Mathematics

22 pages, 858 KiB

Open AccessArticle

Application of an Intensive Longitudinal Functional Model with Multiple Time Scales in Objectively Measured Children’s Physical Activity

by Mostafa Zahed, Trent Lalonde and Maryam Skafyan

Mathematics 2023, 11(8), 1973; https://doi.org/10.3390/math11081973 - 21 Apr 2023

Viewed by 1630

Abstract

This study proposes an intensive longitudinal functional model with multiple time-varying scales and subject-specific random intercepts through mixed model equivalence that includes multiple functional predictors, one or more scalar covariates, and one or more scalar covariates. An estimation framework is proposed for estimating [...] Read more.

This study proposes an intensive longitudinal functional model with multiple time-varying scales and subject-specific random intercepts through mixed model equivalence that includes multiple functional predictors, one or more scalar covariates, and one or more scalar covariates. An estimation framework is proposed for estimating a time-varying coefficient function that is modeled as a linear combination of time-invariant functions with time-varying coefficients. The model takes advantage of the information structure of the penalty, while the estimation procedure utilizes the equivalence between penalized least squares estimation and linear mixed models. A number of simulations are conducted in order to empirically evaluate the process. In the simulation, it was observed that mean square errors for functional coefficients decreased with increasing sample size and level of association. Additionally, sample size had a greater impact on a smaller level of association, and level of association also had a greater impact on a smaller sample size. These results provide empirical evidence that ILFMM estimates of functional coefficients are close to the true functional estimate (basically unchanged). In addition, the results indicated that the AIC could be used to guide the choice of ridge weights. Moreover, when ridge weight ratios were sufficiently large, there was minimal impact on estimation performance. Studying two time scales is important in a wide range of fields, including physics, chemistry, biology, engineering, economics, and more. It allows researchers to gain a better understanding of complex systems and processes that operate over different time frames. Consequently, studying physical activities with two time scales is critical for advancing our understanding of human performance and health and for developing effective strategies to optimize physical activity and exercise programs. Therefore, the proposed model was applied to analyze the physical activity data from the Active Schools Institute of the University of Northern Colorado to determine what kind of time-structure patterns of activities could adequately describe the relationship between daily total magnitude and kids’ daily and weekly physical activity. Full article

(This article belongs to the Special Issue New Advance in Operations Research and Analytics)

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13 pages, 5619 KiB

Open AccessArticle

A Semi-Federated Active Learning Framework for Unlabeled Online Network Data

by Yuwen Zhou, Yuhan Hu, Jing Sun, Rui He and Wenjie Kang

Mathematics 2023, 11(8), 1972; https://doi.org/10.3390/math11081972 - 21 Apr 2023

Viewed by 1406

Abstract

Federated Learning (FL) is a newly emerged federated optimization technique for distributed data in a federated network. The participants in FL that train the model locally are classified into client nodes. The server node assumes the responsibility to aggregate local models from client [...] Read more.

Federated Learning (FL) is a newly emerged federated optimization technique for distributed data in a federated network. The participants in FL that train the model locally are classified into client nodes. The server node assumes the responsibility to aggregate local models from client nodes without data moving. In this regard, FL is an ideal solution to protect data privacy at each node of the network. However, the raw data generated on each node are unlabeled, making it impossible for FL to apply these data directly to train a model. The large volume of data annotating work prevents FL from being widely applied in the real world, especially for online scenarios, where the data are generated continuously. Meanwhile, the data generated on different nodes tend to be differently distributed. It has been proved theoretically and experimentally that non-independent and identically distributed (non-IID) data harm the performance of FL. In this article, we design a semi-federated active learning (semi-FAL) framework to tackle the annotation and non-IID problems jointly. More specifically, the server node can provide (i) a pre-trained model to help each client node annotate the local data uniformly and (ii) an estimation of the global gradient to help correct the local gradient. The evaluation results demonstrate our semi-FAL framework can efficiently handle unlabeled online network data and achieves high accuracy and fast convergence. Full article

(This article belongs to the Special Issue Artificial Intelligence and Machine Learning Based Methods and Applications)

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22 pages, 5114 KiB

Open AccessArticle

Base Station Planning Based on Region Division and Mean Shift Clustering

by Jian Chen, Yongkun Shi, Jiaquan Sun, Jiangkuan Li and Jing Xu

Mathematics 2023, 11(8), 1971; https://doi.org/10.3390/math11081971 - 21 Apr 2023

Cited by 5 | Viewed by 1802

Abstract

The problem of insufficient signal coverage of 5G base stations can be solved by building new base stations in areas with weak signal coverage. However, due to construction costs and other factors, it is not possible to cover all areas. In general, areas [...] Read more.

The problem of insufficient signal coverage of 5G base stations can be solved by building new base stations in areas with weak signal coverage. However, due to construction costs and other factors, it is not possible to cover all areas. In general, areas with high traffic and weak coverage should be given priority. Although many scientists have carried out research, it is not possible to make the large-scale calculation accurately due to the lack of data support. It is necessary to search for the central point through continuous hypothesis testing, so there is a large systematic error. In addition, it is difficult to give a unique solution. In this paper, the weak signal coverage points were divided into three categories according to the number of users and traffic demand. With the lowest cost as the target, and constraints such as the distance requirement of base station construction, the proportion of the total signal coverage business, and so on, a single objective nonlinear programming model was established to solve the base station layout problem. Through traversal search, the optimal threshold of the traffic and the number of base stations was obtained, and then, a kernel function was added to the mean shift clustering algorithm. The center point of the new macro station was determined in the dense area, the location of the micro base station was determined from the scattered and abnormal areas, and finally the unique optimal planning scheme was obtained. Based on the assumptions made in this paper, the minimum total cost is 3752 when the number of macro and micro base stations were determined to be 31 and 3442 respectively, and the signal coverage rate can reach 91.43%. Compared with the existing methods, such as K-means clustering, K-medoids clustering, and simulated annealing algorithms, etc., the method proposed in this paper can achieve good economic benefits; when the traffic threshold and the number of base stations threshold are determined, the unique solution can be obtained. Full article

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15 pages, 4235 KiB

Open AccessArticle

Optimization of LFP Pouch Cell Tab Design for Uniform Temperature Distribution

by Jun Lee, Hyukkyun Chang and Chang-Wan Kim

Mathematics 2023, 11(8), 1970; https://doi.org/10.3390/math11081970 - 21 Apr 2023

Cited by 3 | Viewed by 3029

Abstract

An increase in the size of large-format Li-ion batteries (LIBs) may lead to nonuniform temperature distribution, which degrades the performance and lifespan of the LIBs. To address this issue, we performed design optimization using a 3D electrochemical-thermal coupled model for 55 Ah LFP/graphite [...] Read more.

An increase in the size of large-format Li-ion batteries (LIBs) may lead to nonuniform temperature distribution, which degrades the performance and lifespan of the LIBs. To address this issue, we performed design optimization using a 3D electrochemical-thermal coupled model for 55 Ah LFP/graphite large-format pouch cells. To minimize temperature differences in Normal Tab (NT), Lateral Tab (LT), and Counter Tab (CT) types of LIBs, design optimization was performed on the width, height, and attachment position of each positive and negative Table The upper and lower limits of each design variable were set as constraints without exceeding the sum of the total area of the tabs of the initial NT type. Owing to the optimization of the NT, LT, and CT types, the temperature difference in the optimized CT type was 79.2% less than in the initial NT type. Additionally, the potential difference decreased by 37.1%, minimizing ohmic heat. Aging analysis of 2500 cycles was performed to analyze the improvement in the lifespan due to the uniform temperature distribution. Consequently, the capacity retention rate of the optimized CT type was 6.5% higher than that of the initial NT type. Thus, the temperature distribution and lifespan of LIBs were improved by design optimization. Full article

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18 pages, 3920 KiB

Open AccessEditor’s ChoiceArticle

Multi-Scale Annulus Clustering for Multi-Label Classification

by Yan Liu, Changshun Liu, Jingjing Song, Xibei Yang, Taihua Xu and Pingxin Wang

Mathematics 2023, 11(8), 1969; https://doi.org/10.3390/math11081969 - 21 Apr 2023

Cited by 2 | Viewed by 1393

Abstract

Label-specific feature learning has become a hot topic as it induces classification models by accounting for the underlying features of each label. Compared with single-label annotations, multi-label annotations can describe samples from more comprehensive perspectives. It is generally believed that the compelling classification [...] Read more.

Label-specific feature learning has become a hot topic as it induces classification models by accounting for the underlying features of each label. Compared with single-label annotations, multi-label annotations can describe samples from more comprehensive perspectives. It is generally believed that the compelling classification features of a data set often exist in the aggregation of label distribution. In this in-depth study of a multi-label data set, we find that the distance between all samples and the sample center is a Gaussian distribution, which means that the label distribution has the tendency to cluster from the center and spread to the surroundings. Accordingly, the double annulus field based on this distribution trend, named DEPT for double annulusfield and label-specific features for multi-label classification, is proposed in this paper. The double annulus field emphasizes that samples of a specific size can reflect some unique features of the data set. Through intra-annulus clustering for each layer of annuluses, the distinctive feature space of these labels is captured and formed. Then, the final classification model is obtained by training the feature space. Contrastive experiments on 10 benchmark multi-label data sets verify the effectiveness of the proposed algorithm. Full article

(This article belongs to the Section Fuzzy Sets, Systems and Decision Making)

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19 pages, 10382 KiB

Open AccessArticle

Lattice Boltzmann Numerical Study on Mesoscopic Seepage Characteristics of Soil–Rock Mixture Considering Size Effect

by Peichen Cai, Xuesong Mao, Ke Lou and Zhihui Yun

Mathematics 2023, 11(8), 1968; https://doi.org/10.3390/math11081968 - 21 Apr 2023

Cited by 1 | Viewed by 1583

Abstract

One of the hot topics in the study of rock and soil hydraulics is the size effect of a soil–rock mixture’s (SRM) seepage characteristics. The seepage process of the SRM was simulated from the pore scale through the lattice Boltzmann method (LBM) in [...] Read more.

One of the hot topics in the study of rock and soil hydraulics is the size effect of a soil–rock mixture’s (SRM) seepage characteristics. The seepage process of the SRM was simulated from the pore scale through the lattice Boltzmann method (LBM) in this paper to explore the internal influence mechanism of sample size effect on the SRM seepage characteristics. SRM samples were generated using the improved Monte Carlo method (IMCM), and through 342 simulation test conditions the influence of size feature parameters such as resolution (R), segmentation type, model feature size (S), feature length ratio (F), and soil/rock particle size feature ratio (P) was examined. The study demonstrated that as R increases, the permeability of the SRM gradually rises and tends to stabilize when R reaches 60 ppi. At the same S, the dispersion degree of model permeability obtained by the four segmentation types is in the order of center < random < equal < top. With an increase in S, the permeability (k) of the SRM gradually decreases, conforming to the dimensionless mathematical model,

k = a_{0} \cdot S^{- b_{0}}

, and tends to stabilize at S = 80 mm. With an increase in F and an increase in S, the permeability of the SRM exhibits a linear “zonal” distribution that declines in order. When F is greater than 12, the dispersion of the permeability value distribution is especially small. With an increase in P, the permeability of the SRM decreases gradually before rising abruptly. P is crucial for the grading and structural makeup of the SRM. Overall, this paper concludes that the conditions of R = 60 ppi, center segmentation type, S = 80 mm, F ≥ 12, and P set by demand can be used to select and generate the size of the SRM optimal representative elementary volume (REV) numerical calculation model. The SRM can serve as a general reference for test and engineering construction as a common geotechnical engineering material. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis)

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16 pages, 5110 KiB

Open AccessArticle

Preparation and Analysis of Experimental Findings on the Thermal and Mechanical Characteristics of Pulsating Gas Flows in the Intake System of a Piston Engine for Modelling and Machine Learning

by Leonid Plotnikov

Mathematics 2023, 11(8), 1967; https://doi.org/10.3390/math11081967 - 21 Apr 2023

Cited by 5 | Viewed by 1596

Abstract

Today, reciprocating internal combustion engines are used in many branches of the economy (power engineering, machine engineering, transportation, and others). In order for piston engines to meet stringent environmental and economic regulations, it is necessary to develop complex and accurate control systems for [...] Read more.

Today, reciprocating internal combustion engines are used in many branches of the economy (power engineering, machine engineering, transportation, and others). In order for piston engines to meet stringent environmental and economic regulations, it is necessary to develop complex and accurate control systems for the physical processes in engine elements based on digital twins, machine learning, and artificial intelligence algorithms. This article is aimed at preparing and analysing experimental data on the gas dynamics and heat transfer of pulsating air flows in a piston engine’s intake system for modelling and machine learning. The key studies were carried out on a full-scale model of a single-cylinder piston engine under dynamic conditions. Some experimental findings on the gas-dynamic and heat-exchange characteristics of the flows were obtained with the thermal anemometry method and a corresponding measuring system. The effects of the inlet channel diameter on the air flow, the intensity of turbulence, and the heat transfer coefficient of pulsating air flows in a piston engine’s inlet system are shown. A mathematical description of the dependences of the turbulence intensity, heat transfer coefficient, and Nusselt number on operation factors (crankshaft speed, air flow velocity, Reynolds number) and the inlet channel’s geometric dimensions are proposed. Based on the mathematical modelling of the thermodynamic cycle, the operational and environmental performance of a piston engine with intake systems containing channels with different diameters were assessed. The presented data could be useful for refining engineering calculations and mathematical models, as well as for developing digital twins and engine control systems. Full article

(This article belongs to the Special Issue Methodological and Applied Contributions on Engineering Applications of Artificial Intelligence)

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12 pages, 2123 KiB

Open AccessArticle

Superposition Formulas and Evolution Behaviors of Multi-Solutions to the (3+1)-Dimensional Generalized Shallow Water Wave-like Equation

by Sudao Bilige, Leilei Cui and Xiaomin Wang

Mathematics 2023, 11(8), 1966; https://doi.org/10.3390/math11081966 - 21 Apr 2023

Cited by 1 | Viewed by 1139

Abstract

The superposition formulas of multi-solutions to the (3+1)-dimensional generalized shallow water wave-like Equation (GSWWLE) are proposed. There are arbitrary test functions in the superposition formulas of the mixed solutions and the interaction solutions, and we generalized to the sum of any N terms. [...] Read more.

The superposition formulas of multi-solutions to the (3+1)-dimensional generalized shallow water wave-like Equation (GSWWLE) are proposed. There are arbitrary test functions in the superposition formulas of the mixed solutions and the interaction solutions, and we generalized to the sum of any N terms. By freely selecting the test functions and the positive integer N, we have obtained abundant solutions for the GSWWLE. First, we introduced new mixed solutions between two arbitrary functions and the multi-kink solitons, and the abundant mixed solutions were obtained through symbolic computation. Next, we constructed the multi-localized wave solutions which are the superposition of N-even power functions. Finally, the novel interaction solutions between the multi-localized wave solutions and the multi-arbitrary function solutions for the GSWWLE were obtained. The evolution behaviors of the obtained solutions are shown through 3D, contour and density plots. The received results have immensely enriched the exact solutions of the GSWWLE in the available literature. Full article

(This article belongs to the Section Mathematical Physics)

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41 pages, 10917 KiB

Open AccessReview

Review of GrabCut in Image Processing

by Zhaobin Wang, Yongke Lv, Runliang Wu and Yaonan Zhang

Mathematics 2023, 11(8), 1965; https://doi.org/10.3390/math11081965 - 21 Apr 2023

Cited by 10 | Viewed by 3619

Abstract

As an image-segmentation method based on graph theory, GrabCut has attracted more and more researchers to pay attention to this new method because of its advantages of simple operation and excellent segmentation. In order to clarify the research status of GrabCut, we begin [...] Read more.

As an image-segmentation method based on graph theory, GrabCut has attracted more and more researchers to pay attention to this new method because of its advantages of simple operation and excellent segmentation. In order to clarify the research status of GrabCut, we begin with the original GrabCut model, review the improved algorithms that are new or important based on GrabCut in recent years, and classify them in terms of pre-processing based on superpixel, saliency map, energy function modification, non-interactive improvement and some other improved algorithms. The application status of GrabCut in various fields is also reviewed. We also experiment with some classical improved algorithms, including GrabCut, LazySnapping, OneCut, Saliency Cuts, DenseCut and Deep GrabCut, and objectively analyze the experimental results using five evaluation indicators to verify the performance of GrabCut. Finally, some existing problems are pointed out and we also propose some future work. Full article

(This article belongs to the Special Issue Mathematical Methods for Pattern Recognition)

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18 pages, 3994 KiB

Open AccessArticle

Coordinated Slip Control of Multi-Axle Distributed Drive Vehicle Based on HLQR

by Yutong Bao, Changqing Du, Dongmei Wu, Huan Liu, Wei Liu and Jun Li

Mathematics 2023, 11(8), 1964; https://doi.org/10.3390/math11081964 - 21 Apr 2023

Cited by 3 | Viewed by 1596

Abstract

For multi-axle distributed drive (MADD) vehicles, the complexity of the longitudinal dynamics control system increases with the number of driven wheels, which presents a huge challenge to control the multi-motor drive vehicle with more than four wheels. To reduce the control system complexity, [...] Read more.

For multi-axle distributed drive (MADD) vehicles, the complexity of the longitudinal dynamics control system increases with the number of driven wheels, which presents a huge challenge to control the multi-motor drive vehicle with more than four wheels. To reduce the control system complexity, this paper proposes a coordinated slip control algorithm using the hierarchical linear quadratic regulator (HLQR) scheme for a 12 × 12 MADD vehicle. The 12-wheel driving system is decoupled based on the wheel load and simplified to a double local subsystem. First, the 12 × 12 MADD vehicle dynamics model is established. Then, the optimal slip ratio is obtained on the basis of the road friction coefficient estimation through a fuzzy control algorithm when the wheel slips. Afterwards, the wheel slip ratio is controlled based on the HLQR program for anti-slip regulation. Furthermore, the driving torque control allocation based on quadratic programming (QR) is coordinated with the anti-slip control. Simulink results show that the proposed coordinated slip control based on HLQR can improve slip control accuracy by more than 30% and greatly reduce the calculation load. The torque control allocation is also limited by the slip control results to ensure wheel dynamic stability and smoothly satisfy the driver’s demand. Full article

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14 pages, 317 KiB

Open AccessArticle

Multiplication Algorithms for Approximate Optimal Distributions with Cost Constraints

by Lianyan Fu, Faming Ma, Zhuoxi Yu and Zhichuan Zhu

Mathematics 2023, 11(8), 1963; https://doi.org/10.3390/math11081963 - 21 Apr 2023

Cited by 1 | Viewed by 940

Abstract

In this paper, we study the D- and A-optimal assignment problems for regression models with experimental cost constraints. To solve these two problems, we propose two multiplicative algorithms for obtaining optimal designs and establishing extended D-optimal (

E D

-optimal) [...] Read more.

In this paper, we study the D- and A-optimal assignment problems for regression models with experimental cost constraints. To solve these two problems, we propose two multiplicative algorithms for obtaining optimal designs and establishing extended D-optimal (

E D

-optimal) and A-optimal (

E A

-optimal) criteria. In addition, we give proof of the convergence of the

E D

-optimal algorithm and draw conjectures about some properties of the

E A

-optimal algorithm. Compared with the classical D- and A-optimal algorithms, the

E D

- and

E A

-optimal algorithms consider not only the accuracy of parameter estimation, but also the experimental cost constraint. The proposed methods work well in the digital example. Full article

15 pages, 3535 KiB

Open AccessArticle

Mathematical Solution of Temperature Field in Non-Hollow Frozen Soil Cylinder Formed by Annular Layout of Freezing Pipes

by Zequn Hong, Rongjian Shi, Fengtian Yue, Jiaguang Yang and Yuanhao Wu

Mathematics 2023, 11(8), 1962; https://doi.org/10.3390/math11081962 - 21 Apr 2023

Viewed by 1282

Abstract

With the wide application of the artificial ground freezing method in municipal underground engineering, the annular layout of freezing pipes is often adopted to form a frozen soil cylinder. However, there is still no analytical solution that can calculate the temperature distribution of [...] Read more.

With the wide application of the artificial ground freezing method in municipal underground engineering, the annular layout of freezing pipes is often adopted to form a frozen soil cylinder. However, there is still no analytical solution that can calculate the temperature distribution of frozen soil formed in this case. In this paper, a mathematical model of a steady-state temperature field of single-circle freezing is established, in which the inside of the freeze ring is completely frozen; that is, the temperature of all excavation sections is below the freezing point. Then, the analytical solution of temperature distribution is deduced through the complex variable method and potential superposition method. Comparison results of the analytical solution with those of the numerical simulation show that the analytical solution is precise enough. The temperature distribution of the main section and the intersection is approximately the same on the inner side, but the freezing effect of the main section is relatively better near the freezing pipes and outside the freeze ring. Besides, according to the derived analytical solution and common freezing parameters, a simplified formula to calculate the temperature field with enough accuracy was proposed, and the error can be controlled below 1‰. Finally, based on the simplified formula, a calculation method for frozen soil thickness and the average temperature is also given in this paper. Full article

(This article belongs to the Section Engineering Mathematics)

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18 pages, 12040 KiB

Open AccessArticle

Seaweeds Arising from Brauer Configuration Algebras

by Agustín Moreno Cañadas and Odette M. Mendez

Mathematics 2023, 11(8), 1961; https://doi.org/10.3390/math11081961 - 21 Apr 2023

Viewed by 1311

Abstract

Seaweeds or seaweed Lie algebras are subalgebras of the full-matrix algebra

Mat (n)

introduced by Dergachev and Kirillov to give an example of algebras for which it is possible to compute the Dixmier index via combinatorial methods. It is worth noting [...] Read more.

Seaweeds or seaweed Lie algebras are subalgebras of the full-matrix algebra

Mat (n)

introduced by Dergachev and Kirillov to give an example of algebras for which it is possible to compute the Dixmier index via combinatorial methods. It is worth noting that finding such an index for general Lie algebras is a cumbersome problem. On the other hand, Brauer configuration algebras are multiserial and symmetric algebras whose representation theory can be described using combinatorial data. It is worth pointing out that the set of integer partitions and compositions of a fixed positive integer give rise to Brauer configuration algebras. However, giving a closed formula for the dimension of these kinds of algebras or their centers for all positive integer is also a tricky problem. This paper gives formulas for the dimension of Brauer configuration algebras (and their centers) induced by some restricted compositions. It is also proven that some of these algebras allow defining seaweeds of Dixmier index one. Full article

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

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8 pages, 1467 KiB

Open AccessArticle

A Note on the Geometry of Closed Loops

by Nir Shvalb, Mark Frenkel, Shraga Shoval and Edward Bormashenko

Mathematics 2023, 11(8), 1960; https://doi.org/10.3390/math11081960 - 21 Apr 2023

Cited by 1 | Viewed by 1403

Abstract

In this paper, we utilize the Ramsey theory to investigate the geometrical characteristics of closed contours. We begin by examining a set of six points arranged on a closed contour and connected as a complete graph. We assign the downward-pointing edges a red [...] Read more.

In this paper, we utilize the Ramsey theory to investigate the geometrical characteristics of closed contours. We begin by examining a set of six points arranged on a closed contour and connected as a complete graph. We assign the downward-pointing edges a red color, while coloring the remaining edges green. Our analysis establishes that the curve must contain at least one monochromatic triangle. This finding has practical applications in the study of dynamical billiards. Our second result is derived from the Jordan curve theorem and the Ramsey theorem. Finally, we discuss Ramsey constructions arising from differential geometry. Applications of the Ramsey theory are discussed. Full article

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21 pages, 11197 KiB

Open AccessArticle

Study on the Mechanical Behavior and Constitutive Model of Layered Sandstone under Triaxial Dynamic Loading

by Wenbing Fan, Junwen Zhang, Yang Yang, Yang Zhang, Xukai Dong and Yulong Xing

Mathematics 2023, 11(8), 1959; https://doi.org/10.3390/math11081959 - 21 Apr 2023

Cited by 4 | Viewed by 1388

Abstract

In construction engineering, rock is an important building material. During the construction process, layered rock masses are typically subjected to varying dynamic load disturbances under triaxial loads. It is thus essential to investigate the mechanical response of layered rocks under various disturbances of [...] Read more.

In construction engineering, rock is an important building material. During the construction process, layered rock masses are typically subjected to varying dynamic load disturbances under triaxial loads. It is thus essential to investigate the mechanical response of layered rocks under various disturbances of the triaxial loads. By using a three-dimensional SHPB, triaxial dynamic compression tests with various impact dynamic load disturbances and identical triaxial static loads were carried out on sandstones with differing bedding angles. The impact pressures were 0.8, 1.2, and 1.6 MPa, and the bedding angles were 0°, 30°, 45°, 60°, and 90°. The results showed that the ductility of the sandstone considerably increased under triaxial static loading. With the increasing bedding angle, the sandstone’s dynamic strength and coupling strength first declined and subsequently rose. As the impact pressure increased, the reflective energy ratio, peak strain, and dynamic growth factor of the sandstone essentially rose progressively. The bedding angles and dynamic loads had a major impact on the damage pattern of the layered sandstones. Additionally, a constitutive model considering bedding angle, dynamic load, and static load was established and verified. The constitutive model was able to accurately characterize the dynamic behavior of the rock under load disturbances. Full article

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15 pages, 320 KiB

Open AccessArticle

The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres

by Ibrahim Al-Dayel and Sharief Deshmukh

Mathematics 2023, 11(8), 1958; https://doi.org/10.3390/math11081958 - 21 Apr 2023

Viewed by 1146

Abstract

We studied the random variable

V_{t} = {vol}_{S^{2}} (g_{t} B \cap B)

, where B is a disc on the sphere

S^{2}

centered at the north pole and

{(g_{t})}_{t \geq 0}

is [...] Read more.

We studied the random variable

V_{t} = {vol}_{S^{2}} (g_{t} B \cap B)

, where B is a disc on the sphere

S^{2}

centered at the north pole and

{(g_{t})}_{t \geq 0}

is the Brownian motion on the special orthogonal group

S O (3)

starting at the identity. We applied the results of the theory of compact Lie groups to evaluate the expectation of

V_{t}

for

0 \leq t \leq τ

, where

τ

is the first time when

V_{t}

vanishes. We obtained an integral formula using the heat equation on some Riemannian submanifold

Γ_{B}

seen as the support of the function

f (g) = {vol}_{S^{2}} (g B \cap B)

immersed in

S O (3)

. The integral formula depends on the mean curvature of

Γ_{B}

and the diameter of B. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Submanifolds)

25 pages, 398 KiB

Open AccessArticle

Estimations of the Jensen Gap and Their Applications Based on 6-Convexity

by Muhammad Adil Khan, Asadullah Sohail, Hidayat Ullah and Tareq Saeed

Mathematics 2023, 11(8), 1957; https://doi.org/10.3390/math11081957 - 20 Apr 2023

Cited by 4 | Viewed by 1672

Abstract

The main purpose of this manuscript is to present some new estimations of the Jensen gap in a discrete sense along with their applications. The proposed estimations for the Jensen gap are provided with the help of the notion of 6-convex functions. Some [...] Read more.

The main purpose of this manuscript is to present some new estimations of the Jensen gap in a discrete sense along with their applications. The proposed estimations for the Jensen gap are provided with the help of the notion of 6-convex functions. Some numerical experiments are performed to determine the significance and correctness of the intended estimates. Several outcomes of the main results are discussed for the Hölder inequality and the power and quasi-arithmetic means. Furthermore, some applications are presented in information theory, which provide some bounds for the divergences, Bhattacharyya coefficient, Shannon entropy, and Zipf–Mandelbrot entropy. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

15 pages, 349 KiB

Open AccessArticle

Sequentially Ordered Sobolev Inner Product and Laguerre–Sobolev Polynomials

by Abel Díaz-González, Juan Hernández and Héctor Pijeira-Cabrera

Mathematics 2023, 11(8), 1956; https://doi.org/10.3390/math11081956 - 20 Apr 2023

Cited by 2 | Viewed by 1406

Abstract

We study the sequence of polynomials

{S_{n}}_{n \geq 0}

that are orthogonal with respect to the general discrete Sobolev-type inner product [...] Read more.

We study the sequence of polynomials

{S_{n}}_{n \geq 0}

that are orthogonal with respect to the general discrete Sobolev-type inner product

{⟨ f, g ⟩}_{s} = \int f (x) g (x) d μ (x) + \sum_{j = 1}^{N} \sum_{k = 0}^{d_{j}} λ_{j, k} f^{(k)} (c_{j}) g^{(k)} (c_{j}),

where

μ

is a finite Borel measure whose support

supp (μ)

is an infinite set of the real line,

λ_{j, k} \geq 0

, and the mass points

c_{i}

,

i = 1, \dots, N

are real values outside the interior of the convex hull of

supp (μ)

(

c_{i} \in R \ C_{h} {(supp (μ))}^{\circ})

. Under some restriction of order in the discrete part of

{⟨ \cdot, \cdot ⟩}_{s}

, we prove that

S_{n}

has at least

n - d^{*}

zeros on

C_{h} {(supp (μ))}^{\circ}

, being

d^{*}

the number of terms in the discrete part of

{⟨ \cdot, \cdot ⟩}_{s}

. Finally, we obtain the outer relative asymptotic for

{S_{n}}

in the case that the measure

μ

is the classical Laguerre measure, and for each mass point, only one order derivative appears in the discrete part of

{⟨ \cdot, \cdot ⟩}_{s}

. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions: Recent Trends and Their Applications)

26 pages, 407 KiB

Open AccessArticle

Towards a Proof of Bahri–Coron’s Type Theorem for Mixed Boundary Value Problems

by Azeb Alghanemi, Slim Chaabane, Hichem Chtioui and Abdellahi Soumaré

Mathematics 2023, 11(8), 1955; https://doi.org/10.3390/math11081955 - 20 Apr 2023

Viewed by 1155

Abstract

We consider a nonlinear variational elliptic problem with critical nonlinearity on a bounded domain of

R^{n}, n \geq 3

and mixed Dirichlet–Neumann boundary conditions. We study the effect of the domain’s topology on the existence of solutions as Bahri–Coron did in [...] Read more.

We consider a nonlinear variational elliptic problem with critical nonlinearity on a bounded domain of

R^{n}, n \geq 3

and mixed Dirichlet–Neumann boundary conditions. We study the effect of the domain’s topology on the existence of solutions as Bahri–Coron did in their famous work on the homogeneous Dirichlet problem. However, due to the influence of the part of the boundary where the Neumann condition is prescribed, the blow-up picture in the present setting is more complicated and makes the mixed boundary problems different with respect to the homogeneous ones. Such complexity imposes modification of the argument of Bahri–Coron and demands new constructions and extra ideas. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications III)

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20 pages, 4047 KiB

Open AccessArticle

Grouping Maintenance Policy for Improving Reliability of Wind Turbine Systems Considering Variable Cost

by Hongyan Dui, Yulu Zhang and Yun-An Zhang

Mathematics 2023, 11(8), 1954; https://doi.org/10.3390/math11081954 - 20 Apr 2023

Cited by 5 | Viewed by 1443

Abstract

Wind farms have gained wide attention due to unlimited resources and clean energy. Considering that wind turbine systems are always in harsh conditions, subsystem failures could reduce the reliability of wind turbine systems. At present, the maintenance behaviors for wind turbine systems are [...] Read more.

Wind farms have gained wide attention due to unlimited resources and clean energy. Considering that wind turbine systems are always in harsh conditions, subsystem failures could reduce the reliability of wind turbine systems. At present, the maintenance behaviors for wind turbine systems are various (e.g., corrective maintenance, preventive maintenance) when reliability is reduced below the threshold. Considering the maintenance cost and downtime, it is impossible to repair each component in a timely manner. One of the key problems is dividing components into maintenance groups to improve maintenance efficiency. In this paper, a grouping maintenance policy considering the variable cost (GMP-VC) is proposed to improve direct-drive permanent magnet (DPM) turbine systems. Grouping modes are proposed to fully consider the stated transition probability of turbine components and the variable cost of turbine systems. A maintenance model is formulated to select components as members of the group based on a RIM-VC index. An instance is given to verify the proposed GMP-VC method. The result indicates that the proposed maintenance policy may save maintenance costs over baseline plans. Full article

(This article belongs to the Special Issue System Reliability and Quality Management in Industrial Engineering)

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106 pages, 942 KiB

Open AccessReview

A Comprehensive Review of the Hermite–Hadamard Inequality Pertaining to Fractional Integral Operators

by Muhammad Tariq, Sotiris K. Ntouyas and Asif Ali Shaikh

Mathematics 2023, 11(8), 1953; https://doi.org/10.3390/math11081953 - 20 Apr 2023

Cited by 10 | Viewed by 1262

Abstract

In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Hermite–Hadamard (H-H)-type inequalities involving a variety of classes of [...] Read more.

In the frame of fractional calculus, the term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus and objective of this review paper is to present Hermite–Hadamard (H-H)-type inequalities involving a variety of classes of convexities pertaining to fractional integral operators. Included in the various classes of convexities are classical convex functions, m-convex functions, r-convex functions,

(α, m)

-convex functions,

(α, m)

-geometrically convex functions, harmonically convex functions, harmonically symmetric functions, harmonically

(θ, m)

-convex functions, m-harmonic harmonically convex functions,

(s, r)

-convex functions, arithmetic–geometric convex functions, logarithmically convex functions,

(α, m)

-logarithmically convex functions, geometric–arithmetically s-convex functions, s-convex functions, Godunova–Levin-convex functions, differentiable

ϕ

-convex functions,

M T

-convex functions,

(s, m)

-convex functions, p-convex functions, h-convex functions,

σ

-convex functions, exponential-convex functions, exponential-type convex functions, refined exponential-type convex functions, n-polynomial convex functions,

σ, s

-convex functions, modified

(p, h)

-convex functions, co-ordinated-convex functions, relative-convex functions, quasi-convex functions,

(α, h - m) - p

-convex functions, and preinvex functions. Included in the fractional integral operators are Riemann–Liouville (R-L) fractional integral, Katugampola fractional integral, k-R-L fractional integral,

(k, s)

-R-L fractional integral, Caputo-Fabrizio (C-F) fractional integral, R-L fractional integrals of a function with respect to another function, Hadamard fractional integral, and Raina fractional integral operator. Full article

(This article belongs to the Section Difference and Differential Equations)

14 pages, 6790 KiB

Open AccessArticle

Parameter-Independent Event-Triggered Implicit UKF for the Celestial Navigation Using Time Delay Measurement

by Mingzhen Gui, Caisheng Wei, Yifeng Wei, Kai Xiong, Chengxi Zhang and Mingzhe Dai

Mathematics 2023, 11(8), 1952; https://doi.org/10.3390/math11081952 - 20 Apr 2023

Cited by 1 | Viewed by 1285

Abstract

Celestial navigation using time delay measurement is an innovative autonomous navigation method. To calculate the equivalent measurement, the numerical method needs to be applied, which is time-consuming. The event-triggered mechanism intermittently and aperiodically processes measurements by judging if the update error has changed [...] Read more.

Celestial navigation using time delay measurement is an innovative autonomous navigation method. To calculate the equivalent measurement, the numerical method needs to be applied, which is time-consuming. The event-triggered mechanism intermittently and aperiodically processes measurements by judging if the update error has changed drastically. However, its performance is greatly affected by the constant threshold. To solve this problem, a parameter-independent event-triggered implicit unscented Kalman filter (UKF) is proposed and applied to the celestial navigation using time delay measurement. The innovation at the current moment and the updated estimate covariance at the last moment are compared with the previous value instead of the constant threshold. The event is automatically triggered when the accuracy of the state estimate is low. Simulation results indicate that the proposed parameter-independent event-triggered implicit UKF can reduce the running time by reducing unnecessary measurement updates, whose performance will not be affected by any parameter or window size. In a word, the proposed method substitutes the dynamic threshold for the constant threshold, ensuring that its performance will not be affected by any parameter or window size. Full article

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

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12 pages, 302 KiB

Open AccessArticle

On Oscillations in a Gene Network with Diffusion

by Vladimir Golubyatnikov, Natalia Ayupova and Natalia Kirillova

Mathematics 2023, 11(8), 1951; https://doi.org/10.3390/math11081951 - 20 Apr 2023

Cited by 2 | Viewed by 1070

Abstract

We consider one system of partial derivative equations of the parabolic type as a model of a simple 3D gene network in the presence of diffusion of its three components. Using discretization of the phase portrait of this system, comparison theorems, and other [...] Read more.

We consider one system of partial derivative equations of the parabolic type as a model of a simple 3D gene network in the presence of diffusion of its three components. Using discretization of the phase portrait of this system, comparison theorems, and other methods of the qualitative theory of differential equations, we show uniqueness of the equilibrium solution to this system and find conditions of instability of this equilibrium. Then, we obtain sufficient conditions of existence of at least one oscillating functioning regime of this gene network. An estimate of lower and upper bounds for periods of these oscillations is given as well. In quite a similar way, these results on the existence of cycles in 3D gene networks can be extended to higher-dimensional systems of parabolic or other evolution equations in order to construct mathematical models of more complicated molecular–genetic systems. Full article

(This article belongs to the Special Issue Mathematical Modeling of Evolutionary Dynamics)

18 pages, 993 KiB

Open AccessArticle

Finite-Element Method for the Simulation of Lipid Vesicle/Fluid Interactions in a Quasi–Newtonian Fluid Flow

by Aymen Laadhari

Mathematics 2023, 11(8), 1950; https://doi.org/10.3390/math11081950 - 20 Apr 2023

Cited by 2 | Viewed by 1238

Abstract

We present a computational framework for modeling an inextensible single vesicle driven by the Helfrich force in an incompressible, non-Newtonian extracellular Carreau fluid. The vesicle membrane is captured with a level set strategy. The local inextensibility constraint is relaxed by introducing a penalty [...] Read more.

We present a computational framework for modeling an inextensible single vesicle driven by the Helfrich force in an incompressible, non-Newtonian extracellular Carreau fluid. The vesicle membrane is captured with a level set strategy. The local inextensibility constraint is relaxed by introducing a penalty which allows computational savings and facilitates implementation. A high-order Galerkin finite element approximation allows accurate calculations of the membrane force with high-order derivatives. The time discretization is based on the double composition of the one-step backward Euler scheme, while the time step size is flexibly controlled using a time integration error estimation. Numerical examples are presented with particular attention paid to the validation and assessment of the model’s relevance in terms of physiological significance. Optimal convergence rates of the time discretization are obtained. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

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25 pages, 1831 KiB

Open AccessArticle

Connectivity Status of Intuitionistic Fuzzy Graph and Its Application to Merging of Banks

by Jayanta Bera, Kinkar Chandra Das, Sovan Samanta and Jeong-Gon Lee

Mathematics 2023, 11(8), 1949; https://doi.org/10.3390/math11081949 - 20 Apr 2023

Cited by 6 | Viewed by 2003

Abstract

Intuitionistic fuzzy graph theory is used to represent ambiguous networks, such as financial and social networks. The connectivity of such networks has a significant role in analyzing the network characteristics. This study investigates the connectivity status of vertices in an intuitionistic fuzzy graph. [...] Read more.

Intuitionistic fuzzy graph theory is used to represent ambiguous networks, such as financial and social networks. The connectivity of such networks has a significant role in analyzing the network characteristics. This study investigates the connectivity status of vertices in an intuitionistic fuzzy graph. Few properties have been established. Some areas of applications are shown for merging different banks and finding the central affected nodes by any infectious diseases. Full article

(This article belongs to the Special Issue Recent Advances in Fuzzy Deep Learning for Uncertain Medicine Data)

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22 pages, 2513 KiB

Open AccessArticle

Minimizing Energy Usage and Makespan of Elevator Operation in Rush Hour Using Multi-Objective Variable Neighborhood Strategy Adaptive Search with a Mobile Application

by Rojanee Homchalee, Rapeepan Pitakaso and Orawich Kumphon

Mathematics 2023, 11(8), 1948; https://doi.org/10.3390/math11081948 - 20 Apr 2023

Cited by 2 | Viewed by 2898

Abstract

The purpose of this study is to address two major issues: (1) the spread of epidemics such as COVID-19 due to long waiting times caused by a large number of waiting for customers, and (2) excessive energy consumption resulting from the elevator patterns [...] Read more.

The purpose of this study is to address two major issues: (1) the spread of epidemics such as COVID-19 due to long waiting times caused by a large number of waiting for customers, and (2) excessive energy consumption resulting from the elevator patterns used by various customers. The first issue is addressed through the development of a mobile application, while the second issue is tackled by implementing two strategies: (1) determining optimal stopping strategies for elevators based on registered passengers and (2) assigning passengers to elevators in a way that minimizes the number of floors the elevators need to stop at. The mobile application serves as an input parameter for the optimization toolbox, which employs the exact method and multi-objective variable neighborhood strategy adaptive search (M-VaNSAS) to find the optimal plan for passenger assignment and elevator scheduling. The proposed method, which adopts an even-odd floor strategy, outperforms the currently practiced procedure and leads to a 42.44% reduction in waiting time and a 29.61% reduction in energy consumption. Computational results confirmed the effectiveness of the proposed approach. Full article

(This article belongs to the Section Engineering Mathematics)

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18 pages, 5202 KiB

Open AccessArticle

Optimization of Drone Base Station Location for the Next-Generation Internet-of-Things Using a Pre-Trained Deep Learning Algorithm and NOMA

by Hadeel Alsolai, Wafa Mtouaa, Mashael S. Maashi, Mahmoud Othman, Ishfaq Yaseen, Amani A. Alneil, Azza Elneil Osman and Mohamed Ibrahim Alsaid

Mathematics 2023, 11(8), 1947; https://doi.org/10.3390/math11081947 - 20 Apr 2023

Cited by 2 | Viewed by 1442

Abstract

Next-generation Internet-of-Things applications pose challenges for sixth-generation (6G) mobile networks, involving large bandwidth, increased network capabilities, and remarkably low latency. The possibility of using ultra-dense connectivity to address the existing problem was previously well-acknowledged. Therefore, placing base stations (BSs) is economically challenging. Drone-based [...] Read more.

Next-generation Internet-of-Things applications pose challenges for sixth-generation (6G) mobile networks, involving large bandwidth, increased network capabilities, and remarkably low latency. The possibility of using ultra-dense connectivity to address the existing problem was previously well-acknowledged. Therefore, placing base stations (BSs) is economically challenging. Drone-based stations can efficiently address Next-generation Internet-of-Things requirements while accelerating growth and expansion. Due to their versatility, they can also manage brief network development or offer on-demand connectivity in emergency scenarios. On the other hand, identifying a drone stations are a complex procedure due to the limited energy supply and rapid signal quality degradation in air-to-ground links. The proposed method uses a two-layer optimizer based on a pre-trained VGG-19 model to overcome these issues. The non-orthogonal multiple access protocol improves network performance. Initially, it uses a powerful two-layer optimizer that employs a population of micro-swarms. Next, it automatically develops a lightweight deep model with a few VGG-19 convolutional filters. Finally, non-orthogonal multiple access is used to schedule radio and power resources to devices, which improves network performance. We specifically examine how three scenarios execute when various Cuckoo Search, Grey Wolf Optimization, and Particle Swarm Optimization techniques are used. To measure the various methodologies, we also run non-parametric statistical tests, such as the Friedman and Wilcoxon tests. The proposed method also evaluates the accuracy level for network performance of DBSs using number of Devices. The proposed method achieves better performance of 98.44% compared with other methods. Full article

(This article belongs to the Section Mathematics and Computer Science)

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13 pages, 3097 KiB

Open AccessArticle

Parametric Study on the Sensitivity of Different Properties of Waves Propagating through an Incompressible Medium

by Muhammad Aamir, Weaam Alhejaili, Khalid Lotfy, Alaa A. El-Bary and Adnan Jahangir

Mathematics 2023, 11(8), 1946; https://doi.org/10.3390/math11081946 - 20 Apr 2023

Viewed by 1058

Abstract

This article deals with the study of disturbance that travels through the transversely isotropic medium in the form of waves. The particles of the considered medium have an additional property of small-scale internal rotation along with macroscopic translational deformation. This extra translational freedom [...] Read more.

This article deals with the study of disturbance that travels through the transversely isotropic medium in the form of waves. The particles of the considered medium have an additional property of small-scale internal rotation along with macroscopic translational deformation. This extra translational freedom causes the medium to be micropolar in nature. Along with this, the medium is incompressible, and the dispersion relation of waves propagating through the medium is obtained under specific plan-strain conditions. From the dispersion relation, we can conclude that because of incompressibility, three transverse waves propagate through the medium. The velocity profile, attenuation coefficient, and specific heat loss for these waves are discussed for a particular medium. Later, the special normalized local sensitivity analysis (NLSA) technique is used to depict the effects of parameters on the outcomes of the mathematical model. The obtained results are represented graphically for a particular medium. The proposed model is used to model the mechanical behavior of complex materials with microstructural heterogeneity, such as composites and biological tissues. Full article

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17 pages, 3880 KiB

Open AccessArticle

Unconditional Superconvergence Error Estimates of Semi-Implicit Low-Order Conforming Mixed Finite Element Method for Time-Dependent Navier–Stokes Equations

by Xiaoling Meng and Huaijun Yang

Mathematics 2023, 11(8), 1945; https://doi.org/10.3390/math11081945 - 20 Apr 2023

Cited by 1 | Viewed by 1325

Abstract

In this paper, the unconditional superconvergence error analysis of the semi-implicit Euler scheme with low-order conforming mixed finite element discretization is investigated for time-dependent Navier–Stokes equations. In terms of the high-accuracy error estimates of the low-order finite element pair on the rectangular mesh [...] Read more.

In this paper, the unconditional superconvergence error analysis of the semi-implicit Euler scheme with low-order conforming mixed finite element discretization is investigated for time-dependent Navier–Stokes equations. In terms of the high-accuracy error estimates of the low-order finite element pair on the rectangular mesh and the unconditional boundedness of the numerical solution in

L^{\infty}

-norm, the superclose error estimates for velocity in

H^{1}

-norm and pressure in

L^{2}

-norm are derived firstly by dealing with the trilinear term carefully and skillfully. Then, the global superconvergence results are obtained with the aid of the interpolation post-processing technique. Finally, some numerical experiments are carried out to support the theoretical findings. Full article

(This article belongs to the Special Issue Advanced Computational Methods for Fluid Dynamics and Applications)

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16 pages, 577 KiB

Open AccessArticle

The Regulator Problem to the Convection–Diffusion Equation

by Andrés A. Ramírez and Francisco Jurado

Mathematics 2023, 11(8), 1944; https://doi.org/10.3390/math11081944 - 20 Apr 2023

Viewed by 1569

Abstract

In this paper, from linear operator, semigroup and Sturm–Liouville problem theories, an abstract system model for the convection–diffusion (C–D) equation is proposed. The state operator for this abstract system model is here defined as given in the form of the Sturm–Liouville differential operator [...] Read more.

In this paper, from linear operator, semigroup and Sturm–Liouville problem theories, an abstract system model for the convection–diffusion (C–D) equation is proposed. The state operator for this abstract system model is here defined as given in the form of the Sturm–Liouville differential operator (SLDO) plus an integral term of the same SLDO. Our aim is to achieve the trajectory tracking task in the presence of external disturbances to the C–D equation invoking the regulator problem theory, where the state from a finite-dimensional exosystem is the state to the feedback law. In this context, the regulator (Francis) equations, established from the abstract system model for the C–D equation, here are solved; i.e., the state feedback regulator problem (SFRP) for the C–D system has a solution. Our proposal is validated via numerical simulation results. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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