Editor's Choice Articles

Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to authors, or important in this field. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.

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Article
Modeling Soil Water Redistribution under Gravity Irrigation with the Richards Equation
Mathematics 2020, 8(9), 1581; https://doi.org/10.3390/math8091581 - 13 Sep 2020
Cited by 6
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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Article
A New Extended Two-Parameter Distribution: Properties, Estimation Methods, and Applications in Medicine and Geology
Mathematics 2020, 8(9), 1578; https://doi.org/10.3390/math8091578 - 12 Sep 2020
Cited by 9
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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Article
Do Science, Technology, Engineering and Mathematics (STEM) Experimentation Outreach Programs Affect Attitudes towards Mathematics and Science? A Quasi-Experiment in Primary Education
Mathematics 2020, 8(9), 1490; https://doi.org/10.3390/math8091490 - 03 Sep 2020
Cited by 4
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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Article
Comparison of Ensemble Machine Learning Methods for Automated Classification of Focal and Non-Focal Epileptic EEG Signals
Mathematics 2020, 8(9), 1481; https://doi.org/10.3390/math8091481 - 02 Sep 2020
Cited by 6
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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Article
AHP-TOPSIS Inspired Shopping Mall Site Selection Problem with Fuzzy Data
Mathematics 2020, 8(8), 1380; https://doi.org/10.3390/math8081380 - 17 Aug 2020
Cited by 12
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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Article
Waste Segregation FMEA Model Integrating Intuitionistic Fuzzy Set and the PAPRIKA Method
Mathematics 2020, 8(8), 1375; https://doi.org/10.3390/math8081375 - 17 Aug 2020
Cited by 6
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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Article
Analyzing the Impact of the Renewable Energy Sources on Economic Growth at the EU Level Using an ARDL Model
Mathematics 2020, 8(8), 1367; https://doi.org/10.3390/math8081367 - 14 Aug 2020
Cited by 9
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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Article
Robust Three-Step Regression Based on Comedian and Its Performance in Cell-Wise and Case-Wise Outliers
Mathematics 2020, 8(8), 1259; https://doi.org/10.3390/math8081259 - 01 Aug 2020
Cited by 9
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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Article
A Novel Methodology to Calculate the Probability of Volatility Clusters in Financial Series: An Application to Cryptocurrency Markets
Mathematics 2020, 8(8), 1216; https://doi.org/10.3390/math8081216 - 24 Jul 2020
Cited by 5
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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Article
A Mathematical Model of Epidemics—A Tutorial for Students
Mathematics 2020, 8(7), 1174; https://doi.org/10.3390/math8071174 - 17 Jul 2020
Cited by 5
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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Article
Supportiveness of Low-Carbon Energy Technology Policy Using Fuzzy Multicriteria Decision-Making Methodologies
Mathematics 2020, 8(7), 1178; https://doi.org/10.3390/math8071178 - 17 Jul 2020
Cited by 4
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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Article
Birnbaum-Saunders Quantile Regression Models with Application to Spatial Data
Mathematics 2020, 8(6), 1000; https://doi.org/10.3390/math8061000 - 18 Jun 2020
Cited by 13
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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Article
A Regularity Criterion in Weak Spaces to Boussinesq Equations
Mathematics 2020, 8(6), 920; https://doi.org/10.3390/math8060920 - 05 Jun 2020
Cited by 31
Abstract
In this paper, we study the regularity of weak solutions to the incompressible Boussinesq equations in R 3 × ( 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)
Article
COVID-19 Pandemic Prediction for Hungary; A Hybrid Machine Learning Approach
Mathematics 2020, 8(6), 890; https://doi.org/10.3390/math8060890 - 02 Jun 2020
Cited by 51
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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Article
On Estimating the Number of Deaths Related to Covid-19
Mathematics 2020, 8(5), 655; https://doi.org/10.3390/math8050655 - 26 Apr 2020
Cited by 13
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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Article
Fibonacci Numbers with a Prescribed Block of Digits
Mathematics 2020, 8(4), 639; https://doi.org/10.3390/math8040639 - 21 Apr 2020
Cited by 9
Abstract
In this paper, we prove that F 22 = 17711 is the largest Fibonacci number whose decimal expansion is of the form a b b c 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)
Article
Global Research Trends in Financial Transactions
Mathematics 2020, 8(4), 614; https://doi.org/10.3390/math8040614 - 16 Apr 2020
Cited by 14
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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Article
Power Aggregation Operators and VIKOR Methods for Complex q-Rung Orthopair Fuzzy Sets and Their Applications
Mathematics 2020, 8(4), 538; https://doi.org/10.3390/math8040538 - 05 Apr 2020
Cited by 26
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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Article
Some Results in Green–Lindsay Thermoelasticity of Bodies with Dipolar Structure
Mathematics 2020, 8(4), 497; https://doi.org/10.3390/math8040497 - 02 Apr 2020
Cited by 12
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)
Article
Existence of Weak Solutions for a New Class of Fractional p-Laplacian Boundary Value Systems
Mathematics 2020, 8(4), 475; https://doi.org/10.3390/math8040475 - 31 Mar 2020
Cited by 12
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
Article
A New Fuzzy MARCOS Method for Road Traffic Risk Analysis
Mathematics 2020, 8(3), 457; https://doi.org/10.3390/math8030457 - 24 Mar 2020
Cited by 36
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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Article
An Integrated Approach of Best-Worst Method (BWM) and Triangular Fuzzy Sets for Evaluating Driver Behavior Factors Related to Road Safety
Mathematics 2020, 8(3), 414; https://doi.org/10.3390/math8030414 - 13 Mar 2020
Cited by 19
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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Article
Robust Design Optimization for Low-Cost Concrete Box-Girder Bridge
Mathematics 2020, 8(3), 398; https://doi.org/10.3390/math8030398 - 11 Mar 2020
Cited by 10
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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Article
Regarding New Wave Patterns of the Newly Extended Nonlinear (2+1)-Dimensional Boussinesq Equation with Fourth Order
Mathematics 2020, 8(3), 341; https://doi.org/10.3390/math8030341 - 04 Mar 2020
Cited by 25
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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Article
Improved Decentralized Fractional PD Control of Structure Vibrations
Mathematics 2020, 8(3), 326; https://doi.org/10.3390/math8030326 - 02 Mar 2020
Cited by 14
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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Article
Good (and Not So Good) Practices in Computational Methods for Fractional Calculus
Mathematics 2020, 8(3), 324; https://doi.org/10.3390/math8030324 - 02 Mar 2020
Cited by 7
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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Article
An Alternative Approach to Measure Co-Movement between Two Time Series
Mathematics 2020, 8(2), 261; https://doi.org/10.3390/math8020261 - 17 Feb 2020
Cited by 8
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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Article
Qualitative Analysis of Multi-Terms Fractional Order Delay Differential Equations via the Topological Degree Theory
Mathematics 2020, 8(2), 218; https://doi.org/10.3390/math8020218 - 08 Feb 2020
Cited by 13
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)
Article
Fractional Derivatives and Integrals: What Are They Needed For?
Mathematics 2020, 8(2), 164; https://doi.org/10.3390/math8020164 - 25 Jan 2020
Cited by 9
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”)
Article
Energy Efficiency and Fuel Economy of a Fuel Cell/Renewable Energy Sources Hybrid Power System with the Load-Following Control of the Fueling Regulators
Mathematics 2020, 8(2), 151; https://doi.org/10.3390/math8020151 - 21 Jan 2020
Cited by 15
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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Article
Fractional Derivatives for Economic Growth Modelling of the Group of Twenty: Application to Prediction
Mathematics 2020, 8(1), 50; https://doi.org/10.3390/math8010050 - 01 Jan 2020
Cited by 8
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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Article
Formative Transcendence of Flipped Learning in Mathematics Students of Secondary Education
Mathematics 2019, 7(12), 1226; https://doi.org/10.3390/math7121226 - 12 Dec 2019
Cited by 18
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
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Article
A New Criterion for Model Selection
Mathematics 2019, 7(12), 1215; https://doi.org/10.3390/math7121215 - 10 Dec 2019
Cited by 17
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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Article
A 2D Non-Linear Second-Order Differential Model for Electrostatic Circular Membrane MEMS Devices: A Result of Existence and Uniqueness
Mathematics 2019, 7(12), 1193; https://doi.org/10.3390/math7121193 - 05 Dec 2019
Cited by 15
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
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Article
Some Results on (sq)-Graphic Contraction Mappings in b-Metric-Like Spaces
Mathematics 2019, 7(12), 1190; https://doi.org/10.3390/math7121190 - 04 Dec 2019
Cited by 12
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 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)
Article
Significance of Double Stratification in Stagnation Point Flow of Third-Grade Fluid towards a Radiative Stretching Cylinder
Mathematics 2019, 7(11), 1103; https://doi.org/10.3390/math7111103 - 14 Nov 2019
Cited by 22
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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Article
Topologies on Z n that Are Not Homeomorphic to the n-Dimensional Khalimsky Topological Space
Mathematics 2019, 7(11), 1072; https://doi.org/10.3390/math7111072 - 07 Nov 2019
Cited by 10
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 1 2 -separation axiom. Furthermore, for each n 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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Article
An Optimisation-Driven Prediction Method for Automated Diagnosis and Prognosis
Mathematics 2019, 7(11), 1051; https://doi.org/10.3390/math7111051 - 04 Nov 2019
Cited by 13
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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Article
Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations
Mathematics 2019, 7(10), 992; https://doi.org/10.3390/math7100992 - 18 Oct 2019
Cited by 34
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 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)
Article
Developing an ANFIS-PSO Model to Predict Mercury Emissions in Combustion Flue Gases
Mathematics 2019, 7(10), 965; https://doi.org/10.3390/math7100965 - 14 Oct 2019
Cited by 21
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
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Article
Real-Time Optimization and Control of Nonlinear Processes Using Machine Learning
Mathematics 2019, 7(10), 890; https://doi.org/10.3390/math7100890 - 24 Sep 2019
Cited by 17
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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Article
The Fixed Point Property of Non-Retractable Topological Spaces
Mathematics 2019, 7(10), 879; https://doi.org/10.3390/math7100879 - 21 Sep 2019
Cited by 8
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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Article
A Novel FMEA Model Based on Rough BWM and Rough TOPSIS-AL for Risk Assessment
Mathematics 2019, 7(10), 874; https://doi.org/10.3390/math7100874 - 20 Sep 2019
Cited by 19
Abstract
Failure mode and effects analysis (FMEA) is a risk assessment method that effectively diagnoses a product’s potential failure modes. It is based on expert experience and investigation to determine the potential failure modes of the system or product to develop improvement strategies to [...] Read more.
Failure mode and effects analysis (FMEA) is a risk assessment method that effectively diagnoses a product’s potential failure modes. It is based on expert experience and investigation to determine the potential failure modes of the system or product to develop improvement strategies to reduce the risk of failures. However, the traditional FMEA has many shortcomings that were proposed by many studies. This study proposes a hybrid FMEA and multi-attribute decision-making (MADM) model to establish an evaluation framework, combining the rough best worst method (R-BWM) and rough technique for order preference by similarity to an ideal solution technique (R-TOPSIS) to determine the improvement order of failure modes. In addition, this study adds the concept of aspiration level to R-TOPSIS technology (called R-TOPSIS-AL), which not only optimizes the reliability of the TOPSIS calculation program, but also obtains more potential information. This study then demonstrates the effectiveness and robustness of the proposed model through a multinational audio equipment manufacturing company. The results show that the proposed model can overcome many shortcomings of traditional FMEA, and effectively assist decision-makers and research and development (R&D) departments in improving the reliability of products. Full article
(This article belongs to the Special Issue Recent Advances in Modeling for Reliability Analysis)
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Article
Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations
Mathematics 2019, 7(9), 796; https://doi.org/10.3390/math7090796 - 01 Sep 2019
Cited by 11
Abstract
In this article, we first provide a survey of the exponential option pricing models and show that in the framework of the risk-neutral approach, they are governed by the space-fractional diffusion equation. Then, we introduce a more general class of models based on [...] Read more.
In this article, we first provide a survey of the exponential option pricing models and show that in the framework of the risk-neutral approach, they are governed by the space-fractional diffusion equation. Then, we introduce a more general class of models based on the space-time-fractional diffusion equation and recall some recent results in this field concerning the European option pricing and the risk-neutral parameter. We proceed with an extension of these results to the class of exotic options. In particular, we show that the call and put prices can be expressed in the form of simple power series in terms of the log-forward moneyness and the risk-neutral parameter. Finally, we provide the closed-form formulas for the first and second order risk sensitivities and study the dependencies of the portfolio hedging and profit-and-loss calculations upon the model parameters. Full article
(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)
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Article
Hybrid Control Scheme for Projective Lag Synchronization of Riemann–Liouville Sense Fractional Order Memristive BAM NeuralNetworks with Mixed Delays
Mathematics 2019, 7(8), 759; https://doi.org/10.3390/math7080759 - 19 Aug 2019
Cited by 55
Abstract
This sequel is concerned with the analysis of projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks (FOMBNNs) with mixed time delays via hybrid controller. Firstly, a new type of hybrid control scheme, which is the combination of open loop [...] Read more.
This sequel is concerned with the analysis of projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks (FOMBNNs) with mixed time delays via hybrid controller. Firstly, a new type of hybrid control scheme, which is the combination of open loop control and adaptive state feedback control is designed to guarantee the global projective lag synchronization of the addressed FOMBNNs model. Secondly, by using a Lyapunov–Krasovskii functional and Barbalet’s lemma, a new brand of sufficient criterion is proposed to ensure the projective lag synchronization of the FOMBNNs model considered. Moreover, as special cases by using a hybrid control scheme, some sufficient conditions are derived to ensure the global projective synchronization, global complete synchronization and global anti-synchronization for the FOMBNNs model considered. Finally, numerical simulations are provided to check the accuracy and validity of our obtained synchronization results. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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Article
Development of Public Key Cryptographic Algorithm Using Matrix Pattern for Tele-Ultrasound Applications
Mathematics 2019, 7(8), 752; https://doi.org/10.3390/math7080752 - 17 Aug 2019
Cited by 14
Abstract
A novel public key cryptographic algorithm using a matrix pattern is developed to improve encrypting strength. Compared to the Rivest–Sharmir–Adleman (RSA) and Elliptic Curve Cryptography (ECC) algorithms, our proposed algorithm has superior encrypting strength due to several unknown quantities and one additional sub-equation [...] Read more.
A novel public key cryptographic algorithm using a matrix pattern is developed to improve encrypting strength. Compared to the Rivest–Sharmir–Adleman (RSA) and Elliptic Curve Cryptography (ECC) algorithms, our proposed algorithm has superior encrypting strength due to several unknown quantities and one additional sub-equation during the encrypting process. Our proposed algorithm also provides a faster encoding/decoding speed when the patient’s images for tele-ultrasound applications are transmitted/received, compared to the RSA and ECC encrypting algorithms, because it encodes/decodes the plain memory block by simple addition and multiplication operations of n terms. However, the RSA and ECC algorithms encode/decode each memory block using complex mathematical exponentiation and congruence. To implement encrypting algorithms for tele-ultrasound applications, a streaming server was constructed to transmit the images to the systems using ultrasound machines. Using the obtained ultrasound images from a breast phantom, we compared our developed algorithm, utilizing a matrix pattern, with the RSA and ECC algorithms. The elapsed average time for our proposed algorithm is much faster than that for the RSA and ECC algorithms. Full article
(This article belongs to the Special Issue Information Theory, Cryptography, Randomness and Statistical Modeling)
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Article
The Application of Fractional Calculus in Chinese Economic Growth Models
Mathematics 2019, 7(8), 665; https://doi.org/10.3390/math7080665 - 25 Jul 2019
Cited by 15
Abstract
In this paper, we apply Caputo-type fractional order calculus to simulate China’s gross domestic product (GDP) growth based on R software, which is a free software environment for statistical computing and graphics. Moreover, we compare the results for the fractional model with the [...] Read more.
In this paper, we apply Caputo-type fractional order calculus to simulate China’s gross domestic product (GDP) growth based on R software, which is a free software environment for statistical computing and graphics. Moreover, we compare the results for the fractional model with the integer order model. In addition, we show the importance of variables according to the BIC criterion. The study shows that Caputo fractional order calculus can produce a better model and perform more accurately in predicting the GDP values from 2012–2016. Full article
(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)
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Article
Some New Oscillation Criteria for Second Order Neutral Differential Equations with Delayed Arguments
Mathematics 2019, 7(7), 619; https://doi.org/10.3390/math7070619 - 11 Jul 2019
Cited by 32
Abstract
In this paper, we study the oscillation of second-order neutral differential equations with delayed arguments. Some new oscillatory criteria are obtained by a Riccati transformation. To illustrate the importance of the results, one example is also given. Full article
(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)
Article
Growth Equation of the General Fractional Calculus
Mathematics 2019, 7(7), 615; https://doi.org/10.3390/math7070615 - 11 Jul 2019
Cited by 19
Abstract
We consider the Cauchy problem ( D ( k ) u ) ( t ) = λ u ( t ) , u ( 0 ) = 1 , where D ( k ) is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory 71 (2011), 583–600), λ > 0 . The solution is a generalization of the function t E α ( λ t α ) , where 0 < α < 1 , E α is the Mittag–Leffler function. The asymptotics of this solution, as t , are studied. Full article
(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)
Article
Robust Synchronization of Fractional-Order Uncertain Chaotic Systems Based on Output Feedback Sliding Mode Control
Mathematics 2019, 7(7), 599; https://doi.org/10.3390/math7070599 - 05 Jul 2019
Cited by 54
Abstract
This paper mainly focuses on the robust synchronization issue for drive-response fractional-order chaotic systems (FOCS) when they have unknown parameters and external disturbances. In order to achieve the goal, the sliding mode control scheme only using output information is designed, and at the [...] Read more.
This paper mainly focuses on the robust synchronization issue for drive-response fractional-order chaotic systems (FOCS) when they have unknown parameters and external disturbances. In order to achieve the goal, the sliding mode control scheme only using output information is designed, and at the same time, the structures of a sliding mode surface and a sliding mode controller are also constructed. A sufficient criterion is presented to ensure the robust synchronization of FOCS according to the stability theory of the fractional calculus and sliding mode control technique. In addition, the result can be applied to identical or non-identical chaotic systems with fractional-order. In the end, we build two practical examples to illustrate the feasibility of our theoretical results. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
Article
Some Qualitative Behavior of Solutions of General Class of Difference Equations
Mathematics 2019, 7(7), 585; https://doi.org/10.3390/math7070585 - 01 Jul 2019
Cited by 13
Abstract
In this work, we consider the general class of difference equations (covered many equations that have been studied by other authors or that have never been studied before), as a means of establishing general theorems, for the asymptotic behavior of its solutions. Namely, [...] Read more.
In this work, we consider the general class of difference equations (covered many equations that have been studied by other authors or that have never been studied before), as a means of establishing general theorems, for the asymptotic behavior of its solutions. Namely, we state new necessary and sufficient conditions for local asymptotic stability of these equations. In addition, we study the periodic solution with period two and three. Our results essentially extend and improve the earlier ones. Full article
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Article
Long-Time Asymptotics of a Three-Component Coupled mKdV System
Mathematics 2019, 7(7), 573; https://doi.org/10.3390/math7070573 - 27 Jun 2019
Cited by 47
Abstract
We present an application of the nonlinear steepest descent method to a three-component coupled mKdV system associated with a 4 × 4 matrix spectral problem. An integrable coupled mKdV hierarchy with three potentials is first generated. Based on the corresponding oscillatory Riemann-Hilbert problem, the leading asympototics of the three-component mKdV system is then evaluated by using the nonlinear steepest descent method. Full article
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Article
Back to Basics: Meaning of the Parameters of Fractional Order PID Controllers
Mathematics 2019, 7(6), 530; https://doi.org/10.3390/math7060530 - 11 Jun 2019
Cited by 19
Abstract
The beauty of the proportional-integral-derivative (PID) algorithm for feedback control is its simplicity and efficiency. Those are the main reasons why PID controller is the most common form of feedback. PID combines the three natural ways of taking into account the error: the [...] Read more.
The beauty of the proportional-integral-derivative (PID) algorithm for feedback control is its simplicity and efficiency. Those are the main reasons why PID controller is the most common form of feedback. PID combines the three natural ways of taking into account the error: the actual (proportional), the accumulated (integral), and the predicted (derivative) values; the three gains depend on the magnitude of the error, the time required to eliminate the accumulated error, and the prediction horizon of the error. This paper explores the new meaning of integral and derivative actions, and gains, derived by the consideration of non-integer integration and differentiation orders, i.e., for fractional order PID controllers. The integral term responds with selective memory to the error because of its non-integer order λ , and corresponds to the area of the projection of the error curve onto a plane (it is not the classical area under the error curve). Moreover, for a fractional proportional-integral (PI) controller scheme with automatic reset, both the velocity and the shape of reset can be modified with λ . For its part, the derivative action refers to the predicted future values of the error, but based on different prediction horizons (actually, linear and non-linear extrapolations) depending on the value of the differentiation order, μ . Likewise, in case of a proportional-derivative (PD) structure with a noise filter, the value of μ allows different filtering effects on the error signal to be attained. Similarities and differences between classical and fractional PIDs, as well as illustrative control examples, are given for a best understanding of new possibilities of control with the latter. Examples are given for illustration purposes. Full article
(This article belongs to the Special Issue Fractional Order Systems)
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Article
Economic Machine-Learning-Based Predictive Control of Nonlinear Systems
Mathematics 2019, 7(6), 494; https://doi.org/10.3390/math7060494 - 01 Jun 2019
Cited by 17
Abstract
In this work, a Lyapunov-based economic model predictive control (LEMPC) method is developed to address economic optimality and closed-loop stability of nonlinear systems using machine learning-based models to make predictions. Specifically, an ensemble of recurrent neural network (RNN) models via a k-fold [...] Read more.
In this work, a Lyapunov-based economic model predictive control (LEMPC) method is developed to address economic optimality and closed-loop stability of nonlinear systems using machine learning-based models to make predictions. Specifically, an ensemble of recurrent neural network (RNN) models via a k-fold cross validation is first developed to capture process dynamics in an operating region. Then, the LEMPC using an RNN ensemble is designed to maintain the closed-loop state in a stability region and optimize process economic benefits simultaneously. Parallel computing is employed to improve computational efficiency of real-time implementation of LEMPC with an RNN ensemble. The proposed machine-learning-based LEMPC method is demonstrated using a nonlinear chemical process example. Full article
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Article
Joint Inventory and Pricing Policy for an Online to Offline Closed-Loop Supply Chain Model with Random Defective Rate and Returnable Transport Items
Mathematics 2019, 7(6), 497; https://doi.org/10.3390/math7060497 - 01 Jun 2019
Cited by 14
Abstract
Environmental deterioration is one of the current hot topics of the business world. To cope with the negative environmental impacts of corporate activities, researchers introduced the concept of closed-loop supply chain (CLSC) management and remanufacturing. This paper studies joint inventory and pricing decisions [...] Read more.
Environmental deterioration is one of the current hot topics of the business world. To cope with the negative environmental impacts of corporate activities, researchers introduced the concept of closed-loop supply chain (CLSC) management and remanufacturing. This paper studies joint inventory and pricing decisions in a multi-echelon CLSC model that considers online to offline (O2O) business strategy. An imperfect production process is examined with a random defective rate that follows a probability distribution. The results show that the O2O channel increases the profit of the system. For the defective rate, three different distributions are considered and three examples are solved. The results of the three examples conclude that the highest profit is generated when the defective rate follows a uniform distribution. Furthermore, based on the salvage value of defective items, two cases were studied. Results and sensitivity analysis show that the increase in defective rate does not reduce total profit in every situation, as perceived by the existing literature. Sensitivity analysis and numerical examples are given to show robustness of the model and draw important managerial insights. Full article
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Article
Positive Solutions for a Hadamard Fractional p-Laplacian Three-Point Boundary Value Problem
Mathematics 2019, 7(5), 439; https://doi.org/10.3390/math7050439 - 17 May 2019
Cited by 13
Abstract
This article is to study a three-point boundary value problem of Hadamard fractional p-Laplacian differential equation. When our nonlinearity grows ( p 1 ) -superlinearly and ( p 1 ) -sublinearly, the existence of positive solutions is obtained via fixed point index. Moreover, using an increasing operator fixed-point theorem, the uniqueness of positive solutions and uniform convergence sequences are also established. Full article
Article
Enhancing Elephant Herding Optimization with Novel Individual Updating Strategies for Large-Scale Optimization Problems
Mathematics 2019, 7(5), 395; https://doi.org/10.3390/math7050395 - 30 Apr 2019
Cited by 17
Abstract
Inspired by the behavior of elephants in nature, elephant herd optimization (EHO) was proposed recently for global optimization. Like most other metaheuristic algorithms, EHO does not use the previous individuals in the later updating process. If the useful information in the previous individuals [...] Read more.
Inspired by the behavior of elephants in nature, elephant herd optimization (EHO) was proposed recently for global optimization. Like most other metaheuristic algorithms, EHO does not use the previous individuals in the later updating process. If the useful information in the previous individuals were fully exploited and used in the later optimization process, the quality of solutions may be improved significantly. In this paper, we propose several new updating strategies for EHO, in which one, two, or three individuals are selected from the previous iterations, and their useful information is incorporated into the updating process. Accordingly, the final individual at this iteration is generated according to the elephant generated by the basic EHO, and the selected previous elephants through a weighted sum. The weights are determined by a random number and the fitness of the elephant individuals at the previous iteration. We incorporated each of the six individual updating strategies individually into the basic EHO, creating six improved variants of EHO. We benchmarked these proposed methods using sixteen test functions. Our experimental results demonstrated that the proposed improved methods significantly outperformed the basic EHO. Full article
(This article belongs to the Special Issue Evolutionary Computation)
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Article
A Two-Echelon Supply Chain Management With Setup Time and Cost Reduction, Quality Improvement and Variable Production Rate
Mathematics 2019, 7(4), 328; https://doi.org/10.3390/math7040328 - 03 Apr 2019
Cited by 24
Abstract
This model investigates the variable production cost for a production house; under a two-echelon supply chain management where a single vendor and multi-retailers are involved. This production system goes through a long run system and generates an out-of-control state due to different issues [...] Read more.
This model investigates the variable production cost for a production house; under a two-echelon supply chain management where a single vendor and multi-retailers are involved. This production system goes through a long run system and generates an out-of-control state due to different issues and produces defective items. This model considers the reduction of the defective rate and setup cost through investment. A discrete investment for setup cost reduction and a continuous investment is considered to reduce the defective rate and to increase the quality of products. Setup and processing time are dependent on lead time in this model. The model is solved analytically to find the optimal values of the production rate, safety factors, optimum quantity, lead time length, investment for setup cost reduction, and the probability of the production process going out-of-control. An efficient algorithm is constructed to find the optimal solution numerically and sensitivity analysis is given to show the impact of different parameters. A case study and different cases are also given to validate the model. Full article
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Article
The Bounds of Vertex Padmakar–Ivan Index on k-Trees
Mathematics 2019, 7(4), 324; https://doi.org/10.3390/math7040324 - 01 Apr 2019
Cited by 15
Abstract
The Padmakar–Ivan ( P I ) index is a distance-based topological index and a molecular structure descriptor, which is the sum of the number of vertices over all edges u v of a graph such that these vertices are not equidistant from u and v. In this paper, we explore the results of P I -indices from trees to recursively clustered trees, the k-trees. Exact sharp upper bounds of PI indices on k-trees are obtained by the recursive relationships, and the corresponding extremal graphs are given. In addition, we determine the P I -values on some classes of k-trees and compare them, and our results extend and enrich some known conclusions. Full article
(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)
Article
The Multivariate Theory of Connections
Mathematics 2019, 7(3), 296; https://doi.org/10.3390/math7030296 - 22 Mar 2019
Cited by 15
Abstract
This paper extends the univariate Theory of Connections, introduced in (Mortari, 2017), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons, 1984), by providing analytical expressions, called [...] Read more.
This paper extends the univariate Theory of Connections, introduced in (Mortari, 2017), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons, 1984), by providing analytical expressions, called constrained expressions, representing all possible surfaces with assigned boundary constraints in terms of functions and arbitrary-order derivatives. In two dimensions, these expressions, which contain a freely chosen function, g ( x , y ) , satisfy all constraints no matter what the g ( x , y ) is. The boundary constraints considered in this article are Dirichlet, Neumann, and any combinations of them. Although the focus of this article is on two-dimensional spaces, the final section introduces the Multivariate Theory of Connections, validated by mathematical proof. This represents the multivariate extension of the Theory of Connections subject to arbitrary-order derivative constraints in rectangular domains. The main task of this paper is to provide an analytical procedure to obtain constrained expressions in any space that can be used to transform constrained problems into unconstrained problems. This theory is proposed mainly to better solve PDE and stochastic differential equations. Full article
(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)
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Article
An Efficient Numerical Technique for the Nonlinear Fractional Kolmogorov–Petrovskii–Piskunov Equation
Mathematics 2019, 7(3), 265; https://doi.org/10.3390/math7030265 - 14 Mar 2019
Cited by 35
Abstract
The q -homotopy analysis transform method ( q -HATM) is employed to find the solution for the fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation in the present frame work. To ensure the applicability and efficiency of the proposed algorithm, we consider three distinct initial conditions with two of them having Jacobi elliptic functions. The numerical simulations have been conducted to verify that the proposed scheme is reliable and accurate. Moreover, the uniqueness and convergence analysis for the projected problem is also presented. The obtained results elucidate that the proposed technique is easy to implement and very effective to analyze the complex problems arising in science and technology. Full article
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Article
Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials
Mathematics 2019, 7(3), 224; https://doi.org/10.3390/math7030224 - 27 Feb 2019
Cited by 25
Abstract
The aim of this paper is to solve a class of non-linear fractional variational problems (NLFVPs) using the Ritz method and to perform a comparative study on the choice of different polynomials in the method. The Ritz method has allowed many researchers to [...] Read more.
The aim of this paper is to solve a class of non-linear fractional variational problems (NLFVPs) using the Ritz method and to perform a comparative study on the choice of different polynomials in the method. The Ritz method has allowed many researchers to solve different forms of fractional variational problems in recent years. The NLFVP is solved by applying the Ritz method using different orthogonal polynomials. Further, the approximate solution is obtained by solving a system of nonlinear algebraic equations. Error and convergence analysis of the discussed method is also provided. Numerical simulations are performed on illustrative examples to test the accuracy and applicability of the method. For comparison purposes, different polynomials such as 1) Shifted Legendre polynomials, 2) Shifted Chebyshev polynomials of the first kind, 3) Shifted Chebyshev polynomials of the third kind, 4) Shifted Chebyshev polynomials of the fourth kind, and 5) Gegenbauer polynomials are considered to perform the numerical investigations in the test examples. Further, the obtained results are presented in the form of tables and figures. The numerical results are also compared with some known methods from the literature. Full article
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Article
Dynamic Keynesian Model of Economic Growth with Memory and Lag
Mathematics 2019, 7(2), 178; https://doi.org/10.3390/math7020178 - 15 Feb 2019
Cited by 14
Abstract
A mathematical model of economic growth with fading memory and continuous distribution of delay time is suggested. This model can be considered as a generalization of the standard Keynesian macroeconomic model. To take into account the memory and gamma-distributed lag we use the [...] Read more.
A mathematical model of economic growth with fading memory and continuous distribution of delay time is suggested. This model can be considered as a generalization of the standard Keynesian macroeconomic model. To take into account the memory and gamma-distributed lag we use the Abel-type integral and integro-differential operators with the confluent hypergeometric Kummer function in the kernel. These operators allow us to propose an economic accelerator, in which the memory and lag are taken into account. The fractional differential equation, which describes the dynamics of national income in this generalized model, is suggested. The solution of this fractional differential equation is obtained in the form of series of the confluent hypergeometric Kummer functions. The asymptotic behavior of national income, which is described by this solution, is considered. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
Article
A Partial-Consensus Posterior-Aggregation FAHP Method—Supplier Selection Problem as an Example
Mathematics 2019, 7(2), 179; https://doi.org/10.3390/math7020179 - 15 Feb 2019
Cited by 19
Abstract
Existing fuzzy analytic hierarchy process (FAHP) methods usually aggregate the fuzzy pairwise comparison results produced by multiple decision-makers (DMs) rather than the fuzzy weights estimations. This is problematic because fuzzy pairwise comparison results are subject to uncertainty and lack consensus. To address this [...] Read more.
Existing fuzzy analytic hierarchy process (FAHP) methods usually aggregate the fuzzy pairwise comparison results produced by multiple decision-makers (DMs) rather than the fuzzy weights estimations. This is problematic because fuzzy pairwise comparison results are subject to uncertainty and lack consensus. To address this problem, a partial-consensus posterior-aggregation FAHP (PCPA-FAHP) approach is proposed in this study. The PCPA-FAHP approach seeks a partial consensus among most DMs instead of an overall consensus among all DMs, thereby increasing the possibility of reaching a consensus. Subsequently, the aggregation result is defuzzified using the prevalent center-of-gravity method. The PCPA-FAHP approach was applied to a supplier selection problem to validate its effectiveness. According to the experimental results, the PCPA-FAHP approach not only successfully found out the partial consensus among the DMs, but also shrunk the widths of the estimated fuzzy weights to enhance the precision of the FAHP analysis. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications)
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Article
Hankel and Toeplitz Determinants for a Subclass of q-Starlike Functions Associated with a General Conic Domain
Mathematics 2019, 7(2), 181; https://doi.org/10.3390/math7020181 - 15 Feb 2019
Cited by 39
Abstract
By using a certain general conic domain as well as the quantum (or q-) calculus, here we define and investigate a new subclass of normalized analytic and starlike functions in the open unit disk U . In particular, we find the Hankel determinant and the Toeplitz matrices for this newly-defined class of analytic q-starlike functions. We also highlight some known consequences of our main results. Full article
Article
New Integral Inequalities via the Katugampola Fractional Integrals for Functions Whose Second Derivatives Are Strongly η-Convex
Mathematics 2019, 7(2), 183; https://doi.org/10.3390/math7020183 - 15 Feb 2019
Cited by 12
Abstract
In this paper, we introduced some new integral inequalities of the Hermite–Hadamard type for functions whose second derivatives in absolute values at certain powers are strongly η -convex functions via the Katugampola fractional integrals. Full article
(This article belongs to the Special Issue Inequalities)
Article
Lexicographic Orders of Intuitionistic Fuzzy Values and Their Relationships
Mathematics 2019, 7(2), 166; https://doi.org/10.3390/math7020166 - 13 Feb 2019
Cited by 32
Abstract
Intuitionistic fuzzy multiple attribute decision making deals with the issue of ranking alternatives based on the decision information quantified in terms of intuitionistic fuzzy values. Lexicographic orders can serve as efficient and indispensable tools for comparing intuitionistic fuzzy values. This paper introduces a [...] Read more.
Intuitionistic fuzzy multiple attribute decision making deals with the issue of ranking alternatives based on the decision information quantified in terms of intuitionistic fuzzy values. Lexicographic orders can serve as efficient and indispensable tools for comparing intuitionistic fuzzy values. This paper introduces a number of lexicographic orders by means of several measures such as the membership, non-membership, score, accuracy and expectation score functions. Some equivalent characterizations and illustrative examples are provided, from which the relationships among these lexicographic orders are ascertained. We also propose three different compatible properties of preorders with respect to the algebraic sum and scalar product operations of intuitionistic fuzzy values, and apply them to the investigation of compatible properties of various lexicographic orders. In addition, a benchmark problem regarding risk investment is further explored to give a comparative analysis of different lexicographic orders and highlight the practical value of the obtained results for solving real-world decision-making problems. Full article
(This article belongs to the Section Engineering Mathematics)
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Article
Analysis of General Humoral Immunity HIV Dynamics Model with HAART and Distributed Delays
Mathematics 2019, 7(2), 157; https://doi.org/10.3390/math7020157 - 09 Feb 2019
Cited by 38
Abstract
This paper deals with the study of an HIV dynamics model with two target cells, macrophages and CD4 + T cells and three categories of infected cells, short-lived, long-lived and latent in order to get better insights into HIV infection within the body. The model incorporates therapeutic modalities such as reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs). The model is incorporated with distributed time delays to characterize the time between an HIV contact of an uninfected target cell and the creation of mature HIV. The effect of antibody on HIV infection is analyzed. The production and removal rates of the ten compartments of the model are given by general nonlinear functions which satisfy reasonable conditions. Nonnegativity and ultimately boundedness of the solutions are proven. Using the Lyapunov method, the global stability of the equilibria of the model is proven. Numerical simulations of the system are provided to confirm the theoretical results. We have shown that the antibodies can play a significant role in controlling the HIV infection, but it cannot clear the HIV particles from the plasma. Moreover, we have demonstrated that the intracellular time delay plays a similar role as the Highly Active Antiretroviral Therapies (HAAT) drugs in eliminating the HIV particles. Full article
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Article
Fractional Derivatives: The Perspective of System Theory
Mathematics 2019, 7(2), 150; https://doi.org/10.3390/math7020150 - 05 Feb 2019
Cited by 20
Abstract
This paper addresses the present day problem of multiple proposals for operators under the umbrella of “fractional derivatives”. Several papers demonstrated that various of those “novel” definitions are incorrect. Here the classical system theory is applied to develop a unified framework to clarify [...] Read more.
This paper addresses the present day problem of multiple proposals for operators under the umbrella of “fractional derivatives”. Several papers demonstrated that various of those “novel” definitions are incorrect. Here the classical system theory is applied to develop a unified framework to clarify this important topic in Fractional Calculus. Full article
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)
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Article
Desiderata for Fractional Derivatives and Integrals
Mathematics 2019, 7(2), 149; https://doi.org/10.3390/math7020149 - 04 Feb 2019
Cited by 43
Abstract
The purpose of this brief article is to initiate discussions in this special issue by proposing desiderata for calling an operator a fractional derivative or a fractional integral. Our desiderata are neither axioms nor do they define fractional derivatives or integrals uniquely. Instead [...] Read more.
The purpose of this brief article is to initiate discussions in this special issue by proposing desiderata for calling an operator a fractional derivative or a fractional integral. Our desiderata are neither axioms nor do they define fractional derivatives or integrals uniquely. Instead they intend to stimulate the field by providing guidelines based on a small number of time honoured and well established criteria. Full article
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)
Article
A Novel Bat Algorithm with Multiple Strategies Coupling for Numerical Optimization
Mathematics 2019, 7(2), 135; https://doi.org/10.3390/math7020135 - 01 Feb 2019
Cited by 69
Abstract
A bat algorithm (BA) is a heuristic algorithm that operates by imitating the echolocation behavior of bats to perform global optimization. The BA is widely used in various optimization problems because of its excellent performance. In the bat algorithm, the global search capability [...] Read more.
A bat algorithm (BA) is a heuristic algorithm that operates by imitating the echolocation behavior of bats to perform global optimization. The BA is widely used in various optimization problems because of its excellent performance. In the bat algorithm, the global search capability is determined by the parameter loudness and frequency. However, experiments show that each operator in the algorithm can only improve the performance of the algorithm at a certain time. In this paper, a novel bat algorithm with multiple strategies coupling (mixBA) is proposed to solve this problem. To prove the effectiveness of the algorithm, we compared it with CEC2013 benchmarks test suits. Furthermore, the Wilcoxon and Friedman tests were conducted to distinguish the differences between it and other algorithms. The results prove that the proposed algorithm is significantly superior to others on the majority of benchmark functions. Full article
(This article belongs to the Special Issue Evolutionary Computation)
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Article
Calculating Nodal Voltages Using the Admittance Matrix Spectrum of an Electrical Network
Mathematics 2019, 7(1), 106; https://doi.org/10.3390/math7010106 - 20 Jan 2019
Cited by 8
Abstract
Calculating nodal voltages and branch current flows in a meshed network is fundamental to electrical engineering. This work demonstrates how such calculations can be performed using the eigenvalues and eigenvectors of the Laplacian matrix which describes the connectivity of the electrical network. These [...] Read more.
Calculating nodal voltages and branch current flows in a meshed network is fundamental to electrical engineering. This work demonstrates how such calculations can be performed using the eigenvalues and eigenvectors of the Laplacian matrix which describes the connectivity of the electrical network. These insights should permit the functioning of electrical networks to be understood in the context of spectral analysis. Full article
(This article belongs to the Special Issue Mathematical Methods in Applied Sciences)
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Article
ω-Interpolative Ćirić-Reich-Rus-Type Contractions
Mathematics 2019, 7(1), 57; https://doi.org/10.3390/math7010057 - 08 Jan 2019
Cited by 43
Abstract
In this paper, using the concept of ω -admissibility, we prove some fixed point results for interpolate Ćirić-Reich-Rus-type contraction mappings. We also present some consequences and a useful example. Full article
(This article belongs to the Special Issue Recent Advances in Fixed Point Theory and Its Applications)
Article
An Iterative Algorithm for Solving Generalized Variational Inequalities and Fixed Points Problems
Mathematics 2019, 7(1), 61; https://doi.org/10.3390/math7010061 - 08 Jan 2019
Cited by 55
Abstract
In this paper, a generalized variational inequality and fixed points problem is presented. An iterative algorithm is introduced for finding a solution of the generalized variational inequalities and fixed point of two quasi-pseudocontractive operators under a nonlinear transformation. Strong convergence of the suggested [...] Read more.
In this paper, a generalized variational inequality and fixed points problem is presented. An iterative algorithm is introduced for finding a solution of the generalized variational inequalities and fixed point of two quasi-pseudocontractive operators under a nonlinear transformation. Strong convergence of the suggested algorithm is demonstrated. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)

Review

Jump to: Research

Review
Rules for Fractional-Dynamic Generalizations: Difficulties of Constructing Fractional Dynamic Models
Mathematics 2019, 7(6), 554; https://doi.org/10.3390/math7060554 - 18 Jun 2019
Cited by 18
Abstract
This article is a review of problems and difficulties arising in the construction of fractional-dynamic analogs of standard models by using fractional calculus. These fractional generalizations allow us to take into account the effects of memory and non-locality, distributed lag, and scaling. We [...] Read more.
This article is a review of problems and difficulties arising in the construction of fractional-dynamic analogs of standard models by using fractional calculus. These fractional generalizations allow us to take into account the effects of memory and non-locality, distributed lag, and scaling. We formulate rules (principles) for constructing fractional generalizations of standard models, which were described by differential equations of integer order. Important requirements to building fractional generalization of dynamical models (the rules for “fractional-dynamic generalizers”) are represented as the derivability principle, the multiplicity principle, the solvability and correspondence principles, and the interpretability principle. The characteristic properties of fractional derivatives of non-integer order are the violation of standard rules and properties that are fulfilled for derivatives of integer order. These non-standard mathematical properties allow us to describe non-standard processes and phenomena associated with non-locality and memory. However, these non-standard properties lead to restrictions in the sequential and self-consistent construction of fractional generalizations of standard models. In this article, we give examples of problems arising due to the non-standard properties of fractional derivatives in construction of fractional generalizations of standard dynamic models in economics. Full article
(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)
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Review
On History of Mathematical Economics: Application of Fractional Calculus
Mathematics 2019, 7(6), 509; https://doi.org/10.3390/math7060509 - 04 Jun 2019
Cited by 36
Abstract
Modern economics was born in the Marginal revolution and the Keynesian revolution. These revolutions led to the emergence of fundamental concepts and methods in economic theory, which allow the use of differential and integral calculus to describe economic phenomena, effects, and processes. At [...] Read more.
Modern economics was born in the Marginal revolution and the Keynesian revolution. These revolutions led to the emergence of fundamental concepts and methods in economic theory, which allow the use of differential and integral calculus to describe economic phenomena, effects, and processes. At the present moment the new revolution, which can be called “Memory revolution”, is actually taking place in modern economics. This revolution is intended to “cure amnesia” of modern economic theory, which is caused by the use of differential and integral operators of integer orders. In economics, the description of economic processes should take into account that the behavior of economic agents may depend on the history of previous changes in economy. The main mathematical tool designed to “cure amnesia” in economics is fractional calculus that is a theory of integrals, derivatives, sums, and differences of non-integer orders. This paper contains a brief review of the history of applications of fractional calculus in modern mathematical economics and economic theory. The first stage of the Memory Revolution in economics is associated with the works published in 1966 and 1980 by Clive W. J. Granger, who received the Nobel Memorial Prize in Economic Sciences in 2003. We divide the history of the application of fractional calculus in economics into the following five stages of development (approaches): ARFIMA; fractional Brownian motion; econophysics; deterministic chaos; mathematical economics. The modern stage (mathematical economics) of the Memory revolution is intended to include in the modern economic theory new economic concepts and notions that allow us to take into account the presence of memory in economic processes. The current stage actually absorbs the Granger approach based on ARFIMA models that used only the Granger–Joyeux–Hosking fractional differencing and integrating, which really are the well-known Grunwald–Letnikov fractional differences. The modern stage can also absorb other approaches by formulation of new economic notions, concepts, effects, phenomena, and principles. Some comments on possible future directions for development of the fractional mathematical economics are proposed. Full article
(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)
Review
Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial
Mathematics 2019, 7(5), 407; https://doi.org/10.3390/math7050407 - 07 May 2019
Cited by 34
Abstract
Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches [...] Read more.
Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann–Liouville and Caputo’s derivatives converge, on long times, to the Grünwald–Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
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