Editorial

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2 pages, 188 KiB

Open AccessEditorial

Preface to “Mathematical Methods, Modelling and Applications”

by Lucas Jódar and Rafael Company

Mathematics 2022, 10(9), 1607; https://doi.org/10.3390/math10091607 - 09 May 2022

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Abstract

The reality is more complex than it seems [...] Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

Research

Jump to: Editorial

20 pages, 591 KiB

Open AccessArticle

Advances in the Approximation of the Matrix Hyperbolic Tangent

by Javier Ibáñez, José M. Alonso, Jorge Sastre, Emilio Defez and Pedro Alonso-Jordá

Mathematics 2021, 9(11), 1219; https://doi.org/10.3390/math9111219 - 27 May 2021

Cited by 7 | Viewed by 2058

Abstract

In this paper, we introduce two approaches to compute the matrix hyperbolic tangent. While one of them is based on its own definition and uses the matrix exponential, the other one is focused on the expansion of its Taylor series. For this second [...] Read more.

In this paper, we introduce two approaches to compute the matrix hyperbolic tangent. While one of them is based on its own definition and uses the matrix exponential, the other one is focused on the expansion of its Taylor series. For this second approximation, we analyse two different alternatives to evaluate the corresponding matrix polynomials. This resulted in three stable and accurate codes, which we implemented in MATLAB and numerically and computationally compared by means of a battery of tests composed of distinct state-of-the-art matrices. Our results show that the Taylor series-based methods were more accurate, although somewhat more computationally expensive, compared with the approach based on the exponential matrix. To avoid this drawback, we propose the use of a set of formulas that allows us to evaluate polynomials in a more efficient way compared with that of the traditional Paterson–Stockmeyer method, thus, substantially reducing the number of matrix products (practically equal in number to the approach based on the matrix exponential), without penalising the accuracy of the result. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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19 pages, 3134 KiB

Open AccessArticle

Applied Machine Learning Algorithms for Courtyards Thermal Patterns Accurate Prediction

by Eduardo Diz-Mellado, Samuele Rubino, Soledad Fernández-García, Macarena Gómez-Mármol, Carlos Rivera-Gómez and Carmen Galán-Marín

Mathematics 2021, 9(10), 1142; https://doi.org/10.3390/math9101142 - 18 May 2021

Cited by 18 | Viewed by 2335

Abstract

Currently, there is a lack of accurate simulation tools for the thermal performance modeling of courtyards due to their intricate thermodynamics. Machine Learning (ML) models have previously been used to predict and evaluate the structural performance of buildings as a means of solving [...] Read more.

Currently, there is a lack of accurate simulation tools for the thermal performance modeling of courtyards due to their intricate thermodynamics. Machine Learning (ML) models have previously been used to predict and evaluate the structural performance of buildings as a means of solving complex mathematical problems. Nevertheless, the microclimatic conditions of the building surroundings have not been as thoroughly addressed by these methodologies. To this end, in this paper, the adaptation of ML techniques as a more comprehensive methodology to fill this research gap, covering not only the prediction of the courtyard microclimate but also the interpretation of experimental data and pattern recognition, is proposed. Accordingly, based on the climate zoning and aspect ratios of 32 monitored case studies located in the South of Spain, the Support Vector Regression (SVR) method was applied to predict the measured temperature inside the courtyard. The results provided by this strategy showed good accuracy when compared to monitored data. In particular, for two representative case studies, if the daytime slot with the highest urban overheating is considered, the relative error is almost below 0.05%. Additionally, values for statistical parameters are in good agreement with other studies in the literature, which use more computationally expensive CFD models and show more accuracy than existing commercial tools. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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18 pages, 4393 KiB

Open AccessArticle

Hybrid Model for Time Series of Complex Structure with ARIMA Components

by Oksana Mandrikova, Nadezhda Fetisova and Yuriy Polozov

Mathematics 2021, 9(10), 1122; https://doi.org/10.3390/math9101122 - 15 May 2021

Cited by 22 | Viewed by 1792

Abstract

A hybrid model for the time series of complex structure (HMTS) was proposed. It is based on the combination of function expansions in a wavelet series with ARIMA models. HMTS has regular and anomalous components. The time series components, obtained after expansion, have [...] Read more.

A hybrid model for the time series of complex structure (HMTS) was proposed. It is based on the combination of function expansions in a wavelet series with ARIMA models. HMTS has regular and anomalous components. The time series components, obtained after expansion, have a simpler structure that makes it possible to identify the ARIMA model if the components are stationary. This allows us to obtain a more accurate ARIMA model for a time series of complicated structure and to extend the area for application. To identify the HMTS anomalous component, threshold functions are applied. This paper describes a technique to identify HMTS and proposes operations to detect anomalies. With the example of an ionospheric parameter time series, we show the HMTS efficiency, describe the results and their application in detecting ionospheric anomalies. The HMTS was compared with the nonlinear autoregression neural network NARX, which confirmed HMTS efficiency. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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24 pages, 3026 KiB

Open AccessArticle

Steady Fluid–Structure Coupling Interface of Circular Membrane under Liquid Weight Loading: Closed-Form Solution for Differential-Integral Equations

by Xue Li, Jun-Yi Sun, Xiao-Chen Lu, Zhi-Xin Yang and Xiao-Ting He

Mathematics 2021, 9(10), 1105; https://doi.org/10.3390/math9101105 - 13 May 2021

Cited by 5 | Viewed by 1504

Abstract

In this paper, the problem of fluid–structure interaction of a circular membrane under liquid weight loading is formulated and is solved analytically. The circular membrane is initially flat and works as the bottom of a cylindrical cup or bucket. The initially flat circular [...] Read more.

In this paper, the problem of fluid–structure interaction of a circular membrane under liquid weight loading is formulated and is solved analytically. The circular membrane is initially flat and works as the bottom of a cylindrical cup or bucket. The initially flat circular membrane will undergo axisymmetric deformation and deflection after a certain amount of liquid is poured into the cylindrical cup. The amount of the liquid poured determines the deformation and deflection of the circular membrane, while in turn, the deformation and deflection of the circular membrane changes the shape and distribution of the liquid poured on the deformed and deflected circular membrane, resulting in the so-called fluid-structure interaction between liquid and membrane. For a given amount of liquid, the fluid-structure interaction will eventually reach a static equilibrium and the fluid-structure coupling interface is steady, resulting in a static problem of axisymmetric deformation and deflection of the circular membrane under the weight of given liquid. The established governing equations for the static problem contain both differential operation and integral operation and the power series method plays an irreplaceable role in solving the differential-integral equations. Finally, the closed-form solutions for stress and deflection are presented and are confirmed to be convergent by the numerical examples conducted. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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19 pages, 707 KiB

Open AccessArticle

Mechanical Models for Hermite Interpolation on the Unit Circle

by Elías Berriochoa, Alicia Cachafeiro, Héctor García Rábade and José Manuel García-Amor

Mathematics 2021, 9(9), 1043; https://doi.org/10.3390/math9091043 - 06 May 2021

Cited by 3 | Viewed by 1640

Abstract

In the present paper, we delve into the study of nodal systems on the unit circle that meet certain separation properties. Our aim was to study the Hermite interpolation process on the unit circle by using these nodal arrays. The target was to [...] Read more.

In the present paper, we delve into the study of nodal systems on the unit circle that meet certain separation properties. Our aim was to study the Hermite interpolation process on the unit circle by using these nodal arrays. The target was to develop the corresponding interpolation theory in order to make practical use of these nodal systems linked to certain mechanical models that fit these distributions. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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16 pages, 2982 KiB

Open AccessArticle

Modeling of Fundus Laser Exposure for Estimating Safe Laser Coagulation Parameters in the Treatment of Diabetic Retinopathy

by Aleksandr Shirokanev, Nataly Ilyasova, Nikita Andriyanov, Evgeniy Zamytskiy, Andrey Zolotarev and Dmitriy Kirsh

Mathematics 2021, 9(9), 967; https://doi.org/10.3390/math9090967 - 26 Apr 2021

Cited by 12 | Viewed by 1938

Abstract

A personalized medical approach can make diabetic retinopathy treatment more effective. To select effective methods of treatment, deep analysis and diagnostic data of a patient’s fundus are required. For this purpose, flat optical coherence tomography images are used to restore the three-dimensional structure [...] Read more.

A personalized medical approach can make diabetic retinopathy treatment more effective. To select effective methods of treatment, deep analysis and diagnostic data of a patient’s fundus are required. For this purpose, flat optical coherence tomography images are used to restore the three-dimensional structure of the fundus. Heat propagation through this structure is simulated via numerical methods. The article proposes algorithms for smooth segmentation of the retina for 3D model reconstruction and mathematical modeling of laser exposure while considering various parameters. The experiment was based on a two-fold improvement in the number of intervals and the calculation of the root mean square deviation between the modeled temperature values and the corresponding coordinates shown for the convergence of the integro-interpolation method (balance method). By doubling the number of intervals for a specific spatial or temporal coordinate, a decrease in the root mean square deviation takes place between the simulated temperature values by a factor of 1.7–5.9. This modeling allows us to estimate the basic parameters required for the actual practice of diabetic retinopathy treatment while optimizing for efficiency and safety. Mathematical modeling is used to estimate retina heating caused by the spread of heat from the vascular layer, where the temperature rose to 45 °C in 0.2 ms. It was identified that the formation of two coagulates is possible when they are located at least 180 μm from each other. Moreover, the distance can be reduced to 160 μm with a 15 ms delay between imaging. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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18 pages, 1550 KiB

Open AccessArticle

Modeling Political Corruption in Spain

by Elena de la Poza, Lucas Jódar and Paloma Merello

Mathematics 2021, 9(9), 952; https://doi.org/10.3390/math9090952 - 24 Apr 2021

Cited by 3 | Viewed by 3518

Abstract

Political corruption is a universal phenomenon. Even though it is a cross-country reality, its level of intensity and the manner of its effect vary worldwide. In Spain, the demonstrated political corruption cases that have been echoed by the media in recent years for [...] Read more.

Political corruption is a universal phenomenon. Even though it is a cross-country reality, its level of intensity and the manner of its effect vary worldwide. In Spain, the demonstrated political corruption cases that have been echoed by the media in recent years for their economic, judicial and social significance are merely the tip of the iceberg as regards a problem hidden by many interested parties, plus the shortage of the means to fight against it. This study models and quantifies the population at risk of committing political corruption in Spain by identifying and quantifying the drivers that explain political corruption. Having quantified the problem, the model allows changes to be made in parameters, as well as fiscal, economic and legal measures being simulated, to quantify and better understand their impact on Spanish citizenship. Our results suggest increasing women’s leadership positions to mitigate this problem, plus changes in the political Parties’ Law in Spain and increasing the judiciary system’s budget. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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17 pages, 1311 KiB

Open AccessArticle

The Dirichlet-to-Neumann Map in a Disk with a One-Step Radial Potential: An Analytical and Numerical Study

by Sagrario Lantarón and Susana Merchán

Mathematics 2021, 9(8), 794; https://doi.org/10.3390/math9080794 - 07 Apr 2021

Cited by 1 | Viewed by 1226

Abstract

Herein, we considered the Schrödinger operator with a potential q on a disk and the map that associates to q the corresponding Dirichlet-to-Neumann (DtN) map. We provide some numerical and analytical results on the range of this map and its stability for the [...] Read more.

Herein, we considered the Schrödinger operator with a potential q on a disk and the map that associates to q the corresponding Dirichlet-to-Neumann (DtN) map. We provide some numerical and analytical results on the range of this map and its stability for the particular class of one-step radial potentials. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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25 pages, 14002 KiB

Open AccessArticle

Non-Stationary Contaminant Plumes in the Advective-Diffusive Regime

by Iván Alhama, Gonzalo García-Ros and Matteo Icardi

Mathematics 2021, 9(7), 725; https://doi.org/10.3390/math9070725 - 27 Mar 2021

Cited by 5 | Viewed by 1579

Abstract

Porous media with low/moderate regional velocities can exhibit a complex dynamic of contamination plumes, in which advection and molecular diffusion are comparable. In this work, we present a two-dimensional scenario with a constant concentration source and impermeable upper and lower boundaries. In order [...] Read more.

Porous media with low/moderate regional velocities can exhibit a complex dynamic of contamination plumes, in which advection and molecular diffusion are comparable. In this work, we present a two-dimensional scenario with a constant concentration source and impermeable upper and lower boundaries. In order to characterise the plume patterns, a detailed discriminated dimensionless technique is used to obtain the dimensionless groups that govern the problem: an aspect ratio of the domain including characteristic lengths, and two others relating time and the horizontal length of the spread of contamination. The monomials are related to each other to enable their dependences to be translated into a set of new universal abacuses. Extensive numerical simulations were carried out to check the monomials and to plot these type curves. The abacuses provide a tool to directly manage the contamination process, covering a wide spectrum of possible real cases. Among other applications of interest, they predict the maximum horizontal and transversal plume extensions and the time-spatial dependences of iso-concentration patterns according to the physical parameters of the problem. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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27 pages, 917 KiB

Open AccessArticle

Conservative Finite Volume Schemes for Multidimensional Fragmentation Problems

by Jitraj Saha and Andreas Bück

Mathematics 2021, 9(6), 635; https://doi.org/10.3390/math9060635 - 17 Mar 2021

Cited by 4 | Viewed by 1619

Abstract

In this article, a new numerical scheme for the solution of the multidimensional fragmentation problem is presented. It is the first that uses the conservative form of the multidimensional problem. The idea to apply the finite volume scheme for solving one-dimensional linear fragmentation [...] Read more.

In this article, a new numerical scheme for the solution of the multidimensional fragmentation problem is presented. It is the first that uses the conservative form of the multidimensional problem. The idea to apply the finite volume scheme for solving one-dimensional linear fragmentation problems is extended over a generalized multidimensional setup. The derivation is given in detail for two-dimensional and three-dimensional problems; an outline for the extension to higher dimensions is also presented. Additionally, the existing one-dimensional finite volume scheme for solving conservative one-dimensional multi-fragmentation equation is extended to solve multidimensional problems. The accuracy and efficiency of both proposed schemes is analyzed for several test problems. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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16 pages, 968 KiB

Open AccessArticle

Stochastic Modeling of Plant Virus Propagation with Biological Control

by Benito Chen-Charpentier

Mathematics 2021, 9(5), 456; https://doi.org/10.3390/math9050456 - 24 Feb 2021

Cited by 7 | Viewed by 2508

Abstract

Plants are vital for man and many species. They are sources of food, medicine, fiber for clothes and materials for shelter. They are a fundamental part of a healthy environment. However, plants are subject to virus diseases. In plants most of the virus [...] Read more.

Plants are vital for man and many species. They are sources of food, medicine, fiber for clothes and materials for shelter. They are a fundamental part of a healthy environment. However, plants are subject to virus diseases. In plants most of the virus propagation is done by a vector. The traditional way of controlling the insects is to use insecticides that have a negative effect on the environment. A more environmentally friendly way to control the insects is to use predators that will prey on the vector, such as birds or bats. In this paper we modify a plant-virus propagation model with delays. The model is written using delay differential equations. However, it can also be expressed in terms of biochemical reactions, which is more realistic for small populations. Since there are always variations in the populations, errors in the measured values and uncertainties, we use two methods to introduce randomness: stochastic differential equations and the Gillespie algorithm. We present numerical simulations. The Gillespie method produces good results for plant-virus population models. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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12 pages, 374 KiB

Open AccessArticle

The Relativistic Harmonic Oscillator in a Uniform Gravitational Field

by Michael M. Tung

Mathematics 2021, 9(4), 294; https://doi.org/10.3390/math9040294 - 03 Feb 2021

Cited by 5 | Viewed by 2145

Abstract

We present the relativistic generalization of the classical harmonic oscillator suspended within a uniform gravitational field measured by an observer in a laboratory in which the suspension point of the spring is fixed. The starting point of this analysis is a variational approach [...] Read more.

We present the relativistic generalization of the classical harmonic oscillator suspended within a uniform gravitational field measured by an observer in a laboratory in which the suspension point of the spring is fixed. The starting point of this analysis is a variational approach based on the Euler–Lagrange formalism. Due to the conceptual differences of mass in the framework of special relativity compared with the classical model, the correct treatment of the relativistic gravitational potential requires special attention. It is proved that the corresponding relativistic equation of motion has unique periodic solutions. Some approximate analytical results including the next-to-leading-order term in the non-relativistic limit are also examined. The discussion is rounded up with a numerical simulation of the full relativistic results in the case of a strong gravity field. Finally, the dynamics of the model is further explored by investigating phase space and its quantitative relativistic features. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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15 pages, 481 KiB

Open AccessArticle

Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness

by María Consuelo Casabán, Rafael Company and Lucas Jódar

Mathematics 2021, 9(3), 206; https://doi.org/10.3390/math9030206 - 20 Jan 2021

Cited by 3 | Viewed by 1257

Abstract

This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the [...] Read more.

This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. After studying positivity and conditional random mean square stability, the computation of the expectation and variance of the approximating stochastic process is not performed directly but through using a set of sampling finite difference schemes coming out by taking realizations of the random scheme and using Monte Carlo technique. Thus, the storage accumulation of symbolic expressions collapsing the approach is avoided keeping reliability. Results are simulated and a procedure for the numerical computation is given. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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17 pages, 384 KiB

Open AccessArticle

Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

by Juan-Carlos Cortés, Elena López-Navarro, José-Vicente Romero and María-Dolores Roselló

Mathematics 2021, 9(3), 204; https://doi.org/10.3390/math9030204 - 20 Jan 2021

Cited by 2 | Viewed by 1701

Abstract

We combine the stochastic perturbation method with the maximum entropy principle to construct approximations of the first probability density function of the steady-state solution of a class of nonlinear oscillators subject to small perturbations in the nonlinear term and driven by a stochastic [...] Read more.

We combine the stochastic perturbation method with the maximum entropy principle to construct approximations of the first probability density function of the steady-state solution of a class of nonlinear oscillators subject to small perturbations in the nonlinear term and driven by a stochastic excitation. The nonlinearity depends both upon position and velocity, and the excitation is given by a stationary Gaussian stochastic process with certain additional properties. Furthermore, we approximate higher-order moments, the variance, and the correlation functions of the solution. The theoretical findings are illustrated via some numerical experiments that confirm that our approximations are reliable. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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16 pages, 1146 KiB

Open AccessArticle

Quadrature Integration Techniques for Random Hyperbolic PDE Problems

by Rafael Company, Vera N. Egorova and Lucas Jódar

Mathematics 2021, 9(2), 160; https://doi.org/10.3390/math9020160 - 14 Jan 2021

Cited by 2 | Viewed by 1700

Abstract

In this paper, we consider random hyperbolic partial differential equation (PDE) problems following the mean square approach and Laplace transform technique. Randomness requires not only the computation of the approximating stochastic processes, but also its statistical moments. Hence, appropriate numerical methods should allow [...] Read more.

In this paper, we consider random hyperbolic partial differential equation (PDE) problems following the mean square approach and Laplace transform technique. Randomness requires not only the computation of the approximating stochastic processes, but also its statistical moments. Hence, appropriate numerical methods should allow for the efficient computation of the expectation and variance. Here, we analyse different numerical methods around the inverse Laplace transform and its evaluation by using several integration techniques, including midpoint quadrature rule, Gauss–Laguerre quadrature and its extensions, and the Talbot algorithm. Simulations, numerical convergence, and computational process time with experiments are shown. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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18 pages, 1069 KiB

Open AccessArticle

Convergence and Stability of a Parametric Class of Iterative Schemes for Solving Nonlinear Systems

by Alicia Cordero, Eva G. Villalba, Juan R. Torregrosa and Paula Triguero-Navarro

Mathematics 2021, 9(1), 86; https://doi.org/10.3390/math9010086 - 03 Jan 2021

Cited by 17 | Viewed by 2170

Abstract

A new parametric class of iterative schemes for solving nonlinear systems is designed. The third- or fourth-order convergence, depending on the values of the parameter being proven. The analysis of the dynamical behavior of this class in the context of scalar nonlinear equations [...] Read more.

A new parametric class of iterative schemes for solving nonlinear systems is designed. The third- or fourth-order convergence, depending on the values of the parameter being proven. The analysis of the dynamical behavior of this class in the context of scalar nonlinear equations is presented. This study gives us important information about the stability and reliability of the members of the family. The numerical results obtained by applying different elements of the family for solving the Hammerstein integral equation and the Fisher’s equation confirm the theoretical results. Full article

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20 pages, 3112 KiB

Open AccessArticle

Developing and Applying a Selection Model for Corrugated Box Precision Printing Machine Suppliers

by Chin-Tsai Lin and Cheng-Yu Chiang

Mathematics 2021, 9(1), 68; https://doi.org/10.3390/math9010068 - 30 Dec 2020

Cited by 4 | Viewed by 2153

Abstract

Corrugated box printing machines are precision equipment produced by markedly few manufacturers. They involve high investment cost and risk. Having a corrugated box precision printing machine (CBPPM) supplier with a good reputation enables a corrugated box manufacturer to maintain its competitive advantage. Accordingly, [...] Read more.

Corrugated box printing machines are precision equipment produced by markedly few manufacturers. They involve high investment cost and risk. Having a corrugated box precision printing machine (CBPPM) supplier with a good reputation enables a corrugated box manufacturer to maintain its competitive advantage. Accordingly, establishing an effective CBPPM supplier selection model is crucial for corrugated box manufacturers. This study established a two-stage CBPPM supplier selection model. The first stage involved the use of a modified Delphi method to construct a supplier selection hierarchy with five criteria and 14 subcriteria. In the second stage, an analytic network process was employed to calculate the weights of criteria and subcriteria and to determine the optimal supplier. According to the results, the five criteria in the model, in descending order of importance, are quality, commitment, cost, service attitude, and reputation. This model can provide insights for corrugated box manufacturers formulating their CBPPM supplier selection strategy. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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14 pages, 1867 KiB

Open AccessArticle

Relating Hydraulic Conductivity Curve to Soil-Water Retention Curve Using a Fractal Model

by Carlos Fuentes, Carlos Chávez and Fernando Brambila

Mathematics 2020, 8(12), 2201; https://doi.org/10.3390/math8122201 - 10 Dec 2020

Cited by 4 | Viewed by 1868

Abstract

In the study of water transference in soil according to Darcy law, the knowledge of hydrodynamic characteristics, formed by the water retention curve θ(ψ), and the hydraulic conductivity curve K(ψ) are of great importance. The first one relates the water volumetric content (θ) [...] Read more.

In the study of water transference in soil according to Darcy law, the knowledge of hydrodynamic characteristics, formed by the water retention curve θ(ψ), and the hydraulic conductivity curve K(ψ) are of great importance. The first one relates the water volumetric content (θ) with the water-soil pressure (ψ); the second one, the hydraulic conductivity (K) with the water-soil pressure. The objective of this work is to establish relationships between both curves using concepts of probability theory and fractal geometry in order to reduce the number of unknown functions. The introduction of four definitions used at the literature of the pore effective radius that is involve in the general model has permitted to establish four new specials models to predict the relative hydraulic conductivity. Some additional considerations related to the definitions of flow effective area and the tortuosity factor have allow us to deduce four classical models that are extensively used in different studies. In particular, we have given some interpretations of its empirical parameters in the fractal geometry context. The resulting functions for hydrodynamic characteristics can be utilized in many studies of water movement in the soil. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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16 pages, 394 KiB

Open AccessArticle

Analysis of Generalized Multistep Collocation Solutions for Oscillatory Volterra Integral Equations

by Hao Chen, Ling Liu and Junjie Ma

Mathematics 2020, 8(11), 2004; https://doi.org/10.3390/math8112004 - 10 Nov 2020

Cited by 4 | Viewed by 1571

Abstract

In this work, we introduce a class of generalized multistep collocation methods for solving oscillatory Volterra integral equations, and study two kinds of convergence analysis. The error estimate with respect to the stepsize is given based on the interpolation remainder, and the nonclassical [...] Read more.

In this work, we introduce a class of generalized multistep collocation methods for solving oscillatory Volterra integral equations, and study two kinds of convergence analysis. The error estimate with respect to the stepsize is given based on the interpolation remainder, and the nonclassical convergence analysis with respect to oscillation is developed by investigating the asymptotic property of highly oscillatory integrals. Besides, the linear stability is analyzed with the help of generalized Schur polynomials. Several numerical tests are given to show that the numerical results coincide with our theoretical estimates. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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30 pages, 928 KiB

Open AccessArticle

Corrected Evolutive Kendall’s τ Coefficients for Incomplete Rankings with Ties: Application to Case of Spotify Lists

by Francisco Pedroche and J. Alberto Conejero

Mathematics 2020, 8(10), 1828; https://doi.org/10.3390/math8101828 - 18 Oct 2020

Cited by 3 | Viewed by 2640

Abstract

Mathematical analysis of rankings is essential for a wide range of scientific, public, and industrial applications (e.g., group decision-making, organizational methods, R&D sponsorship, recommender systems, voter systems, sports competitions, grant proposals rankings, web searchers, Internet streaming-on-demand media providers, etc.). Recently, some methods for [...] Read more.

Mathematical analysis of rankings is essential for a wide range of scientific, public, and industrial applications (e.g., group decision-making, organizational methods, R&D sponsorship, recommender systems, voter systems, sports competitions, grant proposals rankings, web searchers, Internet streaming-on-demand media providers, etc.). Recently, some methods for incomplete aggregate rankings (rankings in which not all the elements are ranked) with ties, based on the classic Kendall’s tau coefficient, have been presented. We are interested in ordinal rankings (that is, we can order the elements to be the first, the second, etc.) allowing ties between the elements (e.g., two elements may be in the first position). We extend a previous coefficient for comparing a series of complete rankings with ties to two new coefficients for comparing a series of incomplete rankings with ties. We make use of the newest definitions of Kendall’s tau extensions. We also offer a theoretical result to interpret these coefficients in terms of the type of interactions that the elements of two consecutive rankings may show (e.g., they preserve their positions, cross their positions, and they are tied in one ranking but untied in the other ranking, etc.). We give some small examples to illustrate all the newly presented parameters and coefficients. We also apply our coefficients to compare some series of Spotify charts, both Top 200 and Viral 50, showing the applicability and utility of the proposed measures. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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11 pages, 463 KiB

Open AccessArticle

Modeling of Artificial Groundwater Recharge by Wells: A Model Stratified Porous Medium

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

Mathematics 2020, 8(10), 1764; https://doi.org/10.3390/math8101764 - 13 Oct 2020

Cited by 8 | Viewed by 2344

Abstract

In recent years, groundwater levels have been decreasing due to the demand in agricultural and industrial activities, as well as the population that has grown exponentially in cities. One method of controlling the progressive lowering of the water table is the artificial recharge [...] Read more.

In recent years, groundwater levels have been decreasing due to the demand in agricultural and industrial activities, as well as the population that has grown exponentially in cities. One method of controlling the progressive lowering of the water table is the artificial recharge of water through wells. With this practice, it is possible to control the amount of water that enters the aquifer through field measurements. However, the construction of these wells is costly in some areas, in addition to the fact that most models only simulate the well as if it were a homogeneous profile and the base equations are restricted. In this work, the amount of infiltrated water by a well is modeled using a stratified media of the porous media methodology. The results obtained can help decision-making by evaluating the cost benefit of the construction of wells to a certain location for the recharge of aquifers. Full article

(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)

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