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# Mathematics, Volume 9, Issue 23 (December-1 2021) – 159 articles

Cover Story (view full-size image): A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in the rectifying plane. A remarkable property of these curves is that the ratio between torsion and curvature is a nonconstant linear function of the arc-length parameter. In this paper, we give a new characterization of rectifying curves; namely, we prove that a curve is a rectifying curve if and only if it has a spherical involute. Consequently, rectifying curves can be constructed as evolutes of spherical twisted curves. We express the curvature and the torsion of a rectifying spherical curve and give the necessary and sufficient conditions for a curve and its involute to be both rectifying curves. View this paper.
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Article
On the Packing Partitioning Problem on Directed Graphs
Mathematics 2021, 9(23), 3148; https://doi.org/10.3390/math9233148 - 06 Dec 2021
Viewed by 1374
Abstract
This work is aimed to continue studying the packing sets of digraphs via the perspective of partitioning the vertex set of a digraph into packing sets (which can be interpreted as a type of vertex coloring of digraphs) and focused on finding the [...] Read more.
This work is aimed to continue studying the packing sets of digraphs via the perspective of partitioning the vertex set of a digraph into packing sets (which can be interpreted as a type of vertex coloring of digraphs) and focused on finding the minimum cardinality among all packing partitions for a given digraph D, called the packing partition number of D. Some lower and upper bounds on this parameter are proven, and their exact values for directed trees are given in this paper. In the case of directed trees, the proof results in a polynomial-time algorithm for finding a packing partition of minimum cardinality. We also consider this parameter in digraph products. In particular, a complete solution to this case is presented when dealing with the rooted products. Full article
Article
On g-Noncommuting Graph of a Finite Group Relative to Its Subgroups
Mathematics 2021, 9(23), 3147; https://doi.org/10.3390/math9233147 - 06 Dec 2021
Cited by 4 | Viewed by 1520
Abstract
Let H be a subgroup of a finite non-abelian group G and $g\in G$. Let $Z\left(H,G\right)=\left\{x\in H:xy=yx,\forall y\in G\right\}$. We introduce [...] Read more.
Let H be a subgroup of a finite non-abelian group G and $g\in G$. Let $Z\left(H,G\right)=\left\{x\in H:xy=yx,\forall y\in G\right\}$. We introduce the graph ${\Delta }_{H,G}^{g}$ whose vertex set is $G\Z\left(H,G\right)$ and two distinct vertices x and y are adjacent if $x\in H$ or $y\in H$ and $\left[x,y\right]\ne g,{g}^{-1}$, where $\left[x,y\right]={x}^{-1}{y}^{-1}xy$. In this paper, we determine whether ${\Delta }_{H,G}^{g}$ is a tree among other results. We also discuss about its diameter and connectivity with special attention to the dihedral groups. Full article
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Article
Control Method for Flexible Joints in Manipulator Based on BP Neural Network Tuning PI Controller
Mathematics 2021, 9(23), 3146; https://doi.org/10.3390/math9233146 - 06 Dec 2021
Cited by 3 | Viewed by 2138
Abstract
With the development of robot technology, integrated joints with small volume and convenient installation have been widely used. Based on the double inertia system, an integrated joint motor servo system model considering gear angle error and friction interference is established, and a joint [...] Read more.
With the development of robot technology, integrated joints with small volume and convenient installation have been widely used. Based on the double inertia system, an integrated joint motor servo system model considering gear angle error and friction interference is established, and a joint control strategy based on BP neural network and pole assignment method is designed to suppress the vibration of the system. Firstly, the dynamic equation of a planetary gear system is derived based on the Lagrange method, and the gear vibration of angular displacement is calculated. Secondly, the vibration displacement of the sun gear is introduced into the motor servo system in the form of the gear angle error, and the double inertia system model including angle error and friction torque is established. Then, the PI controller parameters are determined by pole assignment method, and the PI parameters are adjusted in real time based on the BP neural network, which effectively suppresses the vibration of the system. Finally, the effects of friction torque, pole damping coefficient and control strategy on the system response and the effectiveness of vibration suppression are analyzed. Full article
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Article
Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity
Mathematics 2021, 9(23), 3145; https://doi.org/10.3390/math9233145 - 06 Dec 2021
Cited by 1 | Viewed by 1474
Abstract
We consider the Benjamin–Bona–Mahony (BBM) equation of the form ${u}_{t}+{u}_{x}+u{u}_{x}-{u}_{xxt}=0,\left(x,t\right)\in \mathcal{M}×\mathbb{R}$ where $\mathcal{M}=\mathbb{T}$ or $\mathbb{R}.$ We [...] Read more.
We consider the Benjamin–Bona–Mahony (BBM) equation of the form ${u}_{t}+{u}_{x}+u{u}_{x}-{u}_{xxt}=0,\left(x,t\right)\in \mathcal{M}×\mathbb{R}$ where $\mathcal{M}=\mathbb{T}$ or $\mathbb{R}.$ We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthens several known NI results at zero initial data in ${H}^{s}\left(\mathbb{T}\right)$ established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in ${H}^{s}\left(\mathbb{R}\right).$ Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces ${M}_{s}^{2,1}\left(\mathbb{R}\right)$ for $s\ge 0$. Full article
Article
Attribute Service Performance Index Based on Poisson Process
Mathematics 2021, 9(23), 3144; https://doi.org/10.3390/math9233144 - 06 Dec 2021
Cited by 3 | Viewed by 2106
Abstract
The purpose of a shop enhancing customer satisfaction is to raise its total revenue as the rate of customer purchases in the shop increases. Some studies have pointed out that the amount of customer arrival at a shop is a Poisson process. A [...] Read more.
The purpose of a shop enhancing customer satisfaction is to raise its total revenue as the rate of customer purchases in the shop increases. Some studies have pointed out that the amount of customer arrival at a shop is a Poisson process. A simple and easy-to-use evaluation index proposed for the Poisson process with the attribute characteristic will help various shops evaluate their business performance. In addition, developing an excellent and practical service performance evaluation method will be beneficial to the advancement of shop service quality as well as corporate image, thereby increasing the profitability and competitiveness of the shop. As the surroundings of the internet of things (IoT) are becoming gradually common and mature, various commercial data measurement and collection technologies are constantly being refined to form a huge amount of production data. Efficient data analysis and application can assist enterprises in making wise and efficient decisions within a short time. Thus, following the simple and easy-to-use principle, this paper proposes an attribute service performance index based on a Poisson process. Since the index had unknown parameters, this paper subsequently figured out the best estimator and used the central limit theorem to derive the confidence interval of the service efficiency index based on random samples. Then, we constructed the membership function based on the $\alpha -\mathrm{cuts}$ of the triangular shaped fuzzy number. Finally, we came up with a fuzzy testing model based on the membership function to improve the accuracy of the test when the sample size is small in order to meet enterprises’ needs for quick responses as well as reducing the evaluation cost. Full article
(This article belongs to the Special Issue Fuzzy Applications in Industrial Engineering)
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Article
A Continuous-Time Network Evolution Model Describing 2- and 3-Interactions
Mathematics 2021, 9(23), 3143; https://doi.org/10.3390/math9233143 - 06 Dec 2021
Cited by 1 | Viewed by 1582
Abstract
A continuous-time network evolution model is considered. The evolution of the network is based on 2- and 3-interactions. 2-interactions are described by edges, and 3-interactions are described by triangles. The evolution of the edges and triangles is governed by a multi-type continuous-time branching [...] Read more.
A continuous-time network evolution model is considered. The evolution of the network is based on 2- and 3-interactions. 2-interactions are described by edges, and 3-interactions are described by triangles. The evolution of the edges and triangles is governed by a multi-type continuous-time branching process. The limiting behaviour of the network is studied by mathematical methods. We prove that the number of triangles and edges have the same magnitude on the event of non-extinction, and it is ${e}^{\alpha t}$, where $\alpha$ is the Malthusian parameter. The probability of the extinction and the degree process of a fixed vertex are also studied. The results are illustrated by simulations. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)
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Article
On Interval-Valued Fuzzy Soft Preordered Sets and Associated Applications in Decision-Making
Mathematics 2021, 9(23), 3142; https://doi.org/10.3390/math9233142 - 06 Dec 2021
Cited by 2 | Viewed by 1421
Abstract
Recently, using interval-valued fuzzy soft sets to rank alternatives has become an important research area in decision-making because it provides decision-makers with the best option in a vague and uncertain environment. The present study aims to give an extensive insight into decision-making processes [...] Read more.
Recently, using interval-valued fuzzy soft sets to rank alternatives has become an important research area in decision-making because it provides decision-makers with the best option in a vague and uncertain environment. The present study aims to give an extensive insight into decision-making processes relying on a preference relationship of interval-valued fuzzy soft sets. Firstly, interval-valued fuzzy soft preorderings and an interval-valued fuzzy soft equivalence are established based on the interval-valued fuzzy soft topology. Then, two crisp preordering sets, namely lower crisp and upper crisp preordering sets, are proposed. Next, a score function depending on comparison matrices is expressed in solving multi-group decision-making problems. Finally, a numerical example is given to illustrate the validity and efficacy of the proposed method. Full article
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Article
Application of Machine Learning Model for the Prediction of Settling Velocity of Fine Sediments
Mathematics 2021, 9(23), 3141; https://doi.org/10.3390/math9233141 - 06 Dec 2021
Cited by 2 | Viewed by 1983
Abstract
Sedimentation management is one of the primary factors in achieving sustainable development of water resources. However, due to difficulties in conducting in-situ tests, and the complex nature of fine sediments, it remains a challenging task when dealing with issues related to settling velocity. [...] Read more.
Sedimentation management is one of the primary factors in achieving sustainable development of water resources. However, due to difficulties in conducting in-situ tests, and the complex nature of fine sediments, it remains a challenging task when dealing with issues related to settling velocity. Hence, the machine learning model appears as a suitable tool to predict the settling velocity of fine sediments in water bodies. In this study, three different machine learning-based models, namely, the radial basis function neural network (RBFNN), back propagation neural network (BPNN), and self-organizing feature map (SOFM), were developed with four hydraulic parameters, including the inlet depth, particle size, and the relative x and y particle positions. The five distinct statistical measures, consisting of the root mean square error (RMSE), Nash–Sutcliffe efficiency (NSE), mean absolute error (MAE), mean value accounted for (MVAF), and total variance explained (TVE), were used to assess the performance of the models. The SOFM with the 25 × 25 Kohonen map had shown superior results with RMSE of 0.001307, NSE of 0.7170, MAE of 0.000647, MVAF of 101.25%, and TVE of 71.71%. Full article
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Article
Methods of Ensuring Invariance with Respect to External Disturbances: Overview and New Advances
Mathematics 2021, 9(23), 3140; https://doi.org/10.3390/math9233140 - 06 Dec 2021
Cited by 9 | Viewed by 1357
Abstract
In this paper, we carry out a demonstration and comparative analysis of known methods of the synthesis of various control laws ensuring the invariance of the output (controlled) variable with respect to external disturbances under various assumptions about their type and channels of [...] Read more.
In this paper, we carry out a demonstration and comparative analysis of known methods of the synthesis of various control laws ensuring the invariance of the output (controlled) variable with respect to external disturbances under various assumptions about their type and channels of acting on the control plant. Methods of the synthesis are presented on the example of a third-order nonlinear system with single input and single output (SISO-systems), dynamic feedback synthesis is presented at a descriptive level and the focus is on procedures of static feedback synthesis. For the systems in which the matching conditions are not satisfied, it is concluded that it is expedient to introduce smooth and bounded nonlinear local feedbacks. Within the framework of the block control principle, we developed an iterative procedure of synthesis of S-shaped sigmoid feedbacks for such systems. Nonlinear local feedbacks ensure stabilization of the output variable with the given accuracy and settling time as in a system with traditionally used linear local feedbacks with high gains. However, in contrast to it, sigmoid functions do not lead to a large overshoot of state variables and control actions. Full article
Article
Dynamical Analysis of a Navigation Algorithm
Mathematics 2021, 9(23), 3139; https://doi.org/10.3390/math9233139 - 06 Dec 2021
Cited by 5 | Viewed by 2443
Abstract
There is presently a need for more robust navigation algorithms for autonomous industrial vehicles. These have reasonably guaranteed the adequate reliability of the navigation. In the current work, the stability of a modified algorithm for collision-free guiding of this type of vehicle is [...] Read more.
There is presently a need for more robust navigation algorithms for autonomous industrial vehicles. These have reasonably guaranteed the adequate reliability of the navigation. In the current work, the stability of a modified algorithm for collision-free guiding of this type of vehicle is ensured. A lateral control and a longitudinal control are implemented. To demonstrate their viability, a stability analysis employing the Lyapunov method is carried out. In addition, this mathematical analysis enables the constants of the designed algorithm to be determined. In conjunction with the navigation algorithm, the present work satisfactorily solves the localization problem, also known as simultaneous localization and mapping (SLAM). Simultaneously, a convolutional neural network is managed, which is used to calculate the trajectory to be followed by the AGV, by implementing the artificial vision. The use of neural networks for image processing is considered to constitute the most robust and flexible method for realising a navigation algorithm. In this way, the autonomous vehicle is provided with considerable autonomy. It can be regarded that the designed algorithm is adequate, being able to trace any type of path. Full article
(This article belongs to the Special Issue State-of-the-Art Mathematical Applications in Europe)
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Article
A New Method for Time Normalization Based on the Continuous Phase: Application to Neck Kinematics
Mathematics 2021, 9(23), 3138; https://doi.org/10.3390/math9233138 - 05 Dec 2021
Cited by 2 | Viewed by 2028
Abstract
There is growing interest in analyzing human movement data for clinical, sport, and ergonomic applications. Functional Data Analysis (FDA) has emerged as an advanced statistical method for overcoming the shortcomings of traditional analytic methods, because the information about continuous signals can be assessed [...] Read more.
There is growing interest in analyzing human movement data for clinical, sport, and ergonomic applications. Functional Data Analysis (FDA) has emerged as an advanced statistical method for overcoming the shortcomings of traditional analytic methods, because the information about continuous signals can be assessed over time. This paper takes the current literature a step further by presenting a new time scale normalization method, based on the Hilbert transform, for the analysis of functional data and the assessment of the effect on the variability of human movement waveforms. Furthermore, a quantitative comparison of well-known methods for normalizing datasets of temporal biomechanical waveforms using functional data is carried out, including the linear normalization method and nonlinear registration methods of functional data. This is done using an exhaustive database of human neck flexion-extension movements, which encompasses 423 complete cycles of 31 healthy subjects measured in two trials of the experiment on different days. The results show the advantages of the novel method compared to existing techniques in terms of computational cost and the effectiveness of time-scale normalization on the phase differences of curves and on the amplitude of means, which are assessed by Root Mean Square (RMS) values of functional means of angles, angular velocities, and angular accelerations. Additionally, the confidence intervals are obtained through a bootstrapping process. Full article
(This article belongs to the Special Issue Mathematics, Statistics and Applied Computational Methods)
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Article
XCM: An Explainable Convolutional Neural Network for Multivariate Time Series Classification
Mathematics 2021, 9(23), 3137; https://doi.org/10.3390/math9233137 - 05 Dec 2021
Cited by 22 | Viewed by 5717
Abstract
Multivariate Time Series (MTS) classification has gained importance over the past decade with the increase in the number of temporal datasets in multiple domains. The current state-of-the-art MTS classifier is a heavyweight deep learning approach, which outperforms the second-best MTS classifier only on [...] Read more.
Multivariate Time Series (MTS) classification has gained importance over the past decade with the increase in the number of temporal datasets in multiple domains. The current state-of-the-art MTS classifier is a heavyweight deep learning approach, which outperforms the second-best MTS classifier only on large datasets. Moreover, this deep learning approach cannot provide faithful explanations as it relies on post hoc model-agnostic explainability methods, which could prevent its use in numerous applications. In this paper, we present XCM, an eXplainable Convolutional neural network for MTS classification. XCM is a new compact convolutional neural network which extracts information relative to the observed variables and time directly from the input data. Thus, XCM architecture enables a good generalization ability on both large and small datasets, while allowing the full exploitation of a faithful post hoc model-specific explainability method (Gradient-weighted Class Activation Mapping) by precisely identifying the observed variables and timestamps of the input data that are important for predictions. We first show that XCM outperforms the state-of-the-art MTS classifiers on both the large and small public UEA datasets. Then, we illustrate how XCM reconciles performance and explainability on a synthetic dataset and show that XCM enables a more precise identification of the regions of the input data that are important for predictions compared to the current deep learning MTS classifier also providing faithful explainability. Finally, we present how XCM can outperform the current most accurate state-of-the-art algorithm on a real-world application while enhancing explainability by providing faithful and more informative explanations. Full article
(This article belongs to the Special Issue Data Mining for Temporal Data Analysis)
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Article
E-Learning Platform Assessment and Selection Using Two-Stage Multi-Criteria Decision-Making Approach with Grey Theory: A Case Study in Vietnam
Mathematics 2021, 9(23), 3136; https://doi.org/10.3390/math9233136 - 05 Dec 2021
Cited by 13 | Viewed by 2681
Abstract
Education has changed dramatically due to the severe global pandemic COVID-19, with the phenomenal growth of e-learning, whereby teaching is undertaken remotely and on digital platforms. E-learning is revolutionizing education systems, as it remains the only option during the ongoing crisis and has [...] Read more.
Education has changed dramatically due to the severe global pandemic COVID-19, with the phenomenal growth of e-learning, whereby teaching is undertaken remotely and on digital platforms. E-learning is revolutionizing education systems, as it remains the only option during the ongoing crisis and has tremendous potential to fulfill instructional plans and safeguard students’ learning rights. The selection of e-learning platforms is a multi-criteria decision-making (MCDM) problem. Expert analyses over numerous criteria and alternatives are usually linguistic terms, which can be represented through grey numbers. This article proposes an integrated approach of grey analytic hierarchy process (G-AHP) and grey technique for order preference by similarity to ideal solution (G-TOPSIS) to evaluate the best e-learning website for network teaching. This introduced approach handles the linguistic evaluation of experts based on grey systems theory, estimates the relative importance of evaluation criteria with the G-AHP method, and acquires e-learning websites’ ranking utilizing G-TOPSIS. The applicability and superiority of the presented method are illustrated through a practical e-learning website selection case in Vietnam. From G-AHP analysis, educational level, price, right and understandable content, complete content, and up-to-date were found as the most impactful criteria. From G-TOPSIS, Edumall is the best platform. Comparisons are conducted with other MCDM methods; the priority orders of the best websites are similar, indicating the robust proposed methodology. The proposed integrated model in this study supports the stakeholders in selecting the most effective e-learning environments and could be a reference for further development of e-learning teaching-learning systems. Full article
(This article belongs to the Special Issue Decision Making and Its Applications)
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Article
Prediction of Hydraulic Jumps on a Triangular Bed Roughness Using Numerical Modeling and Soft Computing Methods
Mathematics 2021, 9(23), 3135; https://doi.org/10.3390/math9233135 - 05 Dec 2021
Cited by 14 | Viewed by 2214
Abstract
This study investigates the characteristics of free and submerged hydraulic jumps on the triangular bed roughness in various T/I ratios (i.e., height and distance of roughness) using CFD modeling techniques. The accuracy of numerical modeling outcomes was checked and compared using [...] Read more.
This study investigates the characteristics of free and submerged hydraulic jumps on the triangular bed roughness in various T/I ratios (i.e., height and distance of roughness) using CFD modeling techniques. The accuracy of numerical modeling outcomes was checked and compared using artificial intelligence methods, namely Support Vector Machines (SVM), Gene Expression Programming (GEP), and Random Forest (RF). The results of the FLOW-3D® model and experimental data showed that the overall mean value of relative error is 4.1%, which confirms the numerical model’s ability to predict the characteristics of the free and submerged jumps. The SVM model with a minimum of Root Mean Square Error (RMSE) and a maximum of correlation coefficient (R2), compared with GEP and RF models in the training and testing phases for predicting the sequent depth ratio (y2/y1), submerged depth ratio (y3/y1), tailwater depth ratio (y4/y1), length ratio of jumps (${L}_{j}/{y}_{2}^{*}$) and energy dissipation (ΔE/E1), was recognized as the best model. Moreover, the best result for predicting the length ratio of free jumps $\left({L}_{jf/}{y}_{2}^{*}$) in the optimal gamma is γ = 10 and the length ratio of submerged jumps $\left({L}_{js/}{y}_{2}^{*}$) is γ = 0.60. Based on sensitivity analysis, the Froude number has the greatest effect on predicting the (y3/y1) compared with submergence factors (SF) and T/I. By omitting this parameter, the prediction accuracy is significantly reduced. Finally, the relationships with good correlation coefficients for the mentioned parameters in free and submerged jumps were presented based on numerical results. Full article
(This article belongs to the Special Issue Computational Optimizations for Machine Learning)
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Article
Study of Dynamics of a COVID-19 Model for Saudi Arabia with Vaccination Rate, Saturated Treatment Function and Saturated Incidence Rate
Mathematics 2021, 9(23), 3134; https://doi.org/10.3390/math9233134 - 05 Dec 2021
Cited by 3 | Viewed by 2066
Abstract
This paper proposes, validates and analyzes the dynamics of the susceptible exposed infectious recovered (SEIR) model for the propagation of COVID-19 in Saudi Arabia, which recorded the largest number of cases in the Arab world. The model incorporates a saturated incidence rate, a [...] Read more.
This paper proposes, validates and analyzes the dynamics of the susceptible exposed infectious recovered (SEIR) model for the propagation of COVID-19 in Saudi Arabia, which recorded the largest number of cases in the Arab world. The model incorporates a saturated incidence rate, a constant vaccination rate and a nonlinear treatment function. The rate of treatment is assumed to be proportional to the number of infected persons when this number is low and reaches a fixed value for large number of infected individuals. The expression of the basic reproduction number is derived, and the model basic stability properties are studied. We show that when the basic reproduction number is less than one the model can predict both a Hopf and backward bifurcations. Simulations are also provided to fit the model to COVID-19 data in Saudi Arabia and to study the effects of the parameters of the treatment function and vaccination rate on disease control. Full article
(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)
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Article
Stability Analysis of Parameter Varying Genetic Toggle Switches Using Koopman Operators
Mathematics 2021, 9(23), 3133; https://doi.org/10.3390/math9233133 - 05 Dec 2021
Cited by 2 | Viewed by 2255
Abstract
The genetic toggle switch is a well known model in synthetic biology that represents the dynamic interactions between two genes that repress each other. The mathematical models for the genetic toggle switch that currently exist have been useful in describing circuit dynamics in [...] Read more.
The genetic toggle switch is a well known model in synthetic biology that represents the dynamic interactions between two genes that repress each other. The mathematical models for the genetic toggle switch that currently exist have been useful in describing circuit dynamics in rapidly dividing cells, assuming fixed or time-invariant kinetic rates. There is a growing interest in being able to model and extend synthetic biological function for growth conditions such as stationary phase or during nutrient starvation. As cells transition from one growth phase to another, kinetic rates become time-varying parameters. In this paper, we propose a novel class of parameter varying nonlinear models that can be used to describe the dynamics of genetic circuits, including the toggle switch, as they transition from different phases of growth. We show that there exists unique solutions for this class of systems, as well as for a class of systems that incorporates the microbial phenomena of quorum sensing. Further, we show that the domain of these systems, which is the positive orthant, is positively invariant. We also showcase a theoretical control strategy for these systems that would grant asymptotic monostability of a desired fixed point. We then take the general form of these systems and analyze their stability properties through the framework of time-varying Koopman operator theory. A necessary condition for asymptotic stability is also provided as well as a sufficient condition for instability. A Koopman control strategy for the system is also proposed, as well as an analogous discrete time-varying Koopman framework for applications with regularly sampled measurements. Full article
(This article belongs to the Special Issue Dynamical Systems and Operator Theory)
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Article
Determination of Bounds for the Jensen Gap and Its Applications
Mathematics 2021, 9(23), 3132; https://doi.org/10.3390/math9233132 - 05 Dec 2021
Cited by 11 | Viewed by 2478
Abstract
The Jensen inequality has been reported as one of the most consequential inequalities that has a lot of applications in diverse fields of science. For this reason, the Jensen inequality has become one of the most discussed developmental inequalities in the current literature [...] Read more.
The Jensen inequality has been reported as one of the most consequential inequalities that has a lot of applications in diverse fields of science. For this reason, the Jensen inequality has become one of the most discussed developmental inequalities in the current literature on mathematical inequalities. The main intention of this article is to find some novel bounds for the Jensen difference while using some classes of twice differentiable convex functions. We obtain the proposed bounds by utilizing the power mean and Höilder inequalities, the notion of convexity and the prominent Jensen inequality for concave function. We deduce several inequalities for power and quasi-arithmetic means as a consequence of main results. Furthermore, we also establish different improvements for Hölder inequality with the help of obtained results. Moreover, we present some applications of the main results in information theory. Full article
Article
An Efficient Visually Meaningful Quantum Walks-Based Encryption Scheme for Secure Data Transmission on IoT and Smart Applications
Mathematics 2021, 9(23), 3131; https://doi.org/10.3390/math9233131 - 04 Dec 2021
Cited by 4 | Viewed by 1279
Abstract
Smart systems and technologies have become integral parts of modern society. Their ubiquity makes it paramount to prioritise securing the privacy of data transferred between smart devices. Visual encryption is a technique employed to obscure images by rendering them meaningless to evade attention [...] Read more.
Smart systems and technologies have become integral parts of modern society. Their ubiquity makes it paramount to prioritise securing the privacy of data transferred between smart devices. Visual encryption is a technique employed to obscure images by rendering them meaningless to evade attention during transmission. However, the astounding computing power ascribed to quantum technology implies that even the best visually encrypted systems can be effortlessly violated. Consequently, the physical realisation quantum hardware portends great danger for visually encrypted date on smart systems. To circumvent this, our study proposes the integration of quantum walks (QWs) as a cryptographic mechanism to forestall violation of the integrity of images on smart systems. Specifically, we use QW first to substitute the original image and to subsequently permutate and embed it onto the reference image. Based on this structure, our proposed quantum walks visually meaningful cryptosystem facilities confidential transmission of visual information. Simulation-based experiments validate the performance of the proposed system in terms of visual quality, efficiency, robustness, and key space sensitivity, and by that, its potential to safeguard smart systems now and as we transition to the quantum era. Full article
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Article
Early Prediction of DNN Activation Using Hierarchical Computations
Mathematics 2021, 9(23), 3130; https://doi.org/10.3390/math9233130 - 04 Dec 2021
Viewed by 1611
Abstract
Deep Neural Networks (DNNs) have set state-of-the-art performance numbers in diverse fields of electronics (computer vision, voice recognition), biology, bioinformatics, etc. However, the process of learning (training) from the data and application of the learnt information (inference) process requires huge computational resources. Approximate [...] Read more.
Deep Neural Networks (DNNs) have set state-of-the-art performance numbers in diverse fields of electronics (computer vision, voice recognition), biology, bioinformatics, etc. However, the process of learning (training) from the data and application of the learnt information (inference) process requires huge computational resources. Approximate computing is a common method to reduce computation cost, but it introduces loss in task accuracy, which limits their application. Using an inherent property of Rectified Linear Unit (ReLU), a popular activation function, we propose a mathematical model to perform MAC operation using reduced precision for predicting negative values early. We also propose a method to perform hierarchical computation to achieve the same results as IEEE754 full precision compute. Applying this method on ResNet50 and VGG16 shows that up to 80% of ReLU zeros (which is 50% of all ReLU outputs) can be predicted and detected early by using just 3 out of 23 mantissa bits. This method is equally applicable to other floating-point representations. Full article
(This article belongs to the Special Issue Computational Optimizations for Machine Learning)
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Article
Continuous Stability TS Fuzzy Systems Novel Frame Controlled by a Discrete Approach and Based on SOS Methodology
Mathematics 2021, 9(23), 3129; https://doi.org/10.3390/math9233129 - 04 Dec 2021
Cited by 1 | Viewed by 1020
Abstract
Generally, the continuous and discrete TS fuzzy systems’ control is studied independently. Unlike the discrete systems, stability results for the continuous systems suffer from conservatism because it is still quite difficult to apply non-quadratic Lyapunov functions, something which is much easier for the [...] Read more.
Generally, the continuous and discrete TS fuzzy systems’ control is studied independently. Unlike the discrete systems, stability results for the continuous systems suffer from conservatism because it is still quite difficult to apply non-quadratic Lyapunov functions, something which is much easier for the discrete systems. In this paper and in order to obtain new results for the continuous case, we proposed to connect the continuous with the discrete cases and then check the stability of the continuous TS fuzzy systems by means of the discrete design approach. To this end, a novel frame was proposed using the sum of square approach (SOS) to check the stability of the continuous Takagi Sugeno (TS) fuzzy models based on the discrete controller. Indeed, the control of the continuous TS fuzzy models is ensured by the discrete gains obtained from the Euler discrete form and based on the non-quadratic Lyapunov function. The simulation examples applied for various models, by modifying the order of the Euler discrete fuzzy system, are presented to show the effectiveness of the proposed methodology. Full article
(This article belongs to the Special Issue Automatic Control and Soft Computing in Engineering)
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Article
Resolutions of the Jerk and Snap Vectors for a Quasi Curve in Euclidean 3-Space
Mathematics 2021, 9(23), 3128; https://doi.org/10.3390/math9233128 - 04 Dec 2021
Cited by 4 | Viewed by 1090
Abstract
This work aims at studying resolutions of the jerk and snap vectors of a point particle moving along a quasi curve in Euclidean 3-space ${\mathbb{E}}^{3}$. In particular, we obtain the resolution of the jerk and snap vectors along the quasi vectors [...] Read more.
This work aims at studying resolutions of the jerk and snap vectors of a point particle moving along a quasi curve in Euclidean 3-space ${\mathbb{E}}^{3}$. In particular, we obtain the resolution of the jerk and snap vectors along the quasi vectors and offer an alternative resolution of the jerk and snap vectors along the tangential direction and two special radial directions that lie in the osculating and rectifying planes. This alternative resolution for a quasi plane curve in Euclidean 3-space ${\mathbb{E}}^{3}$ is given as corollary. Moreover, our results are illustrated via some examples. Full article
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Article
Mixture of Species Sampling Models
Mathematics 2021, 9(23), 3127; https://doi.org/10.3390/math9233127 - 04 Dec 2021
Cited by 2 | Viewed by 1056
Abstract
We introduce mixtures of species sampling sequences (mSSS) and discuss how these sequences are related to various types of Bayesian models. As a particular case, we recover species sampling sequences with general (not necessarily diffuse) base measures. These models include some “spike-and-slab” non-parametric [...] Read more.
We introduce mixtures of species sampling sequences (mSSS) and discuss how these sequences are related to various types of Bayesian models. As a particular case, we recover species sampling sequences with general (not necessarily diffuse) base measures. These models include some “spike-and-slab” non-parametric priors recently introduced to provide sparsity. Furthermore, we show how mSSS arise while considering hierarchical species sampling random probabilities (e.g., the hierarchical Dirichlet process). Extending previous results, we prove that mSSS are obtained by assigning the values of an exchangeable sequence to the classes of a latent exchangeable random partition. Using this representation, we give an explicit expression of the Exchangeable Partition Probability Function of the partition generated by an mSSS. Some special cases are discussed in detail—in particular, species sampling sequences with general base measures and a mixture of species sampling sequences with Gibbs-type latent partition. Finally, we give explicit expressions of the predictive distributions of an mSSS. Full article
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Article
Enterprise Compensation System Statistical Modeling for Decision Support System Development
Mathematics 2021, 9(23), 3126; https://doi.org/10.3390/math9233126 - 04 Dec 2021
Cited by 2 | Viewed by 1251
Abstract
This article raises the issue of decision support system (DSS) development in enterprises concerning the compensation system (CS). The topic is relevant as the CS is one of the main components in human resource management in business. A key element of such DSSs [...] Read more.
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Article
Modelling of Heat Transfer Processes in Heat Exchangers for Cardiopulmonary Bypass
Mathematics 2021, 9(23), 3125; https://doi.org/10.3390/math9233125 - 04 Dec 2021
Viewed by 1464
Abstract
A model of the heat exchange process in the heat exchanger of the cardiopulmonary bypass device is proposed which allows for automation of the process of temperature regulation in the cardiopulmonary bypass with an accuracy of ±1 °C during cardiac surgery under controlled [...] Read more.
A model of the heat exchange process in the heat exchanger of the cardiopulmonary bypass device is proposed which allows for automation of the process of temperature regulation in the cardiopulmonary bypass with an accuracy of ±1 °C during cardiac surgery under controlled cooling and warming of the patient’s heart and brain. The purpose of this research is to create a concept and model of the temperature control circuit using the MSC Easy5 system, the creation of mathematical models of blocks of the temperature control circuit, and the description of the principle of temperature control in the cardiopulmonary bypass circuit. The model of the temperature control loop in the heat exchanger of the heart-lung machine was created using the MSC Easy5 system with a programmable microcontroller. The microcontroller implements a specialized temperature control algorithm in the C language. The model allows the creation of a full-fledged virtual prototype of a temperature control device in a heat exchanger, and helps to conduct virtual tests of the developed device at the design stage. The model identifies control system flaws and influences decisions made before producing an official prototype of the product. Full article
(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)
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Article
Reliability Analysis of High Concrete-Face Rockfill Dams and Study of Seismic Performance of Earthquake-Resistant Measures Based on Stochastic Dynamic Analysis
Mathematics 2021, 9(23), 3124; https://doi.org/10.3390/math9233124 - 04 Dec 2021
Cited by 3 | Viewed by 1447
Abstract
The randomness of earthquake excitation has a significant impact on the seismic performance of high earth-rock dams. In this paper, the seismic performance of geosynthetic-reinforced soil structures (GRSS) of high concrete face rockfill dams (CFRDs) is evaluated from the stochastic perspective. Multiple groups [...] Read more.
The randomness of earthquake excitation has a significant impact on the seismic performance of high earth-rock dams. In this paper, the seismic performance of geosynthetic-reinforced soil structures (GRSS) of high concrete face rockfill dams (CFRDs) is evaluated from the stochastic perspective. Multiple groups of seismic ground motions are generated based on spectral expression-random function non-stationary model. Taking Gushui CFRD as an example, this study calculates the failure probability of each damage level of non-reinforce slopes and reinforce slopes based on generalized probability density evolution method (GPDEM) and reliability analysis is presented though multiple evaluation indicators. The result shows that GRSS can reduce the mild damage of CFRDs during earthquake and restrain the moderate and severe damage. The influence of vertical spacing and length of GRSS on the seismic performance is obtained, which provides a reference for the seismic design and risk analysis of CFRDs. Full article
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Article
Scalability of k-Tridiagonal Matrix Singular Value Decomposition
Mathematics 2021, 9(23), 3123; https://doi.org/10.3390/math9233123 - 03 Dec 2021
Cited by 4 | Viewed by 1448
Abstract
Singular value decomposition has recently seen a great theoretical improvement for k-tridiagonal matrices, obtaining a considerable speed up over all previous implementations, but at the cost of not ordering the singular values. We provide here a refinement of this method, proving that [...] Read more.
Singular value decomposition has recently seen a great theoretical improvement for k-tridiagonal matrices, obtaining a considerable speed up over all previous implementations, but at the cost of not ordering the singular values. We provide here a refinement of this method, proving that reordering singular values does not affect performance. We complement our refinement with a scalability study on a real physical cluster setup, offering surprising results. Thus, this method provides a major step up over standard industry implementations. Full article
(This article belongs to the Special Issue Models and Algorithms in Cybersecurity)
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Article
Investigation of the Stochastic Modeling of COVID-19 with Environmental Noise from the Analytical and Numerical Point of View
Mathematics 2021, 9(23), 3122; https://doi.org/10.3390/math9233122 - 03 Dec 2021
Cited by 20 | Viewed by 1720
Abstract
In this article, we propose a novel mathematical model for the spread of COVID-19 involving environmental white noise. The new stochastic model was studied for the existence and persistence of the disease, as well as the extinction of the disease. We noticed that [...] Read more.
In this article, we propose a novel mathematical model for the spread of COVID-19 involving environmental white noise. The new stochastic model was studied for the existence and persistence of the disease, as well as the extinction of the disease. We noticed that the existence and extinction of the disease are dependent on ${\mathcal{R}}_{0}$ (the reproduction number). Then, a numerical scheme was developed for the computational analysis of the model; with the existing values of the parameters in the literature, we obtained the related simulations, which gave us more realistic numerical data for the future prediction. The mentioned stochastic model was analyzed for different values of ${\sigma }_{1},\phantom{\rule{0.166667em}{0ex}}{\sigma }_{2}$ and ${\beta }_{1},{\beta }_{2}$, and both the stochastic and the deterministic models were compared for the future prediction of the spread of COVID-19. Full article
(This article belongs to the Special Issue Functional Differential Equations and Epidemiological Modelling)
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Article
Canonical Coordinates and Natural Equation for Lorentz Surfaces in $R13$
Mathematics 2021, 9(23), 3121; https://doi.org/10.3390/math9233121 - 03 Dec 2021
Viewed by 907
Abstract
We consider Lorentz surfaces in ${\mathbb{R}}_{1}^{3}$ satisfying the condition ${H}^{2}-K\ne 0$, where K and H are the Gaussian curvature and the mean curvature, respectively, and call them Lorentz surfaces of general type. For this class of [...] Read more.
We consider Lorentz surfaces in ${\mathbb{R}}_{1}^{3}$ satisfying the condition ${H}^{2}-K\ne 0$, where K and H are the Gaussian curvature and the mean curvature, respectively, and call them Lorentz surfaces of general type. For this class of surfaces, we introduce special isotropic coordinates, which we call canonical, and show that the coefficient F of the first fundamental form and the mean curvature H, expressed in terms of the canonical coordinates, satisfy a special integro-differential equation which we call a natural equation of the Lorentz surfaces of a general type. Using this natural equation, we prove a fundamental theorem of Bonnet type for Lorentz surfaces of a general type. We consider the special cases of Lorentz surfaces of constant non-zero mean curvature and minimal Lorentz surfaces. Finally, we give examples of Lorentz surfaces illustrating the developed theory. Full article
Article
Artificial Intelligence for Stability Control of Actuated In–Wheel Electric Vehicles with CarSim® Validation
Mathematics 2021, 9(23), 3120; https://doi.org/10.3390/math9233120 - 03 Dec 2021
Cited by 4 | Viewed by 1674
Abstract
This paper presents an active controller for electric vehicles in which active front steering and torque vectoring are control actions combined to improve the vehicle driving safety. The electric powertrain consists of four independent in–wheel electric motors situated on each corner. The control [...] Read more.
This paper presents an active controller for electric vehicles in which active front steering and torque vectoring are control actions combined to improve the vehicle driving safety. The electric powertrain consists of four independent in–wheel electric motors situated on each corner. The control approach relies on an inverse optimal controller based on a neural network identifier of the vehicle plant. Moreover, to minimize the number of sensors needed for control purposes, the authors present a discrete–time reduced–order state observer for the estimation of vehicle lateral and roll dynamics. The use of a neural network identifier presents some interesting advantages. Notably, unlike standard strategies, the proposed approach avoids the use of tire lateral forces or Pacejka’s tire parameters. In fact, the neural identification provides an input–affine model in which these quantities are absorbed by neural synaptic weights adapted online by an extended Kalman filter. From a practical standpoint, this eliminates the need of additional sensors, model tuning, or estimation stages. In addition, the yaw angle command given by the controller is converted into electric motor torques in order to ensure safe driving conditions. The mathematical models used to describe the electric machines are able to reproduce the dynamic behavior of Elaphe M700 in–wheel electric motors. Finally, quality and performances of the proposed control strategy are discussed in simulation, using a CarSim® full vehicle model running through a double–lane change maneuver. Full article
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Article
Global and Local Analysis for a Cournot Duopoly Game with Two Different Objective Functions
Mathematics 2021, 9(23), 3119; https://doi.org/10.3390/math9233119 - 03 Dec 2021
Cited by 1 | Viewed by 1086
Abstract
In this paper, a Cournot game with two competing firms is studied. The two competing firms seek the optimality of their quantities by maximizing two different objective functions. The first firm wants to maximize an average of social welfare and profit, while the [...] Read more.
In this paper, a Cournot game with two competing firms is studied. The two competing firms seek the optimality of their quantities by maximizing two different objective functions. The first firm wants to maximize an average of social welfare and profit, while the second firm wants to maximize their relative profit only. We assume that both firms are rational, adopting a bounded rationality mechanism for updating their production outputs. A two-dimensional discrete time map is introduced to analyze the evolution of the game. The map has four equilibrium points and their stability conditions are investigated. We prove the Nash equilibrium point can be destabilized through flip bifurcation only. The obtained results show that the manifold of the game’s map can be analyzed through a one-dimensional map whose analytical form is similar to the well-known logistic map. The critical curves investigations show that the phase plane of game’s map is divided into three zones and, therefore, the map is not invertible. Finally, the contact bifurcation phenomena are discussed using simulation. Full article
(This article belongs to the Special Issue Decision Making and Its Applications)
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