Mathematics, Volume 12, Issue 1 (January-1 2024) – 168 articles

Deep learning technology has greatly propelled the development of intelligent and information-driven research on ship infrared automatic target recognition (SIATR). In future scenarios, there will be various recognition models with different mechanisms to choose from. However, in complex and dynamic environments, ship infrared (IR) data exhibit rich feature space distribution, resulting in performance variations among SIATR models, thus preventing the existence of a universally superior model for all recognition scenarios. In light of this, this study proposes a model-matching method for SIATR tasks based on bipartite graph theory. This method establishes evaluation criteria based on recognition accuracy and feature learning credibility, uncovering the underlying connections between IR attributes of ships and candidate models. The objective is to selectively recommend the optimal candidate model for a given sample, enhancing the overall recognition performance and applicability of the model. We separately conducted tests for the optimization of accuracy and credibility on high-fidelity simulation data, achieving Accuracy and EDMS (our credibility metric) of 95.86% and 0.7781. Our method improves by 1.06% and 0.0274 for each metric compared to the best candidate models (six in total). Subsequently, we created a recommendation system that balances two tasks, resulting in improvements of 0.43% (accuracy) and 0.0071 (EDMS). Additionally, considering the relationship between model resources and performance, we achieved a 28.35% reduction in memory usage while realizing enhancements of 0.33% (accuracy) and 0.0045 (EDMS). Full article

Difference schemes that approximate dynamic systems are considered discrete models of the same phenomena that are described by continuous dynamic systems. Difference schemes with t-symmetry and midpoint and trapezoid schemes are considered. It is shown that these schemes are dual to each other, and, from this fact, we derive theorems on the inheritance of quadratic integrals by these schemes (Cooper’s theorem and its dual theorem on the trapezoidal scheme). Using examples of nonlinear oscillators, it is shown that these schemes poses challenges for theoretical research and practical application due to the problem of extra roots: these schemes do not allow one to unambiguously determine the final values from the initial values and vice versa. Therefore, we consider difference schemes in which the transitions from layer to layer in time are carried out using birational transformations (Cremona transformations). Such schemes are called reversible. It is shown that reversible schemes with t-symmetry can be easily constructed for any dynamical system with a quadratic right-hand side. As an example of such a dynamic system, a top fixed at its center of gravity is considered in detail. In this case, the discrete theory repeats the continuous theory completely: (1) the points of the approximate solution lie on some elliptic curve, which at $\Delta t\to 0$ turns into an integral curve; (2) the difference scheme can be represented using quadrature; and (3) the approximate solution can be represented using an elliptic function of a discrete argument. The last section considers the general case. The integral curves are replaced with closures of the orbits of the corresponding Cremona transformation as sets in the projective space over $\mathbb{R}$. The problem of the dimension of this set is discussed. Full article

In this paper, we establish the cohomology of relative Rota–Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then, we use this type of cohomology to characterize deformations of relative Rota–Baxter operators on Lie-Yamaguti algebras. We show that if two linear or formal deformations of a relative Rota–Baxter operator are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, an order n deformation of a relative Rota–Baxter operator can be extended to an order $n+1$ deformation if and only if the obstruction class in the second cohomology group is trivial. Full article

The article is devoted to the analytical and numerical study of the pattern of propagation and attenuation, due to Coulomb friction, of shear waves in an infinite elastic thin plate with a circular orifice of radius $r_0$ lying on a rough base. Considering the friction forces and their influence on the sample of wave propagation in extended rods or thin plates is important for calculating the stress–strain state in them and the size of the area of motion. An exact analytical solution of a nonlinear boundary value problem for tangential stresses and velocities is obtained in quadratures by the Laplace transform, with respect to time. It turned out that the complete exhaustion of the wave front of a strong rupture occurs at a finite distance $r*$ from the center of the orifice, and an elementary formula is given for this distance (the case of tangential shock stresses suddenly applied to the orifice boundary is considered). For various ratios of the magnitude of the limiting friction force to the amplitude of the applied load, the stopping (trailing) wave fronts are calculated. After passing them, a state of static equilibrium between the elastic and friction forces with a nonlinear distribution of residual stresses is established in the region $r_0\le r\le r*$. For the first time, a precise analytical solution was obtained for the boundary value problem of the propagation of elastic shear waves in an infinite isotropic space with a cylindrical cavity, when a tangential shock load is set on its surface. Full article

At present, sequence-based models have various applications in recommendation systems; these models recommend the interested items of the user according to the user’s behavioral sequence. However, sequence-based models have a limitation of length. When the length of the user’s behavioral sequence exceeds the limitation of the model, the model cannot take advantage of the complete behavioral sequence of the user and cannot know the user’s holistic interests. The accuracy of the model then goes down. Meanwhile, sequence-based models only pay attention to the sequential signals of the data but do not pay attention to the spatial signals of the data, which will also affect the model’s accuracy. This paper proposes a graph sequence-based model called GSRec that combines Graph Convolutional Network (GCN) and Transformer to solve these problems. In the GCN part we designed a reverse-order graph, and in the Transformer part we introduced the user embedding. The reverse-order graph and the user embedding can make the combination of GCN and Transformer more efficient. Experiments on six datasets show that GSRec outperforms the current state-of-the-art (SOTA) models. Full article

In this paper, the distributed interval estimation problem for networked Cyber-Physical systems suffering from both disturbances and noise is investigated. In the distributed interval observers, there are some connected interval observers built for the corresponding subsystems. Then, due to the communication burden in Cyber-Physical systems, we consider the case where the communication among distributed interval observers is switching topology. A novel approach that combines $L_{\infty }$ methodology with reachable set analysis is proposed to design distributed interval observers. Finally, the performance of the proposed distributed interval observers with switching topology is verified through a simulation example. Full article

Partially linear models find extensive application in biometrics, econometrics, social sciences, and various other fields due to their versatility in accommodating both parametric and nonparametric elements. This study aims to establish statistical inference for the parametric component effects within these models, employing a nonparametric empirical likelihood approach. The proposed method involves a projection step to eliminate the nuisance nonparametric component and utilizes an empirical-likelihood-based technique, along with the Bartlett correction, to enhance the coverage probability of the confidence interval for the parameter of interest. This method demonstrates robustness in handling normally and non-normally distributed errors. The proposed empirical likelihood ratio statistic converges to a limiting chi-square distribution under certain regulations. Simulation studies demonstrate that this method provides better inference in terms of coverage probabilities compared to the conventional normal-approximation-based method. The proposed method is illustrated by analyzing the Boston housing data from a real study. Full article

Real-time evaluation of the damage location and level of rock mass is essential for preventing underground engineering disasters. However, the heterogeneity of rock mass, which results from the presence of layered rock media, faults, and pores, makes it difficult to characterize the damage evolution accurately in real time. To address this issue, an improved method for rock damage characterization is proposed. This method optimizes the solution of the global shortest acoustic wave propagation path in the medium and verifies it with layered and defective media models. Based on this, the relationship between the inversion results of the wave velocity field and the distribution of rock damage is established, thereby achieving quantitative characterization of rock damage distribution and degree. Thus, the improved method is more suitable for heterogeneous rock media. Finally, the proposed method was used to characterize the damage distribution evolution process of rock media during uniaxial compression experiments. The obtained results were compared and analyzed with digital speckle patterns, and the influencing factors during the use of the proposed method are discussed. Full article

We model the spatiotemporal dynamics of a community consisting of competing weed and cultivated plant species and a population of specialized phytophagous insects used as the weed biocontrol agent. The model is formulated as a PDE system of taxis–diffusion–reaction type and computer-implemented for one-dimensional and two-dimensional cases of spatial habitat for the Neumann zero-flux boundary condition. In order to discretize the original continuous system, we applied the method of lines. The obtained system of ODEs is integrated using the Runge–Kutta method with a variable time step and control of the integration accuracy. The numerical simulations provide insights into the mechanism of formation of solitary population waves (SPWs) of the phytophage, revealing the factors that determine the efficacy of combined application of the phytophagous insect (classical biological method) and cultivated plant (phytocenotic method) to suppress weed foci. In particular, the presented results illustrate the stabilizing action of cultivated plants, which fix the SPW effect by occupying the free area behind the wave front so that the weed remains suppressed in the absence of a phytophage. Full article

Ill-posed problems arise in many areas of science and engineering. Tikhonov is a usual regularization which replaces the original problem by a minimization problem with a fidelity term and a regularization term. In this paper, a tensor t-production structure preserved Conjugate-Gradient (tCG) method is presented to solve the regularization minimization problem. We provide a truncated version of regularization parameters for the tCG method and a preprocessed version of the tCG method. The discrepancy principle is used to automatically determine the regularization parameter. Several examples on image and video recover are given to show the effectiveness of the proposed methods by comparing them with some previous algorithms. Full article

In this paper, we applied a chaos control method based on integro-differential equations for stabilization of an unstable cardiac rhythm, which is described by a variation of the modified Van der Pol equation. Chaos control with this method may be useful for stabilization of irregular heartbeat using a small perturbation. This method differs from other stabilization strategies by the absence of adjustable parameters and the lack of rough approximations in determining control functions whose control parameters are fixed by the properties of the unstable system itself. Full article

We define and study enriched Suzuki mappings in Hadamard spaces. The results obtained here are extending fundamental findings previously established in related research. The extension is realized with respect to at least two different aspects: the setting and the class of involved operators. More accurately, Hilbert spaces are particular Hadamard spaces, while enriched Suzuki nonexpansive mappings are natural generalizations of enriched nonexpansive mappings. Next, enriched Suzuki nonexpansive mappings naturally contain Suzuki nonexpansive mappings in Hadamard spaces. Besides technical lemmas, the results of this paper deal with (1) the existence of fixed points for enriched Suzuki nonexpansive mappings and (2) $\mathsf{\Delta }$ and strong (metric) convergence of Picard iterates of the $\alpha$-averaged mapping, which are exactly Krasnoselskij iterates for the original mapping. Full article

This article presents an extended distribution that builds upon the exponential distribution. This extension is based on a scale mixture between the exponential and beta distributions. By utilizing this approach, we obtain a distribution that offers increased flexibility in terms of the kurtosis coefficient. We explore the general density, properties, moments, asymmetry, and kurtosis coefficients of this distribution. Statistical inference is performed using both the moments and maximum likelihood methods. To show the performance of this new model, it is applied to a real dataset with atypical observations. The results indicate that the new model outperforms two other extensions of the exponential distribution. Full article

This paper focuses on the asymptotic behavior of nonautonomous neural networks with delays. We establish criteria for analyzing the asymptotic behavior of nonautonomous recurrent neural networks with delays by means of constructing some new generalized Halanay inequalities. We do not require to constructi any complicated Lyapunov function and our results improve some existing works. Lastly, we provide some illustrative examples to demonstrate the effectiveness of the obtained results. Full article

Since its introduction in 1650, Kepler’s equation has never ceased to fascinate mathematicians, scientists, and engineers. Over the course of five centuries, a large number of different solution strategies have been devised and implemented. Among them, the one originally proposed by J. L. Lagrange and later by F. W. Bessel still continue to be a source of mathematical treasures. Here, the Bessel solution of the elliptic Kepler equation is explored from a new perspective offered by the theory of the Stieltjes series. In particular, it has been proven that a complex Kapteyn series obtained directly by the Bessel expansion is a Stieltjes series. This mathematical result, to the best of our knowledge, is a new integral representation of the KE solution. Some considerations on possible extensions of our results to more general classes of the Kapteyn series are also presented. Full article

The fusion of camera and LiDAR perception has become a research focal point in the autonomous driving field. Existing image–point cloud fusion algorithms are overly complex, and processing large amounts of 3D LiDAR point cloud data requires high computational power, which poses challenges for practical applications. To overcome the above problems, herein, we propose an Instance Segmentation Frustum (ISF)–PointPillars method. Within the framework of our method, input data are derived from both a camera and LiDAR. RGB images are processed using an enhanced 2D object detection network based on YOLOv8, thereby yielding rectangular bounding boxes and edge contours of the objects present within the scenes. Subsequently, the rectangular boxes are extended into 3D space as frustums, and the 3D points located outside them are removed. Afterward, the 2D edge contours are also extended to frustums to filter the remaining points from the preceding stage. Finally, the retained points are sent to our improved 3D object detection network based on PointPillars, and this network infers crucial information, such as object category, scale, and spatial position. In pursuit of a lightweight model, we incorporate attention modules into the 2D detector, thereby refining the focus on essential features, minimizing redundant computations, and enhancing model accuracy and efficiency. Moreover, the point filtering algorithm substantially diminishes the volume of point cloud data while concurrently reducing their dimensionality, thereby ultimately achieving lightweight 3D data. Through comparative experiments on the KITTI dataset, our method outperforms traditional approaches, achieving an average precision (AP) of 88.94% and bird’s-eye view (BEV) accuracy of 90.89% in car detection. Full article

In this paper, an intelligent weather conditions fuzzy adjustment based on spatial features (IWeCASF) is developed. It is indispensable for our regional soil moisture estimation approach, complementing a point estimation model of soil moisture from the literature. The point estimation model requires the weather conditions at the point where an estimate is made. Therefore, IWeCASF’s aim is to determine these weather conditions. The procedure begins measuring them at only one checkpoint, called the primary checkpoint. The model determines the weather conditions anywhere within a region through image processing algorithms and fuzzy inference systems. The results are compared with the measurement records and with a spatial interpolation method. The performance is similar to or better than interpolation, especially in the rain, where the model developed is more accurate due to the certainty of replication. Additionally, IWeCASF does not require more than one measurement point. Therefore, it is a more appropriate approach to complement the point estimation model for enabling a regional soil moisture estimation. Full article

This paper primarily focuses on the chaos synchronisation analysis of neural networks (NNs) under a hybrid controller. Firstly, we design a suitable hybrid controller with saturated impulse control, combined with time-dependent intermittent control. Both controls are low-energy consumption and discrete, aligning well with industrial development needs. Secondly, the saturation function in the chaotic neural network is addressed using the polyhedral representation method and the sector nonlinearity method, respectively. By integrating the Lyapunov stability theory, Jensen’s inequality, the mathematical induction method, and the inequality reduction technique, we establish suitable time-dependent Lyapunov generalised equations. This leads to the estimation of the domain of attraction and the derivation of local exponential stability conditions for the error system. The validity of the achieved theoretical criteria is eventually demonstrated through numerical experiment simulations. Full article

Based on the published papers in this Special Issue of the elite scientific journal Mathematics, we herein present the Editorial for “Differential Equations of Mathematical Physics and Related Problems of Mechanics”, the main topics of which are fundamental and applied research on differential equations in mathematical physics and mechanics [...] Full article

Dependent random variables play a crucial role in various fields, from finance and statistics to engineering and environmental sciences. Often, interest lies in understanding the aggregate sum of a collection of dependent variables with random weights. In this paper, we provide a comprehensive study of the distribution of the aggregate sum with random weights. Expressions derived include those for the cumulative distribution function, probability density function, conditional expectation, moment generating function, characteristic function, cumulant generating function, moments, variance, skewness, kurtosis, cumulants, value at risk and the expected shortfall. Real data applications are discussed. Full article

A distributed-power-generating source (DPGS) is intended to locally supply the increased power demand at a load bus. When applied in small amounts, a DPGS offers many technical and economic benefits. However, with large DPGS penetrations, the stability of the transmission system becomes a significant issue. This paper investigates the stability of a transmission system equipped with a DPGS at load centres supplying power to both a constant power (CP) and induction motor (IM) load. The DPGSs considered in the present study are microturbine and diesel turbine power generators (MTGS and DTGS), both interfaced with synchronous generators. The influence of an IM load supplied by the DPGS on small-signal stability is studied by a critical damping ratio analysis. On the other hand, time-domain indicators of the transient response following a short circuit are employed in the analysis. Further, a variance analysis test (VAT) is performed to determine the contribution of IM and CP loads on the system stability. The study revealed that large penetration levels of IM loads significantly affect the stability and depend on the kind of DPGS technology used. Full article

Dealing with municipal sludge in an effective way is crucial for urban development and environmental protection. Co-processing the sludge by burning it in a decomposition furnace in the cement production line has been found to be a viable solution. This work aims to analyze the effects of the co-disposal of municipal sludge on the decomposition reactions and NOx emissions in the decomposing furnaces. Specifically, a practical 6000 t/d decomposition furnace was taken as the research object. To achieve this, ANSYS FLUENT with a UDF (user-defined function) was applied to establish a numerical model coupling the limestone decomposition reaction, fuel combustion, and NOx generation and reduction reactions. The flow, temperature, and component field distributions within the furnace with no sludge were firstly simulated with this model. Compared with site test results, the model was validated. Then, with sludge involved, the structure and operation parameters of the decomposition furnace for the co-disposal of municipal sludge were investigated by simulating the flow field, temperature field, and component field distributions. Parametric studies were carried out in three perspectives, i.e., sludge mixing ratio, preheating furnace arrangement height, and sludge particle size. The results show that all three aspects have great importance in the discomposing process. A set of preferable values, including a sludge mixing ratio of 10%, preheating furnace height of 21.5 m, and sludge particle diameter of 1.0 mm, was obtained, which resulted in a raw material decomposition rate of 89.9% and a NO volume fraction of 251 ppm at the furnace outlet. Full article

We introduce De Morgan Heyting logic for Heyting algebras with De Morgan negation (DH-algebras). The variety $\mathcal{DH}$ of all DH-algebras is congruence distributive. The lattice of all subvarieties of $\mathcal{DH}$ is distributive. We show the discrete dualities between De Morgan frames and DH-algebras. The Kripke completeness and finite approximability of some DH-logics are proven. Some conservativity of DH expansion of a Kripke complete superintuitionistic logic is shown by the construction of frame expansion. Finally, a cut-free terminating Gentzen sequent calculus for the DH-logic of De Morgan Boolean algebras is developed. Full article

This paper theoretically investigates the influence of homo- and hetero-junctions on the propagation characteristics of radially propagated cylindrical surface acoustic waves in a piezoelectric semiconductor semi-infinite medium. First, the basic equations of the piezoelectric semiconductor semi-infinite medium are mathematically derived. Then, based on these basic equations and the transfer matrix method, two equivalent mathematical models are established concerning the propagation of radially propagated cylindrical surface acoustic waves in this piezoelectric semiconductor semi-infinite medium. Based on the surface and interface effect theory, the homo- or hetero-junction is theoretically treated as a two-dimensional electrically imperfect interface in the first mathematical model. To legitimately confirm the interface characteristic lengths that appear in the electrically imperfect interface conditions, the homo- or hetero-junction is equivalently treated as a functional gradient thin layer in the second mathematical model. Finally, based on these two mathematical models, the dispersion and attenuation curves of radially propagated cylindrical surface acoustic waves are numerically calculated to discuss the influence of the homo- and hetero-junctions on the dispersion and attenuation characteristics of radially propagated cylindrical surface acoustic waves. The interface characteristic lengths are legitimately confirmed through the comparison of dispersion and attenuation curves calculated using the two equivalent mathematical models. As piezoelectric semiconductor energy harvesters usually work under elastic deformation, the establishment of mathematical models and the revelation of physical mechanisms are both fundamental to the analysis and optimization of micro-scale surface acoustic wave resonators, energy harvesters, and acoustic wave amplification based on the propagation of surface acoustic waves. Full article

We continue studying the $\sigma$-Ricci vector field $\mathbf{u}$ on a Riemannian manifold $(N^m,g)$, which is not necessarily closed. A Riemannian manifold with Ricci operator T, a Coddazi-type tensor, is called a T-manifold. In the first result of this paper, we show that a complete and simply connected T-manifold$(N^m,g)$, $m>1$, of positive scalar curvature $\tau$, admits a closed $\sigma$-Ricci vector field $\mathbf{u}$ such that the vector $\mathbf{u}-\nabla \sigma$ is an eigenvector of T with eigenvalue $\tau m^{-1}$, if and only if it is isometric to the m-sphere $S_{\alpha }^m$. In the second result, we show that if a compact and connected T-manifold$(N^m,g)$, $m>2$, admits a $\sigma$-Ricci vector field $\mathbf{u}$ with $\sigma \ne 0$ and is an eigenvector of a rough Laplace operator with the integral of the Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$ that has a suitable lower bound, then $(N^m,g)$ is isometric to the m-sphere $S_{\alpha }^m$, and the converse also holds. Finally, we show that a compact and connected Riemannian manifold $(N^m,g)$ admits a $\sigma$-Ricci vector field $\mathbf{u}$ with $\sigma$ as a nontrivial solution of the static perfect fluid equation, and the integral of the Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$ has a lower bound depending on a positive constant $\alpha$, if and only if $(N^m,g)$ is isometric to the m-sphere $S_{\alpha }^m$. Full article

As part of the research in the recently ended project SANET II, we were trying to create a new machine-learning system without a teacher. This system was designed to recognize DDoS attacks in real time, based on adaptation to real-time arbitrary traffic and with the ability to be embedded into the hardware implementation of network probes. The reason for considering this goal was our hands-on experience with the high-speed SANET network, which interconnects Slovak universities and high schools and also provides a connection to the Internet. Similar to any other public-facing infrastructure, it is often the target of DDoS attacks. In this article, we are extending our previous research, mainly by dealing with the use of various statistical parameters for DDoS attack detection. We tested the coefficients of Variation, Kurtosis, Skewness, Autoregression, Correlation, Hurst exponent, and Kullback–Leibler Divergence estimates on traffic captures of different types of DDoS attacks. For early machine recognition of the attack, we have proposed several detection functions that use the response of the investigated statistical parameters to the start of a DDoS attack. The proposed detection methods are easily implementable for monitoring actual IP traffic. Full article

LIDAR is central to the perception systems of autonomous vehicles, but its performance is sensitive to adverse weather. An object detector trained by deep learning with the LIDAR point clouds in clear weather is not able to achieve satisfactory accuracy in adverse weather. Considering the fact that collecting LIDAR data in adverse weather like dusty storms is a formidable task, we propose a novel data augmentation framework based on physical simulation. Our model takes into account finite laser pulse width and beam divergence. The discrete dusty particles are distributed randomly in the surrounding of LIDAR sensors. The attenuation effects of scatters are represented implicitly with extinction coefficients. The coincidentally returned echoes from multiple particles are evaluated by explicitly superimposing their power reflected from each particle. Based on the above model, the position and intensity of real point clouds collected from dusty weather can be modified. Numerical experiments are provided to demonstrate the effectiveness of the method. Full article

Statistical mechanics of walks defined on the spatial graphs of the city of Lubbock (10,421 nodes) and the Texas Tech University (TTU) campus pedestrian network (1466 nodes) are used for evaluating structural isolation and the integration of graph nodes, assessing their accessibility and navigability in the graph, and predicting possible graph structural modifications driving the campus evolution. We present the betweenness and closeness maps of the campus, the first passage times to the different campus areas by isotropic and anisotropic random walks, as well as the first passage times under the conditions of traffic noise. We further show the isolation and integration indices of all areas on the campus, as well as their navigability and strive scores, and energy and fugacity scores. The TTU university campus, a large pedestrian zone located close to the historical city center of Lubbock, mediates between the historical city going downhill and its runaway sprawling body. Full article

The study of differential equation theory has come a long way, with applications in various fields. In 1961, Zygmund and Calderón introduced the notion of derivatives to metric $L^r$, which proved to be better in applications than approximate derivatives. However, most of the studies available are on Fuzzy Set Theory. In view of this, intuitionistic fuzzy $L^r$-norm-based derivatives deserve study. In this study, the $L^r$-norm-based derivative for intuitionistic fuzzy number valued functions is introduced. Some of its basic properties are also discussed, along with numerical examples. The results obtained show that the proposed derivative is not dependent on the existence of the Hukuhara difference. Lastly, the Cauchy problem for the intuitionistic fuzzy differential equation is discussed. Full article