Mathematics

23 pages, 908 KiB

Open AccessEditor’s ChoiceArticle

Some New Facts about the Unit-Rayleigh Distribution with Applications

by Rashad A. R. Bantan, Christophe Chesneau, Farrukh Jamal, Mohammed Elgarhy, Muhammad H. Tahir, Aqib Ali, Muhammad Zubair and Sania Anam

Mathematics 2020, 8(11), 1954; https://doi.org/10.3390/math8111954 - 04 Nov 2020

Cited by 30 | Viewed by 2846

Abstract

The unit-Rayleigh distribution is a one-parameter distribution with support on the unit interval. It is defined as the so-called unit-Weibull distribution with a shape parameter equal to two. As a particular case among others, it seems that it has not been given special [...] Read more.

The unit-Rayleigh distribution is a one-parameter distribution with support on the unit interval. It is defined as the so-called unit-Weibull distribution with a shape parameter equal to two. As a particular case among others, it seems that it has not been given special attention. This paper shows that the unit-Rayleigh distribution is much more interesting than it might at first glance, revealing closed-form expressions of important functions, and new desirable properties for application purposes. More precisely, on the theoretical level, we contribute to the following aspects: (i) we bring new characteristics on the form analysis of its main probabilistic and reliability functions, and show that the possible mode has a simple analytical expression, (ii) we prove new stochastic ordering results, (iii) we expose closed-form expressions of the incomplete and probability weighted moments at the basis of various probability functions and measures, (iv) we investigate distributional properties of the order statistics, (v) we show that the reliability coefficient can have a simple ratio expression, (vi) we provide a tractable expansion for the Tsallis entropy and (vii) we propose some bivariate unit-Rayleigh distributions. On a practical level, we show that the maximum likelihood estimate has a quite simple closed-form. Three data sets are analyzed and adjusted, revealing that the unit-Rayleigh distribution can be a better alternative to standard one-parameter unit distributions, such as the one-parameter Kumaraswamy, Topp–Leone, one-parameter beta, power and transmuted distributions. Full article

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18 pages, 352 KiB

Open AccessEditor’s ChoiceArticle

Existence of Positive Solutions for a System of Singular Fractional Boundary Value Problems with p-Laplacian Operators

by Ahmed Alsaedi, Rodica Luca and Bashir Ahmad

Mathematics 2020, 8(11), 1890; https://doi.org/10.3390/math8111890 - 31 Oct 2020

Cited by 13 | Viewed by 1736

Abstract

We investigate the existence and multiplicity of positive solutions for a system of Riemann–Liouville fractional differential equations with singular nonnegative nonlinearities and p-Laplacian operators, subject to nonlocal boundary conditions which contain fractional derivatives and Riemann–Stieltjes integrals. Full article

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions)

11 pages, 287 KiB

Open AccessEditor’s ChoiceArticle

Boundary Value Problems for Hilfer Fractional Differential Inclusions with Nonlocal Integral Boundary Conditions

by Athasit Wongcharoen, Sotiris K. Ntouyas and Jessada Tariboon

Mathematics 2020, 8(11), 1905; https://doi.org/10.3390/math8111905 - 31 Oct 2020

Cited by 19 | Viewed by 1449

Abstract

In this paper, we study boundary value problems for differential inclusions, involving Hilfer fractional derivatives and nonlocal integral boundary conditions. New existence results are obtained by using standard fixed point theorems for multivalued analysis. Examples illustrating our results are also presented. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)

34 pages, 1562 KiB

Open AccessEditor’s ChoiceArticle

On Semi-Analytical Solutions for Linearized Dispersive KdV Equations

by Appanah Rao Appadu and Abey Sherif Kelil

Mathematics 2020, 8(10), 1769; https://doi.org/10.3390/math8101769 - 14 Oct 2020

Cited by 14 | Viewed by 1960

Abstract

The most well-known equations both in the theory of nonlinearity and dispersion, KdV equations, have received tremendous attention over the years and have been used as model equations for the advancement of the theory of solitons. In this paper, some semi-analytic methods are [...] Read more.

The most well-known equations both in the theory of nonlinearity and dispersion, KdV equations, have received tremendous attention over the years and have been used as model equations for the advancement of the theory of solitons. In this paper, some semi-analytic methods are applied to solve linearized dispersive KdV equations with homogeneous and inhomogeneous source terms. These methods are the Laplace-Adomian decomposition method (LADM), Homotopy perturbation method (HPM), Bernstein-Laplace-Adomian Method (BALDM), and Reduced Differential Transform Method (RDTM). Three numerical experiments are considered. As the main contribution, we proposed a new scheme, known as BALDM, which involves Bernstein polynomials, Laplace transform and Adomian decomposition method to solve inhomogeneous linearized dispersive KdV equations. Besides, some modifications of HPM are also considered to solve certain inhomogeneous KdV equations by first constructing a newly modified homotopy on the source term and secondly by modifying Laplace’s transform with HPM to build HPTM. Both modifications of HPM numerically confirm the efficiency and validity of the methods for some test problems of dispersive KdV-like equations. We also applied LADM and RDTM to both homogeneous as well as inhomogeneous KdV equations to compare the obtained results and extended to higher dimensions. As a result, RDTM is applied to a 3D-dispersive KdV equation. The proposed iterative schemes determined the approximate solution without any discretization, linearization, or restrictive assumptions. The performance of the four methods is gauged over short and long propagation times and we compute absolute and relative errors at a given time for some spatial nodes. Full article

(This article belongs to the Special Issue Numerical Modeling and Analysis)

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17 pages, 2054 KiB

Open AccessEditor’s ChoiceArticle

Capital City as a Factor of Multi-Criteria Decision Analysis—Application on Transport Companies in the Czech Republic

by Roman Vavrek and Jiří Bečica

Mathematics 2020, 8(10), 1765; https://doi.org/10.3390/math8101765 - 13 Oct 2020

Cited by 17 | Viewed by 2111

Abstract

The manuscript applied multi-criteria analysis using several indicators to evaluate 18 transport companies established on the level of the Czech statutory towns during period of 2001–2016 that provided for a mass commuting system. Transport companies were chosen for evaluation in the towns being [...] Read more.

The manuscript applied multi-criteria analysis using several indicators to evaluate 18 transport companies established on the level of the Czech statutory towns during period of 2001–2016 that provided for a mass commuting system. Transport companies were chosen for evaluation in the towns being company establishers in the area of mass commuting systems. Based on the prepared analysis outcomes, we suppose that transport companies in big Czech cities and towns using combination of various transport means within the mass commuting system reached lower effectiveness. The Transport Company of the Czech capital city Prague only one operates subway, i.e., it works with specific requirements laid on assurance of this public transport type. Nevertheless, its inclusion in the analysis didn’t affect total results, thus we are able to work with a complete group of transport companies in the Czech Republic when evaluating their economic effectiveness. Full article

(This article belongs to the Special Issue Advances in Multiple Criteria Decision Analysis)

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19 pages, 2454 KiB

Open AccessEditor’s ChoiceArticle

A Comparative Performance Assessment of Ensemble Learning for Credit Scoring

by Yiheng Li and Weidong Chen

Mathematics 2020, 8(10), 1756; https://doi.org/10.3390/math8101756 - 13 Oct 2020

Cited by 64 | Viewed by 7652

Abstract

Extensive research has been performed by organizations and academics on models for credit scoring, an important financial management activity. With novel machine learning models continue to be proposed, ensemble learning has been introduced into the application of credit scoring, several researches have addressed [...] Read more.

Extensive research has been performed by organizations and academics on models for credit scoring, an important financial management activity. With novel machine learning models continue to be proposed, ensemble learning has been introduced into the application of credit scoring, several researches have addressed the supremacy of ensemble learning. In this research, we provide a comparative performance evaluation of ensemble algorithms, i.e., random forest, AdaBoost, XGBoost, LightGBM and Stacking, in terms of accuracy (ACC), area under the curve (AUC), Kolmogorov–Smirnov statistic (KS), Brier score (BS), and model operating time in terms of credit scoring. Moreover, five popular baseline classifiers, i.e., neural network (NN), decision tree (DT), logistic regression (LR), Naïve Bayes (NB), and support vector machine (SVM) are considered to be benchmarks. Experimental findings reveal that the performance of ensemble learning is better than individual learners, except for AdaBoost. In addition, random forest has the best performance in terms of five metrics, XGBoost and LightGBM are close challengers. Among five baseline classifiers, logistic regression outperforms the other classifiers over the most of evaluation metrics. Finally, this study also analyzes reasons for the poor performance of some algorithms and give some suggestions on the choice of credit scoring models for financial institutions. Full article

(This article belongs to the Special Issue Mathematical Analysis in Economics and Management)

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13 pages, 2192 KiB

Open AccessEditor’s ChoiceArticle

Modeling Soil Water Redistribution under Gravity Irrigation with the Richards Equation

by Sebastián Fuentes, Josué Trejo-Alonso, Antonio Quevedo, Carlos Fuentes and Carlos Chávez

Mathematics 2020, 8(9), 1581; https://doi.org/10.3390/math8091581 - 13 Sep 2020

Cited by 12 | Viewed by 3846

Abstract

Soil water movement is important in fields such as soil mechanics, irrigation, drainage, hydrology, and agriculture. The Richards equation describes the flow of water in an unsaturated porous medium, and analytical solutions exist only for simplified cases. However, numerous practical situations require a [...] Read more.

Soil water movement is important in fields such as soil mechanics, irrigation, drainage, hydrology, and agriculture. The Richards equation describes the flow of water in an unsaturated porous medium, and analytical solutions exist only for simplified cases. However, numerous practical situations require a numerical solution (1D, 2D, or 3D) depending on the complexity of the studied problem. In this paper, numerical solution of the equation describing water infiltration into soil using the finite difference method is studied. The finite difference solution is made via iterative schemes of local balance, including explicit, implicit, and intermediate methods; as a special case, the Laasonen method is shown. The found solution is applied to water transfer problems during and after gravity irrigation to observe phenomena of infiltration, evaporation, transpiration, and percolation. Full article

(This article belongs to the Special Issue Mathematical Modelling in Applied Sciences)

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20 pages, 2160 KiB

Open AccessEditor’s ChoiceArticle

A New Extended Two-Parameter Distribution: Properties, Estimation Methods, and Applications in Medicine and Geology

by Hazem Al-Mofleh, Ahmed Z. Afify and Noor Akma Ibrahim

Mathematics 2020, 8(9), 1578; https://doi.org/10.3390/math8091578 - 12 Sep 2020

Cited by 26 | Viewed by 3356

Abstract

In this paper, a new two-parameter generalized Ramos–Louzada distribution is proposed. The proposed model provides more flexibility in modeling data with increasing, decreasing, J-shaped, and reversed-J shaped hazard rate functions. Several statistical properties of the model were derived. The unknown parameters of the [...] Read more.

In this paper, a new two-parameter generalized Ramos–Louzada distribution is proposed. The proposed model provides more flexibility in modeling data with increasing, decreasing, J-shaped, and reversed-J shaped hazard rate functions. Several statistical properties of the model were derived. The unknown parameters of the new distribution were explored using eight frequentist estimation approaches. These approaches are important for developing guidelines to choose the best method of estimation for the model parameters, which would be of great interest to practitioners and applied statisticians. Detailed numerical simulations are presented to examine the bias and the mean square error of the proposed estimators. The best estimation method and ordering performance of the estimators were determined using the partial and overall ranks of all estimation methods for various parameter combinations. The performance of the proposed distribution is illustrated using two real datasets from the fields of medicine and geology, and both datasets show that the new model is more appropriate as compared to the Marshall–Olkin exponential, exponentiated exponential, beta exponential, gamma, Poisson–Lomax, Lindley geometric, generalized Lindley, and Lindley distributions, among others. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications)

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14 pages, 627 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Do Science, Technology, Engineering and Mathematics (STEM) Experimentation Outreach Programs Affect Attitudes towards Mathematics and Science? A Quasi-Experiment in Primary Education

by Raquel Fernández-Cézar, Dunia Garrido and Natalia Solano-Pinto

Mathematics 2020, 8(9), 1490; https://doi.org/10.3390/math8091490 - 03 Sep 2020

Cited by 12 | Viewed by 4324

Abstract

Science, technology, engineering and mathematics (STEM) outreach programs have been widely studied in recent years considering their possible influence on future STEM career election aiming to counteract the observed decline in enrollment at university. Nonetheless, the presumed effect is not clear due to [...] Read more.

Science, technology, engineering and mathematics (STEM) outreach programs have been widely studied in recent years considering their possible influence on future STEM career election aiming to counteract the observed decline in enrollment at university. Nonetheless, the presumed effect is not clear due to a lack of comparison with control groups. In order to fill this gap, a quasi-experimental design was adopted to analyze the effect of a STEM experimentation outreach program on 5th and 6th graders. The sample was composed by 453 students, (274 experimental group and 179 control group). The Auzmendi Scale of Attitude towards Mathematics Modified (ASMAm), and the attitude towards school science (ASSci), were used as instruments, and were administered before and after the intervention. The analysis was run with sex, type of school (state and state-funded schools), school environment (rural/urban), and teacher as potential factors. The results show that there is a program effect on the attitude towards mathematics, but not on the attitude towards school science. Regarding the factors, the program effect is associated neither with sex nor with rural/urban schools. However, the program had a more positive effect on the ASSci than on the ASMAm in the state schools, and is mediated by the teacher. Full article

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16 pages, 868 KiB

Open AccessEditor’s ChoiceArticle

Comparison of Ensemble Machine Learning Methods for Automated Classification of Focal and Non-Focal Epileptic EEG Signals

by Samed Jukic, Muzafer Saracevic, Abdulhamit Subasi and Jasmin Kevric

Mathematics 2020, 8(9), 1481; https://doi.org/10.3390/math8091481 - 02 Sep 2020

Cited by 41 | Viewed by 3512

Abstract

This research presents the epileptic focus region localization during epileptic seizures by applying different signal processing and ensemble machine learning techniques in intracranial recordings of electroencephalogram (EEG). Multi-scale Principal Component Analysis (MSPCA) is used for denoising EEG signals and the autoregressive (AR) algorithm [...] Read more.

This research presents the epileptic focus region localization during epileptic seizures by applying different signal processing and ensemble machine learning techniques in intracranial recordings of electroencephalogram (EEG). Multi-scale Principal Component Analysis (MSPCA) is used for denoising EEG signals and the autoregressive (AR) algorithm will extract useful features from the EEG signal. The performances of the ensemble machine learning methods are measured with accuracy, F-measure, and the area under the receiver operating characteristic (ROC) curve (AUC). EEG-based focus area localization with the proposed methods reaches 98.9% accuracy using the Rotation Forest classifier. Therefore, our results suggest that ensemble machine learning methods can be applied to differentiate the EEG signals from epileptogenic brain areas and signals recorded from non-epileptogenic brain regions with high accuracy. Full article

(This article belongs to the Special Issue Bioinspired Computation: Recent Advances in Theory and Applications)

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36 pages, 12863 KiB

Open AccessEditor’s ChoiceReview

Fourier-Spectral Method for the Phase-Field Equations

by Sungha Yoon, Darae Jeong, Chaeyoung Lee, Hyundong Kim, Sangkwon Kim, Hyun Geun Lee and Junseok Kim

Mathematics 2020, 8(8), 1385; https://doi.org/10.3390/math8081385 - 18 Aug 2020

Cited by 22 | Viewed by 6323

Abstract

In this paper, we review the Fourier-spectral method for some phase-field models: Allen–Cahn (AC), Cahn–Hilliard (CH), Swift–Hohenberg (SH), phase-field crystal (PFC), and molecular beam epitaxy (MBE) growth. These equations are very important parabolic partial differential equations and are applicable to many interesting scientific [...] Read more.

In this paper, we review the Fourier-spectral method for some phase-field models: Allen–Cahn (AC), Cahn–Hilliard (CH), Swift–Hohenberg (SH), phase-field crystal (PFC), and molecular beam epitaxy (MBE) growth. These equations are very important parabolic partial differential equations and are applicable to many interesting scientific problems. The AC equation is a reaction-diffusion equation modeling anti-phase domain coarsening dynamics. The CH equation models phase segregation of binary mixtures. The SH equation is a popular model for generating patterns in spatially extended dissipative systems. A classical PFC model is originally derived to investigate the dynamics of atomic-scale crystal growth. An isotropic symmetry MBE growth model is originally devised as a method for directly growing high purity epitaxial thin film of molecular beams evaporating on a heated substrate. The Fourier-spectral method is highly accurate and simple to implement. We present a detailed description of the method and explain its connection to MATLAB usage so that the interested readers can use the Fourier-spectral method for their research needs without difficulties. Several standard computational tests are done to demonstrate the performance of the method. Furthermore, we provide the MATLAB codes implementation in the Appendix A. Full article

(This article belongs to the Special Issue Open Source Codes for Numerical Analysis)

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22 pages, 3836 KiB

Open AccessEditor’s ChoiceArticle

AHP-TOPSIS Inspired Shopping Mall Site Selection Problem with Fuzzy Data

by Neha Ghorui, Arijit Ghosh, Ebrahem A. Algehyne, Sankar Prasad Mondal and Apu Kumar Saha

Mathematics 2020, 8(8), 1380; https://doi.org/10.3390/math8081380 - 17 Aug 2020

Cited by 41 | Viewed by 5158

Abstract

In the consumerist world, there is an ever-increasing demand for consumption in urban life. Thus, the demand for shopping malls is growing. For a developer, site selection is an important issue as the optimal selection involves several complex factors and sub-factors for a [...] Read more.

In the consumerist world, there is an ever-increasing demand for consumption in urban life. Thus, the demand for shopping malls is growing. For a developer, site selection is an important issue as the optimal selection involves several complex factors and sub-factors for a successful investment venture. Thus, these tangible and intangible factors can be best solved by the Multi Criteria Decision Making (MCDM) models. In this study, optimal site selection has been done out of multiple alternative locations in and around the city of Kolkata, West Bengal, India. The Fuzzy Analytic Hierarchy Process (FAHP) and Fuzzy Technique for Order of Preference by Similarity to Ideal Solution (FTOPSIS) has been applied for shopping mall site selection. The AHP is used to obtain the crispified weight of factors. Imprecise linguistic terms used by the decision-maker are converted to Triangular Fuzzy Numbers (TFNs). This research used integrated sub-factors fuzzy weights using FAHP to FTOPSIS for ranking of the alternatives. Hardly any research is done with the use of sub-factors. In this study, seven factors and seventeen sub-factors are considered, the authors collected data from different locations with the help of municipal authorities and architects. This work further provides useful guidelines for shopping mall selection in different states and countries. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)

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29 pages, 2895 KiB

Open AccessEditor’s ChoiceArticle

Waste Segregation FMEA Model Integrating Intuitionistic Fuzzy Set and the PAPRIKA Method

by María Carmen Carnero

Mathematics 2020, 8(8), 1375; https://doi.org/10.3390/math8081375 - 17 Aug 2020

Cited by 25 | Viewed by 5551

Abstract

Segregation is an important step in health care waste management. If done incorrectly, the risk of preventable infections, toxic effects, and injuries to care and non-care staff, waste handlers, patients, visitors, and the community at large, is increased. It also increases the risk [...] Read more.

Segregation is an important step in health care waste management. If done incorrectly, the risk of preventable infections, toxic effects, and injuries to care and non-care staff, waste handlers, patients, visitors, and the community at large, is increased. It also increases the risk of environmental pollution and prevents recyclable waste from being recovered. Despite its importance, it is acknowledged that poor waste segregation occurs in most health care organizations. This study therefore intends to produce, for the first time, a classification of failure modes related to segregation in the Nuclear Medicine Department of a health care organization. This will be done using Failure Mode and Effects Analysis (FMEA), by combining an intuitionistic fuzzy hybrid weighted Euclidean distance operator, and the multicriteria method Potentially All Pairwise RanKings of all possible Alternatives (PAPRIKA). Subjective and objective weights of risk factors were considered simultaneously. The failure modes identified in the top three positions are: improper storage of waste (placing items in the wrong bins), improper labeling of containers, and bad waste management (inappropriate collection periods and bin set-up). Full article

(This article belongs to the Special Issue Applications of Fuzzy Optimization and Fuzzy Decision Making)

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18 pages, 1553 KiB

Open AccessEditor’s ChoiceArticle

Analyzing the Impact of the Renewable Energy Sources on Economic Growth at the EU Level Using an ARDL Model

by Mihail Busu

Mathematics 2020, 8(8), 1367; https://doi.org/10.3390/math8081367 - 14 Aug 2020

Cited by 31 | Viewed by 4903

Abstract

Energy is one of the most important drivers of economic growth, but as the population is increasing, in normal circumstances, in all countries of the world, there is a demand for energy produced from conventional resources. Increasing prices of conventional energy and the [...] Read more.

Energy is one of the most important drivers of economic growth, but as the population is increasing, in normal circumstances, in all countries of the world, there is a demand for energy produced from conventional resources. Increasing prices of conventional energy and the negative impact on the environment are two of the main reasons for switching to renewable energy sources (RESs). The aim of the paper is to quantify the impact of the RESs, by type, on the sustainable economic growth at the European Union (EU) level. The research was performed for all 28 EU member states, for a time frame from 2004 to 2017, through a panel autoregressive distributed lag (ARDL) approach and causality analysis. Furthermore, Hausman test was performed on the regression model. By estimating the panel data regression model with random effects, we reveal through our results that RESs, namely wind, solar, biomass, geothermal, and hydropower energy, have a positive influence on economic growth at EU level. Moreover, biomass has the highest impact on economic growth among all RES. In fact, a 1% increase in biomass primary production would impact the economic growth by 0.15%. Based on econometric analysis, our findings suggest that public policies at the EU level should be focused on investment in RESs. Full article

(This article belongs to the Special Issue Modeling and Numerical Analysis of Energy and Environment)

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18 pages, 445 KiB

Open AccessEditor’s ChoiceArticle

Robust Three-Step Regression Based on Comedian and Its Performance in Cell-Wise and Case-Wise Outliers

by Henry Velasco, Henry Laniado, Mauricio Toro, Víctor Leiva and Yuhlong Lio

Mathematics 2020, 8(8), 1259; https://doi.org/10.3390/math8081259 - 01 Aug 2020

Cited by 18 | Viewed by 3528

Abstract

Both cell-wise and case-wise outliers may appear in a real data set at the same time. Few methods have been developed in order to deal with both types of outliers when formulating a regression model. In this work, a robust estimator is proposed [...] Read more.

Both cell-wise and case-wise outliers may appear in a real data set at the same time. Few methods have been developed in order to deal with both types of outliers when formulating a regression model. In this work, a robust estimator is proposed based on a three-step method named 3S-regression, which uses the comedian as a highly robust scatter estimate. An intensive simulation study is conducted in order to evaluate the performance of the proposed comedian 3S-regression estimator in the presence of cell-wise and case-wise outliers. In addition, a comparison of this estimator with recently developed robust methods is carried out. The proposed method is also extended to the model with continuous and dummy covariates. Finally, a real data set is analyzed for illustration in order to show potential applications. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation)

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15 pages, 637 KiB

Open AccessEditor’s ChoiceArticle

A Novel Methodology to Calculate the Probability of Volatility Clusters in Financial Series: An Application to Cryptocurrency Markets

by Venelina Nikolova, Juan E. Trinidad Segovia, Manuel Fernández-Martínez and Miguel Angel Sánchez-Granero

Mathematics 2020, 8(8), 1216; https://doi.org/10.3390/math8081216 - 24 Jul 2020

Cited by 14 | Viewed by 3448

Abstract

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 [...] Read more.

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 emphasis on cryptocurrencies. With this aim, we calculate the Hurst exponent of a volatility series by means of the FD4 approach. An explicit criterion to computationally determine whether there exist volatility clusters of a fixed size is described. We found that the probabilities of volatility clusters of an index (S&P500) and a stock (Apple) showed a similar profile, whereas the probability of volatility clusters of a forex pair (Euro/USD) became quite lower. On the other hand, a similar profile appeared for Bitcoin/USD, Ethereum/USD, and Ripple/USD cryptocurrencies, with the probabilities of volatility clusters of all such cryptocurrencies being much greater than the ones of the three traditional assets. Our results suggest that the volatility in cryptocurrencies changes faster than in traditional assets, and much faster than in forex pairs. Full article

(This article belongs to the Special Issue Quantitative Methods for Economics and Finance)

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16 pages, 443 KiB

Open AccessEditor’s ChoiceArticle

A Mathematical Model of Epidemics—A Tutorial for Students

by Yutaka Okabe and Akira Shudo

Mathematics 2020, 8(7), 1174; https://doi.org/10.3390/math8071174 - 17 Jul 2020

Cited by 16 | Viewed by 27099

Abstract

This is a tutorial for the mathematical model of the spread of epidemic diseases. Beginning with the basic mathematics, we introduce the susceptible-infected-recovered (SIR) model. Subsequently, we present the numerical and exact analytical solutions of the SIR model. The analytical solution is emphasized. [...] Read more.

This is a tutorial for the mathematical model of the spread of epidemic diseases. Beginning with the basic mathematics, we introduce the susceptible-infected-recovered (SIR) model. Subsequently, we present the numerical and exact analytical solutions of the SIR model. The analytical solution is emphasized. Additionally, we treat the generalization of the SIR model including births and natural deaths. Full article

(This article belongs to the Special Issue Mathematical Models in Epidemiology )

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26 pages, 3025 KiB

Open AccessEditor’s ChoiceArticle

Supportiveness of Low-Carbon Energy Technology Policy Using Fuzzy Multicriteria Decision-Making Methodologies

by Konstantinos Kokkinos and Vayos Karayannis

Mathematics 2020, 8(7), 1178; https://doi.org/10.3390/math8071178 - 17 Jul 2020

Cited by 18 | Viewed by 4514

Abstract

The deployment of low-carbon energy (LCE) technologies and management of installations represents an imperative to face climate change. LCE planning is an interminable process affected by a multitude of social, economic, environmental, and health factors. A major challenge for policy makers is to [...] Read more.

The deployment of low-carbon energy (LCE) technologies and management of installations represents an imperative to face climate change. LCE planning is an interminable process affected by a multitude of social, economic, environmental, and health factors. A major challenge for policy makers is to select a future clean energy strategy that maximizes sustainability. Thus, policy formulation and evaluation need to be addressed in an analytical manner including multidisciplinary knowledge emanating from diverse social stakeholders. In the current work, a comparative analysis of LCE planning is provided, evaluating different multicriteria decision-making (MCDM) methodologies. Initially, by applying strengths, weaknesses, opportunities, and threats (SWOT) analysis, the available energy alternative technologies are prioritized. A variety of stakeholders is surveyed for that reason. To deal with the ambiguity that occurred in their judgements, fuzzy goal programming (FGP) is used for the translation into fuzzy numbers. Then, the stochastic fuzzy analytic hierarchical process (SF-AHP) and fuzzy technique for order performance by similarity to ideal solution (F-TOPSIS) are applied to evaluate a repertoire of energy alternative forms including biofuel, solar, hydro, and wind power. The methodologies are estimated based on the same set of tangible and intangible criteria for the case study of Thessaly Region, Greece. The application of FGP ranked the four energy types in terms of feasibility and positioned solar-generated energy as first, with a membership function of 0.99. Among the criteria repertoire used by the stakeholders, the SF-AHP evaluated all the criteria categories separately and selected the most significant category representative. Finally, F-TOPSIS assessed these criteria ordering the energy forms, in terms of descending order of ideal solution, as follows: solar, biofuel, hydro, and wind. Full article

(This article belongs to the Special Issue Applications of Fuzzy Optimization and Fuzzy Decision Making)

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17 pages, 1638 KiB

Open AccessEditor’s ChoiceArticle

Birnbaum-Saunders Quantile Regression Models with Application to Spatial Data

by Luis Sánchez, Víctor Leiva, Manuel Galea and Helton Saulo

Mathematics 2020, 8(6), 1000; https://doi.org/10.3390/math8061000 - 18 Jun 2020

Cited by 32 | Viewed by 2865

Abstract

In the present paper, a novel spatial quantile regression model based on the Birnbaum–Saunders distribution is formulated. This distribution has been widely studied and applied in many fields. To formulate such a spatial model, a parameterization of the multivariate Birnbaum–Saunders distribution, where one [...] Read more.

In the present paper, a novel spatial quantile regression model based on the Birnbaum–Saunders distribution is formulated. This distribution has been widely studied and applied in many fields. To formulate such a spatial model, a parameterization of the multivariate Birnbaum–Saunders distribution, where one of its parameters is associated with the quantile of the respective marginal distribution, is established. The model parameters are estimated by the maximum likelihood method. Finally, a data set is applied for illustrating the formulated model. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation)

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11 pages, 282 KiB

Open AccessEditor’s ChoiceArticle

A Regularity Criterion in Weak Spaces to Boussinesq Equations

by Ravi P. Agarwal, Sadek Gala and Maria Alessandra Ragusa

Mathematics 2020, 8(6), 920; https://doi.org/10.3390/math8060920 - 05 Jun 2020

Cited by 73 | Viewed by 3968

Abstract

In this paper, we study the regularity of weak solutions to the incompressible Boussinesq equations in

R^{3} \times (0, T)

. The main goal is to establish the regularity criterion in terms of one velocity component and the gradient [...] Read more.

In this paper, we study the regularity of weak solutions to the incompressible Boussinesq equations in

R^{3} \times (0, T)

. The main goal is to establish the regularity criterion in terms of one velocity component and the gradient of temperature in Lorentz spaces. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)

20 pages, 3589 KiB

Open AccessEditor’s ChoiceArticle

COVID-19 Pandemic Prediction for Hungary; A Hybrid Machine Learning Approach

by Gergo Pinter, Imre Felde, Amir Mosavi, Pedram Ghamisi and Richard Gloaguen

Mathematics 2020, 8(6), 890; https://doi.org/10.3390/math8060890 - 02 Jun 2020

Cited by 210 | Viewed by 19819

Abstract

Several epidemiological models are being used around the world to project the number of infected individuals and the mortality rates of the COVID-19 outbreak. Advancing accurate prediction models is of utmost importance to take proper actions. Due to the lack of essential data [...] Read more.

Several epidemiological models are being used around the world to project the number of infected individuals and the mortality rates of the COVID-19 outbreak. Advancing accurate prediction models is of utmost importance to take proper actions. Due to the lack of essential data and uncertainty, the epidemiological models have been challenged regarding the delivery of higher accuracy for long-term prediction. As an alternative to the susceptible-infected-resistant (SIR)-based models, this study proposes a hybrid machine learning approach to predict the COVID-19, and we exemplify its potential using data from Hungary. The hybrid machine learning methods of adaptive network-based fuzzy inference system (ANFIS) and multi-layered perceptron-imperialist competitive algorithm (MLP-ICA) are proposed to predict time series of infected individuals and mortality rate. The models predict that by late May, the outbreak and the total morality will drop substantially. The validation is performed for 9 days with promising results, which confirms the model accuracy. It is expected that the model maintains its accuracy as long as no significant interruption occurs. This paper provides an initial benchmarking to demonstrate the potential of machine learning for future research. Full article

(This article belongs to the Section Mathematics and Computer Science)

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18 pages, 4078 KiB

Open AccessEditor’s ChoiceArticle

Improving the Accuracy of Convolutional Neural Networks by Identifying and Removing Outlier Images in Datasets Using t-SNE

by Husein Perez and Joseph H. M. Tah

Mathematics 2020, 8(5), 662; https://doi.org/10.3390/math8050662 - 27 Apr 2020

Cited by 42 | Viewed by 6557

Abstract

In the field of supervised machine learning, the quality of a classifier model is directly correlated with the quality of the data that is used to train the model. The presence of unwanted outliers in the data could significantly reduce the accuracy of [...] Read more.

In the field of supervised machine learning, the quality of a classifier model is directly correlated with the quality of the data that is used to train the model. The presence of unwanted outliers in the data could significantly reduce the accuracy of a model or, even worse, result in a biased model leading to an inaccurate classification. Identifying the presence of outliers and eliminating them is, therefore, crucial for building good quality training datasets. Pre-processing procedures for dealing with missing and outlier data, commonly known as feature engineering, are standard practice in machine learning problems. They help to make better assumptions about the data and also prepare datasets in a way that best expose the underlying problem to the machine learning algorithms. In this work, we propose a multistage method for detecting and removing outliers in high-dimensional data. Our proposed method is based on utilising a technique called t-distributed stochastic neighbour embedding (t-SNE) to reduce high-dimensional map of features into a lower, two-dimensional, probability density distribution and then use a simple descriptive statistical method called interquartile range (IQR) to identifying any outlier values from the density distribution of the features. t-SNE is a machine learning algorithm and a nonlinear dimensionality reduction technique well-suited for embedding high-dimensional data for visualisation in a low-dimensional space of two or three dimensions. We applied this method on a dataset containing images for training a convolutional neural network model (ConvNet) for an image classification problem. The dataset contains four different classes of images: three classes contain defects in construction (mould, stain, and paint deterioration) and a no-defect class (normal). We used the transfer learning technique to modify a pre-trained VGG-16 model. We used this model as a feature extractor and as a benchmark to evaluate our method. We have shown that, when using this method, we can identify and remove the outlier images in the dataset. After removing the outlier images from the dataset and re-training the VGG-16 model, the results have also shown that the accuracy of the classification has significantly improved and the number of misclassified cases has also dropped. While many feature engineering techniques for handling missing and outlier data are common in predictive machine learning problems involving numerical or categorical data, there is little work on developing techniques for handling outliers in high-dimensional data which can be used to improve the quality of machine learning problems involving images such as ConvNet models for image classification and object detection problems. Full article

(This article belongs to the Special Issue Advances in Machine Learning Prediction Models)

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9 pages, 1916 KiB

Open AccessEditor’s ChoiceArticle

On Estimating the Number of Deaths Related to Covid-19

by Hoang Pham

Mathematics 2020, 8(5), 655; https://doi.org/10.3390/math8050655 - 26 Apr 2020

Cited by 30 | Viewed by 10622

Abstract

In this paper, we discuss an explicit model function that can estimate the total number of deaths in the population, and particularly, estimate the cumulative number of deaths in the United States due to the current Covid-19 virus. We compare the modeling results [...] Read more.

In this paper, we discuss an explicit model function that can estimate the total number of deaths in the population, and particularly, estimate the cumulative number of deaths in the United States due to the current Covid-19 virus. We compare the modeling results to two related existing models based on a new criteria and several existing criteria for model selection. The results show the proposed model fits significantly better than the other two related models based on the U.S. Covid-19 death data. We observe that the errors of the fitted data and the predicted data points on the total number of deaths in the U.S. on the last available data point and the next coming day are less than 0.5% and 2.0%, respectively. The results show very encouraging predictability for the model. The new model predicts that the maximum total number of deaths will be approximately 62,100 across the United States due to the Covid-19 virus, and with a 95% confidence that the expected total death toll will be between 60,951 and 63,249 deaths based on the data until 22 April, 2020. If there is a significant change in the coming days due to various testing strategies, social-distancing policies, the reopening of community strategies, or a stay-home policy, the predicted death tolls will definitely change. Future work can be explored further to apply the proposed model to global Covid-19 death data and to other applications, including human population mortality, the spread of disease, and different topics such as movie reviews in recommender systems. Full article

(This article belongs to the Special Issue Reliability and Statistical Learning and Its Applications)

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7 pages, 249 KiB

Open AccessEditor’s ChoiceArticle

Fibonacci Numbers with a Prescribed Block of Digits

by Pavel Trojovský

Mathematics 2020, 8(4), 639; https://doi.org/10.3390/math8040639 - 21 Apr 2020

Cited by 17 | Viewed by 3014

Abstract

In this paper, we prove that

F_{22} = 17711

is the largest Fibonacci number whose decimal expansion is of the form

a b \dots b c \dots c

. The proof uses lower bounds for linear forms in three logarithms of algebraic [...] Read more.

In this paper, we prove that

F_{22} = 17711

is the largest Fibonacci number whose decimal expansion is of the form

a b \dots b c \dots c

. The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation. Full article

(This article belongs to the Section Mathematics and Computer Science)

32 pages, 3229 KiB

Open AccessEditor’s ChoiceArticle

Global Research Trends in Financial Transactions

by Emilio Abad-Segura and Mariana-Daniela González-Zamar

Mathematics 2020, 8(4), 614; https://doi.org/10.3390/math8040614 - 16 Apr 2020

Cited by 28 | Viewed by 6957

Abstract

Traditionally, financial mathematics has been used to solve financial problems. With globalization, financial transactions require new analysis based on tools of probability, statistics, and economic theory. Global research trends in this topic during the period 1935–2019 have been analyzed. With this objective, a [...] Read more.

Traditionally, financial mathematics has been used to solve financial problems. With globalization, financial transactions require new analysis based on tools of probability, statistics, and economic theory. Global research trends in this topic during the period 1935–2019 have been analyzed. With this objective, a bibliometric methodology of 1486 articles from the Scopus database was applied. The obtained results offer data on the scientific activity of countries, institutions, authors, and institutions that promote this research topic. The results reveal an increasing trend, mainly in the last decade. The main subjects of knowledge are social sciences and economics, econometrics, and finance. The author with the most articles is Khare from the Indian Institute of Management Rohtak. The most prolific affiliation is the British University of Oxford. The country with the most academic publications and international collaborations is the United States. In addition, the most used keywords in articles are “financial management”, “financial transaction tax”, “banking”, “financial service”, “blockchain”, “decision making”, and “financial market”. The increase in publications in recent years at the international level confirms the growing trend in research on financial transactions. Full article

(This article belongs to the Special Issue Financial Mathematics)

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34 pages, 1131 KiB

Open AccessEditor’s ChoiceArticle

Power Aggregation Operators and VIKOR Methods for Complex q-Rung Orthopair Fuzzy Sets and Their Applications

by Harish Garg, Jeonghwan Gwak, Tahir Mahmood and Zeeshan Ali

Mathematics 2020, 8(4), 538; https://doi.org/10.3390/math8040538 - 05 Apr 2020

Cited by 73 | Viewed by 3143

Abstract

The aim of this paper is to present the novel concept of Complex q-rung orthopair fuzzy set (Cq-ROFS) which is a useful tool to cope with unresolved and complicated information. It is characterized by a complex-valued membership grade and a complex-valued non-membership grade, [...] Read more.

The aim of this paper is to present the novel concept of Complex q-rung orthopair fuzzy set (Cq-ROFS) which is a useful tool to cope with unresolved and complicated information. It is characterized by a complex-valued membership grade and a complex-valued non-membership grade, the distinction of which is that the sum of q-powers of the real parts (imaginary parts) of the membership and non-membership grades is less than or equal to one. To explore the study, we present some basic operational laws, score and accuracy functions and investigate their properties. Further, to aggregate the given information of Cq-ROFS, we present several weighted averaging and geometric power aggregation operators named as complex q-rung orthopair fuzzy (Cq-ROF) power averaging operator, Cq-ROF power geometric operator, Cq-ROF power weighted averaging operator, Cq-ROF power weighted geometric operator, Cq-ROF hybrid averaging operator and Cq-ROF power hybrid geometric operator. Properties and special cases of the proposed approaches are discussed in detail. Moreover, the VIKOR (“VIseKriterijumska Optimizacija I Kompromisno Resenje”) method for Cq-ROFSs is introduced and its aspects discussed. Furthermore, the above mentioned approaches apply to multi-attribute decision-making problems and VIKOR methods, in which experts state their preferences in the Cq-ROF environment to demonstrate the feasibility, reliability and effectiveness of the proposed approaches. Finally, the proposed approach is compared with existing methods through numerical examples. Full article

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12 pages, 262 KiB

Open AccessEditor’s ChoiceArticle

Some Results in Green–Lindsay Thermoelasticity of Bodies with Dipolar Structure

by Marin Marin, Eduard M. Craciun and Nicolae Pop

Mathematics 2020, 8(4), 497; https://doi.org/10.3390/math8040497 - 02 Apr 2020

Cited by 27 | Viewed by 2305

Abstract

The main concern of this study is an extension of some results, proposed by Green and Lindsay in the classical theory of elasticity, in order to cover the theory of thermoelasticity for dipolar bodies. For dynamical mixed problem we prove a reciprocal theorem, [...] Read more.

The main concern of this study is an extension of some results, proposed by Green and Lindsay in the classical theory of elasticity, in order to cover the theory of thermoelasticity for dipolar bodies. For dynamical mixed problem we prove a reciprocal theorem, in the general case of an anisotropic thermoelastic body. Furthermore, in this general context we have proven a result regarding the uniqueness of the solution of the mixed problem in the dynamical case. We must emphasize that these fundamental results are obtained under conditions that are not very restrictive. Full article

(This article belongs to the Special Issue Applied Mathematics and Solid Mechanics)

18 pages, 316 KiB

Open AccessEditor’s ChoiceArticle

Existence of Weak Solutions for a New Class of Fractional p-Laplacian Boundary Value Systems

by Fares Kamache, Rafik Guefaifia, Salah Boulaaras and Asma Alharbi

Mathematics 2020, 8(4), 475; https://doi.org/10.3390/math8040475 - 31 Mar 2020

Cited by 16 | Viewed by 2155

Abstract

In this paper, at least three weak solutions were obtained for a new class of dual non-linear dual-Laplace systems according to two parameters by using variational methods combined with a critical point theory due to Bonano and Marano. Two examples are given to [...] Read more.

In this paper, at least three weak solutions were obtained for a new class of dual non-linear dual-Laplace systems according to two parameters by using variational methods combined with a critical point theory due to Bonano and Marano. Two examples are given to illustrate our main results applications. Full article

18 pages, 2418 KiB

Open AccessEditor’s ChoiceArticle

A New Fuzzy MARCOS Method for Road Traffic Risk Analysis

by Miomir Stanković, Željko Stević, Dillip Kumar Das, Marko Subotić and Dragan Pamučar

Mathematics 2020, 8(3), 457; https://doi.org/10.3390/math8030457 - 24 Mar 2020

Cited by 193 | Viewed by 9953

Abstract

In this paper, a new fuzzy multi-criteria decision-making model for traffic risk assessment was developed. A part of a main road network of 7.4 km with a total of 38 Sections was analyzed with the aim of determining the degree of risk on [...] Read more.

In this paper, a new fuzzy multi-criteria decision-making model for traffic risk assessment was developed. A part of a main road network of 7.4 km with a total of 38 Sections was analyzed with the aim of determining the degree of risk on them. For that purpose, a fuzzy Measurement Alternatives and Ranking according to the COmpromise Solution (fuzzy MARCOS) method was developed. In addition, a new fuzzy linguistic scale quantified into triangular fuzzy numbers (TFNs) was developed. The fuzzy PIvot Pairwise RElative Criteria Importance Assessment—fuzzy PIPRECIA method—was used to determine the criteria weights on the basis of which the road network sections were evaluated. The results clearly show that there is a dominant section with the highest risk for all road participants, which requires corrective actions. In order to validate the results, a comprehensive validity test was created consisting of variations in the significance of model input parameters, testing the influence of dynamic factors—of reverse rank, and applying the fuzzy Simple Additive Weighing (fuzzy SAW) method and the fuzzy Technique for Order of Preference by Similarity to Ideal Solution (fuzzy TOPSIS). The validation test show the stability of the results obtained and the justification for the development of the proposed model. Full article

(This article belongs to the Special Issue Dynamics under Uncertainty: Modeling Simulation and Complexity)

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20 pages, 1206 KiB

Open AccessEditor’s ChoiceArticle

An Integrated Approach of Best-Worst Method (BWM) and Triangular Fuzzy Sets for Evaluating Driver Behavior Factors Related to Road Safety

by Sarbast Moslem, Muhammet Gul, Danish Farooq, Erkan Celik, Omid Ghorbanzadeh and Thomas Blaschke

Mathematics 2020, 8(3), 414; https://doi.org/10.3390/math8030414 - 13 Mar 2020

Cited by 66 | Viewed by 6530

Abstract

Driver behavior plays a major role in road safety because it is considered as a significant argument in traffic accident avoidance. Drivers mostly face various risky driving factors which lead to fatal accidents or serious injury. This study aims to evaluate and prioritize [...] Read more.

Driver behavior plays a major role in road safety because it is considered as a significant argument in traffic accident avoidance. Drivers mostly face various risky driving factors which lead to fatal accidents or serious injury. This study aims to evaluate and prioritize the significant driver behavior factors related to road safety. In this regard, we integrated a decision-making model of the Best-Worst Method (BWM) with the triangular fuzzy sets as a solution for optimizing our complex decision-making problem, which is associated with uncertainty and ambiguity. Driving characteristics are different in different driving situations which indicate the ambiguous and complex attitude of individuals, and decision-makers (DMs) need to improve the reliability of the decision. Since the crisp values of factors may be inadequate to model the real-world problem considering the vagueness and the ambiguity, and providing the pairwise comparisons with the requirement of less compared data, the BWM integrated with triangular fuzzy sets is used in the study to evaluate risky driver behavior factors for a designed three-level hierarchical structure. The model results provide the most significant driver behavior factors that influence road safety for each level based on evaluator responses on the Driver Behavior Questionnaire (DBQ). Moreover, the model generates a more consistent decision process by the new consistency ratio of F-BWM. An adaptable application process from the model is also generated for future attempts. Full article

(This article belongs to the Special Issue Applications of Fuzzy Optimization and Fuzzy Decision Making)

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14 pages, 2085 KiB

Open AccessEditor’s ChoiceArticle

Robust Design Optimization for Low-Cost Concrete Box-Girder Bridge

by Vicent Penadés-Plà, Tatiana García-Segura and Víctor Yepes

Mathematics 2020, 8(3), 398; https://doi.org/10.3390/math8030398 - 11 Mar 2020

Cited by 24 | Viewed by 5183

Abstract

The design of a structure is generally carried out according to a deterministic approach. However, all structural problems have associated initial uncertain parameters that can differ from the design value. This becomes important when the goal is to reach optimized structures, as a [...] Read more.

The design of a structure is generally carried out according to a deterministic approach. However, all structural problems have associated initial uncertain parameters that can differ from the design value. This becomes important when the goal is to reach optimized structures, as a small variation of these initial uncertain parameters can have a big influence on the structural behavior. The objective of robust design optimization is to obtain an optimum design with the lowest possible variation of the objective functions. For this purpose, a probabilistic optimization is necessary to obtain the statistical parameters that represent the mean value and variation of the objective function considered. However, one of the disadvantages of the optimal robust design is its high computational cost. In this paper, robust design optimization is applied to design a continuous prestressed concrete box-girder pedestrian bridge that is optimum in terms of its cost and robust in terms of structural stability. Furthermore, Latin hypercube sampling and the kriging metamodel are used to deal with the high computational cost. Results show that the main variables that control the structural behavior are the depth of the cross-section and compressive strength of the concrete and that a compromise solution between the optimal cost and the robustness of the design can be reached. Full article

(This article belongs to the Special Issue Optimization for Decision Making II)

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9 pages, 1039 KiB

Open AccessEditor’s ChoiceArticle

Regarding New Wave Patterns of the Newly Extended Nonlinear (2+1)-Dimensional Boussinesq Equation with Fourth Order

by Juan Luis García Guirao, Haci Mehmet Baskonus and Ajay Kumar

Mathematics 2020, 8(3), 341; https://doi.org/10.3390/math8030341 - 04 Mar 2020

Cited by 34 | Viewed by 2849

Abstract

This paper applies the sine-Gordon expansion method to the extended nonlinear (2+1)-dimensional Boussinesq equation. Many new dark, complex and mixed dark-bright soliton solutions of the governing model are derived. Moreover, for better understanding of the results, 2D, 3D and contour graphs under the [...] Read more.

This paper applies the sine-Gordon expansion method to the extended nonlinear (2+1)-dimensional Boussinesq equation. Many new dark, complex and mixed dark-bright soliton solutions of the governing model are derived. Moreover, for better understanding of the results, 2D, 3D and contour graphs under the strain conditions and the suitable values of parameters are also plotted. Full article

(This article belongs to the Special Issue Computational Methods in Applied Analysis and Mathematical Modeling)

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13 pages, 945 KiB

Open AccessEditor’s ChoiceArticle

Improved Decentralized Fractional PD Control of Structure Vibrations

by Kang Xu, Liping Chen, Minwu Wang, António M. Lopes, J. A. Tenreiro Machado and Houzhen Zhai

Mathematics 2020, 8(3), 326; https://doi.org/10.3390/math8030326 - 02 Mar 2020

Cited by 24 | Viewed by 2142

Abstract

This paper presents a new strategy for the control of large displacements in structures under earthquake excitation. Firstly, an improved fractional order proportional-derivative (FOPD) controller is proposed. Secondly, a decentralized strategy is designed by adding a regulator and fault self-regulation to a standard [...] Read more.

This paper presents a new strategy for the control of large displacements in structures under earthquake excitation. Firstly, an improved fractional order proportional-derivative (FOPD) controller is proposed. Secondly, a decentralized strategy is designed by adding a regulator and fault self-regulation to a standard decentralized controller. A new control architecture is obtained by combining the improved FOPD and the decentralized strategy. The parameters of the control system are tuned using an intelligent optimization algorithm. Simulation results demonstrate the performance and reliability of the proposed method. Full article

(This article belongs to the Special Issue Advances in Fractional Order Control and Applications)

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21 pages, 403 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Good (and Not So Good) Practices in Computational Methods for Fractional Calculus

by Kai Diethelm, Roberto Garrappa and Martin Stynes

Mathematics 2020, 8(3), 324; https://doi.org/10.3390/math8030324 - 02 Mar 2020

Cited by 35 | Viewed by 4575

Abstract

The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware [...] Read more.

The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results. Full article

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

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24 pages, 810 KiB

Open AccessEditor’s ChoiceArticle

An Alternative Approach to Measure Co-Movement between Two Time Series

by José Pedro Ramos-Requena, Juan Evangelista Trinidad-Segovia and Miguel Ángel Sánchez-Granero

Mathematics 2020, 8(2), 261; https://doi.org/10.3390/math8020261 - 17 Feb 2020

Cited by 11 | Viewed by 4074

Abstract

The study of the dependences between different assets is a classic topic in financial literature. To understand how the movements of one asset affect to others is critical for derivatives pricing, portfolio management, risk control, or trading strategies. Over time, different methodologies were [...] Read more.

The study of the dependences between different assets is a classic topic in financial literature. To understand how the movements of one asset affect to others is critical for derivatives pricing, portfolio management, risk control, or trading strategies. Over time, different methodologies were proposed by researchers. ARCH, GARCH or EGARCH models, among others, are very popular to model volatility autocorrelation. In this paper, a new simple method called HP is introduced to measure the co-movement between two time series. This method, based on the Hurst exponent of the product series, is designed to detect correlation, even if the relationship is weak, but it also works fine with cointegration as well as non linear correlations or more complex relationships given by a copula. This method and different variations thereaof are tested in statistical arbitrage. Results show that HP is able to detect the relationship between assets better than the traditional correlation method. Full article

(This article belongs to the Section Financial Mathematics)

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13 pages, 298 KiB

Open AccessEditor’s ChoiceArticle

Qualitative Analysis of Multi-Terms Fractional Order Delay Differential Equations via the Topological Degree Theory

by Muhammad Sher, Kamal Shah, Michal Fečkan and Rahmat Ali Khan

Mathematics 2020, 8(2), 218; https://doi.org/10.3390/math8020218 - 08 Feb 2020

Cited by 26 | Viewed by 2367

Abstract

With the help of the topological degree theory in this manuscript, we develop qualitative theory for a class of multi-terms fractional order differential equations (FODEs) with proportional delay using the Caputo derivative. In the same line, we will also study various forms of [...] Read more.

With the help of the topological degree theory in this manuscript, we develop qualitative theory for a class of multi-terms fractional order differential equations (FODEs) with proportional delay using the Caputo derivative. In the same line, we will also study various forms of Ulam stability results. To clarify our theocratical analysis, we provide three different pertinent examples. Full article

(This article belongs to the Special Issue Functional Differential Equations and Applications)

22 pages, 354 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Fractional Derivatives and Integrals: What Are They Needed For?

by Vasily E. Tarasov and Svetlana S. Tarasova

Mathematics 2020, 8(2), 164; https://doi.org/10.3390/math8020164 - 25 Jan 2020

Cited by 28 | Viewed by 4318

Abstract

The question raised in the title of the article is not philosophical. We do not expect general answers of the form “to describe the reality surrounding us”. The question should actually be formulated as a mathematical problem of applied mathematics, a task for [...] Read more.

The question raised in the title of the article is not philosophical. We do not expect general answers of the form “to describe the reality surrounding us”. The question should actually be formulated as a mathematical problem of applied mathematics, a task for new research. This question should be answered in mathematically rigorous statements about the interrelations between the properties of the operator’s kernels and the types of phenomena. This article is devoted to a discussion of the question of what is fractional operator from the point of view of not pure mathematics, but applied mathematics. The imposed restrictions on the kernel of the fractional operator should actually be divided by types of phenomena, in addition to the principles of self-consistency of mathematical theory. In applications of fractional calculus, we have a fundamental question about conditions of kernels of fractional operator of non-integer orders that allow us to describe a particular type of phenomenon. It is necessary to obtain exact correspondences between sets of properties of kernel and type of phenomena. In this paper, we discuss the properties of kernels of fractional operators to distinguish the following types of phenomena: fading memory (forgetting) and power-law frequency dispersion, spatial non-locality and power-law spatial dispersion, distributed lag (time delay), distributed scaling (dilation), depreciation, and aging. Full article

(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)

22 pages, 2995 KiB

Open AccessEditor’s ChoiceArticle

Energy Efficiency and Fuel Economy of a Fuel Cell/Renewable Energy Sources Hybrid Power System with the Load-Following Control of the Fueling Regulators

by Nicu Bizon and Phatiphat Thounthong

Mathematics 2020, 8(2), 151; https://doi.org/10.3390/math8020151 - 21 Jan 2020

Cited by 24 | Viewed by 3346

Abstract

Two Hybrid Power System (HPS) topologies are proposed in this paper based on the Renewable Energy Sources (RESs) and a Fuel Cell (FC) system-based backup energy source. Photovoltaic arrays and wind turbines are modeled as RESs power flow. Hydrogen and air needed for [...] Read more.

Two Hybrid Power System (HPS) topologies are proposed in this paper based on the Renewable Energy Sources (RESs) and a Fuel Cell (FC) system-based backup energy source. Photovoltaic arrays and wind turbines are modeled as RESs power flow. Hydrogen and air needed for FC stack to generate the power requested by the load are achieved through the Load-Following control loop. This control loop will regulate the fueling flow rate to load level. A real-time optimization strategy for RES/FC HPS based on Extremum Seeking Control will find the Maximum Efficiency Point or best fuel economy point by control of the boost converter. Therefore, two HPS configurations and associated strategies based on Load-Following and optimization loops of the fueling regulators were studied here and compared using the following performance indicators: the FC net power generated on the DC bus, the FC energy efficiency, the fuel consumption efficiency, and the total fuel consumption. An increase in the FC system’s electrical efficiency and fuel economy of up to 2% and 12% respectively has been obtained using the proposed optimization strategies compared with a baseline strategy. Full article

(This article belongs to the Special Issue Mathematical Methods applied in Power Systems)

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21 pages, 1057 KiB

Open AccessEditor’s ChoiceArticle

Fractional Derivatives for Economic Growth Modelling of the Group of Twenty: Application to Prediction

by Inés Tejado, Emiliano Pérez and Duarte Valério

Mathematics 2020, 8(1), 50; https://doi.org/10.3390/math8010050 - 01 Jan 2020

Cited by 17 | Viewed by 3604

Abstract

This paper studies the economic growth of the countries in the Group of Twenty (G20) in the period 1970–2018. It presents dynamic models for the world’s most important national economies, including for the first time several economies which are not highly developed. Additional [...] Read more.

This paper studies the economic growth of the countries in the Group of Twenty (G20) in the period 1970–2018. It presents dynamic models for the world’s most important national economies, including for the first time several economies which are not highly developed. Additional care has been devoted to the number of years needed for an accurate short-term prediction of future outputs. Integer order and fractional order differential equation models were obtained from the data. Their output is the gross domestic product (GDP) of a G20 country. Models are multi-input; GDP is found from all or some of the following variables: country’s land area, arable land, population, school attendance, gross capital formation (GCF), exports of goods and services, general government final consumption expenditure (GGFCE), and broad money (M3). Results confirm the better performance of fractional models. This has been established employing several summary statistics. Fractional models do not require increasing the number of parameters, neither do they sacrifice the ability to predict GDP evolution in the short-term. It was found that data over 15 years allows building a model with a satisfactory prediction of the evolution of the GDP. Full article

(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)

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14 pages, 813 KiB

Open AccessEditor’s ChoiceArticle

Formative Transcendence of Flipped Learning in Mathematics Students of Secondary Education

by Jesús López Belmonte, Arturo Fuentes Cabrera, Juan Antonio López Núñez and Santiago Pozo Sánchez

Mathematics 2019, 7(12), 1226; https://doi.org/10.3390/math7121226 - 12 Dec 2019

Cited by 42 | Viewed by 6622

Abstract

Educational technology is achieving great potential in the formative processes of today’s society. Flipped learning is considered as a pedagogical innovation derived from the technological influence in learning spaces. The general objective of the research is to analyze the effectiveness of flipped learning [...] Read more.

Educational technology is achieving great potential in the formative processes of today’s society. Flipped learning is considered as a pedagogical innovation derived from the technological influence in learning spaces. The general objective of the research is to analyze the effectiveness of flipped learning on a traditional teaching and learning approach in the subject of Mathematics. To achieve this objective, an experimental design of a descriptive and correlational type has been followed through a quantitative research method. Two study groups have been set up. In the control group, the contents have been imparted from a traditional perspective, and in the experimental group, innovation has been applied through the use of flipped learning. The sample of participants has been chosen by means of intentional sampling and reached the figure of 60 students in the 4th year of Secondary Education at an educational center in Ceuta (Spain). A questionnaire has been used for data collection. The results reflect that the application of flipped learning has obtained better assessment in established attitudinal and mathematical indicators. It is concluded that with the use of flipped learning, motivation and skills are increased in the analysis and representation of graphs. Full article

(This article belongs to the Special Issue Futuring Mathematics Education—Core Teaching Practices to Prepare Students for the 21st Century: An International Perspective)

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12 pages, 677 KiB

Open AccessEditor’s ChoiceArticle

A New Criterion for Model Selection

by Hoang Pham

Mathematics 2019, 7(12), 1215; https://doi.org/10.3390/math7121215 - 10 Dec 2019

Cited by 105 | Viewed by 6788

Abstract

Selecting the best model from a set of candidates for a given set of data is obviously not an easy task. In this paper, we propose a new criterion that takes into account a larger penalty when adding too many coefficients (or estimated [...] Read more.

Selecting the best model from a set of candidates for a given set of data is obviously not an easy task. In this paper, we propose a new criterion that takes into account a larger penalty when adding too many coefficients (or estimated parameters) in the model from too small a sample in the presence of too much noise, in addition to minimizing the sum of squares error. We discuss several real applications that illustrate the proposed criterion and compare its results to some existing criteria based on a simulated data set and some real datasets including advertising budget data, newly collected heart blood pressure health data sets and software failure data. Full article

(This article belongs to the Special Issue Statistics and Modeling in Reliability Engineering)

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18 pages, 472 KiB

Open AccessEditor’s ChoiceArticle

A 2D Non-Linear Second-Order Differential Model for Electrostatic Circular Membrane MEMS Devices: A Result of Existence and Uniqueness

by Paolo Di Barba, Luisa Fattorusso and Mario Versaci

Mathematics 2019, 7(12), 1193; https://doi.org/10.3390/math7121193 - 05 Dec 2019

Cited by 19 | Viewed by 2512

Abstract

In the framework of 2D circular membrane Micro-Electric-Mechanical-Systems (MEMS), a new non-linear second-order differential model with singularity in the steady-state case is presented in this paper. In particular, starting from the fact that the electric field magnitude is locally proportional to the curvature [...] Read more.

In the framework of 2D circular membrane Micro-Electric-Mechanical-Systems (MEMS), a new non-linear second-order differential model with singularity in the steady-state case is presented in this paper. In particular, starting from the fact that the electric field magnitude is locally proportional to the curvature of the membrane, the problem is formalized in terms of the mean curvature. Then, a result of the existence of at least one solution is achieved. Finally, two different approaches prove that the uniqueness of the solutions is not ensured. Full article

(This article belongs to the Special Issue Mathematical Modeling in Industrial Engineering and Electrical Engineering)

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10 pages, 794 KiB

Open AccessEditor’s ChoiceArticle

Some Results on (s − q)-Graphic Contraction Mappings in b-Metric-Like Spaces

by Manuel De la Sen, Nebojša Nikolić, Tatjana Došenović, Mirjana Pavlović and Stojan Radenović

Mathematics 2019, 7(12), 1190; https://doi.org/10.3390/math7121190 - 04 Dec 2019

Cited by 19 | Viewed by 2191

Abstract

In this paper we consider

(s - q)

-graphic contraction mapping in b-metric like spaces. By using our new approach for the proof that a Picard sequence is Cauchy in the context of b-metric-like space, our results generalize, improve [...] Read more.

In this paper we consider

(s - q)

-graphic contraction mapping in b-metric like spaces. By using our new approach for the proof that a Picard sequence is Cauchy in the context of b-metric-like space, our results generalize, improve and complement several approaches in the existing literature. Moreover, some examples are presented here to illustrate the usability of the obtained theoretical results. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

17 pages, 493 KiB

Open AccessEditor’s ChoiceArticle

Significance of Double Stratification in Stagnation Point Flow of Third-Grade Fluid towards a Radiative Stretching Cylinder

by Anum Shafiq, Ilyas Khan, Ghulam Rasool, Asiful H. Seikh and El-Sayed M. Sherif

Mathematics 2019, 7(11), 1103; https://doi.org/10.3390/math7111103 - 14 Nov 2019

Cited by 35 | Viewed by 3236

Abstract

The present article is devoted to examine the significance of double stratification in third grade stagnation point flow towards a radiative stretching cylinder. The stagnation point is discussed categorically. Analysis is scrutinized in the presence of Thermophoresis, Brownian diffusion, double stratification and heat [...] Read more.

The present article is devoted to examine the significance of double stratification in third grade stagnation point flow towards a radiative stretching cylinder. The stagnation point is discussed categorically. Analysis is scrutinized in the presence of Thermophoresis, Brownian diffusion, double stratification and heat source/sink. Suitable typical transformations are used to drive the system of ordinary differential equation. The governing system is subjected to optimal homotopy analysis method (OHAM) for convergent series solutions. The impact of pertinent fluid parameters on the velocity field, temperature distribution and concentration of the nanoparticles is shown graphically. Numerical data is compiled in tabulare form for skin friction, Nusselt and Sherwood numbers to analyze the variation caused by the present model and to see the impact for industrial and engineering point of view. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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12 pages, 326 KiB

Open AccessEditor’s ChoiceArticle

Topologies on Z n that Are Not Homeomorphic to the n-Dimensional Khalimsky Topological Space

by Sang-Eon Han, Saeid Jafari and Jeong Min Kang

Mathematics 2019, 7(11), 1072; https://doi.org/10.3390/math7111072 - 07 Nov 2019

Cited by 16 | Viewed by 2962

Abstract

The present paper deals with two types of topologies on the set of integers,

Z

: a quasi-discrete topology and a topology satisfying the

T_{\frac{1}{2}}

-separation axiom. Furthermore, for each

n \in N

, we develop countably many topologies on [...] Read more.

The present paper deals with two types of topologies on the set of integers,

Z

: a quasi-discrete topology and a topology satisfying the

T_{\frac{1}{2}}

-separation axiom. Furthermore, for each

n \in N

, we develop countably many topologies on

Z^{n}

which are not homeomorphic to the typical n-dimensional Khalimsky topological space. Based on these different types of new topological structures on

Z^{n}

, many new mathematical approaches can be done in the fields of pure and applied sciences, such as fixed point theory, rough set theory, and so on. Full article

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20 pages, 341 KiB

Open AccessEditor’s ChoiceArticle

An Optimisation-Driven Prediction Method for Automated Diagnosis and Prognosis

by Valentino Santucci, Alfredo Milani and Fabio Caraffini

Mathematics 2019, 7(11), 1051; https://doi.org/10.3390/math7111051 - 04 Nov 2019

Cited by 18 | Viewed by 2571

Abstract

This article presents a novel hybrid classification paradigm for medical diagnoses and prognoses prediction. The core mechanism of the proposed method relies on a centroid classification algorithm whose logic is exploited to formulate the classification task as a real-valued optimisation problem. A novel [...] Read more.

This article presents a novel hybrid classification paradigm for medical diagnoses and prognoses prediction. The core mechanism of the proposed method relies on a centroid classification algorithm whose logic is exploited to formulate the classification task as a real-valued optimisation problem. A novel metaheuristic combining the algorithmic structure of Swarm Intelligence optimisers with the probabilistic search models of Estimation of Distribution Algorithms is designed to optimise such a problem, thus leading to high-accuracy predictions. This method is tested over 11 medical datasets and compared against 14 cherry-picked classification algorithms. Results show that the proposed approach is competitive and superior to the state-of-the-art on several occasions. Full article

(This article belongs to the Special Issue Numerical Optimization and Applications)

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9 pages, 253 KiB

Open AccessEditor’s ChoiceArticle

Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations

by Boris Hanin

Mathematics 2019, 7(10), 992; https://doi.org/10.3390/math7100992 - 18 Oct 2019

Cited by 135 | Viewed by 10621

Abstract

This article concerns the expressive power of depth in neural nets with ReLU activations and a bounded width. We are particularly interested in the following questions: What is the minimal width

w_{\min} (d)

so that ReLU nets of width [...] Read more.

This article concerns the expressive power of depth in neural nets with ReLU activations and a bounded width. We are particularly interested in the following questions: What is the minimal width

w_{\min} (d)

so that ReLU nets of width

w_{\min} (d)

(and arbitrary depth) can approximate any continuous function on the unit cube

{[0, 1]}^{d}

arbitrarily well? For ReLU nets near this minimal width, what can one say about the depth necessary to approximate a given function? We obtain an essentially complete answer to these questions for convex functions. Our approach is based on the observation that, due to the convexity of the ReLU activation, ReLU nets are particularly well suited to represent convex functions. In particular, we prove that ReLU nets with width

d + 1

can approximate any continuous convex function of d variables arbitrarily well. These results then give quantitative depth estimates for the rate of approximation of any continuous scalar function on the d-dimensional cube

{[0, 1]}^{d}

by ReLU nets with width

d + 3

. Full article

(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)

16 pages, 2867 KiB

Open AccessEditor’s ChoiceArticle

Developing an ANFIS-PSO Model to Predict Mercury Emissions in Combustion Flue Gases

by Shahaboddin Shamshirband, Masoud Hadipoor, Alireza Baghban, Amir Mosavi, Jozsef Bukor and Annamária R. Várkonyi-Kóczy

Mathematics 2019, 7(10), 965; https://doi.org/10.3390/math7100965 - 14 Oct 2019

Cited by 45 | Viewed by 4472

Abstract

Accurate prediction of mercury content emitted from fossil-fueled power stations is of the utmost importance for environmental pollution assessment and hazard mitigation. In this paper, mercury content in the output gas of power stations’ boilers was predicted using an adaptive neuro-fuzzy inference system [...] Read more.

Accurate prediction of mercury content emitted from fossil-fueled power stations is of the utmost importance for environmental pollution assessment and hazard mitigation. In this paper, mercury content in the output gas of power stations’ boilers was predicted using an adaptive neuro-fuzzy inference system (ANFIS) method integrated with particle swarm optimization (PSO). The input parameters of the model included coal characteristics and the operational parameters of the boilers. The dataset was collected from 82 sample points in power plants and employed to educate and examine the proposed model. To evaluate the performance of the proposed hybrid model of the ANFIS-PSO, the statistical meter of MARE% was implemented, which resulted in 0.003266 and 0.013272 for training and testing, respectively. Furthermore, relative errors between the acquired data and predicted values were between −0.25% and 0.1%, which confirm the accuracy of the model to deal non-linearity and represent the dependency of flue gas mercury content into the specifications of coal and the boiler type. Full article

(This article belongs to the Special Issue Hybrid Machine Learning Techniques Applied in Real Engineering Applications)

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25 pages, 906 KiB

Open AccessEditor’s ChoiceArticle

Real-Time Optimization and Control of Nonlinear Processes Using Machine Learning

by Zhihao Zhang, Zhe Wu, David Rincon and Panagiotis D. Christofides

Mathematics 2019, 7(10), 890; https://doi.org/10.3390/math7100890 - 24 Sep 2019

Cited by 49 | Viewed by 7099

Abstract

Machine learning has attracted extensive interest in the process engineering field, due to the capability of modeling complex nonlinear process behavior. This work presents a method for combining neural network models with first-principles models in real-time optimization (RTO) and model predictive control (MPC) [...] Read more.

Machine learning has attracted extensive interest in the process engineering field, due to the capability of modeling complex nonlinear process behavior. This work presents a method for combining neural network models with first-principles models in real-time optimization (RTO) and model predictive control (MPC) and demonstrates the application to two chemical process examples. First, the proposed methodology that integrates a neural network model and a first-principles model in the optimization problems of RTO and MPC is discussed. Then, two chemical process examples are presented. In the first example, a continuous stirred tank reactor (CSTR) with a reversible exothermic reaction is studied. A feed-forward neural network model is used to approximate the nonlinear reaction rate and is combined with a first-principles model in RTO and MPC. An RTO is designed to find the optimal reactor operating condition balancing energy cost and reactant conversion, and an MPC is designed to drive the process to the optimal operating condition. A variation in energy price is introduced to demonstrate that the developed RTO scheme is able to minimize operation cost and yields a closed-loop performance that is very close to the one attained by RTO/MPC using the first-principles model. In the second example, a distillation column is used to demonstrate an industrial application of the use of machine learning to model nonlinearities in RTO. A feed-forward neural network is first built to obtain the phase equilibrium properties and then combined with a first-principles model in RTO, which is designed to maximize the operation profit and calculate optimal set-points for the controllers. A variation in feed concentration is introduced to demonstrate that the developed RTO scheme can increase operation profit for all considered conditions. Full article

(This article belongs to the Special Issue Mathematics and Engineering)

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12 pages, 311 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

The Fixed Point Property of Non-Retractable Topological Spaces

by Jeong Min Kang, Sang-Eon Han and Sik Lee

Mathematics 2019, 7(10), 879; https://doi.org/10.3390/math7100879 - 21 Sep 2019

Cited by 8 | Viewed by 2605

Abstract

Unlike the study of the fixed point property (FPP, for brevity) of retractable topological spaces, the research of the FPP of non-retractable topological spaces remains. The present paper deals with the issue. Based on order-theoretic foundations and fixed point theory for [...] Read more.

Unlike the study of the fixed point property (FPP, for brevity) of retractable topological spaces, the research of the FPP of non-retractable topological spaces remains. The present paper deals with the issue. Based on order-theoretic foundations and fixed point theory for Khalimsky (K-, for short) topological spaces, the present paper studies the product property of the FPP for K-topological spaces. Furthermore, the paper investigates the FPP of various types of connected K-topological spaces such as non-K-retractable spaces and some points deleted K-topological (finite) planes, and so on. To be specific, after proving that not every one point deleted subspace of a finite K-topological plane X is a K-retract of X, we study the FPP of a non-retractable topological space Y, such as one point deleted space

Y ∖ {p}

. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Nonlinear Problems with Applications)

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