Mathematics, Volume 12, Issue 9 (May-1 2024) – 145 articles

Under many interactive environments in the real world, there is often a need to evaluate the minimization effects and subsequent allocation outcomes derived from these interactions under multiple considerations. For instance, in the context of product sales, it is necessary to evaluate how to minimize the manufacturing costs of various producing factors, and sometimes, from a holistic perspective, it may even be necessary to evaluate situations with minimal sales benefits. On the other hand, in order to evaluate related effects derived from interactions and subsequent allocation outcomes, many game-theoretical studies are based on interactive models to formulate evaluating mechanisms, and then they apply axiomatic processes to analyze the rationality of these mechanisms. Therefore, this study first proposes a mechanism for evaluating the minimization effects and subsequent allocation outcomes under multiple considerations. Additionally, considering that different environmental impacts result from varying participation factors, this study also presents several weighted derivatives based on participation factors and their behaviors. Concurrently, we utilize axiomatic results to demonstrate the mathematical correctness and practicality for these evaluating mechanisms. Full article

For complex equipment, it is easy to over-evaluate the impact of failure on production by estimating the reliability level only through failure probability. To remedy this problem, this paper proposes a statistical evaluation method based on fuzzy failure data considering the multi-state characteristics of equipment failures. In this method, the new reliability-evaluation scheme is firstly presented based on the traditional statistical analysis method using the Weibull distribution function. For this scheme, the failure-grade index is defined, and a fuzzy-evaluation method is also proposed by comprehensively considering failure severity, failure maintenance, time, and cost; this is then combined with the time between failures to characterize the failure state. Based on the fuzzy failure data, an improved adaptive-failure small-sample-expansion method is proposed based on the classical bootstrap method and the deviation judgment between distributions of the original and newborn samples. Finally, a novel reliability-evaluation model, related to the failure grade and its membership degree, is established to quantify the reliability level of equipment more realistically. Example cases for three methods of the scheme (the failure-grade fuzzy-evaluation method, the sample-expansion method, and the reliability-evaluation modeling method) are presented, respectively, to validate the effectiveness and significance of the proposed reliability-evaluation technology. Full article

Multiple natural frequencies may be encountered when analyzing the essential natural vibration of a symmetric mechanical system or sub-structure system or a system with special parameters. The transfer matrix method (TMM) is a useful tool for analyzing the natural vibration characteristics of mechanical or structural systems. It derives a nonlinear eigen-problem (NEP) in general, even a transcendental eigen-problem. This investigation addresses the NEP in TMM and proposes a novel method, called the determinant-differentiation-based method, for calculating multiple natural frequencies and determining their multiplicities. Firstly, the characteristic determinant is differentiated with respect to frequency, transforming the even multiple natural frequencies into the odd multiple zeros of the differentiation of the characteristic determinant. The odd multiple zeros of the first derivative of the characteristic determinant and the odd multiple natural frequencies can be obtained using the bisection method. Among the odd multiple zeros, the even multiple natural frequencies are picked out by the proposed judgment criteria. Then, the natural frequency multiplicities are determined by the higher-order derivatives of the characteristic determinant. Finally, several numerical simulations including the multiple natural frequencies show that the proposed method can effectively calculate the multiple natural frequencies and determine their multiplicities. Full article

As an immensely important characteristic of natural images, the nonlocal self-similarity (NSS) prior has demonstrated great promise in a variety of inverse problems. Unfortunately, most current methods utilize either the internal or the external NSS prior learned from the degraded image or training images. The former is inevitably disturbed by degradation, while the latter is not adapted to the image to be restored. To mitigate such problems, this work proposes to learn a hybrid NSS prior from both internal images and external training images and employs it in image restoration tasks. To achieve our aims, we first learn internal and external NSS priors from the measured image and high-quality image sets, respectively. Then, with the learned priors, an efficient method, involving only singular value decomposition (SVD) and a simple weighting method, is developed to learn the HNSS prior for patch groups. Subsequently, taking the learned HNSS prior as the dictionary, we formulate a structural sparse representation model with adaptive regularization parameters called HNSS-SSR for image restoration, and a general and efficient image restoration algorithm is developed via an alternating minimization strategy. The experimental results indicate that the proposed HNSS-SSR-based restoration method exceeds many existing competition algorithms in PSNR and SSIM values. Full article

The $H_2{O}_2$-assisted oxidation by a peroxidase enzyme takes place to help plants maintain the concentrations of organic compounds at physiological levels. Cells regulate the oxidation rate by inhibiting the action of this enzyme. The cells use two inhibitory processes to regulate the enzyme: a noncompetitive substrate inhibitory process and a competitive substrate inhibitory process. Numerous applications of peroxidase have been developed in clinical biochemistry, enzyme immunoassays, the treatment of waste water containing phenolic compounds, the synthesis of various aromatic chemicals, and the removal of peroxide from industrial wastes. The kinetic mechanism of the Spanish broom peroxidase enzyme is a Ping Pong Bi Bi mechanism with the presence of competitive inhibition by substrates. A mathematical model may help in identifying the key mechanism from amongst a set of competing mechanisms. In this study, we developed a fractional mathematical model to describe the $H_2{O}_2$-supported oxidation by the enzyme Spanish broom peroxidase. Numerical simulations of the model produced results that are consistent with the known behaviour of Spanish broom peroxidase. Finally, some future investigations of the study are briefly indicated as well. Full article

As the core component of the wind turbine generation gearbox, the gear-rotor-bearing transmission system typically operates in harsh environments, inevitably leading to the occurrence of composite faults in the system, which exacerbates system vibration. Therefore, it is necessary to study the vibration characteristics of wind turbine helical gear-rotor-bearing transmission systems with composite faults. This paper uses an improved energy method to calculate the theoretical time-varying mesh stiffness of a helical gear with a root crack failure. On the premise of considering the time-varying meshing stiffness of the faulty helical gear, the gear eccentric fault, and the nonlinear support force of the faulty bearing, a multi-degree-of-freedom helical gear-rotor-bearing transmission system with compound faults was established by using the lumped parameter method. The dynamic model of the system was solved based on the Runge–Kutta method, and the vibration response of the system under healthy conditions, single faults with gear eccentricity, single faults with tooth root cracks, and coupled bearing composite faults were simulated and analyzed. The results show that the simulation results based on KISSsoft software 2018 version verify the effectiveness of the improved energy method; the existence of single faults and composite faults will cause the fault characteristics in the time domain and frequency domain responses. In this paper, the influence of a single fault and a complex fault on the time domain and frequency domain of the system is mainly discovered through the fault study of the helical rotor-bearing system, and the influence of the fault degree on the vibration of the gear motion system is discussed. The greater the degree of the fault, the more vibration of the system occurs; accordingly, when the system is under the coupling of tooth root crack and bearing fault, there is a significant difference compared with the healthy system and the single fault system. The system vibration has obvious time domain and frequency domain signal characteristics, including periodic pulse impacts caused by gear faults and time domain impact caused by bearing. The fault characteristic frequencies can also be found in the frequency domain. In this paper, the fault study of a helical gear of wind turbine generation provides a reference for the theoretical analysis of the vibration characteristics of the helical gear-rotor-bearing system under various fault conditions, lays a solid foundation for the simulation and subsequent diagnosis of the composite fault signal of the system, and provides help for the fault diagnosis of wind turbine gearboxes in the future. Full article

This paper focuses on the event-triggered synchronization of coupled neural networks with reaction–diffusion terms. At first, an effective event-triggered controller was designed based on time sampling. It is worth noting that the data of the controller for this type can be updated only when corresponding triggering conditions are satisfied, which can significantly reduce the communication burden of the control systems compared to other control strategies. Furthermore, some sufficient criteria were obtained to ensure the event-triggered synchronization of the considered systems through the use of an inequality techniques as well as the designed controller. Finally, the validity of the theoretical results was confirmed using numerical examples. Full article

The variability and intermittency inherent in renewable energy sources poses significant challenges to balancing power supply and demand, often leading to wind and solar energy curtailment. To address these challenges, this paper focuses on enhancing Time of Use (TOU) electricity pricing strategies. We propose a novel method based on equivalent load, which leverages typical power grid load and incorporates a responsibility weight for renewable energy consumption. The responsibility weight acts as an equivalent coefficient that accurately reflects renewable energy output, which facilitates the division of time periods and the development of a demand response model. Subsequently, we formulate an optimized TOU electricity pricing model to increase the utilization rate of renewable energy and reduce the peak–valley load difference of the power grid. To solve the TOU pricing optimization model, we employ the Social Network Search (SNS) algorithm, a metaheuristic algorithm simulating users’ social network interactions to gain popularity. By incorporating the users’ mood when expressing opinions, this algorithm efficiently identifies optimal pricing solutions. Our results demonstrate that the equivalent load-based method not only encourages renewable energy consumption but also reduces power generation costs, stabilizes the power grid load, and benefits power generators, suppliers, and consumers without increasing end users’ electricity charges. Full article

This study introduces a deep neural network approach that utilizes radial basis functions (RBFs) to solve forward and inverse problems in the process of diffusion. The input layer incorporates multiquadric (MQ) RBFs, symbolizing the radial distance between the boundary points on the spacetime boundary and the source points positioned outside the spacetime boundary. The output layer is the initial and boundary data given by analytical solutions of the diffusion equation. Utilizing the concept of the spacetime coordinates, the approximations for forward and backward diffusion problems involve assigning initial data on the bottom or top spacetime boundaries, respectively. As the need for discretization of the governing equation is eliminated, our straightforward approach uses only the provided boundary data and MQ RBFs. To validate the proposed method, various diffusion scenarios, including forward, backward, and inverse problems with noise, are examined. Results indicate that the method can achieve high-precision numerical solutions for solving diffusion problems. Notably, only 1/4 of the initial and boundary conditions are known, yet the method still yields precise results. Full article

Volatility estimation and quantile regression are relevant active research areas in statistics, machine learning and econometrics. In this work, we propose two procedures to estimate the local variances in generic regression problems by using kernel smoothers. The proposed schemes can be applied in multidimensional scenarios (not just for time series analysis) and easily in a multi-output framework as well. Moreover, they enable the possibility of providing uncertainty estimation using a generic kernel smoother technique. Several numerical experiments show the benefits of the proposed methods, even compared with the benchmark techniques. One of these experiments involves a real dataset analysis. Full article

Electro-hydraulic actuators have witnessed significant development over recent years due to their remarkable abilities to perform complex and dynamic movements. Integrating such an actuator in humanoids is highly beneficial, leading to a humanoid capable of performing complex tasks requiring high force. This highlights the importance of safety, especially since high power output and safe interaction seem to be contradictory; the greater the robot’s ability to generate high dynamic movements, the more difficult it is to achieve safety, as this requires managing a large amount of motor energy before, during, and after the collision. No matter what technology or algorithm is used to achieve safety, none can be implemented without a stable control system. Hence, one of the main parameters remains the quality and reliability of the robot’s control architecture through handling a huge amount of data without system failure. This paper addresses the development of a stable control architecture that ensures, in later stages, that the safety algorithm is implemented correctly. The optimum control architecture to utilize and ensure the maximum benefit of electro-hydraulic actuators in humanoid robots is one of the important subjects in this field. For a stable and safe functioning of the humanoid, the development of the control architecture and the communication between the different components should adhere to some requirements such as stability, robustness, speed, and reduced complexity, ensuring the easy addition of numerous components. This paper presents the developed control architecture for an underdeveloped electro-hydraulic actuated humanoid. The proposed solution has the advantage of being a distributed, real-time, open-source, modular, and adaptable control architecture, enabling simple integration of numerous sensors and actuators to emulate human actions and safely interact with them. The contribution of this paper is an enhancement of the updated rate compared to other humanoids by 20% and by 40 % in the latency of the master. The results demonstrate the potential of using EtherCAT fieldbus and open-source software to develop a stable robot control architecture capable of integrating safety and security algorithms in later stages. Full article

In this paper, we explore the trade-offs between public and private investment in autonomous driving technologies. Utilizing an evolutionary game model, we delve into the complex interaction mechanisms between governments and auto manufacturers, focusing on how strategic decisions impact overall outcomes. Specifically, we predict that governments may opt for strategies such as constructing and maintaining infrastructure for Roadside Infrastructure-based Vehicles (RIVs) or subsidizing high-level Autonomous Driving Vehicles (ADVs) without additional road infrastructure. Manufacturers’ choices involve deciding whether to invest in RIVs or ADVs, depending on governmental policies and market conditions. Our simulation results, based on scenarios derived from existing economic data and forecasts on technology development costs, suggest that government subsidy policies need to dynamically adjust in response to manufacturers’ shifting strategies and market behavior. This dynamic adjustment is crucial as it addresses the evolving economic environment and technological advancements, ensuring that subsidies effectively incentivize the desired outcomes in autonomous vehicle development. The findings of this paper could serve as valuable decision-making tools for governments and auto manufacturers, guiding investment strategies that align with the dynamic landscape of autonomous driving technology. Full article

This paper concentrates on examining the characterization of nonlinear mixed bi-skew Lie triple ∗- derivations within an ∗-algebra denoted by $\mathcal{A}$ which contains a nontrivial projection with a unit I. Additionally, we expand this investigation to applications by describing these derivations within prime ∗-algebras, von Neumann algebras, and standard operator algebras. Full article

Load forecast is the foundation of power system operation and planning. The forecast results can guide the power system economic dispatch and security analysis. In order to improve the accuracy of load forecast, this paper proposes a forecasting model based on the combination of the cuckoo search (CS) algorithm and the long short-term memory (LSTM) neural network. Load data are specific data with time series characteristics and periodicity, and the LSTM algorithm can control the information added or discarded through the forgetting gate, so as to realize the function of forgetting or memorizing. Therefore, the use of the LSTM algorithm for load forecast is more effective. The CS algorithm can perform global search better and does not easily fall into local optima. The CS-LSTM forecasting model, where CS algorithm is used to optimize the hyper-parameters of the LSTM model, has a better forecasting effect and is more feasible. Simulation results show that the CS-LSTM model has higher forecasting accuracy than the standard LSTM model, the PSO-LSTM model, and the GA-LSTM model. Full article

With the development of artificial intelligence (AI), deep learning is widely used in various industries. At CRYPTO 2019, researchers used deep learning to analyze the block cipher for the first time and constructed a differential neural network distinguisher to meet a certain accuracy. In this paper, a mixture differential neural network distinguisher using ResNet is proposed to further improve the accuracy by exploring the mixture differential properties. Experiments are conducted on SIMON32/64, and the accuracy of the 8-round mixture differential neural network distinguisher is improved from 74.7% to 92.3%, compared with that of the previous differential neural network distinguisher. The prediction accuracy of the differential neural network distinguisher is susceptible to the choice of the specified input differentials, whereas the mixture differential neural network distinguisher is less affected by the input difference and has greater robustness. Furthermore, by combining the probabilistic expansion of rounds and the neutral bit, the obtained mixture differential neural network distinguisher is extended to 11 rounds, which can realize the 12-round actual key recovery attack on SIMON32/64. With an appropriate increase in the time complexity and data complexity, the key recovery accuracy of the mixture differential neural network distinguisher can be improved to 55% as compared to 52% of the differential neural network distinguisher. The mixture differential neural network distinguisher proposed in this paper can also be applied to other lightweight block ciphers. Full article

In this paper, we study the average Kolmogorov $\sigma$–widths and the average linear $\sigma$–widths of multivariate isotropic and anisotropic Besov classes in Orlicz spaces and give the weak asymptotic estimates of these two widths. At the same time, we also give the asymptotic property of the optimal recovery of isotropic Besov classes in Orlicz spaces. Full article

In this article, we mainly generalize the results in the literature for a fractional q-difference equation. Our study considers a more comprehensive type of fractional $\left(p,q\right)$-difference system of nonlinear equations. By the Banach contraction mapping principle, we obtain a unique solution. By Krasnoselskii’s fixed-point theorem, we prove the existence of solutions. In addition, finite stability has been established too. The main results in the literature have been proven to be a particular corollary of our work. Full article

Selecting features associated with patient-centered outcomes is of major relevance yet the importance given depends on the method. We aimed to compare stepwise selection, least absolute shrinkage and selection operator, random forest, Boruta, extreme gradient boosting and generalized maximum entropy estimation and suggest an aggregated evaluation. We also aimed to describe outcomes in people with chronic obstructive pulmonary disease (COPD). Data from 42 patients were collected at baseline and at 5 months. Acute exacerbations were the aggregated most important feature in predicting the difference in the handgrip muscle strength (dHMS) and the COVID-19 lockdown group had an increased dHMS of 3.08 kg (CI95 ≈ [0.04, 6.11]). Pack-years achieved the highest importance in predicting the difference in the one-minute sit-to-stand test and no clinical change during lockdown was detected. Charlson comorbidity index was the most important feature in predicting the difference in the COPD assessment test (dCAT) and participants with severe values are expected to have a decreased dCAT of 6.51 points (CI95 ≈ [2.52, 10.50]). Feature selection methods yield inconsistent results, particularly extreme gradient boosting and random forest with the remaining. Models with features ordered by median importance had a meaningful clinical interpretation. Lockdown seem to have had a negative impact in the upper-limb muscle strength. Full article

In this paper, the scale mixture of the Gleser (SMG) distribution is introduced. This new distribution is the product of a scale mixture between the Gleser (G) distribution and the Beta$(a,1)$ distribution. The SMG distribution is an alternative to distributions with two parameters and a heavy right tail. We study its representation and some basic properties, maximum likelihood inference, and Fisher’s information matrix. We present an application to a real dataset in which the SMG distribution shows a better fit than two other known distributions. Full article

A PLS-based quality prediction method is proposed for batch processes using two-dimensional extended windows. To realize the adoption of information in the directions of sampling time and batch, a newly defined region of support (ROS), called the k-i-back-extended region of support (KIBROS), is proposed; it establishes an extended window by adding two regions of data to the traditional ROS to include all possible important data for quality prediction. Based on the new ROS, extended windows are established, and different models are proposed using the extended windows for batch process quality prediction. Furthermore, using the typical injection molding batch process as an example, the proposed quality prediction method is experimentally verified, proving that the proposed methods have higher prediction accuracy than the traditional method and that the prediction stability is also improved. Full article

Let $(M,\nabla ,g)$ be a statistical manifold and $TM$ be its tangent bundle endowed with a twisted Sasaki metric G. This paper serves two primary objectives. The first objective is to investigate the curvature properties of the tangent bundle $TM$. The second objective is to explore conformal vector fields and Ricci, Yamabe, and gradient Ricci–Yamabe solitons on the tangent bundle $TM$ according to the twisted Sasaki metric G. Full article

Searching over a sorted list is a classical problem in computer science. Binary Search takes at most $⌊{\log }_{2}n⌋+1$ tries to find an item in a sorted list of size n. Interpolation Search achieves an average time complexity of $\mathsf{O}\left(\log \log n\right)$ for uniformly distributed data. Hybrids of Binary Search and Interpolation Search are also available to handle data with unknown distributions. This paper analyzes the computation cost of these methods and shows that interpolation can significantly affect their performance—accordingly, a new method, Interpolation Once Binary Search (IOBS), is proposed. The experimental results show that IOBS outperforms the hybrids of Binary Search and Interpolation Search for nonuniformly distributed data. Full article

This research stems from the increasing use of hand gestures in various applications, such as sign language recognition to electronic device control. The focus is the importance of accuracy and robustness in recognizing hand gestures to avoid misinterpretation and instruction errors. However, many experiments on hand gesture recognition are conducted in limited laboratory environments, which do not fully reflect the everyday use of hand gestures. Therefore, the importance of an ideal background in hand gesture recognition, involving only the signer without any distracting background, is highlighted. In the real world, the use of hand gestures involves various unique environmental conditions, including differences in background colors, varying lighting conditions, and different hand gesture positions. However, the datasets available to train hand gesture recognition models often lack sufficient variability, thereby hindering the development of accurate and adaptable systems. This research aims to develop a robust hand gesture recognition model capable of operating effectively in diverse real-world environments. By leveraging deep learning-based image augmentation techniques, the study seeks to enhance the accuracy of hand gesture recognition by simulating various environmental conditions. Through data duplication and augmentation methods, including background, geometric, and lighting adjustments, the diversity of the primary dataset is expanded to improve the effectiveness of model training. It is important to note that the utilization of the green screen technique, combined with geometric and lighting augmentation, significantly contributes to the model’s ability to recognize hand gestures accurately. The research results show a significant improvement in accuracy, especially with implementing the proposed green screen technique, underscoring its effectiveness in adapting to various environmental contexts. Additionally, the study emphasizes the importance of adjusting augmentation techniques to the dataset’s characteristics for optimal performance. These findings provide valuable insights into the practical application of hand gesture recognition technology and pave the way for further research in tailoring techniques to datasets with varying complexities and environmental variations. Full article

This paper provides a computationally efficient and novel four-step methodological approach for predicting volatility estimators derived from bitcoin prices. In the first step, open, high, low, and close bitcoin prices are transformed into volatility estimators using Brownian motion assumptions and logarithmic transformations. The second step determines the optimal number of time-series lags required for converting the series into an autoregressive model. This selection process utilizes random forest regression, evaluating the importance of each lag using the Mean Decrease in Impurity (MDI) criterion and optimizing the number of lags considering an 85% cumulative importance threshold. The third step of the developed methodological approach fits the Elastic Net Regression (ENR) to the volatility estimator’s dataset, while the final fourth step assesses the predictive accuracy of ENR, compared to decision tree (DTR), random forest (RFR), and support vector regression (SVR). The results reveal that the ENR prevails in its predictive accuracy for open and close prices, as these prices may be linear and less susceptible to sudden, non-linear shifts typically seen during trading hours. On the other hand, SVR prevails for high and low prices as these prices often experience spikes and drops driven by transient news and intra-day market sentiments, forming complex patterns that do not align well with linear modelling. Full article

This study investigates the integration of quantum circuits with classical neural networks for enhancing credit scoring for small- and medium-sized enterprises (SMEs). We introduce a hybrid quantum–classical model, focusing on the synergy between quantum and classical rather than comparing the performance of separate quantum and classical models. Our model incorporates a quantum layer into a traditional neural network, achieving notable reductions in training time. We apply this innovative framework to a binary classification task with a proprietary real-world classical credit default dataset for SMEs in Singapore. The results indicate that our hybrid model achieves efficient training, requiring significantly fewer epochs (350) compared to its classical counterpart (3500) for a similar predictive accuracy. However, we observed a decrease in performance when expanding the model beyond 12 qubits or when adding additional quantum classifier blocks. This paper also considers practical challenges faced when deploying such models on quantum simulators and actual quantum computers. Overall, our quantum–classical hybrid model for credit scoring reveals its potential in industry, despite encountering certain scalability limitations and practical challenges. Full article

The main objective of this research is to establish a new formulation and mathematical model based on graph theory to create dynamic equations and provide clarity on the fundamental formulation. We have employed graph theory as a new approach to develop a new representation and formulate the vertical dynamics of a motorcycle with four degrees of freedom, including a suspension and tire model. We have outlined the principal procedural steps required to generate the mathematical and dynamic equations. This systematic approach ensures clarity and precision in our formulation process and representation. Subsequently, we implemented the dynamics equations to examine the dynamic behavior of both the sprung and unsprung masses’ vertical displacements, while considering the varying conditions of the road profile. Full article

This paper explores the application of Random Matrix Theory (RMT) as a methodological enhancement for portfolio selection within financial markets. Traditional approaches to portfolio optimization often rely on historical estimates of correlation matrices, which are particularly susceptible to instabilities. To address this challenge, we combine a data preprocessing technique based on the Hilbert transformation of returns with RMT to refine the accuracy and robustness of correlation matrix estimation. By comparing empirical correlations with those generated through RMT, we reveal non-random properties and uncover underlying relationships within financial data. We then utilize this methodology to construct the correlation network dependence structure used in portfolio optimization. The empirical analysis presented in this paper validates the effectiveness of RMT in enhancing portfolio diversification and risk management strategies. This research contributes by offering investors and portfolio managers with methodological insights to construct portfolios that are more stable, robust, and diversified. At the same time, it advances our comprehension of the intricate statistical principles underlying multivariate financial data. Full article

The Least Absolute Shrinkage and Selection Operator (LASSO) regression technique has proven to be a valuable tool for fitting and reducing linear models. The trend of applying LASSO to compositional data is growing, thereby expanding its applicability to diverse scientific domains. This paper aims to contribute to this evolving landscape by undertaking a comprehensive exploration of the $L^1$-norm for the penalty term of a LASSO regression in a compositional context. This implies first introducing a rigorous definition of the compositional $L^p$-norm, as the particular geometric structure of the compositional sample space needs to be taken into account. The focus is subsequently extended to a meticulous data-driven analysis of the dimension reduction effects on linear models, providing valuable insights into the interplay between penalty term norms and model performance. An analysis of a microbial dataset illustrates the proposed approach. Full article

In this paper, we consider the Cauchy problem for a three-component Novikov system on the line. We give a construction of the initial data $({\rho }_0,u_0,v_0)\in B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)$ with $\sigma >max\left\{3+\frac{1}{p},\frac{7}{2}\right\},1\le p\le \infty$, such that the corresponding solution to the three-component Novikov system starting from $({\rho }_0,u_0,v_0)$ is discontinuous at $t=0$ in the metric of $B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)$, which implies the ill-posedness for this system in $B_{p,\infty }^{\sigma -1}\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)\times B_{p,\infty }^{\sigma }\left(\mathbb{R}\right)$. Full article

Indoor wireless positioning has long been a dynamic field of research due to its broad application range. While many commercial products have been developed, they often are not open source or require substantial and costly infrastructure. Academically, research has extensively explored Bluetooth Low Energy (BLE) for positioning, yet there are a noticeable lack of studies that comprehensively compare traditional algorithms under these conditions. This research aims to fill this gap by evaluating classical positioning algorithms such as K-Nearest Neighbor (KNN), Weighted K-Nearest Neighbor (WKNN), Naïve Bayes (NB), and a Received Signal Strength-based Neural Network (RSS-NN) using BLE technology. We also introduce a novel method using Convolutional Neural Networks (CNN), specifically tailored to process RSS data structured in an image-like format. This approach helps overcome the limitations of traditional RSS fingerprinting by effectively managing the environmental dynamics within indoor settings. In our tests, all algorithms performed well, consistently achieving an average accuracy of less than two meters. Remarkably, the CNN method outperformed others, achieving an accuracy of 1.22 m. These results establish a solid basis for future research, particularly towards enhancing the precision of indoor positioning systems using deep learning for cost-effective, easy to set up applications. Full article