Mathematics

22 pages, 1007 KiB

Open AccessEditor’s ChoiceArticle

Heat and Mass Transfer Analysis for the Viscous Fluid Flow: Dual Approximate Solutions

by Remus-Daniel Ene, Nicolina Pop and Rodica Badarau

Mathematics 2023, 11(7), 1648; https://doi.org/10.3390/math11071648 - 29 Mar 2023

Cited by 3 | Viewed by 1861

The aim of this paper is to investigate effective and accurate dual analytic approximate solutions, while taking into account thermal effects. The heat and mass transfer problem in a viscous fluid flow are analytically explored by using the modified Optimal Homotopy Asymptotic Method [...] Read more.

The aim of this paper is to investigate effective and accurate dual analytic approximate solutions, while taking into account thermal effects. The heat and mass transfer problem in a viscous fluid flow are analytically explored by using the modified Optimal Homotopy Asymptotic Method (OHAM). By using similarity transformations, the motion equations are reduced to a set of nonlinear ordinary differential equations. Based on the numerical results, it was revealed that there are dual analytic approximate solutions within the mass transfer problem. The variation of the physical parameters (the Prandtl number and the temperature distribution parameter) over the temperature profile is analytically explored and graphically depicted for the first approximate and the corresponding dual solution, respectively. The advantage of the proposed method arises from using only one iteration for obtaining the dual analytical solutions. The presented results are effective, accurate and in good agreement with the corresponding numerical results with relevance for further engineering applications of heat and mass transfer problems. Full article

(This article belongs to the Special Issue Analysis and Applications of Mathematical Fluid Dynamics)

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18 pages, 1326 KiB

Open AccessEditor’s ChoiceArticle

Facilitating Conditions as the Biggest Factor Influencing Elementary School Teachers’ Usage Behavior of Dynamic Mathematics Software in China

by Zhiqiang Yuan, Jing Liu, Xi Deng, Tianzi Ding and Tommy Tanu Wijaya

Mathematics 2023, 11(6), 1536; https://doi.org/10.3390/math11061536 - 22 Mar 2023

Cited by 12 | Viewed by 3225

Abstract

Dynamic mathematics software, such as GeoGebra, is one of the most important teaching and learning media. This kind of software can help teachers teach mathematics, especially geometry, at the elementary school level. However, the use of dynamic mathematics software of elementary school teachers [...] Read more.

Dynamic mathematics software, such as GeoGebra, is one of the most important teaching and learning media. This kind of software can help teachers teach mathematics, especially geometry, at the elementary school level. However, the use of dynamic mathematics software of elementary school teachers is still very limited so far. This study analyzed the factors influencing elementary school teachers’ usage behavior of dynamic mathematics software. Four independent variables, namely performance expectancy (PE), effort expectancy (EE), social influence (SI), and facilitating conditions (FC) from the united theory of acceptance and use of technology (UTAUT), were used to understand elementary school teachers’ usage behavior of dynamic mathematics software. A questionnaire survey was conducted in the Hunan and Guangdong provinces of China. Two hundred and sixty-six elementary school mathematics teachers provided valid questionnaire data. The partial least squares structural equation modeling (PLS-SEM) approach was used to analyze the data. The results showed that facilitating conditions and effort expectancy significantly affect elementary school teachers’ usage behavior of dynamic mathematics software, and facilitating conditions were the biggest factor that affected user behavior. The moderating effects of gender, major, and training on all relationships in the dynamic mathematics software usage conceptual model were not significant. This study contributes by developing a model and providing new knowledge to elementary school principals and the government about factors that can increase the adoption of dynamic mathematics software. Full article

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15 pages, 1201 KiB

Open AccessEditor’s ChoiceArticle

Fixed-Time Synchronization of Reaction-Diffusion Fuzzy Neural Networks with Stochastic Perturbations

by Hayrengul Sadik, Abdujelil Abdurahman and Rukeya Tohti

Mathematics 2023, 11(6), 1493; https://doi.org/10.3390/math11061493 - 18 Mar 2023

Cited by 5 | Viewed by 1328

Abstract

In this paper, we investigated the fixed-time synchronization problem of a type of reaction-diffusion fuzzy neural networks with stochastic perturbations by developing simple control schemes. First, some generalized fixed-time stability results are introduced for stochastic nonlinear systems. Based on these results, some generic [...] Read more.

In this paper, we investigated the fixed-time synchronization problem of a type of reaction-diffusion fuzzy neural networks with stochastic perturbations by developing simple control schemes. First, some generalized fixed-time stability results are introduced for stochastic nonlinear systems. Based on these results, some generic fixed-time stability criteria are established and upper bounds of settling time are directly calculated by using several special functions. Then, the fixed-time synchronization of a type of reaction-diffusion fuzzy neural networks with stochastic perturbations is analysed by designing a type of controller which is more simple and thus have a better applicability. Finally, one numerical example with its Matlab simulations is provided to show the feasibility of developed theoretical results. Full article

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13 pages, 317 KiB

Open AccessEditor’s ChoiceArticle

The Global Property of Generic Conformally Flat Hypersurfaces in

R^{4}

by Yayun Chen and Tongzhu Li

Mathematics 2023, 11(6), 1435; https://doi.org/10.3390/math11061435 - 16 Mar 2023

Viewed by 1076

Abstract

A conformally flat hypersurface

f : M^{3} \to R^{4}

in the four-dimensional Euclidean space

R^{4}

is said to be generic if the hypersurface has three distinct principal curvatures everywhere. In this paper, we study the generic conformally flat hypersurfaces in [...] Read more.

A conformally flat hypersurface

f : M^{3} \to R^{4}

in the four-dimensional Euclidean space

R^{4}

is said to be generic if the hypersurface has three distinct principal curvatures everywhere. In this paper, we study the generic conformally flat hypersurfaces in

R^{4}

using the framework of Möbius geometry. First, we classify locally the generic conformally flat hypersurfaces with a vanishing Möbius form under the Möbius transformation group of

R^{4}

. Second, we investigate the global behavior of the compact generic conformally flat hypersurfaces and give some integral formulas about the Möbius invariant of these hypersurfaces. Full article

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

21 pages, 690 KiB

Open AccessEditor’s ChoiceArticle

New Results on Finite-Time Synchronization of Complex-Valued BAM Neural Networks with Time Delays by the Quadratic Analysis Approach

by Zhen Yang and Zhengqiu Zhang

Mathematics 2023, 11(6), 1378; https://doi.org/10.3390/math11061378 - 12 Mar 2023

Cited by 3 | Viewed by 1338

Abstract

In this paper, we are interested in the finite-time synchronization of complex-valued BAM neural networks with time delays. Without applying Lyapunov–Krasovskii functional theory, finite-time convergence theorem, graph-theoretic method, the theory of complex functions or the integral inequality method, by using the quadratic analysis [...] Read more.

In this paper, we are interested in the finite-time synchronization of complex-valued BAM neural networks with time delays. Without applying Lyapunov–Krasovskii functional theory, finite-time convergence theorem, graph-theoretic method, the theory of complex functions or the integral inequality method, by using the quadratic analysis approach, inequality techniques and designing two classes of novel controllers, two novel sufficient conditions are achieved to guarantee finite-time synchronization between the master system and the slave system. The quadratic analysis method used in our paper is a different study approach of finite-time synchronization from those in existing papers. Therefore the controllers designed in our paper are fully novel. Full article

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20 pages, 5607 KiB

Open AccessEditor’s ChoiceArticle

Machine Learning Techniques Applied to the Harmonic Analysis of Railway Power Supply

by Manuela Panoiu, Caius Panoiu, Sergiu Mezinescu, Gabriel Militaru and Ioan Baciu

Mathematics 2023, 11(6), 1381; https://doi.org/10.3390/math11061381 - 12 Mar 2023

Cited by 8 | Viewed by 3841

Abstract

Harmonic generation in power system networks presents significant issues that arise in power utilities. This paper describes a machine learning technique that was used to conduct a research study on the harmonic analysis of railway power stations. The research was an investigation of [...] Read more.

Harmonic generation in power system networks presents significant issues that arise in power utilities. This paper describes a machine learning technique that was used to conduct a research study on the harmonic analysis of railway power stations. The research was an investigation of a time series whose values represented the total harmonic distortion (THD) for the electric current. This study was based on information collected at a railway power station. In an electrified substation, measurements of currents and voltages were made during a certain interval of time. From electric current values, the THD was calculated using a fast Fourier transform analysis (FFT) and the results were used to train an adaptive ANN—GMDH (artificial neural network–group method of data handling) algorithm. Following the training, a prediction model was created, the performance of which was investigated in this study. The model was based on the ANN—GMDH method and was developed for the prediction of the THD. The performance of this model was studied based on its parameters. The model’s performance was evaluated using the regression coefficient (R), root-mean-square error (RMSE), and mean absolute error (MAE). The model’s performance was very good, with an RMSE (root-mean-square error) value of less than 0.01 and a regression coefficient value higher than 0.99. Another conclusion from our research was that the model also performed very well in terms of the training time (calculation speed). Full article

(This article belongs to the Special Issue Advanced Methods in Intelligent Transportation Systems)

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16 pages, 789 KiB

Open AccessEditor’s ChoiceArticle

A Full-Body Relative Orbital Motion of Spacecraft Using Dual Tensor Algebra and Dual Quaternions

by Daniel Condurache

Mathematics 2023, 11(6), 1366; https://doi.org/10.3390/math11061366 - 11 Mar 2023

Cited by 1 | Viewed by 1421

Abstract

This paper proposes a new non-linear differential equation for the six degrees of freedom (6-DOF) relative rigid bodies motion. A representation theorem is provided for the 6-DOF differential equation of motion in the arbitrary non-inertial reference frame. The problem of the 6-DOF relative [...] Read more.

This paper proposes a new non-linear differential equation for the six degrees of freedom (6-DOF) relative rigid bodies motion. A representation theorem is provided for the 6-DOF differential equation of motion in the arbitrary non-inertial reference frame. The problem of the 6-DOF relative motion of two spacecraft in the specific case of Keplerian confocal orbits is proposed. The result is an analytical method without secular terms and singularities. Tensors dual algebra and dual quaternions play a fundamental role, with the solution representation being the relative problem. Furthermore, the representation theorems for the rotation and translation parts of the 6-DOF relative orbital motion problems are obtained. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications II)

14 pages, 582 KiB

Open AccessEditor’s ChoiceArticle

Optimal Homotopy Asymptotic Method for an Anharmonic Oscillator: Application to the Chen System

by Remus-Daniel Ene and Nicolina Pop

Mathematics 2023, 11(5), 1124; https://doi.org/10.3390/math11051124 - 23 Feb 2023

Viewed by 1478

Abstract

The aim of our work is to obtain the analytic solutions for a new nonlinear anharmonic oscillator by means of the Optimal Homotopy Asymptotic Method (OHAM), using only one iteration. The accuracy of the obtained results comes from the comparison with the corresponding [...] Read more.

The aim of our work is to obtain the analytic solutions for a new nonlinear anharmonic oscillator by means of the Optimal Homotopy Asymptotic Method (OHAM), using only one iteration. The accuracy of the obtained results comes from the comparison with the corresponding numerical ones for specified physical parameters. Moreover, the OHAM method has a greater degree of flexibility than an iterative method as is presented in this paper. Based on these results, the analytically solutions of the Chen system were obtained for a special case (just one analytic first integral). The chaotic behaviors were excluded here. The provided solutions are usefully for many engineering applications. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications II)

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13 pages, 1124 KiB

Open AccessEditor’s ChoiceArticle

Facing a Risk: To Insure or Not to Insure—An Analysis with the Constant Relative Risk Aversion Utility Function

by M. Mercè Claramunt, Maite Mármol and Xavier Varea

Mathematics 2023, 11(5), 1070; https://doi.org/10.3390/math11051070 - 21 Feb 2023

Cited by 1 | Viewed by 1825

Abstract

The decision to transfer or share an insurable risk is critical for the decision maker’s economy. This paper deals with this decision, starting with the definition of a function that represents the difference between the expected utility of insuring, with or without deductibles, [...] Read more.

The decision to transfer or share an insurable risk is critical for the decision maker’s economy. This paper deals with this decision, starting with the definition of a function that represents the difference between the expected utility of insuring, with or without deductibles, and the expected utility of not insuring. Considering a constant relative risk aversion (CRRA) utility function, we provide a decision pattern for the potential policyholders as a function of their wealth level. The obtained rule applies to any premium principle, any per-claim deductible and any risk distribution. Furthermore, numerical results are presented based on the mean principle, a per-claim absolute deductible and a Poisson-exponential model, and a sensitivity analysis regarding the deductible parameter and the insurer security loading was performed. One of the main conclusions of the paper is that the initial level of wealth is the main variable that determines the decision to insure or not to insure; thus, for high levels of wealth, the decision is always not to insure regardless of the risk aversion of the decision maker. Moreover, the parameters defining the deductible and the premium only have an influence at low levels of wealth. Full article

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

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12 pages, 2888 KiB

Open AccessEditor’s ChoiceArticle

Multimodal Movie Recommendation System Using Deep Learning

by Yongheng Mu and Yun Wu

Mathematics 2023, 11(4), 895; https://doi.org/10.3390/math11040895 - 10 Feb 2023

Cited by 33 | Viewed by 11433

Abstract

Recommendation systems, the best way to deal with information overload, are widely utilized to provide users with personalized content and services with high efficiency. Many recommendation algorithms have been researched and deployed extensively in various e-commerce applications, including the movie streaming services over [...] Read more.

Recommendation systems, the best way to deal with information overload, are widely utilized to provide users with personalized content and services with high efficiency. Many recommendation algorithms have been researched and deployed extensively in various e-commerce applications, including the movie streaming services over the last decade. However, sparse data cold-start problems are often encountered in many movie recommendation systems. In this paper, we reported a personalized multimodal movie recommendation system based on multimodal data analysis and deep learning. The real-world MovieLens datasets were selected to test the effectiveness of our new recommendation algorithm. With the input information, the hidden features of the movies and the users were mined using deep learning to build a deep-learning network algorithm model for training to further predict movie scores. With a learning rate of 0.001, the root mean squared error (RMSE) scores achieved 0.9908 and 0.9096 for test sets of MovieLens 100 K and 1 M datasets, respectively. The scoring prediction results show improved accuracy after incorporating the potential features and connections in multimodal data with deep-learning technology. Compared with the traditional collaborative filtering algorithms, such as user-based collaborative filtering (User-CF), item-based content-based filtering (Item-CF), and singular-value decomposition (SVD) approaches, the multimodal movie recommendation system using deep learning could provide better personalized recommendation results. Meanwhile, the sparse data problem was alleviated to a certain degree. We suggest that the recommendation system can be improved through the combination of the deep-learning technology and the multimodal data analysis. Full article

(This article belongs to the Special Issue Nature Inspired Computing and Optimisation)

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9 pages, 3640 KiB

Open AccessEditor’s ChoiceArticle

Identifying Combination of Dark–Bright Binary–Soliton and Binary–Periodic Waves for a New Two-Mode Model Derived from the (2 + 1)-Dimensional Nizhnik–Novikov–Veselov Equation

by Marwan Alquran and Imad Jaradat

Mathematics 2023, 11(4), 861; https://doi.org/10.3390/math11040861 - 8 Feb 2023

Cited by 30 | Viewed by 1623

Abstract

In this paper, we construct a new two-mode model derived from the

(2 + 1)

-dimensional Nizhnik–Novikov–Veselov (TMNNV) equation. We generalize the concept of Korsunsky to accommodate the derivation of higher-dimensional two-mode equations. Since the TMNNV is presented here, for the [...] Read more.

In this paper, we construct a new two-mode model derived from the

(2 + 1)

-dimensional Nizhnik–Novikov–Veselov (TMNNV) equation. We generalize the concept of Korsunsky to accommodate the derivation of higher-dimensional two-mode equations. Since the TMNNV is presented here, for the first time, we identify some of its solutions by means of two recent and effective schemes. As a result, the Kudryashov-expansion method exports a combination of bright–dark binary solitons, which simulate many applications in optics, photons, and plasma. The modified rational sine and cosine functions export binary–periodic waves that arise in the field of surface water waves. Moreover, by using 2D and 3D graphs, some physical properties of the TMNNV were investigated by means of studying the effect of the following parameters of the model: nonlinearity, dispersion, and phase–velocity. Finally, we checked the validity of the obtained solutions by verifying the correctness of the original governing model. Full article

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23 pages, 1258 KiB

Open AccessEditor’s ChoiceArticle

Train Delay Predictions Using Markov Chains Based on Process Time Deviations and Elastic State Boundaries

by Thomas Spanninger, Beda Büchel and Francesco Corman

Mathematics 2023, 11(4), 839; https://doi.org/10.3390/math11040839 - 7 Feb 2023

Cited by 3 | Viewed by 2837

Abstract

Train delays are inconvenient for passengers and major problems in railway operations. When delays occur, it is vital to provide timely information to passengers regarding delays at their departing, interchanging, and final stations. Furthermore, real-time traffic control requires information on how delays propagate [...] Read more.

Train delays are inconvenient for passengers and major problems in railway operations. When delays occur, it is vital to provide timely information to passengers regarding delays at their departing, interchanging, and final stations. Furthermore, real-time traffic control requires information on how delays propagate throughout the network. Among a multitude of applied models to predict train delays, Markov chains have proven to be stochastic benchmark approaches due to their simplicity, interpretability, and solid performances. In this study, we introduce an advanced Markov chain setting to predict train delays using historical train operation data. Therefore, we applied Markov chains based on process time deviations instead of absolute delays and we relaxed commonly used stationarity assumptions for transition probabilities in terms of direction, train line, and location. Additionally, we defined the state space elastically and analyzed the benefit of an increasing state space dimension. We show (via a test case in the Swiss railway network) that our proposed advanced Markov chain model achieves a prediction accuracy gain of 56% in terms of mean absolute error (MAE) compared to state-of-the-art Markov chain models based on absolute delays. We also illustrate the prediction performance advantages of our proposed model in the case of training data sparsity. Full article

(This article belongs to the Special Issue Advanced Methods in Intelligent Transportation Systems)

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20 pages, 4679 KiB

Open AccessEditor’s ChoiceArticle

Application of Solar Activity Time Series in Machine Learning Predictive Modeling of Precipitation-Induced Floods

by Slavica Malinović-Milićević, Milan M. Radovanović, Sonja D. Radenković, Yaroslav Vyklyuk, Boško Milovanović, Ana Milanović Pešić, Milan Milenković, Vladimir Popović, Marko Petrović, Petro Sydor and Mirjana Gajić

Mathematics 2023, 11(4), 795; https://doi.org/10.3390/math11040795 - 4 Feb 2023

Cited by 4 | Viewed by 3358

Abstract

This research is devoted to the determination of hidden dependencies between the flow of particles that come from the Sun and precipitation-induced floods in the United Kingdom (UK). The analysis covers 20 flood events during the period from October 2001 to December 2019. [...] Read more.

This research is devoted to the determination of hidden dependencies between the flow of particles that come from the Sun and precipitation-induced floods in the United Kingdom (UK). The analysis covers 20 flood events during the period from October 2001 to December 2019. The parameters of solar activity were used as model input data, while precipitations data in the period 10 days before and during each flood event were used as model output. The time lag of 0–9 days was taken into account in the research. Correlation analysis was conducted to determine the degree of randomness for the time series of input and output parameters. For establishing a potential causative link, machine learning classification predictive modeling was applied. Two approaches, the decision tree, and the random forest were used. We analyzed the accuracy of classification models forecast from 0 to 9 days in advance. It was found that the most important factors for flood forecasting are proton density with a time lag of 9, differential proton flux in the range of 310–580 keV, and ion temperature. Research in this paper has shown that the decision tree model is more accurate and adequate in predicting the appearance of precipitation-induced floods up to 9 days ahead with an accuracy of 91%. The results of this study confirmed that by increasing technical capabilities, using improved machine learning techniques and large data sets, it is possible to improve the understanding of the physical link between the solar wind and tropospheric weather and help improve severe weather forecasting. Full article

(This article belongs to the Special Issue Complex Network Analysis of Nonlinear Time Series)

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36 pages, 558 KiB

Open AccessEditor’s ChoiceArticle

Fuzzy Property Grammars for Gradience in Natural Language

by Adrià Torrens-Urrutia, Vilém Novák and María Dolores Jiménez-López

Mathematics 2023, 11(3), 735; https://doi.org/10.3390/math11030735 - 1 Feb 2023

Cited by 2 | Viewed by 1702

Abstract

This paper introduces a new grammatical framework, Fuzzy Property Grammars (FPGr). This is a model based on Property Grammars and Fuzzy Natural Logic. Such grammatical framework is constraint-based and provides a new way to formally characterize gradience by representing grammaticality degrees regarding linguistic [...] Read more.

This paper introduces a new grammatical framework, Fuzzy Property Grammars (FPGr). This is a model based on Property Grammars and Fuzzy Natural Logic. Such grammatical framework is constraint-based and provides a new way to formally characterize gradience by representing grammaticality degrees regarding linguistic competence (without involving speakers judgments). The paper provides a formal-logical characterization of FPGr. A test of the framework is presented by implementing an FPGr for Spanish. FPGr is a formal theory that may serve linguists, computing scientists, and mathematicians since it can capture infinite grammatical structures within the variability of a language. Full article

(This article belongs to the Special Issue FSTA: Fuzzy Set Theory and Applications)

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15 pages, 307 KiB

Open AccessEditor’s ChoiceArticle

Reduced Clustering Method Based on the Inversion Formula Density Estimation

by Mantas Lukauskas and Tomas Ruzgas

Mathematics 2023, 11(3), 661; https://doi.org/10.3390/math11030661 - 28 Jan 2023

Cited by 4 | Viewed by 1947

Abstract

Unsupervised learning is one type of machine learning with an exceptionally high number of applications in various fields. The most popular and best-known group of unsupervised machine learning methods is clustering methods. The main goal of clustering is to find hidden relationships between [...] Read more.

Unsupervised learning is one type of machine learning with an exceptionally high number of applications in various fields. The most popular and best-known group of unsupervised machine learning methods is clustering methods. The main goal of clustering is to find hidden relationships between individual observations. There is great interest in different density estimation methods, especially when there are outliers in the data. Density estimation also can be applied to data clustering methods. This paper presents the extension to the clustering method based on the modified inversion formula density estimation to solve previous method limitations. This new method’s extension works within higher dimensions (d > 15) cases, which was the limitation of the previous method. More than 20 data sets are used in comparative data analysis to prove the effectiveness of the developed method improvement. The results showed that the new method extension positively affects the data clustering results. The new reduced clustering method, based on the modified inversion formula density estimation, outperforms popular data clustering methods on test data sets. In cases when the accuracy is not the best, the data clustering accuracy is close to the best models’ obtained accuracies. Lower dimensionality data were used to compare the standard clustering based on the inversion formula density estimation method with the extended method. The new modification method has better results than the standard method in all cases, which confirmed the hypothesis about the new method’s positive impact on clustering results. Full article

(This article belongs to the Special Issue Advances in Computational Statistics and Applications)

16 pages, 727 KiB

Open AccessEditor’s ChoiceArticle

Stochastic Configuration Based Fuzzy Inference System with Interpretable Fuzzy Rules and Intelligence Search Process

by Wei Zhou, Hongxing Li and Menghong Bao

Mathematics 2023, 11(3), 614; https://doi.org/10.3390/math11030614 - 26 Jan 2023

Cited by 3 | Viewed by 1646

Abstract

In this paper, a stochastic configuration based fuzzy inference system with interpretable fuzzy rules (SCFS-IFRs) is proposed to improve the interpretability and performance of the fuzzy inference system and determine autonomously an appropriate model structure. The proposed SCFS-IFR first accomplishes a fuzzy system [...] Read more.

In this paper, a stochastic configuration based fuzzy inference system with interpretable fuzzy rules (SCFS-IFRs) is proposed to improve the interpretability and performance of the fuzzy inference system and determine autonomously an appropriate model structure. The proposed SCFS-IFR first accomplishes a fuzzy system through interpretable linguistic fuzzy rules (ILFRs), which endows the system with clear semantic interpretability. Meanwhile, using an incremental learning method based on stochastic configuration, the appropriate architecture of the system is determined by incremental generation of ILFRs under a supervision mechanism. In addition, the particle swarm optimization (PSO) algorithm, an intelligence search technique, is used in the incremental learning process of ILFRs to obtain better random parameters and improve approximation accuracy. The performance of SCFS-IFRs is verified by regression and classification benchmark datasets. Regression experiments show that the proposed SCFS-IFRs perform best on 10 of the 20 data sets, statistically significantly outperforming the other eight state-of-the-art algorithms. Classification experiments show that, compared with the other six fuzzy classifiers, SCFS-IFRs achieve higher classification accuracy and better interpretation with fewer rules. Full article

(This article belongs to the Special Issue Intelligent and Fuzzy Systems in Engineering and Technology)

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29 pages, 13692 KiB

Open AccessEditor’s ChoiceArticle

Peristalsis of Nanofluids via an Inclined Asymmetric Channel with Hall Effects and Entropy Generation Analysis

by Abdulwahed Muaybid A. Alrashdi

Mathematics 2023, 11(2), 458; https://doi.org/10.3390/math11020458 - 15 Jan 2023

Cited by 3 | Viewed by 1517

Abstract

This study deals with the entropy investigation of the peristalsis of a water–copper nanofluid through an asymmetric inclined channel. The asymmetric channel is anticipated to be filled with a uniform permeable medium, with a constant magnetic field impinging on the wall of the [...] Read more.

This study deals with the entropy investigation of the peristalsis of a water–copper nanofluid through an asymmetric inclined channel. The asymmetric channel is anticipated to be filled with a uniform permeable medium, with a constant magnetic field impinging on the wall of the channel. The physical effects, such as Hall current, mixed convection, Ohmic heating, and heat generation/annihilation, are also considered. Mathematical modeling from the given physical description is formulated while employing the “long wavelength, low Reynolds number” approximations. Analytical and numerical procedures allow for the determination of flow behavior in the resulting system, the results of which are presented in the form of tables and graphs, in order to facilitate the physical analysis. The results indicate that the growth of nanoparticle volume fraction corresponds to a reduction in temperature, entropy generation, velocity, and pressure gradient. The enhanced Hall and Brinkman parameters reduce the entropy generation and temperature for such flows, whereas the enhanced permeability parameter reduces the velocity and pressure gradient considerably. Furthermore, a comparison of the heat transfer rates for the two results, for different physical parameters, indicates that these results agree well. The significance of the underlying study lies in the fact that it analyzes the peristalsis of a non-Newtonian nanofluid, where the rheological characteristics of the fluid are predicted using the Carreau-Yasuda model and by considering the various physical effects. Full article

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12 pages, 432 KiB

Open AccessEditor’s ChoiceArticle

Adaptive Nonparametric Density Estimation with B-Spline Bases

by Yanchun Zhao, Mengzhu Zhang, Qian Ni and Xuhui Wang

Mathematics 2023, 11(2), 291; https://doi.org/10.3390/math11020291 - 5 Jan 2023

Cited by 8 | Viewed by 2573

Abstract

Learning density estimation is important in probabilistic modeling and reasoning with uncertainty. Since B-spline basis functions are piecewise polynomials with local support, density estimation with B-splines shows its advantages when intensive numerical computations are involved in the subsequent applications. To obtain an optimal [...] Read more.

Learning density estimation is important in probabilistic modeling and reasoning with uncertainty. Since B-spline basis functions are piecewise polynomials with local support, density estimation with B-splines shows its advantages when intensive numerical computations are involved in the subsequent applications. To obtain an optimal local density estimation with B-splines, we need to select the bandwidth (i.e., the distance of two adjacent knots) for uniform B-splines. However, the selection of bandwidth is challenging, and the computation is costly. On the other hand, nonuniform B-splines can improve on the approximation capability of uniform B-splines. Based on this observation, we perform density estimation with nonuniform B-splines. By introducing the error indicator attached to each interval, we propose an adaptive strategy to generate the nonuniform knot vector. The error indicator is an approximation of the information entropy locally, which is closely related to the number of kernels when we construct the nonuniform estimator. The numerical experiments show that, compared with the uniform B-spline, the local density estimation with nonuniform B-splines not only achieves better estimation results but also effectively alleviates the overfitting phenomenon caused by the uniform B-splines. The comparison with the existing estimation procedures, including the state-of-the-art kernel estimators, demonstrates the accuracy of our new method. Full article

(This article belongs to the Section Probability and Statistics)

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17 pages, 888 KiB

Open AccessEditor’s ChoiceArticle

Delayed Impulsive Control for μ-Synchronization of Nonlinear Multi-Weighted Complex Networks with Uncertain Parameter Perturbation and Unbounded Delays

by Hongguang Fan, Jiahui Tang, Kaibo Shi, Yi Zhao and Hui Wen

Mathematics 2023, 11(1), 250; https://doi.org/10.3390/math11010250 - 3 Jan 2023

Cited by 6 | Viewed by 1745

Abstract

The global

μ

-synchronization problem for nonlinear multi-weighted complex dynamical networks with uncertain parameter perturbation and mixed time-varying delays is investigated in this paper. Unlike other existing works, all delays, including sampling and internal and coupling delays, are assumed to be unbounded, making [...] Read more.

The global

μ

-synchronization problem for nonlinear multi-weighted complex dynamical networks with uncertain parameter perturbation and mixed time-varying delays is investigated in this paper. Unlike other existing works, all delays, including sampling and internal and coupling delays, are assumed to be unbounded, making the considered model more general and practical. Based on the generalized impulsive comparison principles, a time-varying impulsive controller with sampling delays is designed, and some new sufficient conditions are obtained to make drive–response multi-weighted networks reach

μ

-synchronization. In addition, the external coupling matrices do not need to meet the requirement of zero-row sum, and the limitation of time delay on pulse interval is weakened. The results obtained in this article can be seen as extensions of previous related research. Full article

(This article belongs to the Topic Engineering Mathematics)

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21 pages, 4493 KiB

Open AccessEditor’s ChoiceArticle

A Combined Model Based on Recurrent Neural Networks and Graph Convolutional Networks for Financial Time Series Forecasting

by Ana Lazcano, Pedro Javier Herrera and Manuel Monge

Mathematics 2023, 11(1), 224; https://doi.org/10.3390/math11010224 - 2 Jan 2023

Cited by 52 | Viewed by 6419

Abstract

Accurate and real-time forecasting of the price of oil plays an important role in the world economy. Research interest in forecasting this type of time series has increased considerably in recent decades, since, due to the characteristics of the time series, it was [...] Read more.

Accurate and real-time forecasting of the price of oil plays an important role in the world economy. Research interest in forecasting this type of time series has increased considerably in recent decades, since, due to the characteristics of the time series, it was a complicated task with inaccurate results. Concretely, deep learning models such as Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs) have appeared in this field with promising results compared to traditional approaches. To improve the performance of existing networks in time series forecasting, in this work two types of neural networks are brought together, combining the characteristics of a Graph Convolutional Network (GCN) and a Bidirectional Long Short-Term Memory (BiLSTM) network. This is a novel evolution that improves existing results in the literature and provides new possibilities in the analysis of time series. The results confirm a better performance of the combined BiLSTM-GCN approach compared to the BiLSTM and GCN models separately, as well as to the traditional models, with a lower error in all the error metrics used: the Root Mean Squared Error (RMSE), the Mean Squared Error (MSE), the Mean Absolute Percentage Error (MAPE) and the R-squared (R²). These results represent a smaller difference between the result returned by the model and the real value and, therefore, a greater precision in the predictions of this model. Full article

(This article belongs to the Special Issue Advanced Methods in the Mathematical Modeling of Economics, Econometrics, and Financial Management)

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12 pages, 316 KiB

Open AccessEditor’s ChoiceArticle

Practical Exponential Stability of Impulsive Stochastic Food Chain System with Time-Varying Delays

by Yuxiao Zhao and Linshan Wang

Mathematics 2023, 11(1), 147; https://doi.org/10.3390/math11010147 - 28 Dec 2022

Cited by 43 | Viewed by 2323

Abstract

This paper studies the practical exponential stability of an impulsive stochastic food chain system with time-varying delays (ISOFCSs). By constructing an auxiliary system equivalent to the original system and comparison theorem, the existence of global positive solutions to the suggested system is discussed. [...] Read more.

This paper studies the practical exponential stability of an impulsive stochastic food chain system with time-varying delays (ISOFCSs). By constructing an auxiliary system equivalent to the original system and comparison theorem, the existence of global positive solutions to the suggested system is discussed. Moreover, we investigate the sufficient conditions for the exponential stability and practical exponential stability of the system, which is given by Razumikhin technique and the Lyapunov method. In addition, when Razumikhin’s condition holds, the exponential stability and practical exponential stability of species are independent of time delay. Finally, numerical simulation finds the validity of the method. Full article

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19 pages, 2470 KiB

Open AccessEditor’s ChoiceArticle

Quasar Identification Using Multivariate Probability Density Estimated from Nonparametric Conditional Probabilities

by Jenny Farmer, Eve Allen and Donald J. Jacobs

Mathematics 2023, 11(1), 155; https://doi.org/10.3390/math11010155 - 28 Dec 2022

Cited by 2 | Viewed by 1895

Abstract

Nonparametric estimation for a probability density function that describes multivariate data has typically been addressed by kernel density estimation (KDE). A novel density estimator recently developed by Farmer and Jacobs offers an alternative high-throughput automated approach to univariate nonparametric density estimation based on [...] Read more.

Nonparametric estimation for a probability density function that describes multivariate data has typically been addressed by kernel density estimation (KDE). A novel density estimator recently developed by Farmer and Jacobs offers an alternative high-throughput automated approach to univariate nonparametric density estimation based on maximum entropy and order statistics, improving accuracy over univariate KDE. This article presents an extension of the single variable case to multiple variables. The univariate estimator is used to recursively calculate a product array of one-dimensional conditional probabilities. In combination with interpolation methods, a complete joint probability density estimate is generated for multiple variables. Good accuracy and speed performance in synthetic data are demonstrated by a numerical study using known distributions over a range of sample sizes from 100 to

10^{6}

for two to six variables. Performance in terms of speed and accuracy is compared to KDE. The multivariate density estimate developed here tends to perform better as the number of samples and/or variables increases. As an example application, measurements are analyzed over five filters of photometric data from the Sloan Digital Sky Survey Data Release 17. The multivariate estimation is used to form the basis for a binary classifier that distinguishes quasars from galaxies and stars with up to 94% accuracy. Full article

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

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11 pages, 290 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Symmetries and Solutions for a Class of Advective Reaction-Diffusion Systems with a Special Reaction Term

by Mariano Torrisi and Rita Tracinà

Mathematics 2023, 11(1), 160; https://doi.org/10.3390/math11010160 - 28 Dec 2022

Cited by 2 | Viewed by 1554

Abstract

This paper is devoted to apply the Lie methods to a class of reaction diffusion advection systems of two interacting species u and v with two arbitrary constitutive functions f and g. The reaction term appearing in the equation for the species [...] Read more.

This paper is devoted to apply the Lie methods to a class of reaction diffusion advection systems of two interacting species u and v with two arbitrary constitutive functions f and g. The reaction term appearing in the equation for the species v is a logistic function of Lotka-Volterra type. Once obtained the Lie algebra for any form of f and g a Lie classification is carried out. Interesting reduced systems are derived admitting wide classes of exact solutions. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering, 2nd Edition)

9 pages, 261 KiB

Open AccessEditor’s ChoiceArticle

Hopf Differential Graded Galois Extensions

by Bo-Ye Zhang

Mathematics 2023, 11(1), 128; https://doi.org/10.3390/math11010128 - 27 Dec 2022

Cited by 1 | Viewed by 1348

Abstract

We introduce the concept of Hopf dg Galois extensions. For a finite dimensional semisimple Hopf algebra H and an H-module dg algebra R, we show that

D (R # H) ≅ D (R^{H})

is equivalent to [...] Read more.

We introduce the concept of Hopf dg Galois extensions. For a finite dimensional semisimple Hopf algebra H and an H-module dg algebra R, we show that

D (R # H) ≅ D (R^{H})

is equivalent to that

R / R^{H}

is a Hopf differential graded Galois extension. We present a weaker version of Hopf differential graded Galois extensions and show the relationships between Hopf differential graded Galois extensions and Hopf Galois extensions. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

17 pages, 384 KiB

Open AccessEditor’s ChoiceArticle

On the Regularity of Weak Solutions to Time-Periodic Navier–Stokes Equations in Exterior Domains

by Thomas Eiter

Mathematics 2023, 11(1), 141; https://doi.org/10.3390/math11010141 - 27 Dec 2022

Viewed by 1514

Abstract

Consider the time-periodic viscous incompressible fluid flow past a body with non-zero velocity at infinity. This article gives sufficient conditions such that weak solutions to this problem are smooth. Since time-periodic solutions do not have finite kinetic energy in general, the well-known regularity [...] Read more.

Consider the time-periodic viscous incompressible fluid flow past a body with non-zero velocity at infinity. This article gives sufficient conditions such that weak solutions to this problem are smooth. Since time-periodic solutions do not have finite kinetic energy in general, the well-known regularity results for weak solutions to the corresponding initial-value problem cannot be transferred directly. The established regularity criterion demands a certain integrability of the purely periodic part of the velocity field or its gradient, but it does not concern the time mean of these quantities. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics II)

14 pages, 7952 KiB

Open AccessEditor’s ChoiceArticle

Heterogeneous Feature Fusion Module Based on CNN and Transformer for Multiview Stereo Reconstruction

by Rui Gao, Jiajia Xu, Yipeng Chen and Kyungeun Cho

Mathematics 2023, 11(1), 112; https://doi.org/10.3390/math11010112 - 26 Dec 2022

Cited by 3 | Viewed by 2930

Abstract

For decades, a vital area of computer vision research has been multiview stereo (MVS), which creates 3D models of a scene using photographs. This study presents an effective MVS network for 3D reconstruction utilizing multiview pictures. Alternative learning-based reconstruction techniques work well, because [...] Read more.

For decades, a vital area of computer vision research has been multiview stereo (MVS), which creates 3D models of a scene using photographs. This study presents an effective MVS network for 3D reconstruction utilizing multiview pictures. Alternative learning-based reconstruction techniques work well, because CNNs (convolutional neural network) can extract only the image’s local features; however, they contain many artifacts. Herein, a transformer and CNN are used to extract the global and local features of the image, respectively. Additionally, hierarchical aggregation and heterogeneous interaction modules were used to improve these features. They are based on the transformer and can extract dense features with 3D consistency and global context that are necessary to provide accurate matching for MVS. Full article

(This article belongs to the Special Issue Advances of Mathematical Image Processing)

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18 pages, 1450 KiB

Open AccessEditor’s ChoiceArticle

Explicit Gaussian Variational Approximation for the Poisson Lognormal Mixed Model

by Xiaoping Shi, Xiang-Sheng Wang and Augustine Wong

Mathematics 2022, 10(23), 4542; https://doi.org/10.3390/math10234542 - 1 Dec 2022

Viewed by 1404

Abstract

In recent years, the Poisson lognormal mixed model has been frequently used in modeling count data because it can accommodate both the over-dispersion of the data and the existence of within-subject correlation. Since the likelihood function of this model is expressed in terms [...] Read more.

In recent years, the Poisson lognormal mixed model has been frequently used in modeling count data because it can accommodate both the over-dispersion of the data and the existence of within-subject correlation. Since the likelihood function of this model is expressed in terms of an intractable integral, estimating the parameters and obtaining inference for the parameters are challenging problems. Some approximation procedures have been proposed in the literature; however, they are computationally intensive. Moreover, the existing studies of approximate parameter inference using the Gaussian variational approximation method are usually restricted to models with only one predictor. In this paper, we consider the Poisson lognormal mixed model with more than one predictor. By extending the Gaussian variational approximation method, we derive explicit forms for the estimators of the parameters and examine their properties, including the asymptotic distributions of the estimators of the parameters. Accurate inference for the parameters is also obtained. A real-life example demonstrates the applicability of the proposed method, and simulation studies illustrate the accuracy of the proposed method. Full article

(This article belongs to the Special Issue Machine Learning and Statistical Methods to Prediction and Optimal Decision-Making)

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16 pages, 616 KiB

Open AccessEditor’s ChoiceArticle

Poincaré Map for Discontinuous Fractional Differential Equations

by Ivana Eliašová and Michal Fečkan

Mathematics 2022, 10(23), 4476; https://doi.org/10.3390/math10234476 - 27 Nov 2022

Cited by 1 | Viewed by 1279

Abstract

We work with a perturbed fractional differential equation with discontinuous right-hand sides where a discontinuity function crosses a discontinuity boundary transversally. The corresponding Poincaré map in a neighbourhood of a periodic orbit of an unperturbed equation is found. Then, bifurcations of periodic boundary [...] Read more.

We work with a perturbed fractional differential equation with discontinuous right-hand sides where a discontinuity function crosses a discontinuity boundary transversally. The corresponding Poincaré map in a neighbourhood of a periodic orbit of an unperturbed equation is found. Then, bifurcations of periodic boundary solutions are analysed together with a concrete example. Full article

(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications)

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15 pages, 344 KiB

Open AccessEditor’s ChoiceArticle

RKHS Representations for Augmented Quaternion Random Signals: Application to Detection Problems

by Antonia Oya

Mathematics 2022, 10(23), 4432; https://doi.org/10.3390/math10234432 - 24 Nov 2022

Cited by 3 | Viewed by 1411

Abstract

The reproducing kernel Hilbert space (RKHS) methodology has shown to be a suitable tool for the resolution of a wide range of problems in statistical signal processing both in the real and complex domains. It relies on the idea of transforming the original [...] Read more.

The reproducing kernel Hilbert space (RKHS) methodology has shown to be a suitable tool for the resolution of a wide range of problems in statistical signal processing both in the real and complex domains. It relies on the idea of transforming the original functional data into an infinite series representation by projection onto an specific RKHS, which usually simplifies the statistical treatment without any loss of efficiency. Moreover, the advantages of quaternion algebra over real-valued three and four-dimensional vector algebra in the modelling of multidimensional data have been proven useful in much relatively recent research. This paper accordingly proposes a generic RKHS framework for the statistical analysis of augmented quaternion random vectors, which provide a complete description of their second order characteristics. It will allow us to exploit the full advantages of the RKHS theory in widely linear processing applications, such as signal detection. In particular, we address the detection of a quaternion signal disturbed by additive Gaussian noise and the discrimination between two quaternion Gaussian signals in continuous time. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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13 pages, 361 KiB

Open AccessEditor’s ChoiceArticle

Two Approaches to the Traffic Quality Intuitionistic Fuzzy Estimation of Service Compositions

by Stoyan Poryazov, Velin Andonov, Emiliya Saranova and Krassimir Atanassov

Mathematics 2022, 10(23), 4439; https://doi.org/10.3390/math10234439 - 24 Nov 2022

Cited by 12 | Viewed by 1169

Abstract

Recently, intuitionistic fuzzy pairs have been used as uncertainty estimations of the request services in service systems. In the present paper, three intuitionistic fuzzy characterizations of virtual service devices are specified: intuitionistic fuzzy traffic estimation, intuitionistic fuzzy flow estimation and intuitionistic fuzzy estimation [...] Read more.

Recently, intuitionistic fuzzy pairs have been used as uncertainty estimations of the request services in service systems. In the present paper, three intuitionistic fuzzy characterizations of virtual service devices are specified: intuitionistic fuzzy traffic estimation, intuitionistic fuzzy flow estimation and intuitionistic fuzzy estimation about probability. Discussed herein are two approaches to the intuitionistic fuzzy estimation of the uncertainty of compositions of services. One of the approaches is based on the definitions of the intuitionistic fuzzy pairs for one service device. The other approach is based on the aggregation operators over intuitionistic fuzzy pairs. A total of six intuitionistic fuzzy estimations of the uncertainty of comprise service device are proposed. The proposed uncertainty estimations allow for the definition of new Quality of Service (QoS) indicators and can be used to determine the quality of service compositions across a wide range of service systems. Full article

(This article belongs to the Special Issue Modelling of Parallel Processes under Intuitionistic Fuzzy Uncertainty)

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17 pages, 4945 KiB

Open AccessEditor’s ChoiceArticle

A Bibliometric Analysis of the Use of Artificial Intelligence Technologies for Social Sciences

by Tuba Bircan and Almila Alkim Akdag Salah

Mathematics 2022, 10(23), 4398; https://doi.org/10.3390/math10234398 - 22 Nov 2022

Cited by 11 | Viewed by 7102

Abstract

The use of Artificial Intelligence (AI) and Big Data analysis algorithms is complementary to theory-driven analysis approaches and becoming more popular also in social sciences. This paper describes the use of Big Data and computational approaches in social sciences by bibliometric analyses of [...] Read more.

The use of Artificial Intelligence (AI) and Big Data analysis algorithms is complementary to theory-driven analysis approaches and becoming more popular also in social sciences. This paper describes the use of Big Data and computational approaches in social sciences by bibliometric analyses of articles indexed between 2015 and 2020 in Social Sciences Citation Index (SSCI) of the Web of Science repository. We have analysed especially the recent research direction called Computational Social Sciences (CSS) that bridges computer analytical approaches with social science challenges, generating new methodologies of Big Data and AI analytics for social sciences. The results indicate that AI and Big Data practices are not confined to CSS only and are diffused in a wide variety of disciplines under Social Sciences and are made use of in many main research lines as well. Thus, the anticipated overlap between the Social Sciences & AI specialization and CSS has yet to be crystallised. Moreover, the impact of computational social science studies is not permeated to social science citation networks yet. Lastly, we demonstrate that the AI and Big Data publications that appear under the SSCI index are more oriented towards computational studies than addressing social science concepts, concerns, and challenges. Full article

(This article belongs to the Special Issue New Applications of Data Analysis Methodologies and Techniques to the Social Sciences)

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17 pages, 1109 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Estimating Value-at-Risk and Expected Shortfall: Do Polynomial Expansions Outperform Parametric Densities?

by Brenda Castillo-Brais, Ángel León and Juan Mora

Mathematics 2022, 10(22), 4329; https://doi.org/10.3390/math10224329 - 18 Nov 2022

Cited by 3 | Viewed by 2226

Abstract

We assess Value-at-Risk (VaR) and Expected Shortfall (ES) estimates assuming different models for the standardized returns: distributions based on polynomial expansions such as Cornish-Fisher and Gram-Charlier, and well-known parametric densities such as normal, skewed-t and Johnson. This paper aims to analyze whether models [...] Read more.

We assess Value-at-Risk (VaR) and Expected Shortfall (ES) estimates assuming different models for the standardized returns: distributions based on polynomial expansions such as Cornish-Fisher and Gram-Charlier, and well-known parametric densities such as normal, skewed-t and Johnson. This paper aims to analyze whether models based on polynomial expansions outperform the parametric ones. We carry out the model performance comparison in two stages: first, with a backtesting analysis of VaR and ES; and second, using loss functions. Our backtesting results show that all distributions, except for normal ones, perform quite well in VaR and ES estimations. Regarding the loss function analysis, we conclude that polynomial expansions (specifically, the Cornish-Fisher one) usually outperform parametric densities in VaR estimation, but the latter (specifically, the Johnson density) slightly outperform the former in ES estimation; however, the gains of using one approach or the other are modest. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

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20 pages, 465 KiB

Open AccessEditor’s ChoiceArticle

DHGEEP: A Dynamic Heterogeneous Graph-Embedding Method for Evolutionary Prediction

by Libin Chen, Luyao Wang, Chengyi Zeng, Hongfu Liu and Jing Chen

Mathematics 2022, 10(22), 4193; https://doi.org/10.3390/math10224193 - 9 Nov 2022

Cited by 3 | Viewed by 2134

Abstract

Current graph-embedding methods mainly focus on static homogeneous graphs, where the entity type is the same and the topology is fixed. However, in real networks, such as academic networks and shopping networks, there are typically various types of nodes and temporal interactions. The [...] Read more.

Current graph-embedding methods mainly focus on static homogeneous graphs, where the entity type is the same and the topology is fixed. However, in real networks, such as academic networks and shopping networks, there are typically various types of nodes and temporal interactions. The dynamical and heterogeneous components of graphs in general contain abundant information. Currently, most studies on dynamic graphs do not sufficiently consider the heterogeneity of the network in question, and hence the semantic information of the interactions between heterogeneous nodes is missing in the graph embeddings. On the other hand, the overall size of the network tends to accumulate over time, and its growth rate can reflect the ability of the entire network to generate interactions of heterogeneous nodes; therefore, we developed a graph dynamics model to model the evolution of graph dynamics. Moreover, the temporal properties of nodes regularly affect the generation of temporal interaction events with which they are connected. Thus, we developed a node dynamics model to model the evolution of node connectivity. In this paper, we propose DHGEEP, a dynamic heterogeneous graph-embedding method based on the Hawkes process, to predict the evolution of dynamic heterogeneous networks. The model considers the generation of temporal events as an effect of historical events, introduces the Hawkes process to simulate this evolution, and then captures semantic and structural information based on the meta-paths of temporal heterogeneous nodes. Finally, the graph-level dynamics of the network and the node-level dynamics of each node are integrated into the DHGEEP framework. The embeddings of the nodes are automatically obtained by minimizing the value of the loss function. Experiments were conducted on three downstream tasks, static link prediction, temporal event prediction for homogeneous nodes, and temporal event prediction for heterogeneous nodes, on three datasets. Experimental results show that DHGEEP achieves excellent performance in these tasks. In the most significant task, temporal event prediction of heterogeneous nodes, the values of precision@2 and recall@2 can reach 30.23% and 10.48% on the AMiner dataset, and reach 4.56% and 1.61% on the DBLP dataset, so that our method is more accurate at predicting future temporal events than the baseline. Full article

(This article belongs to the Special Issue Complex Network Modeling: Theory and Applications)

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14 pages, 9874 KiB

Open AccessEditor’s ChoiceArticle

Stochastic Modeling of Within-Host Dynamics of Plasmodium Falciparum

by Xiao Sun, James M. McCaw and Pengxing Cao

Mathematics 2022, 10(21), 4057; https://doi.org/10.3390/math10214057 - 1 Nov 2022

Cited by 1 | Viewed by 1585

Abstract

Malaria remains a major public health burden in South-East Asia and Africa. Mathematical models of within-host infection dynamics and drug action, developed in support of malaria elimination initiatives, have significantly advanced our understanding of the dynamics of infection and supported development of effective [...] Read more.

Malaria remains a major public health burden in South-East Asia and Africa. Mathematical models of within-host infection dynamics and drug action, developed in support of malaria elimination initiatives, have significantly advanced our understanding of the dynamics of infection and supported development of effective drug-treatment regimens. However, the mathematical models supporting these initiatives are predominately based on deterministic dynamics and therefore cannot capture stochastic phenomena such as extinction (no parasitized red blood cells) following treatment, with potential consequences for our interpretation of data sets in which recrudescence is observed. Here we develop a stochastic within-host infection model to study the growth, decline and possible stochastic extinction of parasitized red blood cells in malaria-infected human volunteers. We show that stochastic extinction can occur when the inoculation size is small or when the number of parasitized red blood cells reduces significantly after an antimalarial treatment. We further show that the drug related parameters, such as the maximum killing rate and half-maximum effective concentration, are the primary factors determining the probability of stochastic extinction following treatment, highlighting the importance of highly-efficacious antimalarials in increasing the probability of cure for the treatment of malaria patients. Full article

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

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18 pages, 4904 KiB

Open AccessEditor’s ChoiceArticle

A Meshfree Approach for Solving Fractional Galilei Invariant Advection–Diffusion Equation through Weighted–Shifted Grünwald Operator

by Farzaneh Safari, Qingshan Tong, Zhen Tang and Jun Lu

Mathematics 2022, 10(21), 4008; https://doi.org/10.3390/math10214008 - 28 Oct 2022

Cited by 8 | Viewed by 1266

Abstract

Fractional Galilei invariant advection–diffusion (GIADE) equation, along with its more general version that is the GIADE equation with nonlinear source term, is discretized by coupling weighted and shifted Grünwald difference approximation formulae and Crank–Nicolson technique. The new version of the backward substitution method, [...] Read more.

Fractional Galilei invariant advection–diffusion (GIADE) equation, along with its more general version that is the GIADE equation with nonlinear source term, is discretized by coupling weighted and shifted Grünwald difference approximation formulae and Crank–Nicolson technique. The new version of the backward substitution method, a well-established class of meshfree methods, is proposed for a numerical approximation of the consequent equation. In the present approach, the final approximation is given by the summation of the radial basis functions, the primary approximation, and the related correcting functions. Then, the approximation is substituted back to the governing equations where the unknown parameters can be determined. The polynomials, trigonometric functions, multiquadric, or the Gaussian radial basis functions are used in the approximation of the GIADE. Moreover, a quasilinearization technique is employed to transform a nonlinear source term into a linear source term. Finally, three numerical experiments in one and two dimensions are presented to support the method. Full article

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19 pages, 6979 KiB

Open AccessEditor’s ChoiceArticle

Quantification of Model Uncertainty Based on Variance and Entropy of Bernoulli Distribution

by Zdeněk Kala

Mathematics 2022, 10(21), 3980; https://doi.org/10.3390/math10213980 - 26 Oct 2022

Cited by 5 | Viewed by 3437

Abstract

This article studies the role of model uncertainties in sensitivity and probability analysis of reliability. The measure of reliability is failure probability. The failure probability is analysed using the Bernoulli distribution with binary outcomes of success (0) and failure (1). Deeper connections between [...] Read more.

This article studies the role of model uncertainties in sensitivity and probability analysis of reliability. The measure of reliability is failure probability. The failure probability is analysed using the Bernoulli distribution with binary outcomes of success (0) and failure (1). Deeper connections between Shannon entropy and variance are explored. Model uncertainties increase the heterogeneity in the data 0 and 1. The article proposes a new methodology for quantifying model uncertainties based on the equality of variance and entropy. This methodology is briefly called “variance = entropy”. It is useful for stochastic computational models without additional information. The “variance = entropy” rule estimates the “safe” failure probability with the added effect of model uncertainties without adding random variables to the computational model. Case studies are presented with seven variants of model uncertainties that can increase the variance to the entropy value. Although model uncertainties are justified in the assessment of reliability, they can distort the results of the global sensitivity analysis of the basic input variables. The solution to this problem is a global sensitivity analysis of failure probability without added model uncertainties. This paper shows that Shannon entropy is a good sensitivity measure that is useful for quantifying model uncertainties. Full article

(This article belongs to the Special Issue Sensitivity Analysis)

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13 pages, 1494 KiB

Open AccessEditor’s ChoiceArticle

Effect of Inflation and Permitted Three-Slot Payment on Two-Warehouse Inventory System with Stock-Dependent Demand and Partial Backlogging

by Rajamanickam Thilagavathi, Jagadeesan Viswanath, Lenka Cepova and Vladimira Schindlerova

Mathematics 2022, 10(21), 3943; https://doi.org/10.3390/math10213943 - 24 Oct 2022

Cited by 8 | Viewed by 1747

Abstract

This study examines the effect of monetary inflation for a two-warehouse single-product inventory system, in which items are stored in a limited capacity Own Warehouse (OW) and an unlimited capacity Rental Warehouse (RW). Demand for an item is considered [...] Read more.

This study examines the effect of monetary inflation for a two-warehouse single-product inventory system, in which items are stored in a limited capacity Own Warehouse (OW) and an unlimited capacity Rental Warehouse (RW). Demand for an item is considered stock dependent. Items may deteriorate at a different constant rate in both warehouses. Shortages are allowed in the stock-out period and are partially backlogged and satisfied in the next replenishment point. The supplier permits flexible payment options for the retailer to pay the amount in three equal payments at different time points. The retailers’ preferred payment option is as follows: the first payment is prior to the replenishment point with some discount; the second payment is one-third of the total purchasing cost, which is paid at the time of the replenishment epoch; and the third payment is after the replenishment point and before the start of the next cycle, with some penalty. The influence of inflation on the cost calculation is considered, and an analytic expression for optimal minimal cost is explicitly derived from this. We performed arrived sensitivity analysis to discern the effects of the inflation and backlogging rates, as well as the effects of the discount rate on purchasing cost, and the effects of penalties upon the late payment of purchasing costs in optimizing the total cost. Full article

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17 pages, 399 KiB

Open AccessEditor’s ChoiceArticle

Quantum Communication with Polarization-Encoded Qubits under Majorization Monotone Dynamics

by Artur Czerwinski

Mathematics 2022, 10(21), 3932; https://doi.org/10.3390/math10213932 - 23 Oct 2022

Cited by 7 | Viewed by 2381

Abstract

Quantum communication can be realized by transmitting photons that carry quantum information. Due to decoherence, the information encoded in the quantum state of a single photon can be distorted, which leads to communication errors. In particular, we consider the impact of majorization monotone [...] Read more.

Quantum communication can be realized by transmitting photons that carry quantum information. Due to decoherence, the information encoded in the quantum state of a single photon can be distorted, which leads to communication errors. In particular, we consider the impact of majorization monotone dynamical maps on the efficiency of quantum communication. The mathematical formalism of majorization is revised with its implications for quantum systems. The discrimination probability for two arbitrary orthogonal states is used as a figure of merit to track the quality of quantum communication in the time domain. Full article

(This article belongs to the Special Issue Advances in Quantum Optics and Quantum Information)

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20 pages, 573 KiB

Open AccessEditor’s ChoiceArticle

On the Regulated Nuclear Transport of Incompletely Spliced mRNAs by HIV-Rev Protein: A Minimal Dynamic Model

by Jeffrey J. Ishizuka, Delaney A. Soble, Tiffany Y. Chang and Enrique Peacock-López

Mathematics 2022, 10(21), 3922; https://doi.org/10.3390/math10213922 - 22 Oct 2022

Viewed by 1711

Abstract

A kinetic model for the HIV-1 Rev protein is developed by drawing upon mechanistic information from the literature to formulate a set of differential equations modeling the behavior of Rev and its various associated factors over time. A set of results demonstrates the [...] Read more.

A kinetic model for the HIV-1 Rev protein is developed by drawing upon mechanistic information from the literature to formulate a set of differential equations modeling the behavior of Rev and its various associated factors over time. A set of results demonstrates the possibility of oscillations in the concentration of these factors. Finally, the results are analyzed, and future directions are discussed. Full article

(This article belongs to the Special Issue Big Data and Bioinformatics)

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18 pages, 1640 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Open-Source Computational Photonics with Auto Differentiable Topology Optimization

by Benjamin Vial and Yang Hao

Mathematics 2022, 10(20), 3912; https://doi.org/10.3390/math10203912 - 21 Oct 2022

Cited by 8 | Viewed by 2879

Abstract

In recent years, technological advances in nanofabrication have opened up new applications in the field of nanophotonics. To engineer and develop novel functionalities, rigorous and efficient numerical methods are required. In parallel, tremendous advances in algorithmic differentiation, in part pushed by the intensive [...] Read more.

In recent years, technological advances in nanofabrication have opened up new applications in the field of nanophotonics. To engineer and develop novel functionalities, rigorous and efficient numerical methods are required. In parallel, tremendous advances in algorithmic differentiation, in part pushed by the intensive development of machine learning and artificial intelligence, has made possible large-scale optimization of devices with a few extra modifications of the underlying code. We present here our development of three different software libraries for solving Maxwell’s equations in various contexts: a finite element code with a high-level interface for problems commonly encountered in photonics, an implementation of the Fourier modal method for multilayered bi-periodic metasurfaces and a plane wave expansion method for the calculation of band diagrams in two-dimensional photonic crystals. All of them are endowed with automatic differentiation capabilities and we present typical inverse design examples. Full article

(This article belongs to the Special Issue Advanced Mathematical and Numerical Modeling in Electrodynamics and Photonics)

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24 pages, 527 KiB

Open AccessEditor’s ChoiceArticle

Multi-Server Queuing Production Inventory System with Emergency Replenishment

by Dhanya Shajin, Achyutha Krishnamoorthy, Agassi Z. Melikov and Janos Sztrik

Mathematics 2022, 10(20), 3839; https://doi.org/10.3390/math10203839 - 17 Oct 2022

Cited by 6 | Viewed by 1847

Abstract

We consider a multi-server production inventory system with an unlimited waiting line. Arrivals occur according to a non-homogeneous Poisson process and exponentially distributed service time. At the service completion epoch, one unit of an item in the on-hand inventory decreases with probability

δ

[...] Read more.

We consider a multi-server production inventory system with an unlimited waiting line. Arrivals occur according to a non-homogeneous Poisson process and exponentially distributed service time. At the service completion epoch, one unit of an item in the on-hand inventory decreases with probability

δ

, and the customer leaves the system without taking the item with probability

(1 - δ)

. The production inventory system adopts an

(s, S)

policy where the processing of inventory requires a positive random amount of time. The production time for a unit item is phase-type distributed. Furthermore, assume that an emergency replenishment of one item with zero lead time takes place when the on-hand inventory level decreases to zero. The emergency replenishment is incorporated in the system to ensure customer satisfaction. We derive the stationary distribution of the system and some main performance measures, such as the distribution of the production on/off time in a cycle and the mean emergency replenishment cycle time. Numerical experiments are conducted to illustrate the system performance. A cost function is constructed, and we examine the optimal number of servers to be employed. Furthermore, we numerically calculate the optimal values of the production starting level and maximum inventory level. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)

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18 pages, 4634 KiB

Open AccessEditor’s ChoiceArticle

Fast Method Based on Fuzzy Logic for Gaussian-Impulsive Noise Reduction in CT Medical Images

by Josep Arnal and Luis Súcar

Mathematics 2022, 10(19), 3652; https://doi.org/10.3390/math10193652 - 5 Oct 2022

Cited by 5 | Viewed by 2356

Abstract

To remove Gaussian-impulsive mixed noise in CT medical images, a parallel filter based on fuzzy logic is applied. The used methodology is structured in two steps. A method based on a fuzzy metric is applied to remove the impulsive noise at the first [...] Read more.

To remove Gaussian-impulsive mixed noise in CT medical images, a parallel filter based on fuzzy logic is applied. The used methodology is structured in two steps. A method based on a fuzzy metric is applied to remove the impulsive noise at the first step. To reduce Gaussian noise, at the second step, a fuzzy peer group filter is used on the filtered image obtained at the first step. A comparative analysis with state-of-the-art methods is performed on CT medical images using qualitative and quantitative measures evidencing the effectiveness of the proposed algorithm. The parallel method is parallelized on shared memory multiprocessors. After applying parallel computing strategies, the obtained computing times indicate that the introduced filter enables to reduce Gaussian-impulse mixed noise on CT medical images in real-time. Full article

(This article belongs to the Special Issue Fuzzy Logic and Its Applications)

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10 pages, 283 KiB

Open AccessEditor’s ChoiceArticle

The Power Fractional Calculus: First Definitions and Properties with Applications to Power Fractional Differential Equations

by El Mehdi Lotfi, Houssine Zine, Delfim F. M. Torres and Noura Yousfi

Mathematics 2022, 10(19), 3594; https://doi.org/10.3390/math10193594 - 1 Oct 2022

Cited by 5 | Viewed by 2301

Abstract

Using the Laplace transform method and the convolution theorem, we introduce new and more general definitions for fractional operators with non-singular kernels, extending well-known concepts existing in the literature. The new operators are based on a generalization of the Mittag–Leffler function, characterized by [...] Read more.

Using the Laplace transform method and the convolution theorem, we introduce new and more general definitions for fractional operators with non-singular kernels, extending well-known concepts existing in the literature. The new operators are based on a generalization of the Mittag–Leffler function, characterized by the presence of a key parameter p. This power parameter p is important to enable researchers to choose an adequate notion of the derivative that properly represents the reality under study, to provide good mathematical models, and to predict future dynamic behaviors. The fundamental properties of the new operators are investigated and rigorously proved. As an application, we solve a Caputo and a Riemann–Liouville fractional differential equation. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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30 pages, 7042 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Analysis of a Queueing Model with MAP Arrivals and Heterogeneous Phase-Type Group Services

by Srinivas R. Chakravarthy

Mathematics 2022, 10(19), 3575; https://doi.org/10.3390/math10193575 - 30 Sep 2022

Cited by 7 | Viewed by 1938

Abstract

Queueing models have proven to be very useful in real-life applications to enable the practitioners to optimize the limited resources to conduct their businesses as well as offer services efficiently. In general, we can group such applications into two sectors: manufacturing and service. [...] Read more.

Queueing models have proven to be very useful in real-life applications to enable the practitioners to optimize the limited resources to conduct their businesses as well as offer services efficiently. In general, we can group such applications into two sectors: manufacturing and service. These two sectors cover everything we deal with on a day-to-day basis. Queues in which the services are offered in blocks (or groups or batches) are well established in the literature and have a wide variety of applications in practice. In this paper, we look at one such queueing model in which the arrivals occur according to a Markovian arrival process and the services are offered in batches of varying sizes from 1 to a finite pre-determined constant, say, b. The service times are assumed to be of phase type with representation depending on the size of the group. Thus, the distributions considered are heterogeneous from both the representation and rate points of view. The model can be studied as a

G I / M / 1

-type queue or as a

Q B D

-model. The model is analyzed in steady state by establishing results including on the rate matrix and the waiting time distribution and providing a number of illustrative examples. Full article

(This article belongs to the Special Issue Mathematics: 10th Anniversary)

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17 pages, 2923 KiB

Open AccessEditor’s ChoiceArticle

Tensor of Order Two and Geometric Properties of 2D Metric Space

by Tomáš Stejskal, Jozef Svetlík and Marcela Lascsáková

Mathematics 2022, 10(19), 3524; https://doi.org/10.3390/math10193524 - 27 Sep 2022

Cited by 1 | Viewed by 1853

Abstract

A 2D metric space has a limited number of properties through which it can be described. This metric space may comprise objects such as a scalar, a vector, and a rank-2 tensor. The paper provides a comprehensive description of relations between objects in [...] Read more.

A 2D metric space has a limited number of properties through which it can be described. This metric space may comprise objects such as a scalar, a vector, and a rank-2 tensor. The paper provides a comprehensive description of relations between objects in 2D space using the matrix product of vectors, geometric product, and dot product of complex numbers. These relations are also an integral part of the Lagrange’s identity. The entire structure of derived theoretical relationships describing properties of 2D space draws on the Lagrange’s identity. The description of how geometric algebra and tensor calculus are interconnected is given here in a comprehensive and essentially clear manner, which is the main contribution of this paper. A new term in this regard is the total geometric and matrix product, which—in a simple manner—predetermines and defines the existence of differential relations such as the gradient, the divergence, and the curl of a vector field. In addition, geometric interpretation of tensors is pointed out, expressed through angular parameters known from the literature as a tensor glyph. This angular interpretation of the tensor has an unequivocal analytical form, and the paper shows how it is linked to the classical tensor denoted by indices. Full article

(This article belongs to the Section Engineering Mathematics)

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20 pages, 2324 KiB

Open AccessEditor’s ChoiceArticle

A Population Pyramid Dynamics Model and Its Analytical Solution. Application Case for Spain

by Joan C. Micó

Mathematics 2022, 10(19), 3443; https://doi.org/10.3390/math10193443 - 22 Sep 2022

Cited by 2 | Viewed by 2674

Abstract

This paper presents the population pyramid dynamics model (PPDM) to study the evolution of the population pyramid of a determined country or society, deducing as a crucial objective its exact analytical solution. The PPDM is a first-order linear partial differential equation whose unknown [...] Read more.

This paper presents the population pyramid dynamics model (PPDM) to study the evolution of the population pyramid of a determined country or society, deducing as a crucial objective its exact analytical solution. The PPDM is a first-order linear partial differential equation whose unknown variable is the population density (population per age unit) depending on time and age, jointly an initial condition in the initial time and a boundary condition given by the births in the zero age. In addition, the dynamical patterns of the crude birth, death, immigration and emigration rates depending on time, jointly with the mathematical pattern of the initial population pyramid depending on ages, take part of the PPDM. These patterns can be obtained from the historical data. An application case of the PPDM analytical solution is presented: Spain, in the 2007–2021 period for its validation, and in the 2021–2026 period for its future forecasting. This application case also permits to obtain the forecasting limits of the PPDM by comparing the historical data with those provided by the PPDM. Other variables that can be obtained from the historical population pyramids data, such as the dependency ratio and the life expectancy at birth, are considered. Full article

(This article belongs to the Special Issue Mathematical Methods and Models in Nature and Society)

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26 pages, 2529 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Sampling Rate Optimization and Execution Time Analysis for Two-Degrees-of-Freedom Control Systems

by Mircea Şuşcă, Vlad Mihaly, Dora Morar and Petru Dobra

Mathematics 2022, 10(19), 3449; https://doi.org/10.3390/math10193449 - 22 Sep 2022

Cited by 3 | Viewed by 1909

Abstract

The current journal paper proposes an end-to-end analysis for the numerical implementation of a two-degrees-of-freedom (2DOF) control structure, starting from the sampling rate selection mechanism via a quasi-optimal manner, along with the estimation of the worst-case execution time (WCET) for the specified controller. [...] Read more.

The current journal paper proposes an end-to-end analysis for the numerical implementation of a two-degrees-of-freedom (2DOF) control structure, starting from the sampling rate selection mechanism via a quasi-optimal manner, along with the estimation of the worst-case execution time (WCET) for the specified controller. For the sampling rate selection, the classical Shannon–Nyquist sampling theorem is replaced by an optimization problem that encompasses the trade-off between the fidelity of the controllers’ representation, along with the fidelity of the resulting closed-loop systems, and the implementation difficulty of the controllers. Additionally, the WCET analysis can be seen as a verification step before automatic code generation, a computational model being provided. The proposed computational model encompasses infinite-impulse response (IIR) and finite-impulse response (FIR) filter models for the controller implementation, along with additional relevant phenomena being discussed, such as saturation, signal scaling and anti-windup techniques. All proposed results will be illustrated on a DC motor benchmark control problem. Full article

(This article belongs to the Section Engineering Mathematics)

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27 pages, 836 KiB

Open AccessEditor’s ChoiceArticle

The Mathematical Model of Cyclic Signals in Dynamic Systems as a Cyclically Correlated Random Process

by Serhii Lupenko

Mathematics 2022, 10(18), 3406; https://doi.org/10.3390/math10183406 - 19 Sep 2022

Cited by 6 | Viewed by 3029

Abstract

This work is devoted to the procedure for constructing of a cyclically correlated random process of a continuous argument as a mathematical model of cyclic signals in dynamic systems, which makes it possible to consistently describe cyclic stochastic signals, both with regular and [...] Read more.

This work is devoted to the procedure for constructing of a cyclically correlated random process of a continuous argument as a mathematical model of cyclic signals in dynamic systems, which makes it possible to consistently describe cyclic stochastic signals, both with regular and irregular rhythms, not separating them, but complementing them within the framework of a single integrated model. The class of cyclically correlated random processes includes the subclass of cyclostationary (periodically) correlated random processes, which enable the use of a set of powerful methods of analysis and the forecasting of cyclic signals with a stable rhythm. Mathematical structures that model the cyclic, phase and rhythmic structures of a cyclically correlated random process are presented. The sufficient and necessary conditions that the structural function and the rhythm function of the cyclically correlated random process must satisfy have been established. The advantages of the cyclically correlated random process in comparison with other mathematical models of cyclic signals with a variable rhythm are given. The obtained results contribute to the emergence of a more complete and rigorous theory of this class of random processes and increase the validity of the methods of their analysis and computer simulation. Full article

(This article belongs to the Special Issue Dynamical Systems and System Analysis)

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13 pages, 780 KiB

Open AccessEditor’s ChoiceArticle

Analytical Investigations into Anomalous Diffusion Driven by Stress Redistribution Events: Consequences of Lévy Flights

by Josiah D. Cleland and Martin A. K. Williams

Mathematics 2022, 10(18), 3235; https://doi.org/10.3390/math10183235 - 6 Sep 2022

Cited by 2 | Viewed by 1624

Abstract

This research is concerned with developing a generalised diffusion equation capable of describing diffusion processes driven by underlying stress-redistributing type events. The work utilises the development of an appropriate continuous time random walk framework as a foundation to consider a new generalised diffusion [...] Read more.

This research is concerned with developing a generalised diffusion equation capable of describing diffusion processes driven by underlying stress-redistributing type events. The work utilises the development of an appropriate continuous time random walk framework as a foundation to consider a new generalised diffusion equation. While previous work has explored the resulting generalised diffusion equation for jump-timings motivated by stick-slip physics, here non-Gaussian probability distributions of the jump displacements are also considered, specifically Lévy flights. This work illuminates several features of the analytic solution to such a generalised diffusion equation using several known properties of the Fox H function. Specifically demonstrated are the temporal behaviour of the resulting position probability density function, and its normalisation. The reduction of the proposed form to expected known solutions upon the insertion of simplifying parameter values, as well as a demonstration of asymptotic behaviours, is undertaken to add confidence to the validity of this equation. This work describes the analytical solution of such a generalised diffusion equation for the first time, and additionally demonstrates the capacity of the Fox H function and its properties in solving and studying generalised Fokker–Planck equations. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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25 pages, 13123 KiB

Open AccessEditor’s ChoiceArticle

A Study of Learning Issues in Feedforward Neural Networks

by Adrian Teso-Fz-Betoño, Ekaitz Zulueta, Mireya Cabezas-Olivenza, Daniel Teso-Fz-Betoño and Unai Fernandez-Gamiz

Mathematics 2022, 10(17), 3206; https://doi.org/10.3390/math10173206 - 5 Sep 2022

Cited by 2 | Viewed by 2414

Abstract

When training a feedforward stochastic gradient descendent trained neural network, there is a possibility of not learning a batch of patterns correctly that causes the network to fail in the predictions in the areas adjacent to those patterns. This problem has usually been [...] Read more.

When training a feedforward stochastic gradient descendent trained neural network, there is a possibility of not learning a batch of patterns correctly that causes the network to fail in the predictions in the areas adjacent to those patterns. This problem has usually been resolved by directly adding more complexity to the network, normally by increasing the number of learning layers, which means it will be heavier to run on the workstation. In this paper, the properties and the effect of the patterns on the network are analysed and two main reasons why the patterns are not learned correctly are distinguished: the disappearance of the Jacobian gradient on the processing layers of the network and the opposite direction of the gradient of those patterns. A simplified experiment has been carried out on a simple neural network and the errors appearing during and after training have been monitored. Taking into account the data obtained, the initial hypothesis of causes seems to be correct. Finally, some corrections to the network are proposed with the aim of solving those training issues and to be able to offer a sufficiently correct prediction, in order to increase the complexity of the network as little as possible. Full article

(This article belongs to the Section Network Science)

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