Mathematics, Volume 10, Issue 13 (July-1 2022) – 210 articles

In this work, by introducing multiple parameters and utilizing the Euler–Maclaurin summation formula and Abel’s partial summation formula, we first establish a reverse Hardy–Hilbert’s inequality containing one partial sum as the terms of double series. Then, based on the newly proposed inequality, we characterize the equivalent conditions of the best possible constant factor associated with several parameters. At the end of the paper, we illustrate that more new inequalities can be generated from the special cases of the reverse Hardy–Hilbert’s inequality. Full article

It is known that, to more adequately describe expert knowledge, it is necessary to go from the traditional (type-1) fuzzy techniques to higher-order ones: type-2, probably type-3 and even higher. Until recently, only type-1 and type-2 fuzzy sets were used in practical applications. However, lately, it turned out that type-3 fuzzy sets are also useful in some applications. Because of this practical importance, it is necessary to design efficient algorithms for data processing under such type-3 (and higher-order) fuzzy uncertainty. In this paper, we show how we can combine known efficient algorithms for processing type-1 and type-2 uncertainty to come up with a new algorithm for the type-3 case. Full article

Peer tutoring in combination with math digital tools was employed with middle school students learning mathematics. A total of 112 students in 9th grade (14 to 15 years old) participated in the study. A pretest–posttest with control group design was used. Students worked with systems of linear equations during the experience. The effects of the intervention of peer tutoring in combination with math digital tools on students’ mathematics achievement were examined using quantitative methods. The way students in the experimental group learned and their motivation towards using digital tools compared with students in the control group were analysed qualitatively. Statistically significant improvements and a large effect size were reported for students’ mathematics achievement in the experimental group. No statistically significant differences were reported between the pretest and the posttest for the control group. The qualitative analyses revealed that students in the experimental group achieved a higher level of autonomous learning, showed a greater association of mathematical concepts, helped their peers more, did more exercises and problems than students in the control group, and enjoyed the experience. Full article

With the popularization of platform economics, many manufacturers are shifting their operations from offline to online, forming platform supply chains (PSCs), which combine e-commerce with supply chain management. To study the influences of network externalities and fairness concerns on advertising strategies of the platform supply chain (PSC), we construct decentralized decision-making models, with and without fairness concerns. Then, we solve the optimal decisions regarding PSC and use numerical examples to verify the conclusions of the decision models. We further analyze the internal influences of advertising strategies on network externalities in the extended model. We find that the network externalities are beneficial to the PSC system, but the manufacturer’s fairness concerns are not beneficial to the PSC. The advertising strategies of the network platform are not affected by network externalities and fairness concerns. In the extended model, the manufacturer can obtain more profits, but the network platform yields less profit than the decentralized model without fairness concerns. Moreover, the more sensitive the network externalities are to the change in advertising strategies, the greater the profits for the PSC members. Full article

Automated fruit classification is a stimulating problem in the fruit growing and retail industrial chain as it assists fruit growers and supermarket owners to recognize variety of fruits and the status of the container or stock to increase business profit and production efficacy. As a result, intelligent systems using machine learning and computer vision approaches were explored for ripeness grading, fruit defect categorization, and identification over the last few years. Recently, deep learning (DL) methods for classifying fruits led to promising performance that effectively extracts the feature and carries out an end-to-end image classification. This paper introduces an Automated Fruit Classification using Hyperparameter Optimized Deep Transfer Learning (AFC-HPODTL) model. The presented AFC-HPODTL model employs contrast enhancement as a pre-processing step which helps to enhance the quality of images. For feature extraction, the Adam optimizer with deep transfer learning-based DenseNet169 model is used in which the Adam optimizer fine-tunes the initial values of the DenseNet169 model. Moreover, a recurrent neural network (RNN) model is utilized for the identification and classification of fruits. At last, the Aquila optimization algorithm (AOA) is exploited for optimal hyperparameter tuning of the RNN model in such a way that the classification performance gets improved. The design of Adam optimizer and AOA-based hyperparameter optimizers for DenseNet and RNN models show the novelty of the work. The performance validation of the presented AFC-HPODTL model is carried out utilizing a benchmark dataset and the outcomes report the promising performance over its recent state-of-the-art approaches. Full article

In this paper we investigate the criteria for the existence of solutions for single-valued as well as multi-valued boundary value problems involving $(k,\psi )$-Hilfer fractional derivative operator of order in $(1,2],$ equipped with nonlocal integral multi-point boundary conditions. For the single-valued case, we rely on fixed point theorems due to Banach and Krasnosel’skiĭ, and Leray–Schauder alternative to establish the desired results. The existence results for the multi-valued problem are obtained by applying the Leray–Schauder nonlinear alternative for multi-valued maps for convex-valued case, while the nonconvex-valued case is studied with the aid of Covit–Nadler’s fixed point theorem for multi-valued contractions. Numerical examples are presented for the illustration of the obtained results. Full article

The behaviour of magnetic impact on the unsteady separated stagnation-point flow of hybrid nanofluid with the influence of viscous dissipation and Joule heating is investigated numerically in this study. A new mathematical hybrid nanofluid model is developed, and similarity solutions are obtained in the form of ordinary differential equations (ODEs). The bvp4c approach in MATLAB is used to determine the reduced ODEs’ estimated solutions. The influence of various physical parameters is scrutinised. The findings revealed that the skin friction coefficient increases with the increment of the nanoparticle volume fraction and the unsteadiness parameter. This observation is also applied to the heat transfer rate of the fluid. Additionally, the presence of the magnetic and acceleration parameter provides a significant result in the heat transfer performance. The addition of the Eckert number increased the temperature profile distribution, thereby spontaneously decreasing the heat transfer rate. The first solution is declared stable by the analysis of solution stability. Full article

The synchronization problem for impulsive fractional-order Cohen–Grossberg neural networks with generalized proportional Caputo fractional derivatives with changeable lower limit at any point of impulse is studied. We consider the cases when the control input is acting continuously as well as when it is acting instantaneously at the impulsive times. We defined the global Mittag–Leffler synchronization as a generalization of exponential synchronization. We obtained some sufficient conditions for Mittag–Leffler synchronization. Our results are illustrated with examples. Full article

Cracks are widespread in infrastructure that are closely related to human activity. It is very popular to use artificial intelligence to detect cracks intelligently, which is known as crack detection. The noise in the background of crack images, discontinuity of cracks and other problems make the crack detection task a huge challenge. Although many approaches have been proposed, there are still two challenges: (1) cracks are long and complex in shape, making it difficult to capture long-range continuity; (2) most of the images in the crack dataset have noise, and it is difficult to detect only the cracks and ignore the noise. In this paper, we propose a novel method called Transformer-based Multi-scale Fusion Model (TransMF) for crack detection, including an Encoder Module (EM), Decoder Module (DM) and Fusion Module (FM). The Encoder Module uses a hybrid of convolution blocks and Swin Transformer block to model the long-range dependencies of different parts in a crack image from a local and global perspective. The Decoder Module is designed with symmetrical structure to the Encoder Module. In the Fusion Module, the output in each layer with unique scales of Encoder Module and Decoder Module are fused in the form of convolution, which can release the effect of background noise and strengthen the correlations between relevant context in order to enhance the crack detection. Finally, the output of each layer of the Fusion Module is concatenated to achieve the purpose of crack detection. Extensive experiments on three benchmark datasets (CrackLS315, CRKWH100 and DeepCrack) demonstrate that the proposed TransMF in this paper exceeds the best performance of present baselines. Full article

Consider the following problem. In a school with three classes containing n students each, given that their genders are unknown, find the minimum possible number of triples of same-gender students—not all of which are from the same class. Muaengwaeng asked this question and conjectured that the minimum scenario occurs when the classes are all boys, all girls and half-and-half. In this paper, we solve many generalizations of the problem including when the school has more than three classes, when triples are replaced by groups of larger sizes, when the classes are of different sizes, and when gender is replaced by other non-binary attributes. Full article

The study examines three experiments of implementing the “learning through teaching” approach combined with the development of digital technologies and explores the influence of these methods on the quality of education for undergraduate engineering students at an urban technical university in Russia. In the first experiment, small independent groups of students within the same cohort developed individual Java modules with the goal of creating an intelligent system to support solving problems using graphs. In the second experiment, each student peer-taught the topic of their choice to three other students and then administered an oral exam to these students to assess their understanding. In the third experiment, each participant selected a problem to develop a solution and recorded a video that explained this solution to other students. All recorded videos were made available to all students, and the combined collection consisted of 100 videos. During the final exam, students were randomly assigned one of these problems and had to present their solutions to the instructor. Analysis of the experiments demonstrated that integrating “learning through teaching” led to an increase in student interest in the discipline and an improvement in conceptual understanding, more so for students in the role of teacher than in the role of learner. Overall, combining pedagogical and digital technologies improved the quality of education for engineering students. Full article

Feature Selection (FS) is a major preprocessing stage which aims to improve Machine Learning (ML) models’ performance by choosing salient features, while reducing the computational cost. Several approaches are presented to select the most Optimal Features Subset (OFS) in a given dataset. In this paper, we introduce an FS-based approach named Reptile Search Algorithm–Snake Optimizer (RSA-SO) that employs both RSA and SO methods in a parallel mechanism to determine OFS. This mechanism decreases the chance of the two methods to stuck in local optima and it boosts the capability of both of them to balance exploration and explication. Numerous experiments are performed on ten datasets taken from the UCI repository and two real-world engineering problems to evaluate RSA-SO. The obtained results from the RSA-SO are also compared with seven popular Meta-Heuristic (MH) methods for FS to prove its superiority. The results show that the developed RSA-SO approach has a comparative performance to the tested MH methods and it can provide practical and accurate solutions for engineering optimization problems. Full article

In this paper, the interpolation matrix method (IMM) is proposed to solve the buckling critical load of axially functionally graded (FG) Timoshenko beams. Based on Timoshenko beam theory, a set of governing equations coupled by the deflection function and rotation function of the beam are obtained. Then, the deflection function and rotation function are decoupled and transformed into an eigenvalue problem of a variable coefficient fourth-order ordinary differential equation with unknown deflection function. According to the theory of interpolation matrix method, the eigenvalue problem of the variable coefficient fourth-order ordinary differential equation is transformed into an eigenvalue problem of a set of linear algebraic equations, and the critical buckling load and the corresponding deflection function of the axially functionally graded Timoshenko beam can be calculated by the orthogonal triangular (QR) decomposition method, which is the most effective and widely used method for finding all eigenvalues of a matrix. The numerical results are in good agreement with the existing results, which shows the effectiveness and accuracy of the method. Full article

This work aims at investigating the nonlinear transient response of fluid-conveying pipes made of graphene nanoplatelet (GPL)-reinforced composite (GPLRC) under blast loads and in a thermal environment. A modified Halpin–Tsai model is used to approximate the effective Young’s modulus of the GPLRC pipes conveying fluid; the mass density and Poisson’s ratio are determined by using the Voigt model. A slender Euler–Bernoulli beam is considered for modeling the pipes conveying fluid. The vibration control equation of the GPLRC pipes conveying fluid under blast loads is obtained by using Hamilton’s principle. A set of second-order ordinary differential equations are obtained by using the second-order Galerkin discrete method and are solved by using the adaptive Runge–Kutta method. Numerical experiments show that GPL distribution and temperature; GPL weight fraction; pipe length-to-thickness ratio; flow velocity; and blast load parameters have important effects on the nonlinear transient response of the GPLRC pipes conveying fluid. The numerical results also show that due to the fluid–structure interaction, the vibration amplitudes of the GPLRC pipes conveying fluid decay after the impact of blast loads. Full article

The COVID-19 pandemic had an unprecedented impact on the civil aviation passenger transport industry. This study analyzes the scale change and spatial distribution of the Chinese main air transport network (CMATN) and its role in the early spread of the pandemic using a complex network analysis method and econometric model. The result shows that CMATN is mainly located in the economically developed and densely populated central and eastern regions of China. The normalized degree, closeness, and betweenness centralities of CMATN node cities show an overall increasing trend, indicating that the air transport network is constantly improving. There was a significant positive relationship between the centrality of node cities, the duration of the COVID-19 pandemic, and the number of confirmed cases, indicating that air transport networks were crucial in the pandemic’s early spread. Furthermore, social and economic variables such as GDP and population had a significant positive impact on the duration of the pandemic, indicating that higher levels of social and economic development increased the seriousness of the pandemic. Our findings are expected to supplement existing research and serve as a point of reference for pandemic prevention and control. Full article

This paper is concerned with the spatially homogeneous Boltzmann equation, with the assumption of Maxwellian interaction. We consider initial data that belong to a small neighborhood of the equilibrium, which is a Maxwellian distribution. We prove that the solution remains in another small neighborhood with the same center and converges to this equilibrium exponentially fast, with an explicit quantification. Full article

A neural network model to predict the dynamic hysteresis loops and the energy-loss curves (i.e., the energy versus the amplitude of the magnetic induction) of soft ferromagnetic materials at different operating frequencies is proposed herein. Firstly, an innovative Fe-Si magnetic alloy, grade 35H270, is experimentally characterized via an Epstein frame in a wide range of frequencies, from 1 Hz up to 600 Hz. Parts of the dynamic hysteresis loops obtained through the experiments are involved in the training of a feedforward neural network, while the remaining ones are considered to validate the model. The training procedure is accurately designed to, firstly, identify the optimum network architecture (i.e., the number of hidden layers and the number of neurons per layer), and then, to effectively train the network. The model turns out to be capable of reproducing the magnetization processes and predicting the dynamic energy losses of the examined material in the whole range of inductions and frequencies considered. In addition, its computational and memory efficiency make the model a useful tool in the design stage of electrical machines and magnetic components. Full article

The role and the availability of digital technology in society is growing, which is why educators need to increasingly more often decide which types of digital technology to integrate into their teaching and when to integrate them. Thus, those decision-making skills need to be developed and measured especially for pre-service mathematics teachers. Therefore, we conducted an explorative interview study to understand the portfolio of argumentation on whether to use digital technology in different teaching phases and what criteria are used when making those decisions. Our results are based on ten interviews with pre- and in-service mathematics teachers in Germany. The analysis shows that (1) different levels of argumentation can be distinguished and (2) there are indications that teachers need to be aware of digital technology when deciding whether or not to use digital technology in a teaching setting. In addition, (3) besides the teaching-phase perspective and the learner-perspective, we expanded current research by formalizing the educator-perspective in a list of decision criteria. The compiled list of decision criteria was theoretically validated through the literature. In combination with the applied teaching phase framework and taxonomy of digital technology the list could aid in the development of the decision-making skills and potentially could result in a more reflective use of digital technology by pre-service and in-service teachers. Full article

The prime objective of the current study is to propose a novel mathematical framework under the fractional-order derivative, which describes the complex within-host behavior of SARS-CoV-2 by taking into account the effects of memory and carrier. To do this, we formulate a mathematical model of SARS-CoV-2 under the Caputo fractional-order derivative. We derived the conditions for the existence of equilibria of the model and computed the basic reproduction number ${\mathcal{R}}_0$. We used mathematical analysis to establish the proposed model’s local and global stability results. Some numerical resolutions of our theoretical results are presented. The main result of this study is that as the fractional derivative order increases, the approach of the solution to the equilibrium points becomes faster. It is also observed that the value of ${\mathcal{R}}_0$ increases as the value of $\beta$ and ${\pi }_v$ increases. Full article

Over the last few decades, Multi-criteria Decision Analysis (MCDA) techniques have been successfully applied to complex decision-making problems in a wide range of fields, such as economics, finance, logistics, environmental restoration, health or industrial organization, to name but a few, and imprecision and uncertainty have been incorporated into the decision-making process and applied to group decision-making contexts. [...] Full article

The goal of this study is to develop some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. Hypergeometric functions of the kind ${}_2{F}_1\left(z\right)$ are included in all connection coefficients for a specific z. Several new connection formulae between some famous polynomials, such as Fibonacci, Lucas, Pell, Fermat, Pell–Lucas, and Fermat–Lucas polynomials, are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some new formulae linking some celebrated numbers are given and also some newly closed formulae of certain definite weighted integrals are deduced. Based on using the two generalized classes of Fibonacci and Lucas polynomials, some new reduction formulae of certain odd and even radicals are developed. Full article

The empirical likelihood ratio test (ELRT) statistic is constructed for testing the homogeneity of several nonparametric populations in the presence of some auxiliary information. It is shown—under some regularity conditions and under the null hypothesis that all distribution functions of the populations are equal—that the asymptotic distribution of the ELRT is a chi-squared distribution. The proposed ELRT could be more powerful than the Kruskal–Wallis test, as extra information can be efficiently employed by ELRT. The advantage of ELRT over T&P (2006) is that researchers do not need to select approximately normal statistics for inter-group comparisons, and ELRT is more suitable for the multi-population consistency test with a small sample size. Full article

Geosynchronous orbit (GEO) is a very important strategic resource. In order to maximize the utilization of the GEO resources, the use of all-electric propulsion GEO platforms can greatly extend the service life of satellites. Therefore, this paper proposes a control scheme of the north/south station keeping (NSSK) by using electric propulsion with a manipulator. First, on the basis of the traditional calculation method of the semi-diurnal period of the orbital inclination, the calculation method of the semi-monthly period and the semi-annual period of the orbital inclination are proposed. The new method can reduce the fuel consumption and reduce the control amount and control frequency of the station keeping (SK). Secondly, a fuel-optimized NSSK algorithm by using electric propulsion with a manipulator is proposed. The algorithm can not only be applied to a large initial orbital inclination but also can unload the large angular momentum of the asymmetric satellites while keeping the north/south station, thereby avoiding the loss of control of the satellite’s attitude. The research results of this paper provide a new idea for the SK control of the GEO satellites and have great engineering application value. Full article

The application of transformations of the state equations of continuous linear and bilinear systems to the canonical form of controllability allows one to simplify the computation of Gramians of these systems. In this paper, we develop the method and obtain algorithms for computation of the controllability and observability Gramians of continuous linear and bilinear stationary systems with many inputs and one output, based on the method of spectral expansion of the Gramians and the iterative method for computing the bilinear systems Gramians. An important feature of the concept is the idea of separability of the Gramians expansion: separate computation of the scalar and matrix parts of the spectral Gramian expansion reduces the sub-Gramian matrices computation to calculation of numerical sequences of their elements. For the continuous linear systems with one output the method and the algorithm of the spectral decomposition of the controllability Gramian are developed in the form of Xiao matrices. Analytical expressions for the diagonal elements of the Gramian matrices are obtained, and by making use of which the rest of the elements can be calculated. For continuous linear systems with many outputs the spectral decompositions of the Gramians in the form of generalized Xiao matrices are obtained, which allows us to significantly reduce the number of calculations. The obtained results are generalized for continuous bilinear systems with one output. Iterative spectral algorithms for computation of elements of Gramians of these systems are proposed. Examples are given that illustrate theoretical results. Full article

This paper tests the random walk hypothesis in the cryptocurrency market. Based on the well-known Meese–Rogoff puzzle, we evaluate whether cryptocurrency returns are predictable or not. For this purpose, we conduct in-sample and out-of-sample analyses to examine the forecasting power of our model built with autoregressive components and lagged returns of BITCOIN, compared with the random walk benchmark. To this end, we considered the 13 major cryptocurrencies between 2018 and 2022. Our results indicate that our models significantly outperform the random walk benchmark. In particular, cryptocurrencies tend to be far more persistent than regular exchange rates, and BITCOIN (BTC) seems to improve the predictive accuracy of our models for some cryptocurrencies. Furthermore, while the predictive performance is time varying, we find predictive ability in different regimes before and during the pandemic crisis. We think that these results are helpful to policymakers and investors because they open a new perspective on cryptocurrency investing strategies and regulations to improve financial stability. Full article

The Teaching Learning-Based Algorithm (TLBA) is a powerful and effective optimization approach. TLBA mimics the teaching-learning process in a classroom, where TLBA’s iterative computing process is separated into two phases, unlike standard evolutionary algorithms and swarm intelligence algorithms, and each phase conducts an iterative learning operation. Advanced technologies of Voltage Source Converters (VSCs) enable greater active and reactive power regulation in these networks. Various objectives are addressed for optimal energy management, with the goal of attaining economic and technical advantages by decreasing overall production fuel costs and transmission power losses in AC-DC transmission networks. In this paper, the TLBA is applied for various sorts of nonlinear and multimodal functioning of hybrid alternating current (AC) and multi-terminal direct current (DC) power grids. The proposed TLBA is evaluated on modified IEEE 30-bus and IEEE 57-bus AC-DC networks and compared to other published methods in the literature. Numerical results demonstrate that the proposed TLBA has great effectiveness and robustness indices over the others. Economically, the reduction percentages of 13.84 and 21.94% are achieved for the IEEE 30-bus and IEEE 57-bus test systems when the fuel costs are minimized. Technically, significant improvement in the transmission power losses with reduction 28.01% and 69.83%, are found for the IEEE 30-bus and IEEE 57-bus test system compared to the initial case. Nevertheless, TLBA has faster convergence, higher quality for the final optimal solution, and more power for escaping from convergence to local optima compared to other published methods in the literature. Full article

As the backbone of every blockchain application, the consensus protocol is impacted by numerous risks, namely resource requirements and energy consumption, which limit the usage of blockchain. Applications such as IoT/IIoT cannot use these high-cost consensus methods due to limited resources. Therefore, we introduce Delegated Proof of Accessibility (DPoAC), a new consensus technique that employs secret sharing, PoS with random selection, and an interplanetary file system (IPFS).DPoAC is decomposed into two stages. During the initial stage, a secret is generated by a randomly chosen super node and divided into n shares. These shares are encrypted and stored in different n nodes on the IPFS network. The nodes will compete to access these shareholders to reconstruct the secret. The winning node will be awarded block generation rights. PoS with random selection is used in the second stage to compute the appropriate hash value and construct a block with valid transactions. In this novel approach, a node with few computational resources and small stakes can still obtain block generation rights by providing access to secret shares and reconstructing the secret, making the system reasonably fair. We qualitatively analyze and compare our scheme based on performance parameters against existing mainstream consensus protocols in the context of IoT/IIoT networks. Full article

Stock markets foster economic growth through meeting the fund requirements of the firms by individual and institutional investors. Pension funds and insurance companies with their long-term investment horizon are critical institutional investors in capital markets. Therefore, this article explores the effect of pension funds and insurance companies on stock market development in 15 emerging market economies over the 2004–2019 period through panel cointegration and causality tests. The causality analysis revealed that stock market development had a significant impact on pension funds and the insurance sector in the short term. However, the cointegration analysis revealed that pension funds had a positive effect on stock market development in Brazil, Chile, Hungary, Mexico, Peru, and South Africa and the insurance sector had a positive impact on stock market development in Chile, Indonesia, Korea Republic, Philippines, and South Africa in the long term. Full article

The Victorian Curriculum and Assessment Authority (VCAA) introduced the use of Computer Algebra System (CAS) technology (calculator and software) into the senior secondary mathematics curriculum and examination assessment in three phases, starting with a research-based pilot from 2000, followed by parallel implementation of CAS and non-CAS subjects from 2006 and culminating in transition to CAS-assumed subjects in 2010. This paper reports reflections on these developments over two decades from the perspectives of a researcher and the state mathematics manager (the authors) in consultation with four implementing teachers (the consultants). The authors critically examined the strategic design decisions that were made for the initiative over time. Then, with contributions from the four consultants, technical design issues relating to assessment and to teaching and the changes over a decade were investigated. A range of modifications have been made over the two decades, driven by changes in device capability and progressively increasing teaching expertise. The place of CAS in senior mathematics is now widely accepted, partly because an examination component not allowing any technology has been implemented. Examination questions have become more general, which may have added difficulty, but more questions involve setting up a real situation mathematically. Full article