 2 groups: experimental group (aquatic biodance) and control group (stretching). The outcome measures were quality of sleep (Pittsburgh questionnaire), anxiety (State Anxiety Inventory), depression (Center for Epidemiologic Studies Depression Scale), pain (visual analogue scale, pressure algometry and McGill) and quality of life (Fibromyalgia Impact Questionnaire) before and after a 12-week therapy.
Results: After treatment, we observed significant differences in the experimental group (P < .05) on sleep quality (49.7%), anxiety (14.1%), impact of fibromyalgia (18.3%), pain (27.9%), McGill (23.7%) and tender points (34.4%).
Conclusions: Aquatic biodance contributed to improvements in sleep quality, anxiety, pain and other fibromyalgia symptoms.