Mathematics

22 pages, 6193 KiB

Open AccessArticle

Using an Improved Differential Evolution for Scheduling Optimization of Dual-Gantry Multi-Head Surface-Mount Placement Machine

by Cheng-Jian Lin and Chun-Hui Lin

Mathematics 2021, 9(16), 2016; https://doi.org/10.3390/math9162016 - 23 Aug 2021

Cited by 3 | Viewed by 1823

Abstract

The difference between dual-gantry and single-gantry surface-mount placement (SMP) machines is that dual-gantry machines exhibit higher complexity and more problems due to their additional gantry robot, such as component allocation and collision. This paper presents algorithms to prescribe the assembly operations of a [...] Read more.

The difference between dual-gantry and single-gantry surface-mount placement (SMP) machines is that dual-gantry machines exhibit higher complexity and more problems due to their additional gantry robot, such as component allocation and collision. This paper presents algorithms to prescribe the assembly operations of a dual-gantry multi-head surface-mount placement machine. It considers five inter-related problems: (i) component allocation; (ii) automatic nozzle changer assignment; (iii) feeder arrangement; and (iv) pick-and-place sequence; it incorporates a practical restriction related to (v) component height. The paper proposes a solution to each problem: (i) equalizing “workloads” assigned to the gantries, (ii) using quantity ratio method, (iii) using two similarity measurement mechanisms in a modified differential evolution algorithm with a random-key encoding mapping method that addresses component height restriction, (iv) and a combination of nearest-neighbor search and 2-opt method to plan each placing operation. This study reports an experiment that involved the processing of 10 printed circuit boards and compared the performance of a modified differential evolution algorithm with well-known algorithms including differential evolution, particle swarm optimization, and genetic algorithm. The results reveal that the number of picks, moving distance of picking components, and total assembly time with the modified differential evolution algorithm are less than other algorithms. Full article

(This article belongs to the Special Issue Applications of Evolutionary Algorithms)

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19 pages, 504 KiB

Open AccessArticle

Logistic Biplot by Conjugate Gradient Algorithms and Iterated SVD

by Jose Giovany Babativa-Márquez and José Luis Vicente-Villardón

Mathematics 2021, 9(16), 2015; https://doi.org/10.3390/math9162015 - 23 Aug 2021

Cited by 2 | Viewed by 2289

Abstract

Multivariate binary data are increasingly frequent in practice. Although some adaptations of principal component analysis are used to reduce dimensionality for this kind of data, none of them provide a simultaneous representation of rows and columns (biplot). Recently, a technique named logistic biplot [...] Read more.

Multivariate binary data are increasingly frequent in practice. Although some adaptations of principal component analysis are used to reduce dimensionality for this kind of data, none of them provide a simultaneous representation of rows and columns (biplot). Recently, a technique named logistic biplot (LB) has been developed to represent the rows and columns of a binary data matrix simultaneously, even though the algorithm used to fit the parameters is too computationally demanding to be useful in the presence of sparsity or when the matrix is large. We propose the fitting of an LB model using nonlinear conjugate gradient (CG) or majorization–minimization (MM) algorithms, and a cross-validation procedure is introduced to select the hyperparameter that represents the number of dimensions in the model. A Monte Carlo study that considers scenarios with several sparsity levels and different dimensions of the binary data set shows that the procedure based on cross-validation is successful in the selection of the model for all algorithms studied. The comparison of the running times shows that the CG algorithm is more efficient in the presence of sparsity and when the matrix is not very large, while the performance of the MM algorithm is better when the binary matrix is balanced or large. As a complement to the proposed methods and to give practical support, a package has been written in the R language called BiplotML. To complete the study, real binary data on gene expression methylation are used to illustrate the proposed methods. Full article

(This article belongs to the Special Issue Multivariate Statistics: Theory and Its Applications)

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14 pages, 828 KiB

Open AccessArticle

On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations

by Junjiang Lai and Hongyu Liu

Mathematics 2021, 9(16), 2014; https://doi.org/10.3390/math9162014 - 23 Aug 2021

Cited by 5 | Viewed by 1480

Abstract

In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose a Galerkin finite element scheme for both the temporal and spatial discretizations. For the proposed numerical scheme, we derive sharp stability [...] Read more.

In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose a Galerkin finite element scheme for both the temporal and spatial discretizations. For the proposed numerical scheme, we derive sharp stability estimates as well as optimal a priori error estimates. Extensive numerical experiments are conducted to verify the promising features of the newly proposed method. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

14 pages, 15081 KiB

Open AccessArticle

Design of Optimal Controllers for Unknown Dynamic Systems through the Nelder–Mead Simplex Method

by Hsun-Heng Tsai, Chyun-Chau Fuh, Jeng-Rong Ho and Chih-Kuang Lin

Mathematics 2021, 9(16), 2013; https://doi.org/10.3390/math9162013 - 23 Aug 2021

Cited by 1 | Viewed by 1869

Abstract

This paper presents an efficient method for designing optimal controllers. First, we established a performance index according to the system characteristics. In order to ensure that this performance index is applicable even when the state/output of the system is not within the allowable [...] Read more.

This paper presents an efficient method for designing optimal controllers. First, we established a performance index according to the system characteristics. In order to ensure that this performance index is applicable even when the state/output of the system is not within the allowable range, we added a penalty function. When we use a certain controller, if the state/output of the system remains within the allowable range within the preset time interval, the penalty function value is zero. Conversely, if the system state/output is not within the allowable range before the preset termination time, the experiment/simulation is terminated immediately, and the penalty function value is proportional to the time difference between the preset termination time and the time at which the experiment was terminated. Then, we used the Nelder–Mead simplex method to search for the optimal controller parameters. The proposed method has the following advantages: (1) the dynamic equation of the system need not be known; (2) the method can be used regardless of the stability of the open-loop system; (3) this method can be used in nonlinear systems; (4) this method can be used in systems with measurement noise; and (5) the method can improve design efficiency. Full article

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22 pages, 342 KiB

Open AccessArticle

Applications to Boundary Value Problems and Homotopy Theory via Tripled Fixed Point Techniques in Partially Metric Spaces

by Hasanen A. Hammad, Praveen Agarwal and Juan L. G. Guirao

Mathematics 2021, 9(16), 2012; https://doi.org/10.3390/math9162012 - 23 Aug 2021

Cited by 23 | Viewed by 1631

Abstract

In this manuscript, some tripled fixed point results were derived under

(φ, ρ, ℓ)

-contraction in the framework of ordered partially metric spaces. Moreover, we furnish an example which supports our theorem. Furthermore, some results about a homotopy results [...] Read more.

In this manuscript, some tripled fixed point results were derived under

(φ, ρ, ℓ)

-contraction in the framework of ordered partially metric spaces. Moreover, we furnish an example which supports our theorem. Furthermore, some results about a homotopy results are obtained. Finally, theoretical results are involved in some applications, such as finding the unique solution to the boundary value problems and homotopy theory. Full article

(This article belongs to the Special Issue New Trends in Analysis and Geometry)

21 pages, 386 KiB

Open AccessArticle

Valuation of Cliquet-Style Guarantees with Death Benefits in Jump Diffusion Models

by Yaodi Yong and Hailiang Yang

Mathematics 2021, 9(16), 2011; https://doi.org/10.3390/math9162011 - 23 Aug 2021

Viewed by 1418

Abstract

This paper aims to value the cliquet-style equity-linked insurance product with death benefits. Whether the insured dies before the contract maturity or not, a benefit payment to the beneficiary is due. The premium is invested in a financial asset, whose dynamics are assumed [...] Read more.

This paper aims to value the cliquet-style equity-linked insurance product with death benefits. Whether the insured dies before the contract maturity or not, a benefit payment to the beneficiary is due. The premium is invested in a financial asset, whose dynamics are assumed to follow an exponential jump diffusion. In addition, the remaining lifetime of an insured is modelled by an independent random variable whose distribution can be approximated by a linear combination of exponential distributions. We found that the valuation problem reduced to calculating certain discounted expectations. The Laplace inverse transform and techniques from existing literature were implemented to obtain analytical valuation formulae. Full article

(This article belongs to the Special Issue Mathematical Economics and Insurance)

20 pages, 1828 KiB

Open AccessArticle

Financial Development, Saving Rates, and International Economic Volatility: A Simple Model

by Hejie Zhang, Huiming Lv and Shenghau Lin

Mathematics 2021, 9(16), 2010; https://doi.org/10.3390/math9162010 - 22 Aug 2021

Cited by 1 | Viewed by 1776

Abstract

This study constructs a dynamic and open economy model to show that low saving rates are the cause of economic volatility in developed countries, whereas inadequate financial development is identified as the reason for economic volatility in emerging countries. With low saving rates [...] Read more.

This study constructs a dynamic and open economy model to show that low saving rates are the cause of economic volatility in developed countries, whereas inadequate financial development is identified as the reason for economic volatility in emerging countries. With low saving rates or inadequate financial development, countries find it difficult to avoid economic volatility, because it is difficult to alleviate the financing constraints of firms and maintain the stability of investment. Under similar conditions, economic volatility is more severe in developed countries and has spillover effects by triggering interest rate fluctuations in the global capital market and intensifying economic volatility in other countries. By contrast, emerging countries or small economies do not have spillover effects. To avoid dramatic international economic volatility, emerging countries should prompt financial development, and developed countries should increase their saving rates. Full article

(This article belongs to the Special Issue Application of Mathematical Methods to Economics, Management, Finance and Social Problems)

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18 pages, 315 KiB

Open AccessArticle

Fixed Point Results via α-Admissibility in Extended Fuzzy Rectangular b-Metric Spaces with Applications to Integral Equations

by Badshah-e-Rome, Muhammad Sarwar and Rosana Rodríguez-López

Mathematics 2021, 9(16), 2009; https://doi.org/10.3390/math9162009 - 22 Aug 2021

Cited by 7 | Viewed by 1495

Abstract

In this article, the concept of extended fuzzy rectangular b-metric space (EFR_bMS, for short) is initiated, and some fixed point results frequently used in the literature are generalized via α-admissibility in the setting of EFR_bMS. For the [...] Read more.

In this article, the concept of extended fuzzy rectangular b-metric space (EFR_bMS, for short) is initiated, and some fixed point results frequently used in the literature are generalized via α-admissibility in the setting of EFR_bMS. For the illustration of the work presented, some supporting examples and an application to the existence of solutions for a class of integral equations are also discussed. Full article

(This article belongs to the Special Issue New Trends on Boundary Value Problems)

25 pages, 8514 KiB

Open AccessArticle

Leveraging Elasticity to Uncover the Role of Rabinowitsch Suspension through a Wavelike Conduit: Consolidated Blood Suspension Application

by Sara I. Abdelsalam and Abdullah Z. Zaher

Mathematics 2021, 9(16), 2008; https://doi.org/10.3390/math9162008 - 22 Aug 2021

Cited by 59 | Viewed by 2440

Abstract

The present work presents a mathematical investigation of a Rabinowitsch suspension fluid through elastic walls with heat transfer under the effect of electroosmotic forces (EOFs). The governing equations contain empirical stress-strain equations of the Rabinowitsch fluid model and equations of fluid motion along [...] Read more.

The present work presents a mathematical investigation of a Rabinowitsch suspension fluid through elastic walls with heat transfer under the effect of electroosmotic forces (EOFs). The governing equations contain empirical stress-strain equations of the Rabinowitsch fluid model and equations of fluid motion along with heat transfer. It is of interest in this work to study the effects of EOFs, which are rigid spherical particles that are suspended in the Rabinowitsch fluid, the Grashof parameter, heat source, and elasticity on the shear stress of the Rabinowitsch fluid model and flow quantities. The solutions are achieved by taking long wavelength approximation with the creeping flow system. A comparison is set between the effect of pseudoplasticity and dilatation on the behaviour of shear stress, axial velocity, and pressure rise. Physical behaviours have been graphically discussed. It was found that the Rabinowitsch and electroosmotic parameters enhance the shear stress while they reduce the pressure gradient. A biomedical application to the problem is presented. The present analysis is particularly important in biomedicine and physiology. Full article

(This article belongs to the Special Issue Dynamical Systems in Engineering)

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21 pages, 2764 KiB

Open AccessArticle

Dynamical Behavior Analysis of a Time-Delay SIRS-L Model in Rechargeable Wireless Sensor Networks

by Guiyun Liu, Junqiang Li, Zhongwei Liang and Zhimin Peng

Mathematics 2021, 9(16), 2007; https://doi.org/10.3390/math9162007 - 22 Aug 2021

Cited by 10 | Viewed by 1518

Abstract

The traditional SIRS virus propagation model is used to analyze the malware propagation behavior of wireless rechargeable sensor networks (WRSNs) by adding a new concept: the low-energy status nodes. The SIRS-L model has been developed in this article. Furthermore, the influence of time [...] Read more.

The traditional SIRS virus propagation model is used to analyze the malware propagation behavior of wireless rechargeable sensor networks (WRSNs) by adding a new concept: the low-energy status nodes. The SIRS-L model has been developed in this article. Furthermore, the influence of time delay during the charging behavior of the low-energy status nodes needs to be considered. Hopf bifurcation is studied by discussing the time delay that is chosen as the bifurcation parameter. Finally, the properties of the Hopf bifurcation are explored by applying the normal form theory and the center manifold theorem. Full article

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15 pages, 337 KiB

Open AccessArticle

Group Analysis of the Plane Steady Vortex Submodel of Ideal Gas with Varying Entropy

by Salavat Khabirov

Mathematics 2021, 9(16), 2006; https://doi.org/10.3390/math9162006 - 21 Aug 2021

Cited by 3 | Viewed by 1666

Abstract

The submodel of ideal gas motion being invariant with respect to the time translation and the space translation by one direct has 4 integrals in the case of vortex flows with the varying entropy. The system of nonlinear differential equations of the third [...] Read more.

The submodel of ideal gas motion being invariant with respect to the time translation and the space translation by one direct has 4 integrals in the case of vortex flows with the varying entropy. The system of nonlinear differential equations of the third order with one arbitrary element was obtained for a stream function and a specific volume. This element contains from the state equation and arbitrary functions of the integrals. The equivalent transformations were found for arbitrary element. The problem of the group classification was solved when admitted algebra was expanded for 8 cases of arbitrary element. The optimal systems of dissimilar subalgebras were obtained for the Lie algebras from the group classification. The example of the invariant vortex motion from the point source or sink was done. The regular partial invariant submodel was considered for the 2-dimensional subalgebra. It describes the turn of a vortex flow in the strip and on the plane with asymptotes for the stream line. Full article

(This article belongs to the Special Issue Partial Differential and Functional Differential Equations: Exact Solutions, Reductions, Symmetries, and Applications)

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13 pages, 1938 KiB

Open AccessArticle

Covariance Principle for Capital Allocation: A Time-Varying Approach

by Jilber Urbina, Miguel Santolino and Montserrat Guillen

Mathematics 2021, 9(16), 2005; https://doi.org/10.3390/math9162005 - 21 Aug 2021

Cited by 3 | Viewed by 2827

Abstract

The covariance allocation principle is one of the most widely used capital allocation principles in practice. Risks change over time, so capital risk allocations should be time-dependent. In this paper, we propose a dynamic covariance capital allocation principle based on the variance-covariance of [...] Read more.

The covariance allocation principle is one of the most widely used capital allocation principles in practice. Risks change over time, so capital risk allocations should be time-dependent. In this paper, we propose a dynamic covariance capital allocation principle based on the variance-covariance of risks that change over time. The conditional correlation of risks is modeled by means of a dynamic conditional correlation (DCC) model. Unlike the static approach, we show that in our dynamic capital allocation setting, the distribution of risk capital allocations can be estimated, and the expected future allocations of capital can be predicted, providing a deeper understanding of the stochastic multivariate behavior of risks. The methodology presented in the paper is illustrated with an example involving the investment risk in a stock portfolio. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods Applications in Finance)

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12 pages, 1756 KiB

Open AccessArticle

An Improved Variable Kernel Density Estimator Based on L₂ Regularization

by Yi Jin, Yulin He and Defa Huang

Mathematics 2021, 9(16), 2004; https://doi.org/10.3390/math9162004 - 21 Aug 2021

Cited by 5 | Viewed by 2120

Abstract

The nature of the kernel density estimator (KDE) is to find the underlying probability density function (p.d.f) for a given dataset. The key to training the KDE is to determine the optimal bandwidth or Parzen window. All the data points share [...] Read more.

The nature of the kernel density estimator (KDE) is to find the underlying probability density function (p.d.f) for a given dataset. The key to training the KDE is to determine the optimal bandwidth or Parzen window. All the data points share a fixed bandwidth (scalar for univariate KDE and vector for multivariate KDE) in the fixed KDE (FKDE). In this paper, we propose an improved variable KDE (IVKDE) which determines the optimal bandwidth for each data point in the given dataset based on the integrated squared error (ISE) criterion with the

L_{2}

regularization term. An effective optimization algorithm is developed to solve the improved objective function. We compare the estimation performance of IVKDE with FKDE and VKDE based on ISE criterion without

L_{2}

regularization on four univariate and four multivariate probability distributions. The experimental results show that IVKDE obtains lower estimation errors and thus demonstrate the effectiveness of IVKDE. Full article

(This article belongs to the Special Issue Advances in Mathematics and Statistics with Applications in Engineering and Industry)

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27 pages, 1956 KiB

Open AccessArticle

Seasonality in Tourism: Do Senior Programs Mitigate It?

by Paz Rico, Bernardí Cabrer-Borrás and Francisco Morillas-Jurado

Mathematics 2021, 9(16), 2003; https://doi.org/10.3390/math9162003 - 21 Aug 2021

Cited by 6 | Viewed by 2165

Abstract

Seasonality is a widely recognised and accredited phenomenon known to cause an imbalance in tourism activity throughout the year, prompting tourist destinations, both public and private, to consider how best to plan the use of their resources. One way of mitigating the economic [...] Read more.

Seasonality is a widely recognised and accredited phenomenon known to cause an imbalance in tourism activity throughout the year, prompting tourist destinations, both public and private, to consider how best to plan the use of their resources. One way of mitigating the economic imbalances that seasonality can cause is to find strategies for seasonal adjustment, such as travel programmes aimed at the elderly. This paper analyses the seasonality of tourism activity in some EU countries, and in particular in Spain. Different indicators are used to compare the results and carry out a sensitivity analysis. The study then focuses on tourism programmes for the elderly in Spain to see whether this type of programme helps to alleviate the seasonality of tourism activity. To corroborate this, an econometric model is specified and estimated, which enables the scope of these programmes to be compared. Full article

(This article belongs to the Special Issue Applications of Quantitative Methods in Business and Economics Research)

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10 pages, 283 KiB

Open AccessArticle

Comprehensive Interval-Induced Weights Allocation with Bipolar Preference in Multi-Criteria Evaluation

by Xu Jin, Ronald R. Yager, Radko Mesiar, Surajit Borkotokey and Lesheng Jin

Mathematics 2021, 9(16), 2002; https://doi.org/10.3390/math9162002 - 21 Aug 2021

Cited by 3 | Viewed by 1939

Abstract

Preferences-involved evaluation and decision making are the main research subjects in Yager’s decision theory. When the involved bipolar preferences are concerned with interval information, some induced weights allocation and aggregation methods should be reanalyzed and redesigned. This work considers the multi-criteria evaluation situation [...] Read more.

Preferences-involved evaluation and decision making are the main research subjects in Yager’s decision theory. When the involved bipolar preferences are concerned with interval information, some induced weights allocation and aggregation methods should be reanalyzed and redesigned. This work considers the multi-criteria evaluation situation in which originally only the interval-valued absolute importance of each criterion is available. Firstly, based on interval-valued importance, upper bounds, lower bounds, and the mean points of each, we used the basic unit monotonic function-based bipolar preference weights allocation method four times to generate weight vectors. A comprehensive weighting mechanism is proposed after considering the normalization of the given absolute importance information. The bipolar optimism–pessimism preference-based weights allocation will also be applied according to the magnitudes of entries of any given interval input vector. A similar comprehensive weighting mechanism is still performed. With the obtained weight vector for criteria, we adopt the weighted ordered weighted averaging allocation on a convex poset to organically consider both two types of interval-inducing information and propose a further comprehensive weights allocation mechanism. The detailed comprehensive evaluation procedures with a numerical example for education are presented to show that the proposed models are feasible and useful in interval, multi-criteria, and bipolar preferences-involved decisional environments. Full article

(This article belongs to the Special Issue New Trends in Fuzzy Sets Theory and Their Extensions)

14 pages, 1575 KiB

Open AccessArticle

Generalization Second Order Macroscopic Traffic Models via Relative Velocity of the Congestion Propagation

by Yaroslav Kholodov, Andrey Alekseenko, Viktor Kazorin and Alexander Kurzhanskiy

Mathematics 2021, 9(16), 2001; https://doi.org/10.3390/math9162001 - 21 Aug 2021

Cited by 4 | Viewed by 1618

Abstract

This paper presents a generalized second-order hydrodynamic traffic model. Its central piece is the expression for the relative velocity of the congestion (compression wave) propagation. We show that the well-known second-order models of Payne–Whitham, Aw–Rascal and Zhang are all special cases of the [...] Read more.

This paper presents a generalized second-order hydrodynamic traffic model. Its central piece is the expression for the relative velocity of the congestion (compression wave) propagation. We show that the well-known second-order models of Payne–Whitham, Aw–Rascal and Zhang are all special cases of the featured generalized model, and their properties are fully defined by how the relative velocity of the congestion is expressed. The proposed model is verified with traffic data from a segment of the Interstate 580 freeway in California, USA, collected by the California DOT’s Performance Measurement System (PeMS). Full article

(This article belongs to the Special Issue Numerical Methods and Algorithms Applied in Intelligent Transportation Systems)

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12 pages, 285 KiB

Open AccessEditor’s ChoiceArticle

New Applications of Sălăgean and Ruscheweyh Operators for Obtaining Fuzzy Differential Subordinations

by Alina Alb Lupaş and Georgia Irina Oros

Mathematics 2021, 9(16), 2000; https://doi.org/10.3390/math9162000 - 21 Aug 2021

Cited by 23 | Viewed by 1988

Abstract

The present paper deals with notions from the field of complex analysis which have been adapted to fuzzy sets theory, namely, the part dealing with geometric function theory. Several fuzzy differential subordinations are established regarding the operator

L_{α}^{m}

, given by [...] Read more.

The present paper deals with notions from the field of complex analysis which have been adapted to fuzzy sets theory, namely, the part dealing with geometric function theory. Several fuzzy differential subordinations are established regarding the operator

L_{α}^{m}

, given by

L_{α}^{m} : A_{n} \to A_{n}

,

L_{α}^{m} f (z) = (1 - α) R^{m} f (z) + α S^{m} f (z),

where

A_{n} = {f \in H (U), f (z) = z + a_{n + 1} z^{n + 1} + \dots, z \in U}

is the subclass of normalized holomorphic functions and the operators

R^{m} f (z)

and

S^{m} f (z)

are Ruscheweyh and Sălăgean differential operator, respectively. Using the operator

L_{α}^{m}

, a certain fuzzy class of analytic functions denoted by

S L_{F}^{m} (δ, α)

is defined in the open unit disc. Interesting results related to this class are obtained using the concept of fuzzy differential subordination. Examples are also given for pointing out applications of the theoretical results contained in the original theorems and corollaries. Full article

(This article belongs to the Special Issue Fuzzy Logic and Soft Computing – Dedicated to the Centenary of the Birth of Lotfi A. Zadeh (1921-2017))

20 pages, 323 KiB

Open AccessArticle

Traveling Waves Solutions for Delayed Temporally Discrete Non-Local Reaction-Diffusion Equation

by Hongpeng Guo and Zhiming Guo

Mathematics 2021, 9(16), 1999; https://doi.org/10.3390/math9161999 - 20 Aug 2021

Cited by 1 | Viewed by 1362

Abstract

This paper deals with the existence of traveling wave solutions to a delayed temporally discrete non-local reaction diffusion equation model, which has been derived recently for a single species with age structure. When the birth function satisfies monotonic condition, we obtained the traveling [...] Read more.

This paper deals with the existence of traveling wave solutions to a delayed temporally discrete non-local reaction diffusion equation model, which has been derived recently for a single species with age structure. When the birth function satisfies monotonic condition, we obtained the traveling wavefront by using upper and lower solution methods together with monotonic iteration techniques. Otherwise, without the monotonicity assumption for birth function, we constructed two auxiliary equations. By means of the traveling wavefronts of the auxiliary equations, using the Schauder’ fixed point theorem, we proved the existence of a traveling wave solution to the equation under consideration with speed

c > c^{*}

, where

c^{*} > 0

is some constant. We found that the delayed temporally discrete non-local reaction diffusion equation possesses the dynamical consistency with its time continuous counterpart at least in the sense of the existence of traveling wave solutions. Full article

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

14 pages, 1412 KiB

Open AccessArticle

Two Gaussian Bridge Processes for Mapping Continuous Trait Evolution along Phylogenetic Trees

by Dwueng-Chwuan Jhwueng

Mathematics 2021, 9(16), 1998; https://doi.org/10.3390/math9161998 - 20 Aug 2021

Cited by 1 | Viewed by 1989

Abstract

Gaussian processes are powerful tools for modeling trait evolution along phylogenetic trees. As the value of a trait may change randomly throughout the evolution, two Gaussian bridge processes, the Brownian bridge (BB) and the Ornstein–Uhlenbeck bridge (OUB), are proposed for mapping continuous trait [...] Read more.

Gaussian processes are powerful tools for modeling trait evolution along phylogenetic trees. As the value of a trait may change randomly throughout the evolution, two Gaussian bridge processes, the Brownian bridge (BB) and the Ornstein–Uhlenbeck bridge (OUB), are proposed for mapping continuous trait evolution for a group of related species along a phylogenetic tree, respectively. The corresponding traitgrams to the two bridge processes are created to display the evolutionary trajectories. The novel models are applied to study the body mass evolution of a group of marsupial species. Full article

(This article belongs to the Special Issue Stochastic Models and Methods with Applications)

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18 pages, 368 KiB

Open AccessArticle

Sequential Interval Reliability for Discrete-Time Homogeneous Semi-Markov Repairable Systems

by Vlad Stefan Barbu, Guglielmo D’Amico and Thomas Gkelsinis

Mathematics 2021, 9(16), 1997; https://doi.org/10.3390/math9161997 - 20 Aug 2021

Cited by 6 | Viewed by 1629

Abstract

In this paper, a new reliability measure, named sequential interval reliability, is introduced for homogeneous semi-Markov repairable systems in discrete time. This measure is the probability that the system is working in a given sequence of non-overlapping time intervals. Many reliability measures are [...] Read more.

In this paper, a new reliability measure, named sequential interval reliability, is introduced for homogeneous semi-Markov repairable systems in discrete time. This measure is the probability that the system is working in a given sequence of non-overlapping time intervals. Many reliability measures are particular cases of this new reliability measure that we propose; this is the case for the interval reliability, the reliability function and the availability function. A recurrent-type formula is established for the calculation in the transient case and an asymptotic result determines its limiting behaviour. The results are illustrated by means of a numerical example which illustrates the possible application of the measure to real systems. Full article

(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)

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19 pages, 1358 KiB

Open AccessArticle

A New Nonlinear Ninth-Order Root-Finding Method with Error Analysis and Basins of Attraction

by Sania Qureshi, Higinio Ramos and Abdul Karim Soomro

Mathematics 2021, 9(16), 1996; https://doi.org/10.3390/math9161996 - 20 Aug 2021

Cited by 9 | Viewed by 1863

Abstract

Nonlinear phenomena occur in various fields of science, business, and engineering. Research in the area of computational science is constantly growing, with the development of new numerical schemes or with the modification of existing ones. However, such numerical schemes, objectively need to be [...] Read more.

Nonlinear phenomena occur in various fields of science, business, and engineering. Research in the area of computational science is constantly growing, with the development of new numerical schemes or with the modification of existing ones. However, such numerical schemes, objectively need to be computationally inexpensive with a higher order of convergence. Taking into account these demanding features, this article attempted to develop a new three-step numerical scheme to solve nonlinear scalar and vector equations. The scheme was shown to have ninth order convergence and requires six function evaluations per iteration. The efficiency index is approximately 1.4422, which is higher than the Newton’s scheme and several other known optimal schemes. Its dependence on the initial estimates was studied by using real multidimensional dynamical schemes, showing its stable behavior when tested upon some nonlinear models. Based on absolute errors, the number of iterations, the number of function evaluations, preassigned tolerance, convergence speed, and CPU time (sec), comparisons with well-known optimal schemes available in the literature showed a better performance of the proposed scheme. Practical models under consideration include open-channel flow in civil engineering, Planck’s radiation law in physics, the van der Waals equation in chemistry, and the steady-state of the Lorenz system in meteorology. Full article

(This article belongs to the Special Issue Application of Iterative Methods for Solving Nonlinear Equations)

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24 pages, 406 KiB

Open AccessArticle

Families of Solutions of Multitemporal Nonlinear Schrödinger PDE

by Cristian Ghiu, Constantin Udriste and Lavinia Laura Petrescu

Mathematics 2021, 9(16), 1995; https://doi.org/10.3390/math9161995 - 20 Aug 2021

Cited by 2 | Viewed by 1049

Abstract

The multitemporal nonlinear Schrödinger PDE (with oblique derivative) was stated for the first time in our research group as a universal amplitude equation which can be derived via a multiple scaling analysis in order to describe slow modulations of the envelope of a [...] Read more.

The multitemporal nonlinear Schrödinger PDE (with oblique derivative) was stated for the first time in our research group as a universal amplitude equation which can be derived via a multiple scaling analysis in order to describe slow modulations of the envelope of a spatially and temporarily oscillating wave packet in space and multitime (an equation which governs the dynamics of solitons through meta-materials). Now we exploit some hypotheses in order to find important explicit families of exact solutions in all dimensions for the multitime nonlinear Schrödinger PDE with a multitemporal directional derivative term. Using quite effective methods, we discovered families of ODEs and PDEs whose solutions generate solutions of multitime nonlinear Schrödinger PDE. Each new construction involves a relatively small amount of intermediate calculations. Full article

(This article belongs to the Section Computational and Applied Mathematics)

28 pages, 3304 KiB

Open AccessArticle

A Proposal for the Optimisation of Algorithms for the Calculation of the Energy Demands of Residential Housing

by Pamela Hermosilla, Claudio Quiroz, Francisco Cabrejos and Felipe Muñoz-La Rivera

Mathematics 2021, 9(16), 1994; https://doi.org/10.3390/math9161994 - 20 Aug 2021

Viewed by 1432

Abstract

In response to increased energy consumption and CO₂ emissions, various energy efficiency policies, standards and housing certifications have emerged around the world. These aim to measure and quantify energy efficiency and endorse homes for meeting certain standards according to consistent categories that [...] Read more.

In response to increased energy consumption and CO₂ emissions, various energy efficiency policies, standards and housing certifications have emerged around the world. These aim to measure and quantify energy efficiency and endorse homes for meeting certain standards according to consistent categories that vary by continent or country. These energy rating systems correspond to a series of criteria and formulations that, through the calculation and combination of multiple variables, establish the classification values. In Chile, there is the Energy Rating System for Housing (CEV), which performs energy efficiency calculations using dynamic heat balance spreadsheets in Microsoft^® Excel. When applied for everyday use and at a large scale, this system has the disadvantage of requiring a great deal of processing time for each simulation. This research proposes an improvement to the CEV energy demand calculation mechanism by generating a solution that takes advantage of the use of multiprocessors and implements the various algorithms in the C programming language. The results show that the CEV values obtained with the proposed calculation engine are equal to those of the current system but demonstrate a 76.5% improvement in their processing time. Full article

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24 pages, 343 KiB

Open AccessArticle

The Hamilton–Jacobi Theory for Contact Hamiltonian Systems

by Manuel de León, Manuel Lainz and Álvaro Muñiz-Brea

Mathematics 2021, 9(16), 1993; https://doi.org/10.3390/math9161993 - 20 Aug 2021

Cited by 10 | Viewed by 2069

Abstract

The aim of this paper is to develop a Hamilton–Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton–Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given Hamiltonian function. We also analyze the corresponding [...] Read more.

The aim of this paper is to develop a Hamilton–Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton–Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given Hamiltonian function. We also analyze the corresponding formulation on the symplectification of the contact Hamiltonian system, and establish the relations between these two approaches. In the last section, some examples are discussed. Full article

(This article belongs to the Special Issue New Trends in Hamilton-Jacobi Theory: Conservative and Dissipative Dynamics)

18 pages, 318 KiB

Open AccessArticle

Some New Simpson’s and Newton’s Formulas Type Inequalities for Convex Functions in Quantum Calculus

by Pimchana Siricharuanun, Samet Erden, Muhammad Aamir Ali, Hüseyin Budak, Saowaluck Chasreechai and Thanin Sitthiwirattham

Mathematics 2021, 9(16), 1992; https://doi.org/10.3390/math9161992 - 20 Aug 2021

Cited by 10 | Viewed by 1859

Abstract

In this paper, using the notions of

q^{κ_{2}}

-quantum integral and

q^{κ_{2}}

-quantum derivative, we present some new identities that enable us to obtain new quantum Simpson’s and quantum Newton’s type inequalities for quantum differentiable convex functions. This paper, [...] Read more.

In this paper, using the notions of

q^{κ_{2}}

-quantum integral and

q^{κ_{2}}

-quantum derivative, we present some new identities that enable us to obtain new quantum Simpson’s and quantum Newton’s type inequalities for quantum differentiable convex functions. This paper, in particular, generalizes and expands previous findings in the field of quantum and classical integral inequalities obtained by various authors. Full article

(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)

13 pages, 907 KiB

Open AccessArticle

Evaluating the Efficiency of Off-Ball Screens in Elite Basketball Teams via Second-Order Markov Modelling

by Nikolaos Stavropoulos, Alexandra Papadopoulou and Pavlos Kolias

Mathematics 2021, 9(16), 1991; https://doi.org/10.3390/math9161991 - 20 Aug 2021

Cited by 5 | Viewed by 3196

Abstract

In basketball, the offensive movements on both strong and weak sides and tactical behavior play major roles in the effectiveness of a team’s offense. In the literature, studies are mostly focused on offensive actions, such as ball screens on the strong side. In [...] Read more.

In basketball, the offensive movements on both strong and weak sides and tactical behavior play major roles in the effectiveness of a team’s offense. In the literature, studies are mostly focused on offensive actions, such as ball screens on the strong side. In the present paper, for the first time a second-order Markov model is defined to evaluate players’ interactions on the weak side, particularly for exploring the effectiveness of tactical structures and off-ball screens regarding the final outcome. The sample consisted of 1170 possessions of the FIBA Basketball Champions League 2018–2019. The variables of interest were the type of screen on the weak side, the finishing move, and the outcome of the shot. The model incorporates partial non-homogeneity according to the time of the execution (0–24″) and the quarter of playtime, and it is conditioned on the off-ball screen type. Regarding the overall performance, the results indicated that the outcome of each possession was influenced not only by the type of the executed shot, but also by the specific type of screen that took place earlier on the weak side of the offense. Thus, the proposed model could operate as an advisory tool for the coach’s strategic plans. Full article

(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)

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14 pages, 312 KiB

Open AccessArticle

On the A_α-Eigenvalues of Signed Graphs

by Germain Pastén, Oscar Rojo and Luis Medina

Mathematics 2021, 9(16), 1990; https://doi.org/10.3390/math9161990 - 20 Aug 2021

Cited by 2 | Viewed by 1489

Abstract

For

α \in [0, 1]

, let

A_{α} (G_{σ}) = α D (G) + (1 - α) A (G_{σ})

, where G is a simple undirected graph, [...] Read more.

For

α \in [0, 1]

, let

A_{α} (G_{σ}) = α D (G) + (1 - α) A (G_{σ})

, where G is a simple undirected graph,

D (G)

is the diagonal matrix of its vertex degrees and

A (G_{σ})

is the adjacency matrix of the signed graph

G_{σ}

whose underlying graph is G. In this paper, basic properties of

A_{α} (G_{σ})

are obtained, its positive semidefiniteness is studied and some bounds on its eigenvalues are derived—in particular, lower and upper bounds on its largest eigenvalue are obtained. Full article

(This article belongs to the Section Algebra, Geometry and Topology)

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20 pages, 425 KiB

Open AccessArticle

Modeling Proportion Data with Inflation by Using a Power-Skew-Normal/Logit Mixture Model

by Guillermo Martínez-Flórez, Hector W. Gomez and Roger Tovar-Falón

Mathematics 2021, 9(16), 1989; https://doi.org/10.3390/math9161989 - 20 Aug 2021

Cited by 1 | Viewed by 1720

Abstract

Rate or proportion data are modeled by using a regression model. The considered regression model can be used for studying phenomena with a response on the (0, 1), [0, 1), (0, 1], or [0, 1] intervals. To connect the response variable with the [...] Read more.

Rate or proportion data are modeled by using a regression model. The considered regression model can be used for studying phenomena with a response on the (0, 1), [0, 1), (0, 1], or [0, 1] intervals. To connect the response variable with the linear predictor in the regression model, we use a logit link function, which guarantees that the obtained prediction ranges between zero and one in the cases inflated at zero or one (or both). The model is complemented with the assumption that the errors follow a power-skew-normal distribution, resulting in a very flexible model, and with a non-singular information matrix, constituting an advantage over other existing models in the literature. To explain the probability of point mass at the values zero and/or one (inflated part), we used a polytomic logistic model with covariates. The results of two illustrations showed that the proposed model is a better alternative compared to widely known models in the literature. Full article

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10 pages, 280 KiB

Open AccessArticle

Kadomtsev–Petviashvili Hierarchy: Negative Times

by Andrei K. Pogrebkov

Mathematics 2021, 9(16), 1988; https://doi.org/10.3390/math9161988 - 20 Aug 2021

Cited by 2 | Viewed by 1337

Abstract

The Kadomtsev–Petviashvili equation is known to be the leading term of a semi-infinite hierarchy of integrable equations with evolutions given by times with positive numbers. Here, we introduce new hierarchy directed to negative numbers of times. The derivation of such systems, as well [...] Read more.

The Kadomtsev–Petviashvili equation is known to be the leading term of a semi-infinite hierarchy of integrable equations with evolutions given by times with positive numbers. Here, we introduce new hierarchy directed to negative numbers of times. The derivation of such systems, as well as the corresponding hierarchy, is based on the commutator identities. This approach enables introduction of linear differential equations that admit lifts up to nonlinear integrable ones by means of the special dressing procedure. Thus, one can construct not only nonlinear equations, but corresponding Lax pairs as well. The Lax operator of this evolution coincides with the Lax operator of the “positive” hierarchy. We also derive (1 + 1)-dimensional reductions of equations of this hierarchy. Full article

26 pages, 1316 KiB

Open AccessArticle

Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions

by Saadoune Brahimi, Ahcene Merad and Adem Kılıçman

Mathematics 2021, 9(16), 1987; https://doi.org/10.3390/math9161987 - 19 Aug 2021

Viewed by 1698

Abstract

In this paper, we are interested in the study of a Caputo time fractional advection–diffusion equation with nonhomogeneous boundary conditions of integral types

\int_{0}^{1} v (x, t) d x

and [...] Read more.

In this paper, we are interested in the study of a Caputo time fractional advection–diffusion equation with nonhomogeneous boundary conditions of integral types

\int_{0}^{1} v (x, t) d x

and

\int_{0}^{1} x^{n} v (x, t) d x

. The existence and uniqueness of the given problem’s solution is proved using the method of the energy inequalities known as the “a priori estimate” method relying on the range density of the operator generated by the considered problem. The approximate solution for this problem with these new kinds of boundary conditions is established by using a combination of the finite difference method and the numerical integration. Finally, we give some numerical tests to illustrate the usefulness of the obtained results. Full article

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Mathematics, Volume 9, Issue 16 (August-2 2021) – 183 articles

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